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name="searchtype"><option value="all">All fields</option><option value="title">Title</option><option selected value="author">Author(s)</option><option value="abstract">Abstract</option><option value="comments">Comments</option><option value="journal_ref">Journal reference</option><option value="acm_class">ACM classification</option><option value="msc_class">MSC classification</option><option value="report_num">Report number</option><option value="paper_id">arXiv identifier</option><option value="doi">DOI</option><option value="orcid">ORCID</option><option value="license">License (URI)</option><option value="author_id">arXiv author ID</option><option value="help">Help pages</option><option value="full_text">Full text</option></select> <input id="query" name="query" type="text" value="Potrie, R"> <ul id="abstracts"><li><input checked id="abstracts-0" name="abstracts" type="radio" value="show"> <label for="abstracts-0">Show abstracts</label></li><li><input id="abstracts-1" 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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=Potrie%2C+R&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Potrie%2C+R&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Potrie%2C+R&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/2411.10875">arXiv:2411.10875</a> <span> [<a href="https://arxiv.org/pdf/2411.10875">pdf</a>, <a href="https://arxiv.org/format/2411.10875">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"> Quasi-isometric center action in dimension 3 </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Espitia%2C+M">Marcielis Espitia</a>, <a href="/search/math?searchtype=author&query=Martinchich%2C+S">Santiago Martinchich</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="2411.10875v1-abstract-short" style="display: inline;"> We study transitive partially hyperbolic diffeomorphisms in dimension 3 preserving a center foliation on which they act quasi-isometrically. We show that the diffeomorphism is up to finite lift and iterate, either a skew-product or a discretised Anosov flow. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.10875v1-abstract-full" style="display: none;"> We study transitive partially hyperbolic diffeomorphisms in dimension 3 preserving a center foliation on which they act quasi-isometrically. We show that the diffeomorphism is up to finite lift and iterate, either a skew-product or a discretised Anosov flow. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.10875v1-abstract-full').style.display = 'none'; document.getElementById('2411.10875v1-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 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 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">13 pages, 2 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/2411.03509">arXiv:2411.03509</a> <span> [<a href="https://arxiv.org/pdf/2411.03509">pdf</a>, <a href="https://arxiv.org/format/2411.03509">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> <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"> Quasi-isometric free group representations into SL_3(R) </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Carvajales%2C+L">Le贸n Carvajales</a>, <a href="/search/math?searchtype=author&query=Lessa%2C+P">Pablo Lessa</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="2411.03509v1-abstract-short" style="display: inline;"> We study quasi-isometric representations of finitely generated non-abelian free groups into some higher rank semi-simple Lie groups which are not Anosov, nor approximated by Anosov. We show in some cases that these can be perturbed to be non-quasi-isometric, or to have some instability properties with respect to their action on the flag space. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.03509v1-abstract-full" style="display: none;"> We study quasi-isometric representations of finitely generated non-abelian free groups into some higher rank semi-simple Lie groups which are not Anosov, nor approximated by Anosov. We show in some cases that these can be perturbed to be non-quasi-isometric, or to have some instability properties with respect to their action on the flag space. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.03509v1-abstract-full').style.display = 'none'; document.getElementById('2411.03509v1-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> <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</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.06637">arXiv:2410.06637</a> <span> [<a href="https://arxiv.org/pdf/2410.06637">pdf</a>, <a href="https://arxiv.org/format/2410.06637">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 in dimension 3: an outside look </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="2410.06637v1-abstract-short" style="display: inline;"> These notes were intended as support material for a minicourse on Anosov flows in the conference "Symplectic geometry and Anosov flows'' which took place in Heidelberg in July 2024 organized by Peter Albers, Jonathan Bowden and Agust铆n Moreno. I took the invitation to present the subject as asking from an outsider view of the subject, given the fact that my research uses both ideas and results fro… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.06637v1-abstract-full').style.display = 'inline'; document.getElementById('2410.06637v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.06637v1-abstract-full" style="display: none;"> These notes were intended as support material for a minicourse on Anosov flows in the conference "Symplectic geometry and Anosov flows'' which took place in Heidelberg in July 2024 organized by Peter Albers, Jonathan Bowden and Agust铆n Moreno. I took the invitation to present the subject as asking from an outsider view of the subject, given the fact that my research uses both ideas and results from the theory of Anosov flows. The point of view of the course is to provide an overview of the main results and questions in the subject, with emphasis on the interaction with topology, geometry, specially symplectic geometry and contact aspects of the theory. Some detail is given in the presentation of the Barbot-Fenley theory of leaf spaces. Hopefully the notes will contribute in gaining a working knowledge of the theory and its many beautiful connections. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.06637v1-abstract-full').style.display = 'none'; document.getElementById('2410.06637v1-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 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 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">40 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/2311.12979">arXiv:2311.12979</a> <span> [<a href="https://arxiv.org/pdf/2311.12979">pdf</a>, <a href="https://arxiv.org/format/2311.12979">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"> The stability conjecture for geodesic flows of compact manifolds without conjugate points and quasi-convex universal covering </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Potrie%2C+R">Rafael Potrie</a>, <a href="/search/math?searchtype=author&query=Ruggiero%2C+R+O">Rafael O. Ruggiero</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="2311.12979v1-abstract-short" style="display: inline;"> Let $(M,g)$ be a $C^{\infty}$ compact, boudaryless connected manifold without conjugate points with quasi-convex universal covering and divergent geodesic rays. We show that the geodesic flow of $(M,g)$ is $C^{2}$-structurally stable from Ma帽茅's viewpoint if and only if it is an Anosov flow, proving the so-called $C^{1}$-stability conjecture. