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</div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <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/2502.07716">arXiv:2502.07716</a> <span> [<a href="https://arxiv.org/pdf/2502.07716">pdf</a>, <a href="https://arxiv.org/format/2502.07716">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"> A Sm枚rg氓sbord of (bi)contact structures, Reeb flows and 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> </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="2502.07716v1-abstract-short" style="display: inline;"> This note was written for the proceedings of the conference "Symplectic Geometry and Anosov Flows" held in Heideleberg in July 2024. It is meant as an invitation to the study of certain families of contact structures, centering around the following question: ``How much can one relate the dynamics of two distinct Reeb flows of the same contact structure?'' We gather some results as well as state ma… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.07716v1-abstract-full').style.display = 'inline'; document.getElementById('2502.07716v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.07716v1-abstract-full" style="display: none;"> This note was written for the proceedings of the conference "Symplectic Geometry and Anosov Flows" held in Heideleberg in July 2024. It is meant as an invitation to the study of certain families of contact structures, centering around the following question: ``How much can one relate the dynamics of two distinct Reeb flows of the same contact structure?'' We gather some results as well as state many more questions and conjectures around that theme. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.07716v1-abstract-full').style.display = 'none'; document.getElementById('2502.07716v1-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 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </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. Comments welcome</span> </p> </li> <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/2308.02098">arXiv:2308.02098</a> <span> [<a href="https://arxiv.org/pdf/2308.02098">pdf</a>, <a href="https://arxiv.org/format/2308.02098">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"> Anosov flows with the same periodic orbits </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=Fenley%2C+S">Sergio Fenley</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="2308.02098v2-abstract-short" style="display: inline;"> In [Orbit equivalences of pseudo-Anosov flows, arXiv:2211.10505], it was proved that transitive pseudo-Anosov flows on any closed 3-manifold are determined up to orbit equivalence by the set of free homotopy classes represented by periodic orbits, provided their orbit space does not contain a feature called a "tree of scalloped regions." In this article we describe what happens in these exceptiona… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.02098v2-abstract-full').style.display = 'inline'; document.getElementById('2308.02098v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2308.02098v2-abstract-full" style="display: none;"> In [Orbit equivalences of pseudo-Anosov flows, arXiv:2211.10505], it was proved that transitive pseudo-Anosov flows on any closed 3-manifold are determined up to orbit equivalence by the set of free homotopy classes represented by periodic orbits, provided their orbit space does not contain a feature called a "tree of scalloped regions." In this article we describe what happens in these exceptional cases: we show what topological features in the manifold correspond to trees of scalloped regions, completely classify the flows which do have the same free homotopy data, and construct explicit examples of flows with the same free homotopy data that are not orbit equivalent. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.02098v2-abstract-full').style.display = 'none'; document.getElementById('2308.02098v2-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 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 August, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 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">27 pages. Comments welcome. v2: added some justifications for background results in the non-orientable case. minor changes</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2211.10505">arXiv:2211.10505</a> <span> [<a href="https://arxiv.org/pdf/2211.10505">pdf</a>, <a href="https://arxiv.org/format/2211.10505">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"> Orbit equivalences of 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=Frankel%2C+S">Steven Frankel</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="2211.10505v1-abstract-short" style="display: inline;"> We prove a classification theorem for transitive Anosov and pseudo-Anosov flows on closed 3-manifolds, up to orbit equivalence. In many cases, flows on a 3-manifold $M$ are completely determined by the set of free homotopy classes of their (unoriented) periodic orbits. The exceptional cases are flows with a special structure in their orbit space called a ``tree of scalloped regions"; in these ca… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2211.10505v1-abstract-full').style.display = 'inline'; document.getElementById('2211.10505v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2211.10505v1-abstract-full" style="display: none;"> We prove a classification theorem for transitive Anosov and pseudo-Anosov flows on closed 3-manifolds, up to orbit equivalence. In many cases, flows on a 3-manifold $M$ are completely determined by the set of free homotopy classes of their (unoriented) periodic orbits. The exceptional cases are flows with a special structure in their orbit space called a ``tree of scalloped regions"; in these cases the set of free homotopy classes of unoriented periodic orbits together with the additional data of a choice of sign for each $蟺_1(M)$-orbit of tree gives a complete invariant of orbit equivalence classes of flows. The framework for the proof is a more general result about \emph{Anosov-like actions} of abstract groups on bifoliated planes, showing that the homeomorphism type of the bifoliation and the conjugacy class of the action can be recovered from knowledge of which elements of the group act with fixed points. As a consequence, we show that Anosov flows are determined up to orbit equivalence by the action on the ideal boundary of their orbit spaces, and more generally that transitive Anosov-like actions on bifoliated planes are determined up to conjugacy by their actions on the plane's ideal boundary: any conjugacy between two such actions on their ideal circles can be extended uniquely to a conjugacy on the interior of the plane. