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<button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Bonatti%2C+C&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Bonatti%2C+C&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Bonatti%2C+C&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> </ul> </nav> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.03586">arXiv:2411.03586</a> <span> [<a href="https://arxiv.org/pdf/2411.03586">pdf</a>, <a href="https://arxiv.org/format/2411.03586">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> Non-transitive pseudo-Anosov flows </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Barthelm%C3%A9%2C+T">Thomas Barthelm茅</a>, <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Mann%2C+K">Kathryn Mann</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.03586v1-abstract-short" style="display: inline;"> We study (topological) pseudo-Anosov flows from the perspective of the associated group actions on their orbit spaces and boundary at infinity. We extend the definition of Anosov-like action from [BFM22] from the transitive to the general non-transitive context and show that one can recover the basic sets of a flow, the Smale order on basic sets, and their essential features, from such general gro… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.03586v1-abstract-full').style.display = 'inline'; document.getElementById('2411.03586v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.03586v1-abstract-full" style="display: none;"> We study (topological) pseudo-Anosov flows from the perspective of the associated group actions on their orbit spaces and boundary at infinity. We extend the definition of Anosov-like action from [BFM22] from the transitive to the general non-transitive context and show that one can recover the basic sets of a flow, the Smale order on basic sets, and their essential features, from such general group actions. Using these tools, we prove that a pseudo-Anosov flow in a $3$ manifold is entirely determined by the associated action of the fundamental group on the boundary at infinity of its orbit space. We also give a proof that any topological pseudo-Anosov flow on an atoroidal 3-manifold is necessarily transitive, and prove that density of periodic orbits implies transitivity, in the topological rather than smooth case. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.03586v1-abstract-full').style.display = 'none'; document.getElementById('2411.03586v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.18917">arXiv:2406.18917</a> <span> [<a href="https://arxiv.org/pdf/2406.18917">pdf</a>, <a href="https://arxiv.org/format/2406.18917">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Completing Prelaminations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Barthelm%C3%A9%2C+T">Thomas Barthelm茅</a>, <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Mann%2C+K">Kathryn Mann</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.18917v2-abstract-short" style="display: inline;"> Motivated by problems in the study of Anosov and pseudo-Anosov flows on 3-manifolds, we characterize when a pair $(L^+, L^-)$ of subsets of transverse laminations of the circle can be completed to a pair of transverse foliations of the plane or, separately, realized as the endpoints of such a bifoliation of the plane. (We allow also singular bifoliations with simple prongs, such as arise in pseudo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.18917v2-abstract-full').style.display = 'inline'; document.getElementById('2406.18917v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.18917v2-abstract-full" style="display: none;"> Motivated by problems in the study of Anosov and pseudo-Anosov flows on 3-manifolds, we characterize when a pair $(L^+, L^-)$ of subsets of transverse laminations of the circle can be completed to a pair of transverse foliations of the plane or, separately, realized as the endpoints of such a bifoliation of the plane. (We allow also singular bifoliations with simple prongs, such as arise in pseudo-Anosov flows). This program is carried out at a level of generality applicable to bifoliations coming from pseudo-Anosov flows with and without perfect fits, as well as many other examples, and is natural with respect to group actions preserving these structures. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.18917v2-abstract-full').style.display = 'none'; document.getElementById('2406.18917v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">v2: title changed and introduction rewritten</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.12686">arXiv:2405.12686</a> <span> [<a href="https://arxiv.org/pdf/2405.12686">pdf</a>, <a href="https://arxiv.org/format/2405.12686">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Heterodimensional cycles of hyperbolic ergodic measures </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Diaz%2C+L+J">Lorenzo J. Diaz</a>, <a href="/search/math?searchtype=author&query=Gelfert%2C+K">Katrin Gelfert</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.12686v1-abstract-short" style="display: inline;"> We introduce the concept of a heterodimensional cycle of hyperbolic ergodic measures and a special type of them that we call rich. Within a partially hyperbolic context, we prove that if two measures are related by a rich heterodimensional cycle, then the entire segment of probability measures linking them lies within the closure of measures supported on periodic orbits. Motivated by the occurrenc… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.12686v1-abstract-full').style.display = 'inline'; document.getElementById('2405.12686v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.12686v1-abstract-full" style="display: none;"> We introduce the concept of a heterodimensional cycle of hyperbolic ergodic measures and a special type of them that we call rich. Within a partially hyperbolic context, we prove that if two measures are related by a rich heterodimensional cycle, then the entire segment of probability measures linking them lies within the closure of measures supported on periodic orbits. Motivated by the occurrence of robust heterodimensional cycles of hyperbolic basic sets, we study robust rich heterodimensional cycles of measures providing a framework for this phenomenon for diffeomorphisms. In the setting of skew products, we construct an open set of maps having uncountably many measures related by rich heterodimensional cycles. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.12686v1-abstract-full').style.display = 'none'; document.getElementById('2405.12686v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 28A33; 37D30; 37C40; 37C29 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2306.11517">arXiv:2306.11517</a> <span> [<a href="https://arxiv.org/pdf/2306.11517">pdf</a>, <a href="https://arxiv.org/ps/2306.11517">ps</a>, <a href="https://arxiv.org/format/2306.11517">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Non-locally discrete actions on the circle with at most $N$ fixed points </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Carnevale%2C+J">Jo茫o Carnevale</a>, <a href="/search/math?searchtype=author&query=Triestino%2C+M">Michele Triestino</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2306.11517v3-abstract-short" style="display: inline;"> A subgroup of $\mathrm{Homeo}_+(\mathbb{S}^1)$ is M枚bius-like if every element is conjugate to an element of $\mathrm{PSL}(2,\mathbb{R})$. In general, a M枚bius-like subgroup of $\mathrm{Homeo}_+(\mathbb{S}^1)$ is not necessarily (semi-)conjugate to a subgroup of $\mathrm{PSL}(2,\mathbb{R})$, as discovered by N. Kova膷evi膰 [Trans. Amer. Math. Soc. 351 (1999), 4823-4835]. Here we determine simple dyn… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2306.11517v3-abstract-full').style.display = 'inline'; document.getElementById('2306.11517v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2306.11517v3-abstract-full" style="display: none;"> A subgroup of $\mathrm{Homeo}_+(\mathbb{S}^1)$ is M枚bius-like if every element is conjugate to an element of $\mathrm{PSL}(2,\mathbb{R})$. In general, a M枚bius-like subgroup of $\mathrm{Homeo}_+(\mathbb{S}^1)$ is not necessarily (semi-)conjugate to a subgroup of $\mathrm{PSL}(2,\mathbb{R})$, as discovered by N. Kova膷evi膰 [Trans. Amer. Math. Soc. 351 (1999), 4823-4835]. Here we determine simple dynamical criteria for the existence of such a (semi-)conjugacy. We show that M枚bius-like subgroups of $\mathrm{Homeo}_+(\mathbb{S}^1)$ which are elementary (namely, preserving a Borel probability measure), are semi-conjugate to subgroups of $\mathrm{PSL}(2,\mathbb{R})$. On the other hand, we provide an example of elementary subgroup of $\mathrm{Diff}^\infty_+(\mathbb{S}^1)$ satisfying that every non-trivial element fixes at most 2 points, which is not isomorphic to any subgroup of $\mathrm{PSL}(2,\mathbb{R})$. Finally, we show that non-elementary, non-locally discrete subgroups acting with at most $N$ fixed points are conjugate to a dense subgroup of some finite central extension of $\mathrm{PSL}(2,\mathbb{R})$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2306.11517v3-abstract-full').style.display = 'none'; document.getElementById('2306.11517v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 June, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">16 pages. This work is part of the second author's PhD thesis at Universit茅 de Bourgogne; to appear in Math. Z</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary 37C85; 57M60. Secondary 37B05; 37E05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2301.04530">arXiv:2301.04530</a> <span> [<a href="https://arxiv.org/pdf/2301.04530">pdf</a>, <a href="https://arxiv.org/ps/2301.04530">ps</a>, <a href="https://arxiv.org/format/2301.04530">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Action on the circle at infinity of foliations of ${\mathbb R}^2 $ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2301.04530v1-abstract-short" style="display: inline;"> This paper provides a canonical compactification of the plane ${\mathbb R}^2$ by adding a circle at infinity associated to a countable family of singular foliations or laminations (under some hypotheses), generalizing an idea by Mather \cite{Ma}. Moreover any homeomorphism of ${\mathbb R}^2 $ preserving the foliations extends on the circle at infinity. Then this paper provides conditions ensurin… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2301.04530v1-abstract-full').style.display = 'inline'; document.getElementById('2301.04530v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2301.04530v1-abstract-full" style="display: none;"> This paper provides a canonical compactification of the plane ${\mathbb R}^2$ by adding a circle at infinity associated to a countable family of singular foliations or laminations (under some hypotheses), generalizing an idea by Mather \cite{Ma}. Moreover any homeomorphism of ${\mathbb R}^2 $ preserving the foliations extends on the circle at infinity. Then this paper provides conditions ensuring the minimality of the action on the circle at infinity induced by an action on ${\mathbb R}^2 $ preserving one foliation or two transverse foliations. In particular the action on the circle at infinity associated to an Anosov flow $X$ on a closed $3$-manifold is minimal if and only if $X$ is non-$\mathbb R$-covered. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2301.04530v1-abstract-full').style.display = 'none'; document.getElementById('2301.04530v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 January, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">36 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37D20; 37E10; 37E35; 37C86 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2212.06483">arXiv:2212.06483</a> <span> [<a href="https://arxiv.org/pdf/2212.06483">pdf</a>, <a href="https://arxiv.org/ps/2212.06483">ps</a>, <a href="https://arxiv.org/format/2212.06483">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Oriented Birkhoff sections of Anosov flows </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Asaoka%2C+M">Masayuki Asaoka</a>, <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Marty%2C+T">Th茅o Marty</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2212.06483v2-abstract-short" style="display: inline;"> This paper gives 3 different proofs (independently obtained by the 3 authors) of the following fact: given an Anosov flow on an oriented 3 manifold, the existence of a positive Birkhoff section is equivalent to the fact that the flow is $\mathbb{R}$-covered positively twisted. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2212.06483v2-abstract-full" style="display: none;"> This paper gives 3 different proofs (independently obtained by the 3 authors) of the following fact: given an Anosov flow on an oriented 3 manifold, the existence of a positive Birkhoff section is equivalent to the fact that the flow is $\mathbb{R}$-covered positively twisted. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.06483v2-abstract-full').style.display = 'none'; document.getElementById('2212.06483v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">50 pages, 10 figures. New version: minor corrections and modifications, 4 new figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37D20 (Primary) 37C86(Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2209.13249">arXiv:2209.13249</a> <span> [<a href="https://arxiv.org/pdf/2209.13249">pdf</a>, <a href="https://arxiv.org/ps/2209.13249">ps</a>, <a href="https://arxiv.org/format/2209.13249">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Aperiodic chain recurrence classes of $C^1$-generic diffeomorphisms </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Shinohara%2C+K">Katsutoshi Shinohara</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2209.13249v1-abstract-short" style="display: inline;"> We consider the space of $C^1$-diffeomorphims equipped with the $C^1$-topology on a three dimensional closed manifold. It is known that there are open sets in which $C^1$-generic diffeomorphisms display uncountably many chain recurrences classes, while only countably many of them may contain periodic orbits. The classes without periodic orbits, called aperiodic classes, are the main subject of thi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.13249v1-abstract-full').style.display = 'inline'; document.getElementById('2209.13249v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2209.13249v1-abstract-full" style="display: none;"> We consider the space of $C^1$-diffeomorphims equipped with the $C^1$-topology on a three dimensional closed manifold. It is known that there are open sets in which $C^1$-generic diffeomorphisms display uncountably many chain recurrences classes, while only countably many of them may contain periodic orbits. The classes without periodic orbits, called aperiodic classes, are the main subject of this paper. The aim of the paper is to show that aperiodic classes of $C^1$-generic diffeomorphisms can exhibit a variety of topological properties. More specifically, there are $C^1$-generic diffeomorphisms with (1) minimal expansive aperiodic classes, (2) minimal but non-uniquely ergodic aperiodic classes, (3) transitive but non-minimal aperiodic classes, (4) non-transitive, uniquely ergodic aperiodic classes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.13249v1-abstract-full').style.display = 'none'; document.getElementById('2209.13249v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 September, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37C20; 37D30; 57M30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2209.13245">arXiv:2209.13245</a> <span> [<a href="https://arxiv.org/pdf/2209.13245">pdf</a>, <a href="https://arxiv.org/ps/2209.13245">ps</a>, <a href="https://arxiv.org/format/2209.13245">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1017/etds.2023.76">10.1017/etds.2023.76 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A mechanism for ejecting a horseshoe from a partially hyperbolic chain recurrence class </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Shinohara%2C+K">Katsutoshi Shinohara</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2209.13245v1-abstract-short" style="display: inline;"> We give a $C^1$-perturbation technique for ejecting an a priori given finite set of periodic points preserving a given finite set of homo/hetero-clinic intersections from a chain recurrence class of a periodic point. The technique is first stated under a simpler setting called Markov iterated function system, a two dimensional iterated function system in which the compositions are chosen in Markov… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.13245v1-abstract-full').style.display = 'inline'; document.getElementById('2209.13245v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2209.13245v1-abstract-full" style="display: none;"> We give a $C^1$-perturbation technique for ejecting an a priori given finite set of periodic points preserving a given finite set of homo/hetero-clinic intersections from a chain recurrence class of a periodic point. The technique is first stated under a simpler setting called Markov iterated function system, a two dimensional iterated function system in which the compositions are chosen in Markovian way. Then we apply the result to the setting of three dimensional partially hyperbolic diffeomorphisms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.13245v1-abstract-full').style.display = 'none'; document.getElementById('2209.13245v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 September, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37B25; 37D30; 37G35 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Ergod. Th. Dynam. Sys. 44 (2024) 2080-2142 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2101.07391">arXiv:2101.07391</a> <span> [<a href="https://arxiv.org/pdf/2101.07391">pdf</a>, <a href="https://arxiv.org/format/2101.07391">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Upper, down, two-sided Lorenz attractor, collisions, merging and switching </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Barros%2C+D">Diego Barros</a>, <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Pacifico%2C+M+J">Maria Jose Pacifico</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2101.07391v2-abstract-short" style="display: inline;"> We present a slightly modified version of the well known "geometric Lorenz attractor". It consists in a C1 open set O of vector fields in R3 having an attracting region U containing: (1) a unique singular saddle point sigma; (2) a unique attractor Lambda containing the singular point; (3) the maximal invariant in U contains at most 2 chain recurrence classes, which are Lambda and (at most) one hyp… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.07391v2-abstract-full').style.display = 'inline'; document.getElementById('2101.07391v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2101.07391v2-abstract-full" style="display: none;"> We present a slightly modified version of the well known "geometric Lorenz attractor". It consists in a C1 open set O of vector fields in R3 having an attracting region U containing: (1) a unique singular saddle point sigma; (2) a unique attractor Lambda containing the singular point; (3) the maximal invariant in U contains at most 2 chain recurrence classes, which are Lambda and (at most) one hyperbolic horseshoe. The horseshoe and the singular attractor have a collision along the union of 2 co-dimension 1 sub-manifolds which divide O in 3 regions. By crossing this collision locus, the attractor and the horseshoe may merge in a two-sided Lorenz attractor, or they may exchange their nature: the Lorenz attractor expel the singular point sigma and becomes a horseshoe and the horseshoe absorbs sigma becoming a Lorenz attractor. By crossing this collision locus, the attractor and the horseshoe may merge in a two-sided Lorenz attractor, or they may exchange their nature: the Lorenz attractor expel the singular point sigma and becomes a horseshoe and the horseshoe absorbs sigma becoming a Lorenz attractor. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.07391v2-abstract-full').style.display = 'none'; document.getElementById('2101.07391v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 January, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">64 pages and 21 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2011.12332">arXiv:2011.12332</a> <span> [<a href="https://arxiv.org/pdf/2011.12332">pdf</a>, <a href="https://arxiv.org/format/2011.12332">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> On a quadratic form associated with a surface automorphism and its applications to Singularity Theory </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Alan%C3%ADs-L%C3%B3pez%2C+L">Lilia Alan铆s-L贸pez</a>, <a href="/search/math?searchtype=author&query=Bartolo%2C+E+A">Enrique Artal Bartolo</a>, <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=G%C3%B3mez-Mont%2C+X">Xavier G贸mez-Mont</a>, <a href="/search/math?searchtype=author&query=Villa%2C+M+G">Manuel Gonz谩lez Villa</a>, <a href="/search/math?searchtype=author&query=Cuadrado%2C+P+P">Pablo Portilla Cuadrado</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2011.12332v3-abstract-short" style="display: inline;"> We study the nilpotent part $N'$ of a pseudo-periodic automorphism $h$ of a real oriented surface with boundary $危$. We associate a quadratic form $Q$ defined on the first homology group (relative to the boundary) of the surface $危$. Using the twist formula and techniques from mapping class group theory, we prove that the form $\tilde{Q}$ obtained after killing ${\ker N}$ is positive definite if a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.12332v3-abstract-full').style.display = 'inline'; document.getElementById('2011.12332v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2011.12332v3-abstract-full" style="display: none;"> We study the nilpotent part $N'$ of a pseudo-periodic automorphism $h$ of a real oriented surface with boundary $危$. We associate a quadratic form $Q$ defined on the first homology group (relative to the boundary) of the surface $危$. Using the twist formula and techniques from mapping class group theory, we prove that the form $\tilde{Q}$ obtained after killing ${\ker N}$ is positive definite if all the screw numbers associated with certain orbits of annuli are positive. We also prove that the restriction of $\tilde Q$ to the absolute homology group of $危$ is even whenever the quotient of the Nielsen-Thurston graph under the action of the automorphism is a tree. The case of monodromy automorphisms of Milnor fibers $危=F$ of germs of curves on normal surface singularities is discussed in detail, and the aforementioned results are specialized to such situation. Moreover, the form $\tilde{Q}$ is computable in terms of the dual resolution or semistable reduction graph, as illustrated with several examples. Numerical invariants associated with $\tilde{Q}$ are able to distinguish plane curve singularities with different topological types but same spectral pairs. Finally, we discuss a generic linear germ defined on a superisolated surface. In this case the plumbing graph is not a tree and the restriction of $\tilde Q$ to the absolute monodromy of $危=F$ is not even. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.12332v3-abstract-full').style.display = 'none'; document.getElementById('2011.12332v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 January, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">General re-organization of the paper. Some corrections on examples and correction of a secondary result of the paper</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 32S05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2007.11518">arXiv:2007.11518</a> <span> [<a href="https://arxiv.org/pdf/2007.11518">pdf</a>, <a href="https://arxiv.org/format/2007.11518">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Anosov flows on $3$-manifolds: the surgeries and the foliations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Iakovoglou%2C+I">Ioannis Iakovoglou</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2007.11518v1-abstract-short" style="display: inline;"> To any Anosov flow X on a 3-manifold Fe1 associated a bi-foliated plane (a plane endowed with two transverse foliations Fs and Fu) which reflects the normal structure of the flow endowed with the center-stable and center unstable foliations. A flow is R-covered if Fs (or Fu) is trivial. From one Anosov flow one can build infinitely many others by Dehn-Goodman-Fried surgeries. This paper investig… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2007.11518v1-abstract-full').style.display = 'inline'; document.getElementById('2007.11518v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2007.11518v1-abstract-full" style="display: none;"> To any Anosov flow X on a 3-manifold Fe1 associated a bi-foliated plane (a plane endowed with two transverse foliations Fs and Fu) which reflects the normal structure of the flow endowed with the center-stable and center unstable foliations. A flow is R-covered if Fs (or Fu) is trivial. From one Anosov flow one can build infinitely many others by Dehn-Goodman-Fried surgeries. This paper investigates how these surgeries modify the bi-foliated plane. We first noticed that surgeries along some specific periodic orbits do not modify the bi-foliated plane: for instance, - surgeries on families of orbits corresponding to disjoint simple closed geodesics do not affect the bi-foliated plane associated to the geodesic flow of a hyperbolic surface (Theorem 1); - Fe2 associates a (non-empty) finite family of periodic orbits, called pivots, to any non-R-covered Anosov flow. Surgeries on pivots do not affect the branching structure of the bi-foliated plane (Theorem 2) We consider the set Surg(A) of Anosov flows obtained by Dehn-Goodman-Fried surgery from the suspension flows of Anosov automorphisms A in SL(2,Z) of the torus T2. Every such surgery is associated to a finite set of couples (C,m(C)), where the C are periodic orbits and the m(C) integers. When all the m(C) have the same sign, Fenley proved that the induced Anosov flow is R-covered and twisted according to the sign of the surgery. We analyse here the case where the surgeries are positive on a finite set X and negative on another set Y. Among other results, we show that given any flow X in Surg(A) : - there exists e>0 such that for every e-dense periodic orbit C, every flow obtained from X by a non trivial surgery along C is R-covered (Theorem 4). - there exist periodic orbits C+,C- such that every flow obtained from X by surgeries with distinct signs on C+ and C- is non-R-covered (Theorem 5). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2007.11518v1-abstract-full').style.display = 'none'; document.getElementById('2007.11518v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 July, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">40 pages, 15 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37D20-37D40-57M10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2001.05733">arXiv:2001.05733</a> <span> [<a href="https://arxiv.org/pdf/2001.05733">pdf</a>, <a href="https://arxiv.org/format/2001.05733">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/1361-6544/abf8fa">10.1088/1361-6544/abf8fa <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Lorenz attractors and the modular surface </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Pinsky%2C+T">Tali Pinsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2001.05733v1-abstract-short" style="display: inline;"> We define an extension of the geometric Lorenz model, defined on the three sphere. This geometric model has an invariant one dimensional trefoil knot, a union of invariant manifolds of the singularities. It is similar to the invariant trefoil knot arising in the classical Lorenz flow near the classical parameters. We prove that this geometric model is topologically equivalent to the geodesic flow… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2001.05733v1-abstract-full').style.display = 'inline'; document.getElementById('2001.05733v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2001.05733v1-abstract-full" style="display: none;"> We define an extension of the geometric Lorenz model, defined on the three sphere. This geometric model has an invariant one dimensional trefoil knot, a union of invariant manifolds of the singularities. It is similar to the invariant trefoil knot arising in the classical Lorenz flow near the classical parameters. We prove that this geometric model is topologically equivalent to the geodesic flow on the modular surface, once compactifying the latter. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2001.05733v1-abstract-full').style.display = 'none'; document.getElementById('2001.05733v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 January, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">16 pages, 13 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1910.04620">arXiv:1910.04620</a> <span> [<a href="https://arxiv.org/pdf/1910.04620">pdf</a>, <a href="https://arxiv.org/ps/1910.04620">ps</a>, <a href="https://arxiv.org/format/1910.04620">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.2140/agt.2020.20.3183">10.2140/agt.2020.20.3183 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Small $C^1$ actions of semidirect products on compact manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Kim%2C+S">Sang-hyun Kim</a>, <a href="/search/math?searchtype=author&query=Koberda%2C+T">Thomas Koberda</a>, <a href="/search/math?searchtype=author&query=Triestino%2C+M">Michele Triestino</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1910.04620v3-abstract-short" style="display: inline;"> Let $T$ be a compact fibered $3$--manifold, presented as a mapping torus of a compact, orientable surface $S$ with monodromy $蠄$, and let $M$ be a compact Riemannian manifold. Our main result is that if the induced action $蠄^*$ on $H^1(S,\mathbb{R})$ has no eigenvalues on the unit circle, then there exists a neighborhood $\mathcal U$ of the trivial action in the space of $C^1$ actions of $蟺_1(T)$… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1910.04620v3-abstract-full').style.display = 'inline'; document.getElementById('1910.04620v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1910.04620v3-abstract-full" style="display: none;"> Let $T$ be a compact fibered $3$--manifold, presented as a mapping torus of a compact, orientable surface $S$ with monodromy $蠄$, and let $M$ be a compact Riemannian manifold. Our main result is that if the induced action $蠄^*$ on $H^1(S,\mathbb{R})$ has no eigenvalues on the unit circle, then there exists a neighborhood $\mathcal U$ of the trivial action in the space of $C^1$ actions of $蟺_1(T)$ on $M$ such that any action in $\mathcal{U}$ is abelian. We will prove that the same result holds in the generality of an infinite cyclic extension of an arbitrary finitely generated group $H$, provided that the conjugation action of the cyclic group on $H^1(H,\mathbb{R})\neq 0$ has no eigenvalues of modulus one. We thus generalize a result of A. McCarthy, which addressed the case of abelian--by--cyclic groups acting on compact manifolds. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1910.04620v3-abstract-full').style.display = 'none'; document.getElementById('1910.04620v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 March, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 October, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">11 pages; final version to appear in Algebraic & Geometric Topology</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary 37C85. Secondary 20E22; 57K32 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Algebr. Geom. Topol. 20 (2020) 3183-3203 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1904.05295">arXiv:1904.05295</a> <span> [<a href="https://arxiv.org/pdf/1904.05295">pdf</a>, <a href="https://arxiv.org/ps/1904.05295">ps</a>, <a href="https://arxiv.org/format/1904.05295">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s11425-019-1751-2">10.1007/s11425-019-1751-2 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Transitive partially hyperbolic diffeomorphisms with one-dimensional neutral center </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jinhua Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1904.05295v1-abstract-short" style="display: inline;"> In this paper, we study transitive partially hyperbolic diffeomorphisms with one-dimensional topologically neutral center, meaning that the length of the iterate of small center segments remains small. Such systems are dynamically coherent. We show that there exists a continuous metric along the center foliation which is invariant under the dynamics. As an application, we classify the transitive p… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1904.05295v1-abstract-full').style.display = 'inline'; document.getElementById('1904.05295v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1904.05295v1-abstract-full" style="display: none;"> In this paper, we study transitive partially hyperbolic diffeomorphisms with one-dimensional topologically neutral center, meaning that the length of the iterate of small center segments remains small. Such systems are dynamically coherent. We show that there exists a continuous metric along the center foliation which is invariant under the dynamics. As an application, we classify the transitive partially hyperbolic diffeomorphisms on 3-manifolds with topologically neutral center. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1904.05295v1-abstract-full').style.display = 'none'; document.getElementById('1904.05295v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 April, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">29 pages, 2 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> SCIENCE CHINA Mathematics, 2020 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1810.02384">arXiv:1810.02384</a> <span> [<a href="https://arxiv.org/pdf/1810.02384">pdf</a>, <a href="https://arxiv.org/ps/1810.02384">ps</a>, <a href="https://arxiv.org/format/1810.02384">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Robust existence of nonhyperbolic ergodic measures with positive entropy and full support </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=D%C3%ADaz%2C+L+J">Lorenzo J. D铆az</a>, <a href="/search/math?searchtype=author&query=Kwietniak%2C+D">Dominik Kwietniak</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1810.02384v1-abstract-short" style="display: inline;"> We prove that for some manifolds $M$ the set of robustly transitive partially hyperbolic diffeomorphisms of $M$ with one-dimensional nonhyperbolic centre direction contains a $C^1$-open and dense subset of diffeomorphisms with nonhyperbolic measures which are ergodic, fully supported and have positive entropy. To do so, we formulate abstract conditions sufficient for the construction of an ergodic… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.02384v1-abstract-full').style.display = 'inline'; document.getElementById('1810.02384v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1810.02384v1-abstract-full" style="display: none;"> We prove that for some manifolds $M$ the set of robustly transitive partially hyperbolic diffeomorphisms of $M$ with one-dimensional nonhyperbolic centre direction contains a $C^1$-open and dense subset of diffeomorphisms with nonhyperbolic measures which are ergodic, fully supported and have positive entropy. To do so, we formulate abstract conditions sufficient for the construction of an ergodic, fully supported measure $渭$ which has positive entropy and is such that for a continuous function $蠁\colon X\to\mathbb{R}$ the integral $\int蠁\,d渭$ vanishes. The criterion is an extended version of the control at any scale with a long and sparse tail technique coming from the previous works. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.02384v1-abstract-full').style.display = 'none'; document.getElementById('1810.02384v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 October, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">23 pages, 0 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37D25; 37D35; 37D30; 28D99 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1802.03977">arXiv:1802.03977</a> <span> [<a href="https://arxiv.org/pdf/1802.03977">pdf</a>, <a href="https://arxiv.org/ps/1802.03977">ps</a>, <a href="https://arxiv.org/format/1802.03977">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Anosov diffeomorphism with a horseshoe that attracts almost any point </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">C. Bonatti</a>, <a href="/search/math?searchtype=author&query=Minkov%2C+S">S. Minkov</a>, <a href="/search/math?searchtype=author&query=Okunev%2C+A">A. Okunev</a>, <a href="/search/math?searchtype=author&query=Shilin%2C+I">I. Shilin</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1802.03977v2-abstract-short" style="display: inline;"> We present an example of a C1 Anosov diffeomorphism of a two-torus with a physical measure such that its basin has full Lebesgue measure and its support is a horseshoe of zero measure. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1802.03977v2-abstract-full" style="display: none;"> We present an example of a C1 Anosov diffeomorphism of a two-torus with a physical measure such that its basin has full Lebesgue measure and its support is a horseshoe of zero measure. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1802.03977v2-abstract-full').style.display = 'none'; document.getElementById('1802.03977v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 October, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 February, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages, 3 figures. Rewrote the second page of Introduction, some minor changes (to improve readability)</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37D20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1712.07755">arXiv:1712.07755</a> <span> [<a href="https://arxiv.org/pdf/1712.07755">pdf</a>, <a href="https://arxiv.org/ps/1712.07755">ps</a>, <a href="https://arxiv.org/format/1712.07755">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1112/plms.12321">10.1112/plms.12321 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Anomalous Anosov flows revisited </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Barthelm%C3%A9%2C+T">Thomas Barthelm茅</a>, <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Gogolev%2C+A">Andrey Gogolev</a>, <a href="/search/math?searchtype=author&query=Hertz%2C+F+R">Federico Rodriguez Hertz</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1712.07755v1-abstract-short" style="display: inline;"> This paper is devoted to higher dimensional Anosov flows and consists of two parts. In the first part, we investigate fiberwise Anosov flows on affine torus bundles which fiber over 3-dimensional Anosov flows. We provide a dichotomy result for such flows --- they are either suspensions of Anosov diffeomorphisms or the stable and unstable distributions have equal dimensions. In the second part, w… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1712.07755v1-abstract-full').style.display = 'inline'; document.getElementById('1712.07755v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1712.07755v1-abstract-full" style="display: none;"> This paper is devoted to higher dimensional Anosov flows and consists of two parts. In the first part, we investigate fiberwise Anosov flows on affine torus bundles which fiber over 3-dimensional Anosov flows. We provide a dichotomy result for such flows --- they are either suspensions of Anosov diffeomorphisms or the stable and unstable distributions have equal dimensions. In the second part, we give a new surgery type construction of Anosov flows, which yields non-transitive Anosov flows in all odd dimensions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1712.07755v1-abstract-full').style.display = 'none'; document.getElementById('1712.07755v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 December, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">26 pages, 3 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37D30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1710.08292">arXiv:1710.08292</a> <span> [<a href="https://arxiv.org/pdf/1710.08292">pdf</a>, <a href="https://arxiv.org/format/1710.08292">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1215/00127094-2019-0019">10.1215/00127094-2019-0019 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Topological classification of Morse-Smale diffeomorphisms on 3-manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Ch. Bonatti</a>, <a href="/search/math?searchtype=author&query=Grines%2C+V">V. Grines</a>, <a href="/search/math?searchtype=author&query=Pochinka%2C+O">O. Pochinka</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1710.08292v1-abstract-short" style="display: inline;"> Topological classification of even the simplest Morse-Smale diffeomorphisms on 3-manifolds does not fit into the concept of singling out a skeleton consisting of stable and unstable manifolds of periodic orbits. The reason for this lies primarily in the possible "wild" behaviour of separatrices of saddle points. Another difference between Morse-Smale diffeomorphisms in dimension 3 from their surfa… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1710.08292v1-abstract-full').style.display = 'inline'; document.getElementById('1710.08292v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1710.08292v1-abstract-full" style="display: none;"> Topological classification of even the simplest Morse-Smale diffeomorphisms on 3-manifolds does not fit into the concept of singling out a skeleton consisting of stable and unstable manifolds of periodic orbits. The reason for this lies primarily in the possible "wild" behaviour of separatrices of saddle points. Another difference between Morse-Smale diffeomorphisms in dimension 3 from their surface analogues lies in the variety of heteroclinic intersections: a connected component of such an intersection may be not only a point as in the two-dimensional case, but also a curve, compact or non-compact. The problem of a topological classification of Morse-Smale cascades on 3-manifolds either without heteroclinic points (gradient-like cascades) or without heteroclinic curves was solved in a series of papers from 2000 to 2006 by Ch. Bonatti, V. Grines, F. Laudenbach, V. Medvedev, E. Pecou, O. Pochinka. The present paper is devoted to a complete topological classification of the set $MS(M^3)$ of orientation preserving Morse-Smale diffeomorphisms $f$ given on smooth closed orientable 3-manifolds $M^3$. A complete topological invariant for a diffeomorphism $f\in MS(M^3)$ is an equivalent class of its scheme $S_f$, which contains an information on a periodic date and a topology of embedding of two-dimensional invariant manifolds of the saddle periodic points of $f$ into the ambient manifold. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1710.08292v1-abstract-full').style.display = 'none'; document.getElementById('1710.08292v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 October, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">48 pages, 15 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Duke Math. J. 168, no. 13 (2019), 2507-2558 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1710.06743">arXiv:1710.06743</a> <span> [<a href="https://arxiv.org/pdf/1710.06743">pdf</a>, <a href="https://arxiv.org/format/1710.06743">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1112/plms.12342">10.1112/plms.12342 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Existence of common zeros for commuting vector fields on $3$-manifolds II. Solving global difficulties </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Alvarez%2C+S">S茅bastien Alvarez</a>, <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Santiago%2C+B">Bruno Santiago</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1710.06743v4-abstract-short" style="display: inline;"> We address the following conjecture about the existence of common zeros for commuting vector fields in dimension three: if $X,Y$ are two $C^1$ commuting vector fields on a $3$-manifold $M$, and $U$ is a relatively compact open such that $X$ does not vanish on the boundary of $U$ and has a non vanishing Poincar茅-Hopf index in $U$, then $X$ and $Y$ have a common zero inside $U$. We prove this conjec… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1710.06743v4-abstract-full').style.display = 'inline'; document.getElementById('1710.06743v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1710.06743v4-abstract-full" style="display: none;"> We address the following conjecture about the existence of common zeros for commuting vector fields in dimension three: if $X,Y$ are two $C^1$ commuting vector fields on a $3$-manifold $M$, and $U$ is a relatively compact open such that $X$ does not vanish on the boundary of $U$ and has a non vanishing Poincar茅-Hopf index in $U$, then $X$ and $Y$ have a common zero inside $U$. We prove this conjecture when $X$ and $Y$ are of class $C^3$ and every periodic orbit of $Y$ along which $X$ and $Y$ are collinear is partially hyperbolic. We also prove the conjecture, still in the $C^3$ setting, assuming that the flow $Y$ leaves invariant a transverse plane field. These results shed new light on the $C^3$ case of the conjecture. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1710.06743v4-abstract-full').style.display = 'none'; document.getElementById('1710.06743v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 April, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 October, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">50 pages, 16 figures. FInal version to appear in Proceedings of the London Mathematical Society</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1709.02521">arXiv:1709.02521</a> <span> [<a href="https://arxiv.org/pdf/1709.02521">pdf</a>, <a href="https://arxiv.org/format/1709.02521">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Projective cocycles over SL(2,R) actions: measures invariant under the upper triangular group </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Eskin%2C+A">Alex Eskin</a>, <a href="/search/math?searchtype=author&query=Wilkinson%2C+A">Amie Wilkinson</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1709.02521v3-abstract-short" style="display: inline;"> We consider the action of $SL(2,\mathbb{R})$ on a vector bundle $\mathbf{H}$ preserving an ergodic probability measure $谓$ on the base $X$. Under an irreducibility assumption on this action, we prove that if $\hat谓$ is any lift of $谓$ to a probability measure on the projectivized bunde $\mathbb{P}(\mathbf{H})$ that is invariant under the upper triangular subgroup, then $\hat 谓$ is supported in the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.02521v3-abstract-full').style.display = 'inline'; document.getElementById('1709.02521v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1709.02521v3-abstract-full" style="display: none;"> We consider the action of $SL(2,\mathbb{R})$ on a vector bundle $\mathbf{H}$ preserving an ergodic probability measure $谓$ on the base $X$. Under an irreducibility assumption on this action, we prove that if $\hat谓$ is any lift of $谓$ to a probability measure on the projectivized bunde $\mathbb{P}(\mathbf{H})$ that is invariant under the upper triangular subgroup, then $\hat 谓$ is supported in the projectivization $\mathbb{P}(\mathbf{E}_1)$ of the top Lyapunov subspace of the positive diagonal semigroup. We derive two applications. First, the Lyapunov exponents for the Kontsevich-Zorich cocycle depend continuously on affine measures, answering a question in [MMY]. Second, if $\mathbb{P}(\mathbf{V})$ is an irreducible, flat projective bundle over a compact hyperbolic surface $危$, with hyperbolic foliation $\mathcal{F}$ tangent to the flat connection, then the foliated horocycle flow on $T^1\mathcal{F}$ is uniquely ergodic if the top Lyapunov exponent of the foliated geodesic flow is simple. This generalizes results in [BG] to arbitrary dimension. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.02521v3-abstract-full').style.display = 'none'; document.getElementById('1709.02521v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 June, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 September, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Minor corrections. 24 pages, 1 figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37C40; 37A05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1706.05704">arXiv:1706.05704</a> <span> [<a href="https://arxiv.org/pdf/1706.05704">pdf</a>, <a href="https://arxiv.org/format/1706.05704">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.2140/gt.2019.23.1841">10.2140/gt.2019.23.1841 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Hyperbolicity as an obstruction to smoothability for one-dimensional actions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Lodha%2C+Y">Yash Lodha</a>, <a href="/search/math?searchtype=author&query=Triestino%2C+M">Michele Triestino</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1706.05704v4-abstract-short" style="display: inline;"> Ghys and Sergiescu proved in the $80$s that Thompson's group $T$, and hence $F$, admits actions by $C^{\infty}$ diffeomorphisms of the circle . They proved that the standard actions of these groups are topologically conjugate to a group of $C^\infty$ diffeomorphisms. Monod defined a family of groups of piecewise projective homeomorphisms, and Lodha-Moore defined finitely presentable groups of piec… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1706.05704v4-abstract-full').style.display = 'inline'; document.getElementById('1706.05704v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1706.05704v4-abstract-full" style="display: none;"> Ghys and Sergiescu proved in the $80$s that Thompson's group $T$, and hence $F$, admits actions by $C^{\infty}$ diffeomorphisms of the circle . They proved that the standard actions of these groups are topologically conjugate to a group of $C^\infty$ diffeomorphisms. Monod defined a family of groups of piecewise projective homeomorphisms, and Lodha-Moore defined finitely presentable groups of piecewise projective homeomorphisms. These groups are of particular interest because they are nonamenable and contain no free subgroup. In contrast to the result of Ghys-Sergiescu, we prove that the groups of Monod and Lodha-Moore are not topologically conjugate to a group of $C^1$ diffeomorphisms. Furthermore, we show that the group of Lodha-Moore has no nonabelian $C^1$ action on the interval. We also show that many Monod's groups $H(A)$, for instance when $A$ is such that $\mathsf{PSL}(2,A)$ contains a rational homothety $x\mapsto \tfrac{p}{q}x$, do not admit a $C^1$ action on the interval. The obstruction comes from the existence of hyperbolic fixed points for $C^1$ actions. With slightly different techniques, we also show that some groups of piecewise affine homeomorphisms of the interval or the circle are not smoothable. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1706.05704v4-abstract-full').style.display = 'none'; document.getElementById('1706.05704v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 April, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 June, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">26 pages, 1 figure. Arithmetic conditions in the main theorems have been weakened</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37C85; 57M60 (Primary); 43A07; 37D40; 37E05 (Secondary) </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Geom. Topol. 23 (2019) 1841-1876 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1706.04962">arXiv:1706.04962</a> <span> [<a href="https://arxiv.org/pdf/1706.04962">pdf</a>, <a href="https://arxiv.org/format/1706.04962">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.2140/gt.2020.24.1751">10.2140/gt.2020.24.1751 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Anomalous partially hyperbolic diffeomorphisms III: abundance and incoherence </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Gogolev%2C+A">Andrey Gogolev</a>, <a href="/search/math?searchtype=author&query=Hammerlindl%2C+A">Andy Hammerlindl</a>, <a href="/search/math?searchtype=author&query=Potrie%2C+R">Rafael Potrie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1706.04962v2-abstract-short" style="display: inline;"> Let $M$ be a closed 3-manifold which admits an Anosov flow. In this paper we develop a technique for constructing partially hyperbolic representatives in many mapping classes of $M$. We apply this technique both in the setting of geodesic flows on closed hyperbolic surfaces and for Anosov flows which admit transverse tori. We emphasize the similarity of both constructions through the concept of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1706.04962v2-abstract-full').style.display = 'inline'; document.getElementById('1706.04962v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1706.04962v2-abstract-full" style="display: none;"> Let $M$ be a closed 3-manifold which admits an Anosov flow. In this paper we develop a technique for constructing partially hyperbolic representatives in many mapping classes of $M$. We apply this technique both in the setting of geodesic flows on closed hyperbolic surfaces and for Anosov flows which admit transverse tori. We emphasize the similarity of both constructions through the concept of $h$-transversality, a tool which allows us to compose different mapping classes while retaining partial hyperbolicity. In the case of the geodesic flow of a closed hyperbolic surface $S$ we build stably ergodic, partially hyperbolic diffeomorphisms whose mapping classes form a subgroup of the mapping class group $\mathcal{M}(T^1S)$ which is isomorphic to $\mathcal{M}(S)$. At the same time we show that the totality of mapping classes which can be realized by partially hyperbolic diffeomorphisms does not form a subgroup of $\mathcal{M}(T^1S)$. Finally, some of the examples on $T^1S$ are absolutely partially hyperbolic, stably ergodic and robustly non-dynamically coherent, disproving a conjecture by F. Rodriguez Hertz, J. Rodriguez Hertz and R. Ures. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1706.04962v2-abstract-full').style.display = 'none'; document.getElementById('1706.04962v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 November, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 June, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">31 pages, 4 figures. To appear in G&T</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Geom. Topol. 24 (2020) 1751-1790 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1705.05799">arXiv:1705.05799</a> <span> [<a href="https://arxiv.org/pdf/1705.05799">pdf</a>, <a href="https://arxiv.org/format/1705.05799">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Star fows and multisingular hyperbolicity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=da+Luz%2C+A">Adriana da Luz</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1705.05799v2-abstract-short" style="display: inline;"> A vector field X is called a star flow if every periodic orbit, of any vector field C1-close to X, is hyperbolic. It is known that the chain recurrence classes of a generic star flow X on a 3 or 4 manifold are either hyperbolic or singular hyperbolic (see [MPP] for 3-manifolds and [GLW] on 4-manifolds). As it is defined, the notion of singular hyperbolicity forces the singularities in the same cla… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1705.05799v2-abstract-full').style.display = 'inline'; document.getElementById('1705.05799v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1705.05799v2-abstract-full" style="display: none;"> A vector field X is called a star flow if every periodic orbit, of any vector field C1-close to X, is hyperbolic. It is known that the chain recurrence classes of a generic star flow X on a 3 or 4 manifold are either hyperbolic or singular hyperbolic (see [MPP] for 3-manifolds and [GLW] on 4-manifolds). As it is defined, the notion of singular hyperbolicity forces the singularities in the same class to have the same index. However, in higher dimension (i.e $\geq 5$) \cite{BdL} shows that singularities of different indices may be robustly in the same chain recurrence class of a star flow. Therefore the usual notion of singular hyperbolicity is not enough for characterizing the star flows. We present a form of hyperbolicity (called multi-singular hyperbolic) which makes compatible the hyperbolic structure of regular orbits together with the one of singularities even if they have different indices. We show that multisingular hyperbolicity implies that the flow is star, and conversely, there is a C1-open and dense subset of the an open set of star flows which are multisingular hyperbolic. More generally, for most of the hyperbolic structures (dominated splitting, partial hyperbolicity etc...) well defined on regular orbits, we propose a way for generalizing it to a compact set containing singular points. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1705.05799v2-abstract-full').style.display = 'none'; document.getElementById('1705.05799v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 October, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 May, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">There are new results in section 7 compared with the previous version</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1702.04960">arXiv:1702.04960</a> <span> [<a href="https://arxiv.org/pdf/1702.04960">pdf</a>, <a href="https://arxiv.org/format/1702.04960">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> Topological classification of Morse-Smale diffeomorphisms without heteroclinic curves on 3-manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Ch Bonatti</a>, <a href="/search/math?searchtype=author&query=Grines%2C+V">V Grines</a>, <a href="/search/math?searchtype=author&query=Laudenbach%2C+F">F Laudenbach</a>, <a href="/search/math?searchtype=author&query=Pochinka%2C+O">O Pochinka</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1702.04960v2-abstract-short" style="display: inline;"> We show that, up to topological conjugation, the equivalence class of a Morse-Smale diffeomorphism without heteroclinic curves on 3-manifold is completely defined by an em- bedding of two-dimensional stable and unstable heteroclinic laminations to a characteristic space. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1702.04960v2-abstract-full" style="display: none;"> We show that, up to topological conjugation, the equivalence class of a Morse-Smale diffeomorphism without heteroclinic curves on 3-manifold is completely defined by an em- bedding of two-dimensional stable and unstable heteroclinic laminations to a characteristic space. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.04960v2-abstract-full').style.display = 'none'; document.getElementById('1702.04960v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 September, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 February, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2017. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1609.08489">arXiv:1609.08489</a> <span> [<a href="https://arxiv.org/pdf/1609.08489">pdf</a>, <a href="https://arxiv.org/ps/1609.08489">ps</a>, <a href="https://arxiv.org/format/1609.08489">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Periodic measures and partially hyperbolic homoclinic classes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jinhua Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1609.08489v1-abstract-short" style="display: inline;"> In this paper, we give a precise meaning to the following fact, and we prove it: $C^1$-open and densely, all the non-hyperbolic ergodic measures generated by a robust cycle are approximated by periodic measures. We apply our technique to the global setting of partially hyperbolic diffeomorphisms with one dimensional center. When both strong stable and unstable foliations are minimal, we get that… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1609.08489v1-abstract-full').style.display = 'inline'; document.getElementById('1609.08489v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1609.08489v1-abstract-full" style="display: none;"> In this paper, we give a precise meaning to the following fact, and we prove it: $C^1$-open and densely, all the non-hyperbolic ergodic measures generated by a robust cycle are approximated by periodic measures. We apply our technique to the global setting of partially hyperbolic diffeomorphisms with one dimensional center. When both strong stable and unstable foliations are minimal, we get that the closure of the set of ergodic measures is the union of two convex sets corresponding to the two possible $s$-indices; these two convex sets intersect along the closure of the set of non-hyperbolic ergodic measures. That is the case for robustly transitive perturbation of the time one map of a transitive Anosov flow, or of the skew product of an Anosov torus diffeomorphism by rotation of the circle. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1609.08489v1-abstract-full').style.display = 'none'; document.getElementById('1609.08489v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 September, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">48 pages, 5 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Trans. Amer. Math. Soc., 2019 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1609.07764">arXiv:1609.07764</a> <span> [<a href="https://arxiv.org/pdf/1609.07764">pdf</a>, <a href="https://arxiv.org/format/1609.07764">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> A criterion for zero averages and full support of ergodic measures </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Diaz%2C+L+J">Lorenzo J. Diaz</a>, <a href="/search/math?searchtype=author&query=Bochi%2C+J">Jairo Bochi</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1609.07764v1-abstract-short" style="display: inline;"> Consider a homeomorphism $f$ defined on a compact metric space $X$ and a continuous map $蠁\colon X \to \mathbb{R}$. We provide an abstract criterion, called \emph{control at any scale with a long sparse tail} for a point $x\in X$ and the map $蠁$, that guarantees that any weak$\ast$ limit measure $渭$ of the Birkhoff average of Dirac measures $\frac1n\sum_0^{n-1}未(f^i(x))$ is such that $渭$-almost ev… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1609.07764v1-abstract-full').style.display = 'inline'; document.getElementById('1609.07764v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1609.07764v1-abstract-full" style="display: none;"> Consider a homeomorphism $f$ defined on a compact metric space $X$ and a continuous map $蠁\colon X \to \mathbb{R}$. We provide an abstract criterion, called \emph{control at any scale with a long sparse tail} for a point $x\in X$ and the map $蠁$, that guarantees that any weak$\ast$ limit measure $渭$ of the Birkhoff average of Dirac measures $\frac1n\sum_0^{n-1}未(f^i(x))$ is such that $渭$-almost every point $y$ has a dense orbit in $X$ and the Birkhoff average of $蠁$ along the orbit of $y$ is zero. As an illustration of the strength of this criterion, we prove that the diffeomorphisms with nonhyperbolic ergodic measures form a $C^1$-open and dense subset of the set of robustly transitive partially hyperbolic diffeomorphisms with one dimensional nonhyperbolic central direction. We also obtain applications for nonhyperbolic homoclinic classes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1609.07764v1-abstract-full').style.display = 'none'; document.getElementById('1609.07764v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 September, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37D25; 37D30; 37D35; 28D99 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1608.02848">arXiv:1608.02848</a> <span> [<a href="https://arxiv.org/pdf/1608.02848">pdf</a>, <a href="https://arxiv.org/format/1608.02848">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="History and Overview">math.HO</span> </div> </div> <p class="title is-5 mathjax"> What is ... a blender? </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Crovisier%2C+S">Sylvain Crovisier</a>, <a href="/search/math?searchtype=author&query=Diaz%2C+L">Lorenzo Diaz</a>, <a href="/search/math?searchtype=author&query=Wilkinson%2C+A">Amie Wilkinson</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1608.02848v1-abstract-short" style="display: inline;"> What is a blender? In six illustrated pages we define the construction of a blender and the role it plays in the study of smooth dynamical systems. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1608.02848v1-abstract-full" style="display: none;"> What is a blender? In six illustrated pages we define the construction of a blender and the role it plays in the study of smooth dynamical systems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1608.02848v1-abstract-full').style.display = 'none'; document.getElementById('1608.02848v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 August, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">To appear in Notices of the AMS</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37C05; 37C20; 37C70 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1606.06119">arXiv:1606.06119</a> <span> [<a href="https://arxiv.org/pdf/1606.06119">pdf</a>, <a href="https://arxiv.org/ps/1606.06119">ps</a>, <a href="https://arxiv.org/format/1606.06119">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> On the existence of non-hyperbolic ergodic measures as the limit of periodic measures </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jinhua Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1606.06119v1-abstract-short" style="display: inline;"> [GIKN] and [BBD1] propose two very different ways for building non hyperbolic measures, [GIKN] building such a measure as the limit of periodic measures and [BBD1] as the $蠅$-limit set of a single orbit, with a uniformly vanishing Lyapunov exponent. The technique in [GIKN] was essentially used in a generic setting, as the periodic orbits were built by small perturbations. It is not known if the me… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1606.06119v1-abstract-full').style.display = 'inline'; document.getElementById('1606.06119v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1606.06119v1-abstract-full" style="display: none;"> [GIKN] and [BBD1] propose two very different ways for building non hyperbolic measures, [GIKN] building such a measure as the limit of periodic measures and [BBD1] as the $蠅$-limit set of a single orbit, with a uniformly vanishing Lyapunov exponent. The technique in [GIKN] was essentially used in a generic setting, as the periodic orbits were built by small perturbations. It is not known if the measures obtained by the technique in [BBD1] are accumulated by periodic measures. In this paper we use a shadowing lemma from [G]: $\bullet$for getting the periodic orbits in [GIKN] without perturbing the dynamics, $\bullet$for recovering the compact set in [BBD1] with a uniformly vanishing Lyapunov exponent by considering the limit of periodic orbits. As a consequence, we prove that there exists an open and dense subset $\mathcal{U}$ of the set of robustly transitive non-hyperbolic diffeomorphisms far from homoclinic tangencies, such that for any $f\in\mathcal{U}$, there exists a non-hyperbolic ergodic measure with full support and approximated by hyperbolic periodic measures. We also prove that there exists an open and dense subset $\mathcal{V}$ of the set of diffeomorphisms exhibiting a robust cycle, such that for any $f\in\mathcal{V}$, there exists a non-hyperbolic ergodic measure approximated by hyperbolic periodic measures. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1606.06119v1-abstract-full').style.display = 'none'; document.getElementById('1606.06119v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 June, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">37 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Ergodic Theory Dynam. Systems, 2019 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1603.03803">arXiv:1603.03803</a> <span> [<a href="https://arxiv.org/pdf/1603.03803">pdf</a>, <a href="https://arxiv.org/ps/1603.03803">ps</a>, <a href="https://arxiv.org/format/1603.03803">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Many intermingled basins in dimension 3 </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Potrie%2C+R">Rafael Potrie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1603.03803v2-abstract-short" style="display: inline;"> We construct a diffeomorphism of $\mathbb{T}^3$ admitting any finite or countable number of physical measures with intermingled basins. The examples are partially hyperbolic with splitting $T\mathbb{T}^3 = E^{cs} \oplus E^u$ and can be made volume hyperbolic and topologically mixing. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1603.03803v2-abstract-full" style="display: none;"> We construct a diffeomorphism of $\mathbb{T}^3$ admitting any finite or countable number of physical measures with intermingled basins. The examples are partially hyperbolic with splitting $T\mathbb{T}^3 = E^{cs} \oplus E^u$ and can be made volume hyperbolic and topologically mixing. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1603.03803v2-abstract-full').style.display = 'none'; document.getElementById('1603.03803v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 February, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 March, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">20 pages, 4 figures. Some changes made after referee report. To appear in Israel J. of Math</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1602.04355">arXiv:1602.04355</a> <span> [<a href="https://arxiv.org/pdf/1602.04355">pdf</a>, <a href="https://arxiv.org/ps/1602.04355">ps</a>, <a href="https://arxiv.org/format/1602.04355">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Transverse foliations on the torus $\T^2$ and partially hyperbolic diffeomorphisms on 3-manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jinhua Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1602.04355v1-abstract-short" style="display: inline;"> In this paper, we prove that given two $C^1$ foliations $\mathcal{F}$ and $\mathcal{G}$ on $\mathbb{T}^2$ which are transverse, there exists a non-null homotopic loop $\{桅_t\}_{t\in[0,1]}$ in $\diff^{1}(\T^2)$ such that $桅_t(\calF)\pitchfork \calG$ for every $t\in[0,1]$, and $桅_0=桅_1= Id$. As a direct consequence, we get a general process for building new partially hyperbolic diffeomorphisms on cl… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1602.04355v1-abstract-full').style.display = 'inline'; document.getElementById('1602.04355v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1602.04355v1-abstract-full" style="display: none;"> In this paper, we prove that given two $C^1$ foliations $\mathcal{F}$ and $\mathcal{G}$ on $\mathbb{T}^2$ which are transverse, there exists a non-null homotopic loop $\{桅_t\}_{t\in[0,1]}$ in $\diff^{1}(\T^2)$ such that $桅_t(\calF)\pitchfork \calG$ for every $t\in[0,1]$, and $桅_0=桅_1= Id$. As a direct consequence, we get a general process for building new partially hyperbolic diffeomorphisms on closed $3$-manifolds. \cite{BPP} built a new example of dynamically coherent non-transitive partially hyperbolic diffeomorphism on a closed $3$-manifold, the example in \cite{BPP} is obtained by composing the time $t$ map, $t>0$ large enough, of a very specific non-transitive Anosov flow by a Dehn twist along a transverse torus. Our result shows that the same construction holds starting with any non-transitive Anosov flow on an oriented $3$-manifold. Moreover, for a given transverse torus, our result explains which type of Dehn twists lead to partially hyperbolic diffeomorphisms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1602.04355v1-abstract-full').style.display = 'none'; document.getElementById('1602.04355v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 February, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">34 pages, 7 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Comment. Math. Helv. 2017 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1510.05026">arXiv:1510.05026</a> <span> [<a href="https://arxiv.org/pdf/1510.05026">pdf</a>, <a href="https://arxiv.org/ps/1510.05026">ps</a>, <a href="https://arxiv.org/format/1510.05026">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1017/etds.2018.61">10.1017/etds.2018.61 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Foliated hyperbolicity and foliations with hyperbolic leaves </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=G%C3%B3mez-Mont%2C+X">Xavier G贸mez-Mont</a>, <a href="/search/math?searchtype=author&query=Mart%C3%ADnez%2C+M">Matilde Mart铆nez</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1510.05026v1-abstract-short" style="display: inline;"> Given a lamination in a compact space and a laminated vector field $X$ which is hyperbolic when restricted to the leaves of the lamination, we distinguish a class of $X$-invariant probabilities that describe the behaviour of almost every $X$-orbit in every leaf, that we call u-Gibbs states. We apply this to the case of foliations in compact manifolds having leaves with negative curvature, using th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1510.05026v1-abstract-full').style.display = 'inline'; document.getElementById('1510.05026v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1510.05026v1-abstract-full" style="display: none;"> Given a lamination in a compact space and a laminated vector field $X$ which is hyperbolic when restricted to the leaves of the lamination, we distinguish a class of $X$-invariant probabilities that describe the behaviour of almost every $X$-orbit in every leaf, that we call u-Gibbs states. We apply this to the case of foliations in compact manifolds having leaves with negative curvature, using the foliated hyperbolic vector field on the unit tangent bundle to the foliation generating the leaf geodesics. When the Lyapunov exponents of such an ergodic u-Gibbs states are negative, it is an SRB-measure (having a positive Lebesgue basin of attraction). When the foliation is by hyperbolic leaves, this class of probabilities coincide with the classical harmonic measures introduced by L. Garnett. If furthermore the foliation is transversally conformal and does not admit a transverse invariant measure we show that the ergodic u-Gibbs states are finitely many, supported each in one minimal set of the foliation, have negative Lyapunov exponents and the union of their basins of attraction has full Lebesgue measure. The leaf geodesics emanating from a point have a proportion whose asymptotic statistics is described by each of these ergodic u-Gibbs states, giving rise to continuous visibility functions of the attractors. Reversing time, by considering $-X$, we obtain the existence of the same number of repellors of the foliated geodesic flow having the same harmonic measures as projections to $M$. In the case of only 1 attractor, we obtain a North to South pole dynamics. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1510.05026v1-abstract-full').style.display = 'none'; document.getElementById('1510.05026v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 October, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Ergod. Th. Dynam. Sys. 40 (2020) 881-903 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1506.07804">arXiv:1506.07804</a> <span> [<a href="https://arxiv.org/pdf/1506.07804">pdf</a>, <a href="https://arxiv.org/ps/1506.07804">ps</a>, <a href="https://arxiv.org/format/1506.07804">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00222-016-0663-7">10.1007/s00222-016-0663-7 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Anomalous partially hyperbolic diffeomorphisms II: stably ergodic examples </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Gogolev%2C+A">Andrey Gogolev</a>, <a href="/search/math?searchtype=author&query=Potrie%2C+R">Rafael Potrie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1506.07804v2-abstract-short" style="display: inline;"> We construct examples of robustly transitive and stably ergodic partially hyperbolic diffeomorphisms $f$ on compact $3$-manifolds with fundamental groups of exponential growth such that $f^n$ is not homotopic to identity for all $n>0$. These provide counterexamples to a classification conjecture of Pujals. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1506.07804v2-abstract-full" style="display: none;"> We construct examples of robustly transitive and stably ergodic partially hyperbolic diffeomorphisms $f$ on compact $3$-manifolds with fundamental groups of exponential growth such that $f^n$ is not homotopic to identity for all $n>0$. These provide counterexamples to a classification conjecture of Pujals. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1506.07804v2-abstract-full').style.display = 'none'; document.getElementById('1506.07804v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 April, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 25 June, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">30 pages, 7 figures. Updated version after referee's remarks. To appear in Inventiones Math</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1505.07901">arXiv:1505.07901</a> <span> [<a href="https://arxiv.org/pdf/1505.07901">pdf</a>, <a href="https://arxiv.org/ps/1505.07901">ps</a>, <a href="https://arxiv.org/format/1505.07901">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1017/etds.2016.51">10.1017/etds.2016.51 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Volume hyperbolicity and wildness </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Shinohara%2C+K">Katsutoshi Shinohara</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1505.07901v1-abstract-short" style="display: inline;"> It is known that volume hyperbolicity (partial hyperbolicity and uniform expansion or contraction of the volume in the extremal bundles) is a necessary condition for robust transitivity or robust chain recurrence hence for tameness. In this paper, on any 3-manifold we build examples of quasi-attractors which are volume hyperbolic and wild at the same time. As a main corollary, we see that, for any… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.07901v1-abstract-full').style.display = 'inline'; document.getElementById('1505.07901v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1505.07901v1-abstract-full" style="display: none;"> It is known that volume hyperbolicity (partial hyperbolicity and uniform expansion or contraction of the volume in the extremal bundles) is a necessary condition for robust transitivity or robust chain recurrence hence for tameness. In this paper, on any 3-manifold we build examples of quasi-attractors which are volume hyperbolic and wild at the same time. As a main corollary, we see that, for any closed 3-manifold $M$, the space $\mathrm{Diff}^1(M)$ admits a non-empty open set where every $C^1$-generic diffeomorphism has no attractors or repellers. The main tool of our construction is the notion of flexible periodic points introduced by the authors. For ejecting the flexible points from the quasi-attractor, we control the topology of the quasi-attractor using the notion of partially hyperbolic filtrating Markov partition, which we introduce in this paper. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.07901v1-abstract-full').style.display = 'none'; document.getElementById('1505.07901v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 May, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37C05; 37C15; 37C70; 37D45 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1505.06259">arXiv:1505.06259</a> <span> [<a href="https://arxiv.org/pdf/1505.06259">pdf</a>, <a href="https://arxiv.org/ps/1505.06259">ps</a>, <a href="https://arxiv.org/format/1505.06259">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> A spectral-like decomposition for transitive Anosov flows in dimension three </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=B%C3%A9guin%2C+F">Fran莽ois B茅guin</a>, <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Yu%2C+B">Bin Yu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1505.06259v1-abstract-short" style="display: inline;"> Given a (transitive or non-transitive) Anosov vector field $X$ on a closed three-dimensional manifold $M$, one may try to decompose $(M,X)$ by cutting $M$ along two-tori transverse to $X$. We prove that one can find a finite collection $\{T_1,\dots,T_n\}$ of pairwise disjoint, pairwise non-parallel incompressible tori transverse to $X$, such that the maximal invariant sets $螞_1,\dots,螞_m$ of the c… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.06259v1-abstract-full').style.display = 'inline'; document.getElementById('1505.06259v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1505.06259v1-abstract-full" style="display: none;"> Given a (transitive or non-transitive) Anosov vector field $X$ on a closed three-dimensional manifold $M$, one may try to decompose $(M,X)$ by cutting $M$ along two-tori transverse to $X$. We prove that one can find a finite collection $\{T_1,\dots,T_n\}$ of pairwise disjoint, pairwise non-parallel incompressible tori transverse to $X$, such that the maximal invariant sets $螞_1,\dots,螞_m$ of the connected components $V_1,\dots,V_m$ of $M-(T_1\cup\dots\cup T_n)$ satisfy the following properties: 1, each $螞_i$ is a compact invariant locally maximal transitive set for $X$, 2, the collection $\{螞_1,\dots,螞_m\}$ is canonically attached to the pair $(M,X)$ (i.e., it can be defined independently of the collection of tori $\{T_1,\dots,T_n\}$), 3, the $螞_i$'s are the smallest possible: for every (possibly infinite) collection $\{S_i\}_{i\in I}$ of tori transverse to $X$, the $螞_i$'s are contained in the maximal invariant set of $M-\cup_i S_i$. To a certain extent, the sets $螞_1,\dots,螞_m$ are analogs (for Anosov vector field in dimension 3) of the basic pieces which appear in the spectral decomposition of a non-transitive axiom A vector field. Then we discuss the uniqueness of such a decomposition: we prove that the pieces of the decomposition $V_1,\dots,V_m$, equipped with the restriction of the Anosov vector field $X$, are "almost unique up to topological equivalence". <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.06259v1-abstract-full').style.display = 'none'; document.getElementById('1505.06259v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 May, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages, 4 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1504.06104">arXiv:1504.06104</a> <span> [<a href="https://arxiv.org/pdf/1504.06104">pdf</a>, <a href="https://arxiv.org/format/1504.06104">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Existence of common zeros for commuting vector fields on $3$-manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Santiago%2C+B">Bruno Santiago</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1504.06104v2-abstract-short" style="display: inline;"> In $64$ E. Lima proved that commuting vector fields on surfaces with non-zero Euler characteristic have common zeros. Such statement is empty in dimension $3$, since all the Euler characteristics vanish. Nevertheless, \cite{Bonatti_analiticos} proposed a local version, replacing the Euler characteristic by the Poincar茅-Hopf index of a vector field $X$ in a region $U$, denoted by… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1504.06104v2-abstract-full').style.display = 'inline'; document.getElementById('1504.06104v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1504.06104v2-abstract-full" style="display: none;"> In $64$ E. Lima proved that commuting vector fields on surfaces with non-zero Euler characteristic have common zeros. Such statement is empty in dimension $3$, since all the Euler characteristics vanish. Nevertheless, \cite{Bonatti_analiticos} proposed a local version, replacing the Euler characteristic by the Poincar茅-Hopf index of a vector field $X$ in a region $U$, denoted by $\operatorname{Ind}(X,U)$; he asked: \emph{Given commuting vector fields $X,Y$ and a region $U$ where $\operatorname{Ind}(X,U)\neq 0$, does $U$ contain a common zero of $X$ and $Y$?} \cite{Bonatti_analiticos} gave a positive answer in the case where $X$ and $Y$ are real analytic. In this paper, we prove the existence of common zeros for commuting $C^1$ vector fields $X$, $Y$ on a $3$-manifold, in any region $U$ such that $\operatorname{Ind}(X,U)\neq 0$, assuming that the set of collinearity of $X$ and $Y$ is contained in a smooth surface. This is a strong indication that the results in \cite{Bonatti_analiticos} should hold for $C^1$-vector fields. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1504.06104v2-abstract-full').style.display = 'none'; document.getElementById('1504.06104v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 November, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 April, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Final version, to appear in Annales de L'Institut Fourier</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1503.05901">arXiv:1503.05901</a> <span> [<a href="https://arxiv.org/pdf/1503.05901">pdf</a>, <a href="https://arxiv.org/format/1503.05901">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Dominated Pesin theory: convex sum of hyperbolic measures </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bochi%2C+J">Jairo Bochi</a>, <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Gelfert%2C+K">Katrin Gelfert</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1503.05901v3-abstract-short" style="display: inline;"> In the uniformly hyperbolic setting it is well known that the set of all measures supported on periodic orbits is dense in the convex space of all invariant measures. In this paper we consider the converse question, in the non-uniformly hyperbolic setting: assuming that some ergodic measure converges to a convex combination of hyperbolic ergodic measures, what can we deduce about the initial measu… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1503.05901v3-abstract-full').style.display = 'inline'; document.getElementById('1503.05901v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1503.05901v3-abstract-full" style="display: none;"> In the uniformly hyperbolic setting it is well known that the set of all measures supported on periodic orbits is dense in the convex space of all invariant measures. In this paper we consider the converse question, in the non-uniformly hyperbolic setting: assuming that some ergodic measure converges to a convex combination of hyperbolic ergodic measures, what can we deduce about the initial measures? To every hyperbolic measure $渭$ whose stable/unstable Oseledets splitting is dominated we associate canonically a unique class $H(渭)$ of periodic orbits for the homoclinic relation, called its \emph{intersection class}. In a dominated setting, we prove that a measure for which almost every measure in its ergodic decomposition is hyperbolic with the same index such as the dominated splitting is accumulated by ergodic measures if, and only if, almost all such ergodic measures have a common intersection class. We provide examples which indicate the importance of the domination assumption. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1503.05901v3-abstract-full').style.display = 'none'; document.getElementById('1503.05901v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 July, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 March, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">final version, new co-author, to appear in: Israel Journal of Mathematics</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37C29; 37C40; 37C50; 37D25; 37D30; 28A33 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1502.06535">arXiv:1502.06535</a> <span> [<a href="https://arxiv.org/pdf/1502.06535">pdf</a>, <a href="https://arxiv.org/format/1502.06535">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00220-016-2644-5">10.1007/s00220-016-2644-5 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Robust criterion for the existence of nonhyperbolic ergodic measures </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bochi%2C+J">Jairo Bochi</a>, <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=D%C3%ADaz%2C+L+J">Lorenzo J. D铆az</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1502.06535v1-abstract-short" style="display: inline;"> We give explicit $C^1$-open conditions that ensure that a diffeomorphism possesses a nonhyperbolic ergodic measure with positive entropy. Actually, our criterion provides the existence of a partially hyperbolic compact set with one-dimensional center and positive topological entropy on which the center Lyapunov exponent vanishes uniformly. The conditions of the criterion are met on a $C^1$-dense… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1502.06535v1-abstract-full').style.display = 'inline'; document.getElementById('1502.06535v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1502.06535v1-abstract-full" style="display: none;"> We give explicit $C^1$-open conditions that ensure that a diffeomorphism possesses a nonhyperbolic ergodic measure with positive entropy. Actually, our criterion provides the existence of a partially hyperbolic compact set with one-dimensional center and positive topological entropy on which the center Lyapunov exponent vanishes uniformly. The conditions of the criterion are met on a $C^1$-dense and open subset of the set of a diffeomorphisms having a robust cycle. As a corollary, there exists a $C^1$-open and dense subset of the set of non-Anosov robustly transitive diffeomorphisms consisting of systems with nonhyperbolic ergodic measures with positive entropy. The criterion is based on a notion of a blender defined dynamically in terms of strict invariance of a family of discs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1502.06535v1-abstract-full').style.display = 'none'; document.getElementById('1502.06535v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 February, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37D30; 37C40; 37D25; 37A35 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1411.1221">arXiv:1411.1221</a> <span> [<a href="https://arxiv.org/pdf/1411.1221">pdf</a>, <a href="https://arxiv.org/ps/1411.1221">ps</a>, <a href="https://arxiv.org/format/1411.1221">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Anomalous partially hyperbolic diffeomorphisms I: dynamically coherent examples </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Parwani%2C+K">Kamlesh Parwani</a>, <a href="/search/math?searchtype=author&query=Potrie%2C+R">Rafael Potrie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1411.1221v2-abstract-short" style="display: inline;"> We build an example of a non-transitive, dynamically coherent partially hyperbolic diffeomorphism $f$ on a closed $3$-manifold with exponential growth in its fundamental group such that $f^n$ is not isotopic to the identity for all $n\neq 0$. This example contradicts a conjecture in \cite{HHU}. The main idea is to consider a well-understood time-$t$ map of a non-transitive Anosov flow and then car… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1411.1221v2-abstract-full').style.display = 'inline'; document.getElementById('1411.1221v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1411.1221v2-abstract-full" style="display: none;"> We build an example of a non-transitive, dynamically coherent partially hyperbolic diffeomorphism $f$ on a closed $3$-manifold with exponential growth in its fundamental group such that $f^n$ is not isotopic to the identity for all $n\neq 0$. This example contradicts a conjecture in \cite{HHU}. The main idea is to consider a well-understood time-$t$ map of a non-transitive Anosov flow and then carefully compose with a Dehn twist. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1411.1221v2-abstract-full').style.display = 'none'; document.getElementById('1411.1221v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 November, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 5 November, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">14 pages, 1 figure. Small corrections. Similar version will appear in Annales Sci. ENS</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1408.3951">arXiv:1408.3951</a> <span> [<a href="https://arxiv.org/pdf/1408.3951">pdf</a>, <a href="https://arxiv.org/ps/1408.3951">ps</a>, <a href="https://arxiv.org/format/1408.3951">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.2140/gt.2017.21.1837">10.2140/gt.2017.21.1837 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Building Anosov flows on 3-manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=B%C3%A9guin%2C+F">Fran莽ois B茅guin</a>, <a href="/search/math?searchtype=author&query=Yu%2C+B">Bin Yu</a>, <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1408.3951v1-abstract-short" style="display: inline;"> We prove a result allowing to build (transitive or non-transitive) Anosov flows on 3-manifolds by gluing together filtrating neighborhoods of hyperbolic sets. We give several applications; for example: 1. we build a 3-manifold supporting both of a transitive Anosov vector field and a non-transitive Anosov vector field; 2. for any n, we build a 3-manifold M supporting at least n pairwise different… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1408.3951v1-abstract-full').style.display = 'inline'; document.getElementById('1408.3951v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1408.3951v1-abstract-full" style="display: none;"> We prove a result allowing to build (transitive or non-transitive) Anosov flows on 3-manifolds by gluing together filtrating neighborhoods of hyperbolic sets. We give several applications; for example: 1. we build a 3-manifold supporting both of a transitive Anosov vector field and a non-transitive Anosov vector field; 2. for any n, we build a 3-manifold M supporting at least n pairwise different Anosov vector fields; 3. we build transitive attractors with prescribed entrance foliation; in particular, we construct some incoherent transitive attractors; 4. we build a transitive Anosov vector field admitting infinitely many pairwise non-isotopic trans- verse tori. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1408.3951v1-abstract-full').style.display = 'none'; document.getElementById('1408.3951v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 August, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">58 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Geom. Topol. 21 (2017) 1837-1930 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1401.2452">arXiv:1401.2452</a> <span> [<a href="https://arxiv.org/pdf/1401.2452">pdf</a>, <a href="https://arxiv.org/ps/1401.2452">ps</a>, <a href="https://arxiv.org/format/1401.2452">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1017/S1474748015000055">10.1017/S1474748015000055 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Center manifolds for partially hyperbolic set without strong unstable connections </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Crovisier%2C+S">Sylvain Crovisier</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1401.2452v1-abstract-short" style="display: inline;"> We consider compact sets which are invariant and partially hyperbolic under the dynamics of a diffeomorphism of a manifold. We prove that such a set K is contained in a locally invariant center submanifold if and only if each strong stable and strong unstable leaf intersects K at exactly one point. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1401.2452v1-abstract-full" style="display: none;"> We consider compact sets which are invariant and partially hyperbolic under the dynamics of a diffeomorphism of a manifold. We prove that such a set K is contained in a locally invariant center submanifold if and only if each strong stable and strong unstable leaf intersects K at exactly one point. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1401.2452v1-abstract-full').style.display = 'none'; document.getElementById('1401.2452v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 January, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Inst. Math. Jussieu 15 (2016) 785-828 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1312.7418">arXiv:1312.7418</a> <span> [<a href="https://arxiv.org/pdf/1312.7418">pdf</a>, <a href="https://arxiv.org/ps/1312.7418">ps</a>, <a href="https://arxiv.org/format/1312.7418">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Centralizers of $C^1$-contractions of the half line </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Farinelli%2C+%C3%89">脡glantine Farinelli</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1312.7418v1-abstract-short" style="display: inline;"> A subgroup $G\subset Diff^1_+([0,1])$ is $C^1$-close to the identity if there is a sequence $h_n\in Diff^1_+([0,1])$ such that the conjugates $h_n g h_n^{-1}$ tend to the identity for the $C^1$-topology, for every $g\in G$. This is equivalent to the fact that $G$ can be embedded in the $C^1$-centralizer of a $C^1$-contraction of $[0,+\infty)$ (see [Fa] and Theorem 1.1). We first describe the top… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1312.7418v1-abstract-full').style.display = 'inline'; document.getElementById('1312.7418v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1312.7418v1-abstract-full" style="display: none;"> A subgroup $G\subset Diff^1_+([0,1])$ is $C^1$-close to the identity if there is a sequence $h_n\in Diff^1_+([0,1])$ such that the conjugates $h_n g h_n^{-1}$ tend to the identity for the $C^1$-topology, for every $g\in G$. This is equivalent to the fact that $G$ can be embedded in the $C^1$-centralizer of a $C^1$-contraction of $[0,+\infty)$ (see [Fa] and Theorem 1.1). We first describe the topological dynamics of groups $C^1$-close to the identity. Then, we show that the class of groups $C^1$-close to the identity is invariant under some natural dynamical and algebraic extensions. As a consequence, we can describe a large class of groups $G\subset Diff^1_+([0,1])$ whose topological dynamics implies that they are $C^1$-close to the identity. This allows us to show that the free group ${\mathbb F}_2$ admits faithfull actions which are $C^1$-close to the identity. In particular, the $C^1$-centralizer of a $C^1$-contraction may contain free groups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1312.7418v1-abstract-full').style.display = 'none'; document.getElementById('1312.7418v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 December, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2013. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 22F05; 37C85 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1309.5277">arXiv:1309.5277</a> <span> [<a href="https://arxiv.org/pdf/1309.5277">pdf</a>, <a href="https://arxiv.org/ps/1309.5277">ps</a>, <a href="https://arxiv.org/format/1309.5277">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Rigidity for $C^1$ actions on the interval arising from hyperbolicity I: solvable groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">C. Bonatti</a>, <a href="/search/math?searchtype=author&query=Monteverde%2C+I">I. Monteverde</a>, <a href="/search/math?searchtype=author&query=Navas%2C+A">A. Navas</a>, <a href="/search/math?searchtype=author&query=Rivas%2C+C">C. Rivas</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1309.5277v3-abstract-short" style="display: inline;"> We consider Abelian-by-cyclic groups for which the cyclic factor acts by hyperbolic automorphisms on the Abelian subgroup. We show that if such a group acts faithfully by $C^1$ diffeomorphisms of the closed interval with no global fixed point at the interior, then the action is topologically conjugated to that of an affine group. Moreover, in case of non-Abelian image, we show a rigidity result co… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1309.5277v3-abstract-full').style.display = 'inline'; document.getElementById('1309.5277v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1309.5277v3-abstract-full" style="display: none;"> We consider Abelian-by-cyclic groups for which the cyclic factor acts by hyperbolic automorphisms on the Abelian subgroup. We show that if such a group acts faithfully by $C^1$ diffeomorphisms of the closed interval with no global fixed point at the interior, then the action is topologically conjugated to that of an affine group. Moreover, in case of non-Abelian image, we show a rigidity result concerning the multipliers of the homotheties, despite the fact that the conjugacy is not necessarily smooth. Some consequences for non-solvable groups are proposed. In particular, we give new proofs/examples yielding the existence of finitely-generated, locally-indicable groups with no faithful action by $C^1$ diffeomorphisms of the interval. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1309.5277v3-abstract-full').style.display = 'none'; document.getElementById('1309.5277v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 July, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 September, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2013. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">A more detailed proof of Proposition 4.15 added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F16; 22F05; 37C85; 37F15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1306.6391">arXiv:1306.6391</a> <span> [<a href="https://arxiv.org/pdf/1306.6391">pdf</a>, <a href="https://arxiv.org/ps/1306.6391">ps</a>, <a href="https://arxiv.org/format/1306.6391">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.3934/jmd.2013.7.605">10.3934/jmd.2013.7.605 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> The $C^{1+伪}$ hypothesis in Pesin theory revisited </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Crovisier%2C+S">Sylvain Crovisier</a>, <a href="/search/math?searchtype=author&query=Shinohara%2C+K">Katsutoshi Shinohara</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1306.6391v1-abstract-short" style="display: inline;"> We show that for every compact 3-manifold $M$ there exists an open subset of $\diff ^1(M)$ in which every generic diffeomorphism admits uncountably many ergodic probability measures which are hyperbolic while their supports are disjoint and admit a basis of attracting neighborhoods and a basis of repelling neighborhoods. As a consequence, the points in the support of these measures have no stable… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1306.6391v1-abstract-full').style.display = 'inline'; document.getElementById('1306.6391v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1306.6391v1-abstract-full" style="display: none;"> We show that for every compact 3-manifold $M$ there exists an open subset of $\diff ^1(M)$ in which every generic diffeomorphism admits uncountably many ergodic probability measures which are hyperbolic while their supports are disjoint and admit a basis of attracting neighborhoods and a basis of repelling neighborhoods. As a consequence, the points in the support of these measures have no stable and no unstable manifolds. This contrasts with the higher regularity case, where Pesin theory gives us the stable and the unstable manifolds with complementary dimensions at almost every point. We also give such an example in dimension two, without local genericity. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1306.6391v1-abstract-full').style.display = 'none'; document.getElementById('1306.6391v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 June, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2013. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 37D25; 37D30. Secondary: 37C70 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Mod. Dyn. 7 (2013), No. 4, pp. 605--618 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1212.1634">arXiv:1212.1634</a> <span> [<a href="https://arxiv.org/pdf/1212.1634">pdf</a>, <a href="https://arxiv.org/ps/1212.1634">ps</a>, <a href="https://arxiv.org/format/1212.1634">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1017/etds.2013.105">10.1017/etds.2013.105 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Flexible periodic points </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Shinohara%2C+K">Katsutoshi Shinohara</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1212.1634v1-abstract-short" style="display: inline;"> We define the notion of $\varepsilon$-flexible periodic point: it is a periodic point with stable index equal to two whose dynamics restricted to the stable direction admits $\varepsilon$-perturbations both to a homothety and a saddle having an eigenvalue equal to one. We show that $\varepsilon$-perturbation to an $\varepsilon$-flexible point allows to change it in a stable index one periodic poin… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1212.1634v1-abstract-full').style.display = 'inline'; document.getElementById('1212.1634v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1212.1634v1-abstract-full" style="display: none;"> We define the notion of $\varepsilon$-flexible periodic point: it is a periodic point with stable index equal to two whose dynamics restricted to the stable direction admits $\varepsilon$-perturbations both to a homothety and a saddle having an eigenvalue equal to one. We show that $\varepsilon$-perturbation to an $\varepsilon$-flexible point allows to change it in a stable index one periodic point whose (one dimensional) stable manifold is an arbitrarily chosen $C^1$ -curve. We also show that the existence of flexible point is a general phenomenon among systems with a robustly non-hyperbolic two dimensional center-stable bundle. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1212.1634v1-abstract-full').style.display = 'none'; document.getElementById('1212.1634v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 December, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37C29; 37D30 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Ergod. Th. Dynam. Sys. 35 (2014) 1394-1422 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1211.3435">arXiv:1211.3435</a> <span> [<a href="https://arxiv.org/pdf/1211.3435">pdf</a>, <a href="https://arxiv.org/ps/1211.3435">ps</a>, <a href="https://arxiv.org/format/1211.3435">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Robust vanishing of all Lyapunov exponents for iterated function systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bochi%2C+J">Jairo Bochi</a>, <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=D%C3%ADaz%2C+L+J">Lorenzo J. D铆az</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1211.3435v2-abstract-short" style="display: inline;"> Given any compact connected manifold $M$, we describe $C^2$-open sets of iterated functions systems (IFS's) admitting fully-supported ergodic measures whose Lyapunov exponents along $M$ are all zero. Moreover, these measures are approximated by measures supported on periodic orbits. We also describe $C^1$-open sets of IFS's admitting ergodic measures of positive entropy whose Lyapunov exponents al… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1211.3435v2-abstract-full').style.display = 'inline'; document.getElementById('1211.3435v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1211.3435v2-abstract-full" style="display: none;"> Given any compact connected manifold $M$, we describe $C^2$-open sets of iterated functions systems (IFS's) admitting fully-supported ergodic measures whose Lyapunov exponents along $M$ are all zero. Moreover, these measures are approximated by measures supported on periodic orbits. We also describe $C^1$-open sets of IFS's admitting ergodic measures of positive entropy whose Lyapunov exponents along $M$ are all zero. The proofs involve the construction of non-hyperbolic measures for the induced IFS's on the flag manifold. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1211.3435v2-abstract-full').style.display = 'none'; document.getElementById('1211.3435v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 June, 2013; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 November, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Minor corrections were made. This is the final version, to appear in Mathematische Zeitschrift</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1210.2835">arXiv:1210.2835</a> <span> [<a href="https://arxiv.org/pdf/1210.2835">pdf</a>, <a href="https://arxiv.org/ps/1210.2835">ps</a>, <a href="https://arxiv.org/format/1210.2835">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Partially hyperbolic diffeomorphisms with uniformly center foliation: the quotient dynamics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bohnet%2C+D">Doris Bohnet</a>, <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1210.2835v3-abstract-short" style="display: inline;"> We show that a partially hyperbolic $C^1$ -diffeomorphism $f : M \to M$ with a uniformly compact $f$ -invariant center foliation $F^c$ is dynamically coherent. Further, the induced homeomorphism $F : M/F^c \to M/F^c$ on the quotient space of the center foliation has the shadowing property, i.e. for every $\varepsilon> 0$ there exists $未> 0$ such that every $未$-pseudo orbit of center leaves is… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1210.2835v3-abstract-full').style.display = 'inline'; document.getElementById('1210.2835v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1210.2835v3-abstract-full" style="display: none;"> We show that a partially hyperbolic $C^1$ -diffeomorphism $f : M \to M$ with a uniformly compact $f$ -invariant center foliation $F^c$ is dynamically coherent. Further, the induced homeomorphism $F : M/F^c \to M/F^c$ on the quotient space of the center foliation has the shadowing property, i.e. for every $\varepsilon> 0$ there exists $未> 0$ such that every $未$-pseudo orbit of center leaves is $\varepsilon$-shadowed by an orbit of center leaves. Although the shadowing orbit is not necessarily unique, we prove the density of periodic center leaves inside the chain recurrent set of the quotient dynamics. Some other interesting properties of the quotient dynamics are discussed. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1210.2835v3-abstract-full').style.display = 'none'; document.getElementById('1210.2835v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 December, 2014; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 October, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">36 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37D30; 37C15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1209.1601">arXiv:1209.1601</a> <span> [<a href="https://arxiv.org/pdf/1209.1601">pdf</a>, <a href="https://arxiv.org/ps/1209.1601">ps</a>, <a href="https://arxiv.org/format/1209.1601">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1017/etds.2015.3">10.1017/etds.2015.3 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Connectedness of the space of smooth actions of $\Z^n$ on the interval </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Eynard%2C+H">H茅l猫ne Eynard</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1209.1601v1-abstract-short" style="display: inline;"> We prove that the spaces of $\Cinf$ orientation-preserving actions of $\Z^n$ on $[0,1]$ and nonfree actions of $\Z^2$ on the circle are connected. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1209.1601v1-abstract-full" style="display: none;"> We prove that the spaces of $\Cinf$ orientation-preserving actions of $\Z^n$ on $[0,1]$ and nonfree actions of $\Z^2$ on the circle are connected. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1209.1601v1-abstract-full').style.display = 'none'; document.getElementById('1209.1601v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 September, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37E05; 37E10; 37C05; 37C85 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Ergod. Th. Dynam. Sys. 36 (2016) 2076-2106 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1207.2508">arXiv:1207.2508</a> <span> [<a href="https://arxiv.org/pdf/1207.2508">pdf</a>, <a href="https://arxiv.org/ps/1207.2508">ps</a>, <a href="https://arxiv.org/format/1207.2508">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Smooth Conjugacy classes of circle diffeomorphisms with irrational rotation number </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Guelman%2C+N">Nancy Guelman</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1207.2508v1-abstract-short" style="display: inline;"> In this paper we prove the $C^1$-density of every $C^r$-conjugacy class in the closed subset of diffeomorphisms of the circle with a given irrational rotation number. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1207.2508v1-abstract-full" style="display: none;"> In this paper we prove the $C^1$-density of every $C^r$-conjugacy class in the closed subset of diffeomorphisms of the circle with a given irrational rotation number. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1207.2508v1-abstract-full').style.display = 'none'; document.getElementById('1207.2508v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 July, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2012. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1112.1002">arXiv:1112.1002</a> <span> [<a href="https://arxiv.org/pdf/1112.1002">pdf</a>, <a href="https://arxiv.org/ps/1112.1002">ps</a>, <a href="https://arxiv.org/format/1112.1002">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Tame dynamics and robust transitivity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Crovisier%2C+S">Sylvain Crovisier</a>, <a href="/search/math?searchtype=author&query=Gourmelon%2C+N">Nicolas Gourmelon</a>, <a href="/search/math?searchtype=author&query=Potrie%2C+R">Rafael Potrie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1112.1002v1-abstract-short" style="display: inline;"> One main task of smooth dynamical systems consists in finding a good decomposition into elementary pieces of the dynamics. This paper contributes to the study of chain-recurrence classes. It is known that $C^1$-generically, each chain-recurrence class containing a periodic orbit is equal to the homoclinic class of this orbit. Our result implies that in general this property is fragile. We build… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1112.1002v1-abstract-full').style.display = 'inline'; document.getElementById('1112.1002v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1112.1002v1-abstract-full" style="display: none;"> One main task of smooth dynamical systems consists in finding a good decomposition into elementary pieces of the dynamics. This paper contributes to the study of chain-recurrence classes. It is known that $C^1$-generically, each chain-recurrence class containing a periodic orbit is equal to the homoclinic class of this orbit. Our result implies that in general this property is fragile. We build a C1-open set U of tame diffeomorphisms (their dynamics only splits into finitely many chain-recurrence classes) such that for any diffeomorphism in a C-infinity-dense subset of U, one of the chain-recurrence classes is not transitive (and has an isolated point). Moreover, these dynamics are obtained among partially hyperbolic systems with one-dimensional center. R茅sum茅 : Dynamique mod茅r茅e et transitivit茅 robuste. L'un des buts des syst猫mes dynamiques consiste 脿 trouver une bonne d茅composition de la dynamique en pi猫ces 茅l茅mentaires. Cet article contribue 脿 l'茅tude des classes de r茅currence par cha卯nes. On sait que C1-g茅n茅riquement, chaque classe de r茅currence par cha卯nes contenant une orbite p茅riodique coincide avec la classe homocline de cette orbite. Notre r茅sultat montre que cette propri茅t茅 est en g茅n茅rale fragile. Nous construisons un ouvert U de diff茅omorphismes mod茅r茅s (leur dynamique ne se d茅compose qu'en un nombre fini de classes de r茅currence par cha卯nes) tel que pour tout diff茅omorphisme appartenant 脿 un sous-ensemble C-infini-dense de U, une des classes de r茅currence par cha卯nes n'est pas transitive (elle a un point isol茅). De plus, ces dynamiques sont obtenues comme syst猫mes partiellement hyperboliques avec une direction centrale uni-dimensionnelle. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1112.1002v1-abstract-full').style.display = 'none'; document.getElementById('1112.1002v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 December, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2011. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1106.3905">arXiv:1106.3905</a> <span> [<a href="https://arxiv.org/pdf/1106.3905">pdf</a>, <a href="https://arxiv.org/ps/1106.3905">ps</a>, <a href="https://arxiv.org/format/1106.3905">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Dominated chain recurrent class with singularities </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bonatti%2C+C">Christian Bonatti</a>, <a href="/search/math?searchtype=author&query=Gan%2C+S">Shaobo Gan</a>, <a href="/search/math?searchtype=author&query=Yang%2C+D">Dawei Yang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1106.3905v1-abstract-short" style="display: inline;"> We prove that for generic three-dimensional vector fields, domination implies singular hyperbolicity. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1106.3905v1-abstract-full" style="display: none;"> We prove that for generic three-dimensional vector fields, domination implies singular hyperbolicity. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1106.3905v1-abstract-full').style.display = 'none'; document.getElementById('1106.3905v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 June, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2011. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">15 pages, 0 figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37C10 37C20 37C27 37D45 37D05 37D30 </p> </li> </ol> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a 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