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2211.10505v1-abstract-full').style.display = 'none'; document.getElementById('2211.10505v1-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 November, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 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">52 pages, 24 figures, comments welcome</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2012.11811">arXiv:2012.11811</a> <span> [<a href="https://arxiv.org/pdf/2012.11811">pdf</a>, <a href="https://arxiv.org/ps/2012.11811">ps</a>, <a href="https://arxiv.org/format/2012.11811">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/10.2140/gt.2024.28.867">10.2140/gt.2024.28.867 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Orbit equivalences of $\mathbb{R}$-covered Anosov flows and hyperbolic-like actions on the line </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=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="2012.11811v4-abstract-short" style="display: inline;"> We prove a rigidity result for group actions on the line whose elements have what we call "hyperbolic-like" dynamics. Using this, we give a spectral rigidity theorem for $\mathbb{R}$-covered Anosov flows on 3-manifolds, characterizing orbit equivalent flows in terms of the elements of the fundamental group represented by periodic orbits. As consequences of this, we give an efficient criterion to d… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.11811v4-abstract-full').style.display = 'inline'; document.getElementById('2012.11811v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2012.11811v4-abstract-full" style="display: none;"> We prove a rigidity result for group actions on the line whose elements have what we call "hyperbolic-like" dynamics. Using this, we give a spectral rigidity theorem for $\mathbb{R}$-covered Anosov flows on 3-manifolds, characterizing orbit equivalent flows in terms of the elements of the fundamental group represented by periodic orbits. As consequences of this, we give an efficient criterion to determine the isotopy classes of self orbit equivalences of $\mathbb{R}$-covered Anosov flows, and prove finiteness of contact Anosov flows on any given manifold. In the appendix with Jonathan Bowden, we prove that orbit equivalences of contact Anosov flows correspond exactly to isomorphisms of the associated contact structures. This gives a powerful tool to translate results on Anosov flows to contact geometry and vice versa. We illustrate its use by giving two new results in contact geometry: the existence of manifolds with arbitrarily many distinct Anosov contact structures, answering a question of Foulon--Hasselblatt--Vaugon, and a virtual description of the group of contact transformations of a contact Anosov structure, generalizing a result of Giroux and Massot. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.11811v4-abstract-full').style.display = 'none'; document.getElementById('2012.11811v4-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 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 December, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 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">Appendix together with Jonathan Bowden. 20 pages. v2: corrected typo in statement of Theorem 1.8. v3: Section 3 rewritten with significant corrections. New introduction v4: Final accepted version, title changed, to appear in Geometry & Topology</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Geom. Topol. 28 (2024) 867-899 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2008.06547">arXiv:2008.06547</a> <span> [<a href="https://arxiv.org/pdf/2008.06547">pdf</a>, <a href="https://arxiv.org/format/2008.06547">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"> Collapsed Anosov flows and self orbit equivalences </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=Fenley%2C+S+R">Sergio R. Fenley</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="2008.06547v3-abstract-short" style="display: inline;"> We propose a generalization of the concept of discretized Anosov flows that covers a wide class of partially hyperbolic diffeomorphisms in 3-manifolds, and that we call collapsed Anosov flows. They are related with Anosov flows via a self orbit equivalence of the flow. We show that all the examples constructed in a paper by Bonatti, Gogolev, Hammerlindl and the third author belong to this class, a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.06547v3-abstract-full').style.display = 'inline'; document.getElementById('2008.06547v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2008.06547v3-abstract-full" style="display: none;"> We propose a generalization of the concept of discretized Anosov flows that covers a wide class of partially hyperbolic diffeomorphisms in 3-manifolds, and that we call collapsed Anosov flows. They are related with Anosov flows via a self orbit equivalence of the flow. We show that all the examples constructed in a paper by Bonatti, Gogolev, Hammerlindl and the third author belong to this class, and that it is an open and closed class among partially hyperbolic diffeomorphisms. We provide some equivalent definitions which may be more amenable to analysis and are useful in different situations. Conversely we describe the class of partially hyperbolic diffeomorphisms that are collapsed Anosov flows associated with certain types of Anosov flows. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.06547v3-abstract-full').style.display = 'none'; document.getElementById('2008.06547v3-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 June, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 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">81 pages, 1 figure. This version clarifies some hypothesis and provides more details in some arguments. It also points out shortcuts to get some relevant instances of the results. Final version, to appear in Comm. Math. Helv</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2008.04871">arXiv:2008.04871</a> <span> [<a href="https://arxiv.org/pdf/2008.04871">pdf</a>, <a href="https://arxiv.org/format/2008.04871">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/10.2140/gt.2023.27.3095">10.2140/gt.2023.27.3095 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, Part II: Branching foliations </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=Fenley%2C+S+R">Sergio R. Fenley</a>, <a href="/search/math?searchtype=author&query=Frankel%2C+S">Steven Frankel</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="2008.04871v2-abstract-short" style="display: inline;"> We study $3$-dimensional partially hyperbolic diffeomorphisms that are homotopic to the identity, focusing on the geometry and dynamics of Burago and Ivanov's center stable and center unstable \emph{branching} foliations. This extends our study of the true foliations that appear in the dynamically coherent case (see \emph{Partially hyperbolic diffeomorphisms homotopic to the identity in dimension… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.04871v2-abstract-full').style.display = 'inline'; document.getElementById('2008.04871v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2008.04871v2-abstract-full" style="display: none;"> We study $3$-dimensional partially hyperbolic diffeomorphisms that are homotopic to the identity, focusing on the geometry and dynamics of Burago and Ivanov's center stable and center unstable \emph{branching} foliations. This extends our study of the true foliations that appear in the dynamically coherent case (see \emph{Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, Part I: The dynamically coherent case}, arxiv:1908.06227v3). We complete the classification of such diffeomorphisms in Seifert fibered manifolds. In hyperbolic manifolds, we show that any such diffeomorphism is either dynamically coherent and has a power that is a discretized Anosov flow, or is of a new potential class called a \emph{double translation}. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.04871v2-abstract-full').style.display = 'none'; document.getElementById('2008.04871v2-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 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 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">This is the second part of arxiv:1908.06227v1 that was split in two. This version has a new part 2-specific introduction and some improved explanations in the text. v2: This is the final accepted version. Some improvements in the text thanks to referees reports</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Geom. Topol. 27 (2023) 3095-3181 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2002.10315">arXiv:2002.10315</a> <span> [<a href="https://arxiv.org/pdf/2002.10315">pdf</a>, <a href="https://arxiv.org/format/2002.10315">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"> Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms </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=Fenley%2C+S">Sergio Fenley</a>, <a href="/search/math?searchtype=author&query=Frankel%2C+S">Steven Frankel</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="2002.10315v1-abstract-short" style="display: inline;"> We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends work of Bonatti, Gogolev, Hammerlindl and Potrie to the whole isotopy class. We relate the techniques with the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in the p… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.10315v1-abstract-full').style.display = 'inline'; document.getElementById('2002.10315v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2002.10315v1-abstract-full" style="display: none;"> We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends work of Bonatti, Gogolev, Hammerlindl and Potrie to the whole isotopy class. We relate the techniques with the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in the previous paper by the authors. The appendix reviews some consequences of the Nielsen-Thurston classification of surface homeomorphisms to the dynamics of lifts of such maps to the universal cover. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.10315v1-abstract-full').style.display = 'none'; document.getElementById('2002.10315v1-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> 24 February, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 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">18 pages, 1 figure</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1911.05532">arXiv:1911.05532</a> <span> [<a href="https://arxiv.org/pdf/1911.05532">pdf</a>, <a href="https://arxiv.org/ps/1911.05532">ps</a>, <a href="https://arxiv.org/format/1911.05532">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"> Centralizers of partially hyperbolic diffeomorphisms in dimension 3 </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=Gogolev%2C+A">Andrey Gogolev</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="1911.05532v1-abstract-short" style="display: inline;"> In this note we describe centralizers of volume preserving partially hyperbolic diffeomorphisms which are homotopic to identity on Seifert fibered and hyperbolic 3-manifolds. Our proof follows the strategy of Damjanovic, Wilkinson and Xu (arXiv:1902.05201) who recently classified the centralizer for perturbations of time-$1$ maps of geodesic flows in negative curvature. We strongly rely on recent… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1911.05532v1-abstract-full').style.display = 'inline'; document.getElementById('1911.05532v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1911.05532v1-abstract-full" style="display: none;"> In this note we describe centralizers of volume preserving partially hyperbolic diffeomorphisms which are homotopic to identity on Seifert fibered and hyperbolic 3-manifolds. Our proof follows the strategy of Damjanovic, Wilkinson and Xu (arXiv:1902.05201) who recently classified the centralizer for perturbations of time-$1$ maps of geodesic flows in negative curvature. We strongly rely on recent classification results in dimension 3 established in (arXiv:1908.06227). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1911.05532v1-abstract-full').style.display = 'none'; document.getElementById('1911.05532v1-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 November, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 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">5 pages, comments are welcome</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1908.06227">arXiv:1908.06227</a> <span> [<a href="https://arxiv.org/pdf/1908.06227">pdf</a>, <a href="https://arxiv.org/format/1908.06227">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"> Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, Part I: The dynamically coherent case </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=Fenley%2C+S+R">Sergio R. Fenley</a>, <a href="/search/math?searchtype=author&query=Frankel%2C+S">Steven Frankel</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="1908.06227v4-abstract-short" style="display: inline;"> We study 3-dimensional dynamically coherent partially hyperbolic diffeomorphisms that are homotopic to the identity, focusing on the transverse geometry and topology of the center stable and center unstable foliations, and the dynamics within their leaves. We find a structural dichotomy for these foliations, which we use to show that every such diffeomorphism on a hyperbolic or Seifert fibered 3-m… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1908.06227v4-abstract-full').style.display = 'inline'; document.getElementById('1908.06227v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1908.06227v4-abstract-full" style="display: none;"> We study 3-dimensional dynamically coherent partially hyperbolic diffeomorphisms that are homotopic to the identity, focusing on the transverse geometry and topology of the center stable and center unstable foliations, and the dynamics within their leaves. We find a structural dichotomy for these foliations, which we use to show that every such diffeomorphism on a hyperbolic or Seifert fibered 3-manifold is leaf conjugate to the time one map of a (topological) Anosov flow. This proves a classification conjecture of Hertz-Hertz-Ures in hyperbolic 3-manifolds and in the homotopy class of the identity of Seifert manifolds. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1908.06227v4-abstract-full').style.display = 'none'; document.getElementById('1908.06227v4-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 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 August, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 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">v4: Final accepted version. Some improvement in the text thanks to referees</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37D30; 57R30; 37C15; 57M50; 37D20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1902.02896">arXiv:1902.02896</a> <span> [<a href="https://arxiv.org/pdf/1902.02896">pdf</a>, <a href="https://arxiv.org/ps/1902.02896">ps</a>, <a href="https://arxiv.org/format/1902.02896">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="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Geometry and entropies in a fixed conformal class on surfaces </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=Erchenko%2C+A">Alena Erchenko</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="1902.02896v2-abstract-short" style="display: inline;"> We show the flexibility of the metric entropy and obtain additional restrictions on the topological entropy of geodesic flow on closed surfaces of negative Euler characteristic with smooth non-positively curved Riemannian metrics with fixed total area in a fixed conformal class. Moreover, we obtain a collar lemma, a thick-thin decomposition, and precompactness for the considered class of metrics.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1902.02896v2-abstract-full').style.display = 'inline'; document.getElementById('1902.02896v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1902.02896v2-abstract-full" style="display: none;"> We show the flexibility of the metric entropy and obtain additional restrictions on the topological entropy of geodesic flow on closed surfaces of negative Euler characteristic with smooth non-positively curved Riemannian metrics with fixed total area in a fixed conformal class. Moreover, we obtain a collar lemma, a thick-thin decomposition, and precompactness for the considered class of metrics. Also, we extend some of the results to metrics of fixed total area in a fixed conformal class with no focal points and with some integral bounds on the positive part of the Gaussian curvature. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1902.02896v2-abstract-full').style.display = 'none'; document.getElementById('1902.02896v2-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> 6 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 February, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 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">Minor changes to exposition. 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/1801.00214">arXiv:1801.00214</a> <span> [<a href="https://arxiv.org/pdf/1801.00214">pdf</a>, <a href="https://arxiv.org/ps/1801.00214">ps</a>, <a href="https://arxiv.org/format/1801.00214">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"> Partially hyperbolic diffeomorphisms homotopic to the identity on 3-manifolds </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=Fenley%2C+S">Sergio Fenley</a>, <a href="/search/math?searchtype=author&query=Frankel%2C+S">Steven Frankel</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="1801.00214v1-abstract-short" style="display: inline;"> We announce some results towards the classification of partially hyperbolic diffeomorphisms on 3-manifolds, and outline the proofs in the case when the diffeomorphism is dynamically coherent. Detailed proofs are long and technical and will appear later. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1801.00214v1-abstract-full" style="display: none;"> We announce some results towards the classification of partially hyperbolic diffeomorphisms on 3-manifolds, and outline the proofs in the case when the diffeomorphism is dynamically coherent. Detailed proofs are long and technical and will appear later. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1801.00214v1-abstract-full').style.display = 'none'; document.getElementById('1801.00214v1-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> 30 December, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 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">Research Announcement. 15 pages</span> </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/1709.09234">arXiv:1709.09234</a> <span> [<a href="https://arxiv.org/pdf/1709.09234">pdf</a>, <a href="https://arxiv.org/format/1709.09234">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="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> </div> <p class="title is-5 mathjax"> Flexibility of geometrical and dynamical data in fixed conformal classes </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=Erchenko%2C+A">Alena Erchenko</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="1709.09234v1-abstract-short" style="display: inline;"> Consider a smooth closed surface $M$ of fixed genus $\geqslant 2$ with a hyperbolic metric $蟽$ of total area $A$. In this article, we study the behavior of geometric and dynamical characteristics (e.g., diameter, Laplace spectrum, Gaussian curvature and entropies) of nonpositively curved smooth metrics with total area $A$ conformally equivalent to $蟽$. For such metrics, we show that the diameter i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.09234v1-abstract-full').style.display = 'inline'; document.getElementById('1709.09234v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1709.09234v1-abstract-full" style="display: none;"> Consider a smooth closed surface $M$ of fixed genus $\geqslant 2$ with a hyperbolic metric $蟽$ of total area $A$. In this article, we study the behavior of geometric and dynamical characteristics (e.g., diameter, Laplace spectrum, Gaussian curvature and entropies) of nonpositively curved smooth metrics with total area $A$ conformally equivalent to $蟽$. For such metrics, we show that the diameter is bounded above and the Laplace spectrum is bounded below away from zero by constants which depend on $蟽$. On the other hand, we prove that the metric entropy of the geodesic flow with respect to the Liouville measure is flexible. Consequently, we also provide the first known example showing that the bottom of the $L^2$-spectrum of the Laplacian cannot be bounded from above by a function of the metric entropy. We also provide examples showing that our conditions are essential for the established bounds. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.09234v1-abstract-full').style.display = 'none'; document.getElementById('1709.09234v1-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 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">19 pages, 1 figure</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1702.01178">arXiv:1702.01178</a> <span> [<a href="https://arxiv.org/pdf/1702.01178">pdf</a>, <a href="https://arxiv.org/ps/1702.01178">ps</a>, <a href="https://arxiv.org/format/1702.01178">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"> A note on self orbit equivalences of Anosov flows and bundles with fiberwise 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=Gogolev%2C+A">Andrey Gogolev</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="1702.01178v2-abstract-short" style="display: inline;"> We show that a self orbit equivalence of a transitive Anosov flow on a $3$-manifold which is homotopic to identity has to either preserve every orbit or the Anosov flow is $\mathbb{R}$-covered and the orbit equivalence has to be of a specific type. This result shows that one can remove a relatively unnatural assumption in a result of Farrell and Gogolev about the topological rigidity of bundles su… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.01178v2-abstract-full').style.display = 'inline'; document.getElementById('1702.01178v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1702.01178v2-abstract-full" style="display: none;"> We show that a self orbit equivalence of a transitive Anosov flow on a $3$-manifold which is homotopic to identity has to either preserve every orbit or the Anosov flow is $\mathbb{R}$-covered and the orbit equivalence has to be of a specific type. This result shows that one can remove a relatively unnatural assumption in a result of Farrell and Gogolev about the topological rigidity of bundles supporting a fiberwise Anosov flow when the fiber is $3$-dimensional. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.01178v2-abstract-full').style.display = 'none'; document.getElementById('1702.01178v2-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 November, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 February, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 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">Accepted version of the article</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Math. Res. Lett. Volume 26, Number 3, 711--728, 2019 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1508.07474">arXiv:1508.07474</a> <span> [<a href="https://arxiv.org/pdf/1508.07474">pdf</a>, <a href="https://arxiv.org/ps/1508.07474">ps</a>, <a href="https://arxiv.org/format/1508.07474">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Entropy rigidity of Hilbert and Riemannian metrics </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=Marquis%2C+L">Ludovic Marquis</a>, <a href="/search/math?searchtype=author&query=Zimmer%2C+A">Andrew Zimmer</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="1508.07474v2-abstract-short" style="display: inline;"> In this paper we provide two new characterizations of real hyperbolic $n$-space using the Poincar茅 exponent of a discrete group and the volume growth entropy. The first characterization is in the space of Hilbert metrics and generalizes a result of Crampon. The second is in the space of Riemannian metrics with Ricci curvature bounded below and generalizes a result of Ledrappier and Wang. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1508.07474v2-abstract-full" style="display: none;"> In this paper we provide two new characterizations of real hyperbolic $n$-space using the Poincar茅 exponent of a discrete group and the volume growth entropy. The first characterization is in the space of Hilbert metrics and generalizes a result of Crampon. The second is in the space of Riemannian metrics with Ricci curvature bounded below and generalizes a result of Ledrappier and Wang. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1508.07474v2-abstract-full').style.display = 'none'; document.getElementById('1508.07474v2-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 September, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 August, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 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">14 pages, some revisions following the referees remarks. To be published in IMRN</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1505.07999">arXiv:1505.07999</a> <span> [<a href="https://arxiv.org/pdf/1505.07999">pdf</a>, <a href="https://arxiv.org/ps/1505.07999">ps</a>, <a href="https://arxiv.org/format/1505.07999">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"> Counting periodic orbits of Anosov flows in free homotopy classes </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=Fenley%2C+S+R">Sergio R. Fenley</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="1505.07999v1-abstract-short" style="display: inline;"> The main result of this article is that if a $3$-manifold $M$ supports an Anosov flow, then the number of conjugacy classes in the fundamental group of $M$ grows exponentially fast with the length of the shortest orbit representative, hereby answering a question raised by Plante and Thurston in 1972. In fact we show that, when the flow is transitive, the exponential growth rate is exactly the topo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.07999v1-abstract-full').style.display = 'inline'; document.getElementById('1505.07999v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1505.07999v1-abstract-full" style="display: none;"> The main result of this article is that if a $3$-manifold $M$ supports an Anosov flow, then the number of conjugacy classes in the fundamental group of $M$ grows exponentially fast with the length of the shortest orbit representative, hereby answering a question raised by Plante and Thurston in 1972. In fact we show that, when the flow is transitive, the exponential growth rate is exactly the topological entropy of the flow. We also show that taking only the shortest orbit representatives in each conjugacy classes still yields Bowen's version of the measure of maximal entropy. These results are achieved by obtaining counting results on the growth rate of the number of periodic orbits inside a free homotopy class. In the first part of the article, we also construct many examples of Anosov flows having some finite and some infinite free homotopy classes of periodic orbits, and we also give a characterization of algebraic Anosov flows as the only $\mathbb{R}$-covered Anosov flows up to orbit equivalence that do not admit at least one infinite free homotopy class of periodic orbits. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.07999v1-abstract-full').style.display = 'none'; document.getElementById('1505.07999v1-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 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">44 pages, 11 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/1410.1069">arXiv:1410.1069</a> <span> [<a href="https://arxiv.org/pdf/1410.1069">pdf</a>, <a href="https://arxiv.org/ps/1410.1069">ps</a>, <a href="https://arxiv.org/format/1410.1069">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> </div> <p class="title is-5 mathjax"> On deformations of the spectrum of a Finsler--Laplacian that preserve the length spectrum </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> </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="1410.1069v2-abstract-short" style="display: inline;"> In this article, we show that a Finsler--Laplacian introduced previously can detect changes in the Finsler metric that the marked length spectrum cannot. We also construct examples of non-reversible Finsler metrics in negative curvature such that $4位_1 > h^2$, where $位_1$ is the bottom of the $L^2$-spectrum and $h$ the topological entropy of the flow. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1410.1069v2-abstract-full" style="display: none;"> In this article, we show that a Finsler--Laplacian introduced previously can detect changes in the Finsler metric that the marked length spectrum cannot. We also construct examples of non-reversible Finsler metrics in negative curvature such that $4位_1 > h^2$, where $位_1$ is the bottom of the $L^2$-spectrum and $h$ the topological entropy of the flow. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1410.1069v2-abstract-full').style.display = 'none'; document.getElementById('1410.1069v2-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 June, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 October, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 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">Comments welcome. v2: Remove an unnecessary hypothesis in Theorem 1, added Proposition 19, minor changes in the text</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 58J50; Secondary: 37D40; 53B40; 53D25 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1211.6376">arXiv:1211.6376</a> <span> [<a href="https://arxiv.org/pdf/1211.6376">pdf</a>, <a href="https://arxiv.org/ps/1211.6376">ps</a>, <a href="https://arxiv.org/format/1211.6376">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</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/10.2748/tmj/1412783204">10.2748/tmj/1412783204 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Laplacian and spectral gap in regular Hilbert geometries </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=Colbois%2C+B">Bruno Colbois</a>, <a href="/search/math?searchtype=author&query=Crampon%2C+M">Micka毛l Crampon</a>, <a href="/search/math?searchtype=author&query=Verovic%2C+P">Patrick Verovic</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="1211.6376v1-abstract-short" style="display: inline;"> We study the spectrum of the Finsler--Laplace operator for regular Hilbert geometries, defined by convex sets with $C^2$ boundaries. We show that for an $n$-dimensional geometry, the spectral gap is bounded above by $(n-1)^2/4$, which we prove to be the infimum of the essential spectrum. We also construct examples of convex sets with arbitrarily small eigenvalues. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1211.6376v1-abstract-full" style="display: none;"> We study the spectrum of the Finsler--Laplace operator for regular Hilbert geometries, defined by convex sets with $C^2$ boundaries. We show that for an $n$-dimensional geometry, the spectral gap is bounded above by $(n-1)^2/4$, which we prove to be the infimum of the essential spectrum. We also construct examples of convex sets with arbitrarily small eigenvalues. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1211.6376v1-abstract-full').style.display = 'none'; document.getElementById('1211.6376v1-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 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">21 pages, 3 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/1208.6487">arXiv:1208.6487</a> <span> [<a href="https://arxiv.org/pdf/1208.6487">pdf</a>, <a href="https://arxiv.org/ps/1208.6487">ps</a>, <a href="https://arxiv.org/format/1208.6487">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/jtopol/jtt041">10.1112/jtopol/jtt041 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Knot theory of R-covered Anosov flows: homotopy versus isotopy of closed orbits </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=Fenley%2C+S+R">Sergio R. Fenley</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="1208.6487v1-abstract-short" style="display: inline;"> In this article, we study the knots realized by periodic orbits of R-covered Anosov flows in compact 3-manifolds. We show that if two orbits are freely homotopic then in fact they are isotopic. We show that lifts of periodic orbits to the universal cover are unknotted. When the manifold is atoroidal, we deduce some finer properties regarding the existence of embedded cylinders connecting two given… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1208.6487v1-abstract-full').style.display = 'inline'; document.getElementById('1208.6487v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1208.6487v1-abstract-full" style="display: none;"> In this article, we study the knots realized by periodic orbits of R-covered Anosov flows in compact 3-manifolds. We show that if two orbits are freely homotopic then in fact they are isotopic. We show that lifts of periodic orbits to the universal cover are unknotted. When the manifold is atoroidal, we deduce some finer properties regarding the existence of embedded cylinders connecting two given homotopic orbits. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1208.6487v1-abstract-full').style.display = 'none'; document.getElementById('1208.6487v1-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> 31 August, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 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">20 pages, 9 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/1206.1439">arXiv:1206.1439</a> <span> [<a href="https://arxiv.org/pdf/1206.1439">pdf</a>, <a href="https://arxiv.org/ps/1206.1439">ps</a>, <a href="https://arxiv.org/format/1206.1439">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</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/s10455-012-9355-8">10.1007/s10455-012-9355-8 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Eigenvalues control for a Finsler--Laplace operator </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=Colbois%2C+B">Bruno Colbois</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="1206.1439v1-abstract-short" style="display: inline;"> Using the definition of a Finsler--Laplacian given by the first author, we show that two bi-Lipschitz Finsler metrics have a controlled spectrum. We deduce from that several generalizations of Riemannian results. In particular, we show that the spectrum on Finsler surfaces is controlled above by a constant depending on the topology of the surface and on the quasireversibility constant of the metri… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1206.1439v1-abstract-full').style.display = 'inline'; document.getElementById('1206.1439v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1206.1439v1-abstract-full" style="display: none;"> Using the definition of a Finsler--Laplacian given by the first author, we show that two bi-Lipschitz Finsler metrics have a controlled spectrum. We deduce from that several generalizations of Riemannian results. In particular, we show that the spectrum on Finsler surfaces is controlled above by a constant depending on the topology of the surface and on the quasireversibility constant of the metric. In contrast to Riemannian geometry, we then give examples of highly non-reversible metrics on surfaces with arbitrarily large first eigenvalue. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1206.1439v1-abstract-full').style.display = 'none'; document.getElementById('1206.1439v1-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 June, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 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">27 pages, 3 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/1204.0879">arXiv:1204.0879</a> <span> [<a href="https://arxiv.org/pdf/1204.0879">pdf</a>, <a href="https://arxiv.org/ps/1204.0879">ps</a>, <a href="https://arxiv.org/format/1204.0879">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</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"> A new Laplace operator in Finsler geometry and periodic orbits of 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> </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="1204.0879v1-abstract-short" style="display: inline;"> In the first part of this dissertation, we give a new definition of a Laplace operator for Finsler metric as an average, with regard to an angle measure, of the second directional derivatives. This operator is elliptic, symmetric with respect to the Holmes-Thompson volume, and coincides with the usual Laplace--Beltrami operator when the Finsler metric is Riemannian. We compute explicit spectral da… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1204.0879v1-abstract-full').style.display = 'inline'; document.getElementById('1204.0879v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1204.0879v1-abstract-full" style="display: none;"> In the first part of this dissertation, we give a new definition of a Laplace operator for Finsler metric as an average, with regard to an angle measure, of the second directional derivatives. This operator is elliptic, symmetric with respect to the Holmes-Thompson volume, and coincides with the usual Laplace--Beltrami operator when the Finsler metric is Riemannian. We compute explicit spectral data for some Katok-Ziller metrics. When the Finsler metric is negatively curved, we show, thanks to a result of Ancona that the Martin boundary is H枚lder-homeomorphic to the visual boundary. This allow us to deduce the existence of harmonic measures and some ergodic preoperties. In the second part of this dissertation, we study Anosov flows in 3-manifolds, with leaf-spaces homeomorphic to $\mathbb{R}$. When the manifold is hyperbolic, Thurston showed that the (un)stable foliations induces a regulating flow. We use this second flow to study isotopy class of periodic orbits of the Anosov flow and existence of embedded cylinders. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1204.0879v1-abstract-full').style.display = 'none'; document.getElementById('1204.0879v1-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 April, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 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">Ph.D. Dissertation, Universit茅 de Strasbourg, 2012</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 58J60; 37D20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1104.4326">arXiv:1104.4326</a> <span> [<a href="https://arxiv.org/pdf/1104.4326">pdf</a>, <a href="https://arxiv.org/ps/1104.4326">ps</a>, <a href="https://arxiv.org/format/1104.4326">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</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/s11856-012-0168-z">10.1007/s11856-012-0168-z <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A natural Finsler--Laplace operator </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> </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="1104.4326v2-abstract-short" style="display: inline;"> We give a new definition of a Laplace operator for Finsler metric as an average with regard to an angle measure of the second directional derivatives. This definition uses a dynamical approach due to Foulon that does not require the use of connections nor local coordinates. We show using 1-parameter families of Katok--Ziller metrics that this Finsler--Laplace operator admits explicit representatio… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1104.4326v2-abstract-full').style.display = 'inline'; document.getElementById('1104.4326v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1104.4326v2-abstract-full" style="display: none;"> We give a new definition of a Laplace operator for Finsler metric as an average with regard to an angle measure of the second directional derivatives. This definition uses a dynamical approach due to Foulon that does not require the use of connections nor local coordinates. We show using 1-parameter families of Katok--Ziller metrics that this Finsler--Laplace operator admits explicit representations and computations of spectral data. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1104.4326v2-abstract-full').style.display = 'none'; document.getElementById('1104.4326v2-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 February, 2012; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 April, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 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">25 pages, v2: minor modifications, changed the introduction</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 58J60; 53C60 </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul class="nav-spaced"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg> <a href="https://info.arxiv.org/help/contact.html"> Contact</a> </li> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>subscribe to arXiv mailings</title><desc>Click here to subscribe</desc><path d="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z"/></svg> <a href="https://info.arxiv.org/help/subscribe"> Subscribe</a> </li> </ul> </div> </div> </div>   <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/help/license/index.html">Copyright</a></li> <li><a href="https://info.arxiv.org/help/policies/privacy_policy.html">Privacy Policy</a></li> </ul> </div> <div class="column sorry-app-links"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/help/web_accessibility.html">Web Accessibility Assistance</a></li> <li> <p class="help"> <a class="a11y-main-link" href="https://status.arxiv.org" target="_blank">arXiv Operational Status <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 256 512" class="icon filter-dark_grey" role="presentation"><path d="M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z"/></svg></a><br> Get status notifications via <a class="is-link" href="https://subscribe.sorryapp.com/24846f03/email/new" target="_blank"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg>email</a> or <a class="is-link" href="https://subscribe.sorryapp.com/24846f03/slack/new" target="_blank"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512" class="icon filter-black" role="presentation"><path d="M94.12 315.1c0 25.9-21.16 47.06-47.06 47.06S0 341 0 315.1c0-25.9 21.16-47.06 47.06-47.06h47.06v47.06zm23.72 0c0-25.9 21.16-47.06 47.06-47.06s47.06 21.16 47.06 47.06v117.84c0 25.9-21.16 47.06-47.06 47.06s-47.06-21.16-47.06-47.06V315.1zm47.06-188.98c-25.9 0-47.06-21.16-47.06-47.06S139 32 164.9 32s47.06 21.16 47.06 47.06v47.06H164.9zm0 23.72c25.9 0 47.06 21.16 47.06 47.06s-21.16 47.06-47.06 47.06H47.06C21.16 243.96 0 222.8 0 196.9s21.16-47.06 47.06-47.06H164.9zm188.98 47.06c0-25.9 21.16-47.06 47.06-47.06 25.9 0 47.06 21.16 47.06 47.06s-21.16 47.06-47.06 47.06h-47.06V196.9zm-23.72 0c0 25.9-21.16 47.06-47.06 47.06-25.9 0-47.06-21.16-47.06-47.06V79.06c0-25.9 21.16-47.06 47.06-47.06 25.9 0 47.06 21.16 47.06 47.06V196.9zM283.1 385.88c25.9 0 47.06 21.16 47.06 47.06 0 25.9-21.16 47.06-47.06 47.06-25.9 0-47.06-21.16-47.06-47.06v-47.06h47.06zm0-23.72c-25.9 0-47.06-21.16-47.06-47.06 0-25.9 21.16-47.06 47.06-47.06h117.84c25.9 0 47.06 21.16 47.06 47.06 0 25.9-21.16 47.06-47.06 47.06H283.1z"/></svg>slack</a> </p> </li> </ul> </div> </div> </div>  </div> </footer> <script src="https://static.arxiv.org/static/base/1.0.0a5/js/member_acknowledgement.js"></script> </body> </html>

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