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.12979v1-abstract-full" style="display: none;"> Let $(M,g)$ be a $C^{\infty}$ compact, boudaryless connected manifold without conjugate points with quasi-convex universal covering and divergent geodesic rays. We show that the geodesic flow of $(M,g)$ is $C^{2}$-structurally stable from Ma帽茅's viewpoint if and only if it is an Anosov flow, proving the so-called $C^{1}$-stability conjecture. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.12979v1-abstract-full').style.display = 'none'; document.getElementById('2311.12979v1-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 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 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">29 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/2310.05176">arXiv:2310.05176</a> <span> [<a href="https://arxiv.org/pdf/2310.05176">pdf</a>, <a href="https://arxiv.org/format/2310.05176">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="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"> Intersection of transverse foliations in 3-manifolds: Hausdorff leafspace implies leafwise quasi-geodesic </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="2310.05176v2-abstract-short" style="display: inline;"> Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be transverse two dimensional foliations with Gromov hyperbolic leaves in a closed 3-manifold $M$ whose fundamental group is not solvable, and let $\mathcal{G}$ be the one dimensional foliation obtained by intersection. We show that $\mathcal{G}$ is \emph{leafwise quasigeodesic} in $\mathcal{F}_1$ and $\mathcal{F}_2$ if and only if the foliation… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.05176v2-abstract-full').style.display = 'inline'; document.getElementById('2310.05176v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2310.05176v2-abstract-full" style="display: none;"> Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be transverse two dimensional foliations with Gromov hyperbolic leaves in a closed 3-manifold $M$ whose fundamental group is not solvable, and let $\mathcal{G}$ be the one dimensional foliation obtained by intersection. We show that $\mathcal{G}$ is \emph{leafwise quasigeodesic} in $\mathcal{F}_1$ and $\mathcal{F}_2$ if and only if the foliation $\mathcal{G}_L$ induced by $\mathcal{G}$ in the universal cover $L$ of any leaf of $\mathcal{F}_1$ or $\mathcal{F}_2$ has Hausdorff leaf space. We end up with a discussion on the hypothesis of Gromov hyperbolicity of the leaves. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.05176v2-abstract-full').style.display = 'none'; document.getElementById('2310.05176v2-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 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 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">43 pages, 9 figures. Added a section with application to partially hyperbolic dynamics by suggestion of T. Barthelme</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2303.14525">arXiv:2303.14525</a> <span> [<a href="https://arxiv.org/pdf/2303.14525">pdf</a>, <a href="https://arxiv.org/format/2303.14525">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="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"> Transverse minimal foliations on unit tangent bundles and applications </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="2303.14525v2-abstract-short" style="display: inline;"> We show that if $\mathcal{F}_1$ and $\mathcal{F}_2$ are two transverse minimal foliations on $M = T^1S$ then either they intersect in an Anosov foliation or there exists a Reeb-surface in the intersection foliation. The existence of a Reeb surface is incompatible with partially hyperbolic foliations so we deduce from this that certain partially hyperbolic diffeomorphisms in unit tangent bundles ar… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.14525v2-abstract-full').style.display = 'inline'; document.getElementById('2303.14525v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2303.14525v2-abstract-full" style="display: none;"> We show that if $\mathcal{F}_1$ and $\mathcal{F}_2$ are two transverse minimal foliations on $M = T^1S$ then either they intersect in an Anosov foliation or there exists a Reeb-surface in the intersection foliation. The existence of a Reeb surface is incompatible with partially hyperbolic foliations so we deduce from this that certain partially hyperbolic diffeomorphisms in unit tangent bundles are collapsed Anosov flows. We also conclude that every volume preserving partially hyperbolic diffeomorphism of a unit tangent bundle is ergodic. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.14525v2-abstract-full').style.display = 'none'; document.getElementById('2303.14525v2-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 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 25 March, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 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">64 pages, 20 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/2302.12981">arXiv:2302.12981</a> <span> [<a href="https://arxiv.org/pdf/2302.12981">pdf</a>, <a href="https://arxiv.org/format/2302.12981">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"> Geometric properties of partially hyperbolic measures and applications to measure rigidity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Eskin%2C+A">Alex Eskin</a>, <a href="/search/math?searchtype=author&query=Potrie%2C+R">Rafael Potrie</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Z">Zhiyuan 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="2302.12981v2-abstract-short" style="display: inline;"> We give a geometric characterization of the quantitative non-integrability, introduced by Katz, of strong stable and unstable bundles of partially hyperbolic measures and sets in dimension 3. This is done via the use of higher order templates for the invariant bundles. Using the recent work of Katz, we derive some consequences, including the measure rigidity of $uu$-states and the existence of phy… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.12981v2-abstract-full').style.display = 'inline'; document.getElementById('2302.12981v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2302.12981v2-abstract-full" style="display: none;"> We give a geometric characterization of the quantitative non-integrability, introduced by Katz, of strong stable and unstable bundles of partially hyperbolic measures and sets in dimension 3. This is done via the use of higher order templates for the invariant bundles. Using the recent work of Katz, we derive some consequences, including the measure rigidity of $uu$-states and the existence of physical measures. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.12981v2-abstract-full').style.display = 'none'; document.getElementById('2302.12981v2-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">v1</span> submitted 24 February, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 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">52 pages. Substantial revision thanks to input from several colleagues. Added some flowcharts to describe the structure of the proof and included some toy versions to convey the structure of the proof of some points</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2106.03291">arXiv:2106.03291</a> <span> [<a href="https://arxiv.org/pdf/2106.03291">pdf</a>, <a href="https://arxiv.org/ps/2106.03291">ps</a>, <a href="https://arxiv.org/format/2106.03291">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 transitivity and domination for endomorphisms displaying critical points </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lizana%2C+C">C. Lizana</a>, <a href="/search/math?searchtype=author&query=Potrie%2C+R">R. Potrie</a>, <a href="/search/math?searchtype=author&query=Pujals%2C+E+R">E. R. Pujals</a>, <a href="/search/math?searchtype=author&query=Ranter%2C+W">W. Ranter</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="2106.03291v4-abstract-short" style="display: inline;"> We show that robustly transitive endomorphisms of a closed manifolds must have a non-trivial dominated splitting or be a local diffeomorphism. This allows to get some topological obstructions for the existence of robustly transitive endomorphisms. To obtain the result we must understand the structure of the kernel of the differential and the recurrence to the critical set of the endomorphism after… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.03291v4-abstract-full').style.display = 'inline'; document.getElementById('2106.03291v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2106.03291v4-abstract-full" style="display: none;"> We show that robustly transitive endomorphisms of a closed manifolds must have a non-trivial dominated splitting or be a local diffeomorphism. This allows to get some topological obstructions for the existence of robustly transitive endomorphisms. To obtain the result we must understand the structure of the kernel of the differential and the recurrence to the critical set of the endomorphism after perturbation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.03291v4-abstract-full').style.display = 'none'; document.getElementById('2106.03291v4-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, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 June, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 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">We rewrote Lemmas 2.7 and 4.7 in order to be clearer and add some referee's sugentions. We would like to thanks the referee for their suggestion and comments</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2103.14630">arXiv:2103.14630</a> <span> [<a href="https://arxiv.org/pdf/2103.14630">pdf</a>, <a href="https://arxiv.org/format/2103.14630">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"> Accessibility and ergodicity for collapsed Anosov flows </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="2103.14630v1-abstract-short" style="display: inline;"> We consider a class of partially hyperbolic diffeomorphisms introduced in [BFP] which is open and closed and contains all known examples. If in addition the diffeomorphism is non-wandering, then we show it is accessible unless it contains a su-torus. This implies that these systems are ergodic when they preserve volume, confirming a conjecture by Hertz-Hertz-Ures for this class of systems. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2103.14630v1-abstract-full" style="display: none;"> We consider a class of partially hyperbolic diffeomorphisms introduced in [BFP] which is open and closed and contains all known examples. If in addition the diffeomorphism is non-wandering, then we show it is accessible unless it contains a su-torus. This implies that these systems are ergodic when they preserve volume, confirming a conjecture by Hertz-Hertz-Ures for this class of systems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.14630v1-abstract-full').style.display = 'none'; document.getElementById('2103.14630v1-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 March, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 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">17 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/2103.02364">arXiv:2103.02364</a> <span> [<a href="https://arxiv.org/pdf/2103.02364">pdf</a>, <a href="https://arxiv.org/ps/2103.02364">ps</a>, <a href="https://arxiv.org/format/2103.02364">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 remark on uniform expansion </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="2103.02364v1-abstract-short" style="display: inline;"> For every $\mathcal{U} \subset \mathrm{Diff}^\infty_{vol}(\mathbb{T}^2)$ there is a measure of finite support contained in $\mathcal{U}$ which is uniformly expanding. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2103.02364v1-abstract-full" style="display: none;"> For every $\mathcal{U} \subset \mathrm{Diff}^\infty_{vol}(\mathbb{T}^2)$ there is a measure of finite support contained in $\mathcal{U}$ which is uniformly expanding. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.02364v1-abstract-full').style.display = 'none'; document.getElementById('2103.02364v1-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 March, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 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">10 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/2102.02156">arXiv:2102.02156</a> <span> [<a href="https://arxiv.org/pdf/2102.02156">pdf</a>, <a href="https://arxiv.org/format/2102.02156">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"> Partial hyperbolicity and pseudo-Anosov dynamics </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="2102.02156v3-abstract-short" style="display: inline;"> We show that if a hyperbolic 3-manifold admits a partially hyperbolic diffeomorphism then it also admits an Anosov flow. Moreover, we give a complete classification of partially hyperbolic diffeomorphism in hyperbolic 3-manifolds as well as partially hyperbolic diffeomorphisms in Seifert manifolds inducing pseudo-Anosov dynamics in the base. This classification is given in terms of the structure o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2102.02156v3-abstract-full').style.display = 'inline'; document.getElementById('2102.02156v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2102.02156v3-abstract-full" style="display: none;"> We show that if a hyperbolic 3-manifold admits a partially hyperbolic diffeomorphism then it also admits an Anosov flow. Moreover, we give a complete classification of partially hyperbolic diffeomorphism in hyperbolic 3-manifolds as well as partially hyperbolic diffeomorphisms in Seifert manifolds inducing pseudo-Anosov dynamics in the base. This classification is given in terms of the structure of their center (branching) foliations and the notion of collapsed Anosov flows. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2102.02156v3-abstract-full').style.display = 'none'; document.getElementById('2102.02156v3-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 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 February, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 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">66 pages, 9 figures. Some changes to take into accounts suggestions by a referee. To appear in GAFA</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2008.06547">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/2005.10889">arXiv:2005.10889</a> <span> [<a href="https://arxiv.org/pdf/2005.10889">pdf</a>, <a href="https://arxiv.org/format/2005.10889">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 dynamics and 3-manifold topology </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="2005.10889v2-abstract-short" style="display: inline;"> This is an expository note intended to illustrate current research in topological study of partially hyperbolic diffeomorphisms in dimension 3 with a beautiful result due to Margulis and Plante-Thurston on topological obstructions for a manifold to admit an Anosov flow. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2005.10889v2-abstract-full" style="display: none;"> This is an expository note intended to illustrate current research in topological study of partially hyperbolic diffeomorphisms in dimension 3 with a beautiful result due to Margulis and Plante-Thurston on topological obstructions for a manifold to admit an Anosov flow. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2005.10889v2-abstract-full').style.display = 'none'; document.getElementById('2005.10889v2-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, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 May, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 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">10 pages, 4 figures. To appear in Notices of the AMS</span> </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/2002.07015">arXiv:2002.07015</a> <span> [<a href="https://arxiv.org/pdf/2002.07015">pdf</a>, <a href="https://arxiv.org/ps/2002.07015">ps</a>, <a href="https://arxiv.org/format/2002.07015">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"> Eigenvalue gaps for hyperbolic groups and semigroups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Kassel%2C+F">Fanny Kassel</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="2002.07015v3-abstract-short" style="display: inline;"> Given a locally constant linear cocycle over a subshift of finite type, we show that the existence of a uniform gap between the i-th and (i+1)-th Lyapunov exponents for all invariant measures implies the existence of a dominated splitting of index i. We establish a similar result for sofic subshifts coming from word hyperbolic groups, in relation with Anosov representations of such groups. We disc… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.07015v3-abstract-full').style.display = 'inline'; document.getElementById('2002.07015v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2002.07015v3-abstract-full" style="display: none;"> Given a locally constant linear cocycle over a subshift of finite type, we show that the existence of a uniform gap between the i-th and (i+1)-th Lyapunov exponents for all invariant measures implies the existence of a dominated splitting of index i. We establish a similar result for sofic subshifts coming from word hyperbolic groups, in relation with Anosov representations of such groups. We discuss the case of finitely generated semigroups, and propose a notion of Anosov representation in this setting. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.07015v3-abstract-full').style.display = 'none'; document.getElementById('2002.07015v3-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, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 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">42 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Mod. Dyn. 18 (2022) 161-208 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2001.05522">arXiv:2001.05522</a> <span> [<a href="https://arxiv.org/pdf/2001.05522">pdf</a>, <a href="https://arxiv.org/format/2001.05522">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"> Minimality of the action on the universal circle of uniform foliations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fenley%2C+S">Sergio 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="2001.05522v2-abstract-short" style="display: inline;"> Given a uniform foliation by Gromov hyperbolic leaves on a $3$-manifold, we show that the action of the fundamental group on the universal circle is minimal and transitive on pairs of different points. We also prove two other results: we prove that general uniform Reebless foliations are $\mathbb{R}$-covered and we give a new description of the universal circle of $\mathbb{R}$-covered foliations w… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2001.05522v2-abstract-full').style.display = 'inline'; document.getElementById('2001.05522v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2001.05522v2-abstract-full" style="display: none;"> Given a uniform foliation by Gromov hyperbolic leaves on a $3$-manifold, we show that the action of the fundamental group on the universal circle is minimal and transitive on pairs of different points. We also prove two other results: we prove that general uniform Reebless foliations are $\mathbb{R}$-covered and we give a new description of the universal circle of $\mathbb{R}$-covered foliations with Gromov hyperbolic leaves in terms of the JSJ decomposition of $M$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2001.05522v2-abstract-full').style.display = 'none'; document.getElementById('2001.05522v2-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 May, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 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">29 pages, 2 figures, to appear in GGD</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/1906.10029">arXiv:1906.10029</a> <span> [<a href="https://arxiv.org/pdf/1906.10029">pdf</a>, <a href="https://arxiv.org/format/1906.10029">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"> Topology of leaves for minimal laminations by hyperbolic surfaces </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=Brum%2C+J">Joaqu铆n Brum</a>, <a href="/search/math?searchtype=author&query=Mart%C3%ADnez%2C+M">Matilde Mart铆nez</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="1906.10029v2-abstract-short" style="display: inline;"> We construct minimal laminations by hyperbolic surfaces whose generic leaf is a disk and contain any prescribed family of surfaces and with a precise control of the topologies of the surfaces that appear. The laminations are constructed via towers of finite coverings of surfaces for which we need to develop a relative version of residual finiteness which may be of independent interest. The main st… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.10029v2-abstract-full').style.display = 'inline'; document.getElementById('1906.10029v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1906.10029v2-abstract-full" style="display: none;"> We construct minimal laminations by hyperbolic surfaces whose generic leaf is a disk and contain any prescribed family of surfaces and with a precise control of the topologies of the surfaces that appear. The laminations are constructed via towers of finite coverings of surfaces for which we need to develop a relative version of residual finiteness which may be of independent interest. The main step in establishing this relative version of residual finiteness is to obtain finite covers with control on the \emph{second systole} of the surface, which is done in the appendix. In a companion paper, the case of other generic leaves is treated. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.10029v2-abstract-full').style.display = 'none'; document.getElementById('1906.10029v2-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> 1 February, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 June, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 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">With an appendix by the authors and Maxime Wolff. 43 pages. 11 figures. Final version. To appear in Journal of Topology</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1809.02284">arXiv:1809.02284</a> <span> [<a href="https://arxiv.org/pdf/1809.02284">pdf</a>, <a href="https://arxiv.org/format/1809.02284">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"> Ergodicity of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fenley%2C+S">Sergio 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="1809.02284v3-abstract-short" style="display: inline;"> We study conservative partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds. We show that they are always accessible and deduce as a result that every conservative $C^{1+}$ partially hyperbolic in a hyperbolic 3-manifold must be ergodic, giving an afirmative answer to a conjecture of Hertz-Hertz-Ures in the context of hyperbolic 3-manifolds. Some of the intermediary steps are also done fo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1809.02284v3-abstract-full').style.display = 'inline'; document.getElementById('1809.02284v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1809.02284v3-abstract-full" style="display: none;"> We study conservative partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds. We show that they are always accessible and deduce as a result that every conservative $C^{1+}$ partially hyperbolic in a hyperbolic 3-manifold must be ergodic, giving an afirmative answer to a conjecture of Hertz-Hertz-Ures in the context of hyperbolic 3-manifolds. Some of the intermediary steps are also done for general partially hyperbolic diffeomorphisms homotopic to the identity. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1809.02284v3-abstract-full').style.display = 'none'; document.getElementById('1809.02284v3-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 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 September, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 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">43 pages, 4 figures. To appear in Advances in 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/1802.05291">arXiv:1802.05291</a> <span> [<a href="https://arxiv.org/pdf/1802.05291">pdf</a>, <a href="https://arxiv.org/format/1802.05291">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"> Robust dynamics, invariant structures and topological classification </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="1802.05291v1-abstract-short" style="display: inline;"> This text is about geometric structures imposed by robust dynamical behaviour. We explain recent results towards the classification of partially hyperbolic systems in dimension 3 using the theory of foliations and its interaction with topology. We also present recent examples which introduce a challenge in the classification program and we propose some steps to continue this classification. Finall… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1802.05291v1-abstract-full').style.display = 'inline'; document.getElementById('1802.05291v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1802.05291v1-abstract-full" style="display: none;"> This text is about geometric structures imposed by robust dynamical behaviour. We explain recent results towards the classification of partially hyperbolic systems in dimension 3 using the theory of foliations and its interaction with topology. We also present recent examples which introduce a challenge in the classification program and we propose some steps to continue this classification. Finally, we give some suggestions on what to do after classification is achieved. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1802.05291v1-abstract-full').style.display = 'none'; document.getElementById('1802.05291v1-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 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, 2 figures. Invited paper Dynamical Systems session in ICM 2018. http://www.icm2018.org</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/1706.08684">arXiv:1706.08684</a> <span> [<a href="https://arxiv.org/pdf/1706.08684">pdf</a>, <a href="https://arxiv.org/format/1706.08684">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"> Finiteness of partially hyperbolic attractors with one-dimensional center </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Crovisier%2C+S">Sylvain Crovisier</a>, <a href="/search/math?searchtype=author&query=Potrie%2C+R">Rafael Potrie</a>, <a href="/search/math?searchtype=author&query=Sambarino%2C+M">Mart铆n Sambarino</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="1706.08684v2-abstract-short" style="display: inline;"> We prove that the set of diffeomorphisms having at most finitely many attractors contains a dense and open subset of the space of $C^1$ partially hyperbolic diffeomorphisms with one-dimensional center. This is obtained thanks to a robust geometric property of partially hyperbolic laminations that we show to hold after perturbations of the dynamics. This technique also allows to prove that $C^1$-… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1706.08684v2-abstract-full').style.display = 'inline'; document.getElementById('1706.08684v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1706.08684v2-abstract-full" style="display: none;"> We prove that the set of diffeomorphisms having at most finitely many attractors contains a dense and open subset of the space of $C^1$ partially hyperbolic diffeomorphisms with one-dimensional center. This is obtained thanks to a robust geometric property of partially hyperbolic laminations that we show to hold after perturbations of the dynamics. This technique also allows to prove that $C^1$-generic diffeomorphisms far from homoclinic tangencies in dimension $3$ either have at most finitely many attractors, or satisfy Newhouse phenomenon. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1706.08684v2-abstract-full').style.display = 'none'; document.getElementById('1706.08684v2-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 December, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 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">28 pages, 4 figures. Some corrections in section 5 thanks to comments of D.Yang and J. Zhang. To appear in Ann. 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/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/1702.06206">arXiv:1702.06206</a> <span> [<a href="https://arxiv.org/pdf/1702.06206">pdf</a>, <a href="https://arxiv.org/ps/1702.06206">ps</a>, <a href="https://arxiv.org/format/1702.06206">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"> Classification of systems with center-stable tori </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="1702.06206v1-abstract-short" style="display: inline;"> This paper gives a classification of partially hyperbolic systems in dimension 3 which have at least one torus tangent to the center-stable bundle. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1702.06206v1-abstract-full" style="display: none;"> This paper gives a classification of partially hyperbolic systems in dimension 3 which have at least one torus tangent to the center-stable bundle. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.06206v1-abstract-full').style.display = 'none'; document.getElementById('1702.06206v1-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, 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">20 pages, 1 figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37D30; 57R30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1701.01196">arXiv:1701.01196</a> <span> [<a href="https://arxiv.org/pdf/1701.01196">pdf</a>, <a href="https://arxiv.org/format/1701.01196">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="General Topology">math.GN</span> </div> </div> <p class="title is-5 mathjax"> Seifert manifolds admitting partially hyperbolic diffeomorphisms </p> <p class="authors"> <span class="search-hit">Authors:</span> <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>, <a href="/search/math?searchtype=author&query=Shannon%2C+M">Mario Shannon</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="1701.01196v2-abstract-short" style="display: inline;"> We characterize which 3-dimensional Seifert manifolds admit transitive partially hyperbolic diffeomorphisms. In particular, a circle bundle over a higher-genus surface admits a transitive partially hyperbolic diffeomorphism if and only if it admits an Anosov flow. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1701.01196v2-abstract-full" style="display: none;"> We characterize which 3-dimensional Seifert manifolds admit transitive partially hyperbolic diffeomorphisms. In particular, a circle bundle over a higher-genus surface admits a transitive partially hyperbolic diffeomorphism if and only if it admits an Anosov flow. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1701.01196v2-abstract-full').style.display = 'none'; document.getElementById('1701.01196v2-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 April, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 January, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 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">32 pages, 4 figures. Some revisions after referee report. To appear in JMD</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1609.09452">arXiv:1609.09452</a> <span> [<a href="https://arxiv.org/pdf/1609.09452">pdf</a>, <a href="https://arxiv.org/ps/1609.09452">ps</a>, <a href="https://arxiv.org/format/1609.09452">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"> Free orbits for minimal actions on the circle </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Brum%2C+J">Joaqu铆n Brum</a>, <a href="/search/math?searchtype=author&query=Mart%C3%ADnez%2C+M">Matilde Mart铆nez</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="1609.09452v1-abstract-short" style="display: inline;"> We prove that if $螕$ is a countable group without a subgroup isomorphic to $\mathbb{Z}^2$ that acts faithfully and minimally by orientation preserving homeomorphisms on the circle, then it has a free orbit. We give examples showing that this does not hold for actions by homeomorphisms of the line. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1609.09452v1-abstract-full" style="display: none;"> We prove that if $螕$ is a countable group without a subgroup isomorphic to $\mathbb{Z}^2$ that acts faithfully and minimally by orientation preserving homeomorphisms on the circle, then it has a free orbit. We give examples showing that this does not hold for actions by homeomorphisms of the line. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1609.09452v1-abstract-full').style.display = 'none'; document.getElementById('1609.09452v1-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 September, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2016. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1605.01742">arXiv:1605.01742</a> <span> [<a href="https://arxiv.org/pdf/1605.01742">pdf</a>, <a href="https://arxiv.org/format/1605.01742">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Metric Geometry">math.MG</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/10.4171/JEMS/905">10.4171/JEMS/905 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Anosov representations and dominated splittings </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=Potrie%2C+R">Rafael Potrie</a>, <a href="/search/math?searchtype=author&query=Sambarino%2C+A">Andr茅s Sambarino</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="1605.01742v3-abstract-short" style="display: inline;"> We provide a link between Anosov representations introduced by Labourie and dominated splitting of linear cocycles. This allows us to obtain equivalent characterizations for Anosov representations and to recover recent results due to Gu茅ritaud-Guichard-Kassel-Wienhard and Kapovich-Leeb-Porti by different methods. We also give characterizations in terms of multicones and cone-types inspired in the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.01742v3-abstract-full').style.display = 'inline'; document.getElementById('1605.01742v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1605.01742v3-abstract-full" style="display: none;"> We provide a link between Anosov representations introduced by Labourie and dominated splitting of linear cocycles. This allows us to obtain equivalent characterizations for Anosov representations and to recover recent results due to Gu茅ritaud-Guichard-Kassel-Wienhard and Kapovich-Leeb-Porti by different methods. We also give characterizations in terms of multicones and cone-types inspired in the work of Avila-Bochi-Yoccoz and Bochi-Gourmelon. Finally we provide a new proof of the higher rank Morse Lemma of Kapovich-Leeb-Porti. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.01742v3-abstract-full').style.display = 'none'; document.getElementById('1605.01742v3-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> 1 May, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 5 May, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 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">Published version. A wrong claim about cone types in Sec. 5.3 was struck out; this claim wasn't used</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 22E40 (primary); 20F67; 37B15; 37D30; 53C35; 53D25 (secondary) </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Eur. Math. Soc. 21 (2019), no. 11, pp. 3343-3414 </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/1511.04471">arXiv:1511.04471</a> <span> [<a href="https://arxiv.org/pdf/1511.04471">pdf</a>, <a href="https://arxiv.org/ps/1511.04471">ps</a>, <a href="https://arxiv.org/format/1511.04471">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.2016.50">10.1017/etds.2016.50 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Partial hyperbolicity and classification: a survey </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="1511.04471v2-abstract-short" style="display: inline;"> This paper surveys recent results on classifying partially hyperbolic diffeomorphisms. This includes the construction of branching foliations and leaf conjugacies on three-dimensional manifolds with solvable fundamental group. Classification results in higher-dimensional settings are also discussed. The paper concludes with an overview of the construction of new partially hyperbolic examples deriv… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1511.04471v2-abstract-full').style.display = 'inline'; document.getElementById('1511.04471v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1511.04471v2-abstract-full" style="display: none;"> This paper surveys recent results on classifying partially hyperbolic diffeomorphisms. This includes the construction of branching foliations and leaf conjugacies on three-dimensional manifolds with solvable fundamental group. Classification results in higher-dimensional settings are also discussed. The paper concludes with an overview of the construction of new partially hyperbolic examples derived from Anosov flows. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1511.04471v2-abstract-full').style.display = 'none'; document.getElementById('1511.04471v2-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 April, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 November, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 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">49 pages, 6 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37C05; 37D30; 57R30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1511.04434">arXiv:1511.04434</a> <span> [<a href="https://arxiv.org/pdf/1511.04434">pdf</a>, <a href="https://arxiv.org/ps/1511.04434">ps</a>, <a href="https://arxiv.org/format/1511.04434">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.2018.22.2145">10.2140/gt.2018.22.2145 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Rotation intervals and entropy on attracting annular continua </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Passeggi%2C+A">Alejandro Passeggi</a>, <a href="/search/math?searchtype=author&query=Potrie%2C+R">Rafael Potrie</a>, <a href="/search/math?searchtype=author&query=Sambarino%2C+M">Mart铆n Sambarino</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="1511.04434v2-abstract-short" style="display: inline;"> We show that if $f$ is an annular homeomorphism admitting an attractor which is an irreducible annular continua with two different rotation numbers, then the entropy of $f$ is positive. Further, the entropy is shown to be associated to a $C^0$-robust rotational horseshoe. On the other hand, we construct examples of annular homeomorphisms with such attractors so that the rotation interval is unifor… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1511.04434v2-abstract-full').style.display = 'inline'; document.getElementById('1511.04434v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1511.04434v2-abstract-full" style="display: none;"> We show that if $f$ is an annular homeomorphism admitting an attractor which is an irreducible annular continua with two different rotation numbers, then the entropy of $f$ is positive. Further, the entropy is shown to be associated to a $C^0$-robust rotational horseshoe. On the other hand, we construct examples of annular homeomorphisms with such attractors so that the rotation interval is uniformly large but the entropy approaches zero as much as desired. The developed techniques allow us to obtain similar results in the context of Birkhoff attractors. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1511.04434v2-abstract-full').style.display = 'none'; document.getElementById('1511.04434v2-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> 2 October, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 November, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 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">37 pages, 7 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. 22 (2018) 2145-2186 </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/1411.5405">arXiv:1411.5405</a> <span> [<a href="https://arxiv.org/pdf/1411.5405">pdf</a>, <a href="https://arxiv.org/ps/1411.5405">ps</a>, <a href="https://arxiv.org/format/1411.5405">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="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Eigenvalues and Entropy of a Hitchin representation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Potrie%2C+R">Rafael Potrie</a>, <a href="/search/math?searchtype=author&query=Sambarino%2C+A">Andr茅s Sambarino</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="1411.5405v3-abstract-short" style="display: inline;"> We show that the critical exponent of a representation in the Hitchin component of $PSL(d,\mathbb{R})$ is bounded above, the least upper bound being attained only in the Fuchsian locus. This provides a rigid inequality for the area of a minimal surface on $蟻\backslash X,$ where $X$ is the symmetric space of $PSL(d,\mathbb{R}).$ The proof relies in a construction useful to prove a regularity statem… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1411.5405v3-abstract-full').style.display = 'inline'; document.getElementById('1411.5405v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1411.5405v3-abstract-full" style="display: none;"> We show that the critical exponent of a representation in the Hitchin component of $PSL(d,\mathbb{R})$ is bounded above, the least upper bound being attained only in the Fuchsian locus. This provides a rigid inequality for the area of a minimal surface on $蟻\backslash X,$ where $X$ is the symmetric space of $PSL(d,\mathbb{R}).$ The proof relies in a construction useful to prove a regularity statement: if the Frenet equivariant curve of $蟻$ is smooth, then $蟻$ is Fuchsian. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1411.5405v3-abstract-full').style.display = 'none'; document.getElementById('1411.5405v3-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, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 November, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2014. </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/1409.4138">arXiv:1409.4138</a> <span> [<a href="https://arxiv.org/pdf/1409.4138">pdf</a>, <a href="https://arxiv.org/ps/1409.4138">ps</a>, <a href="https://arxiv.org/format/1409.4138">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"> Livsic theorem for low-dimensional diffeomorphism cocycles </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Kocsard%2C+A">Alejandro Kocsard</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="1409.4138v1-abstract-short" style="display: inline;"> We prove a Livsic type theorem for cocycles taking values in groups of diffeomorphisms of low-dimensional manifolds. The results hold without any localization assumption and in very low regularity. We also obtain a general result (in any dimension) which gives necessary and sufficient conditions to be a coboundary. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1409.4138v1-abstract-full" style="display: none;"> We prove a Livsic type theorem for cocycles taking values in groups of diffeomorphisms of low-dimensional manifolds. The results hold without any localization assumption and in very low regularity. We also obtain a general result (in any dimension) which gives necessary and sufficient conditions to be a coboundary. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1409.4138v1-abstract-full').style.display = 'none'; document.getElementById('1409.4138v1-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 September, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 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">19 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37D20; 37H15; 37H05; 37D30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1408.3428">arXiv:1408.3428</a> <span> [<a href="https://arxiv.org/pdf/1408.3428">pdf</a>, <a href="https://arxiv.org/ps/1408.3428">ps</a>, <a href="https://arxiv.org/format/1408.3428">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 a trapping property </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="1408.3428v2-abstract-short" style="display: inline;"> We study partially hyperbolic diffeomorphisms satisfying a trapping property which makes them look as if they were Anosov at large scale. We show that, as expected, they share several properties with Anosov diffeomorphisms. We construct an expansive quotient of the dynamics and study some dynamical consequences related to this quotient. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1408.3428v2-abstract-full" style="display: none;"> We study partially hyperbolic diffeomorphisms satisfying a trapping property which makes them look as if they were Anosov at large scale. We show that, as expected, they share several properties with Anosov diffeomorphisms. We construct an expansive quotient of the dynamics and study some dynamical consequences related to this quotient. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1408.3428v2-abstract-full').style.display = 'none'; document.getElementById('1408.3428v2-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> 2 February, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 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">20 pages. New version incoporates improvements suggested by the referee</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1310.5918">arXiv:1310.5918</a> <span> [<a href="https://arxiv.org/pdf/1310.5918">pdf</a>, <a href="https://arxiv.org/ps/1310.5918">ps</a>, <a href="https://arxiv.org/format/1310.5918">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/s10884-014-9362-5">10.1007/s10884-014-9362-5 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A few remarks on partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ isotopic to Anosov </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="1310.5918v1-abstract-short" style="display: inline;"> We show that a strong partially hyperbolic diffeomorphism of $\mathbb{T}^3$ isotopic to Anosov has a unique quasi-attractor. Moreover, we study the entropy of the diffeomorphism restricted to this quasi-attractor. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1310.5918v1-abstract-full" style="display: none;"> We show that a strong partially hyperbolic diffeomorphism of $\mathbb{T}^3$ isotopic to Anosov has a unique quasi-attractor. Moreover, we study the entropy of the diffeomorphism restricted to this quasi-attractor. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1310.5918v1-abstract-full').style.display = 'none'; document.getElementById('1310.5918v1-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 October, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 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">13 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37C05; 37C20; 37C25; 37C29; 37D30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1307.4631">arXiv:1307.4631</a> <span> [<a href="https://arxiv.org/pdf/1307.4631">pdf</a>, <a href="https://arxiv.org/ps/1307.4631">ps</a>, <a href="https://arxiv.org/format/1307.4631">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/jtv009">10.1112/jtopol/jtv009 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Classification of partially hyperbolic diffeomorphisms in 3-manifolds with solvable fundamental group </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="1307.4631v2-abstract-short" style="display: inline;"> A classification of partially hyperbolic diffeomorphisms on 3-dimensional manifolds with (virtually) solvable fundamental group is obtained. If such a diffeomorphism does not admit a periodic attracting or repelling two-dimensional torus, it is dynamically coherent and leaf conjugate to a known algebraic example. This classification includes manifolds which support Anosov flows, and it confirms co… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1307.4631v2-abstract-full').style.display = 'inline'; document.getElementById('1307.4631v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1307.4631v2-abstract-full" style="display: none;"> A classification of partially hyperbolic diffeomorphisms on 3-dimensional manifolds with (virtually) solvable fundamental group is obtained. If such a diffeomorphism does not admit a periodic attracting or repelling two-dimensional torus, it is dynamically coherent and leaf conjugate to a known algebraic example. This classification includes manifolds which support Anosov flows, and it confirms conjectures by Rodriguez Hertz--Rodriguez Hertz--Ures and Pujals in the specific case of solvable fundamental group. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1307.4631v2-abstract-full').style.display = 'none'; document.getElementById('1307.4631v2-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 February, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 July, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 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">36 pages, 3 figures. To appear in the Journal of Topology</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37C05; 37C20; 37C25; 37C29; 37D30; 57R30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1305.1915">arXiv:1305.1915</a> <span> [<a href="https://arxiv.org/pdf/1305.1915">pdf</a>, <a href="https://arxiv.org/ps/1305.1915">ps</a>, <a href="https://arxiv.org/format/1305.1915">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"> Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fisher%2C+T">Todd Fisher</a>, <a href="/search/math?searchtype=author&query=Potrie%2C+R">Rafael Potrie</a>, <a href="/search/math?searchtype=author&query=Sambarino%2C+M">Mart铆n Sambarino</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="1305.1915v3-abstract-short" style="display: inline;"> We show that partially hyperbolic diffeomorphisms of $d$-dimensional tori isotopic to an Anosov diffeomorphism, where the isotopy is contained in the set of partially hyperbolic diffeomorphisms, are dynamically coherent. Moreover, we show a \textit{global stability result}, i.e. every partially hyperbolic diffeomorphism as above is \textit{leaf-conjugate} to the linear one. As a consequence, we ob… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1305.1915v3-abstract-full').style.display = 'inline'; document.getElementById('1305.1915v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1305.1915v3-abstract-full" style="display: none;"> We show that partially hyperbolic diffeomorphisms of $d$-dimensional tori isotopic to an Anosov diffeomorphism, where the isotopy is contained in the set of partially hyperbolic diffeomorphisms, are dynamically coherent. Moreover, we show a \textit{global stability result}, i.e. every partially hyperbolic diffeomorphism as above is \textit{leaf-conjugate} to the linear one. As a consequence, we obtain intrinsic ergodicity and measure equivalence for partially hyperbolic diffeomorphisms with one-dimensional center direction that are isotopic to Anosov diffeomorphisms through such a path. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1305.1915v3-abstract-full').style.display = 'none'; document.getElementById('1305.1915v3-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 October, 2013; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 May, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 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">31 pages, 1 figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37C05; 37C20; 37C25; 37C29; 37D30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1302.0543">arXiv:1302.0543</a> <span> [<a href="https://arxiv.org/pdf/1302.0543">pdf</a>, <a href="https://arxiv.org/ps/1302.0543">ps</a>, <a href="https://arxiv.org/format/1302.0543">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/jlms/jdu013">10.1112/jlms/jdu013 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Pointwise partial hyperbolicity in 3-dimensional nilmanifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="1302.0543v1-abstract-short" style="display: inline;"> We show the existence of a family of manifolds on which all (pointwise or absolutely) partially hyperbolic systems are dynamically coherent. This family is the set of 3-manifolds with nilpotent, non-abelian fundamental group. We further classify the partially hyperbolic systems on these manifolds up to leaf conjugacy. We also classify those systems on the 3-torus which do not have an attracting or… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1302.0543v1-abstract-full').style.display = 'inline'; document.getElementById('1302.0543v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1302.0543v1-abstract-full" style="display: none;"> We show the existence of a family of manifolds on which all (pointwise or absolutely) partially hyperbolic systems are dynamically coherent. This family is the set of 3-manifolds with nilpotent, non-abelian fundamental group. We further classify the partially hyperbolic systems on these manifolds up to leaf conjugacy. We also classify those systems on the 3-torus which do not have an attracting or repelling periodic 2-torus. These classification results allow us to prove some dynamical consequences, including existence and uniqueness results for measures of maximal entropy and quasi-attractors. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1302.0543v1-abstract-full').style.display = 'none'; document.getElementById('1302.0543v1-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, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 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">28 pages, 1 figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37C05; 37C20; 37C25; 37C29; 37D30; 57R30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1208.4394">arXiv:1208.4394</a> <span> [<a href="https://arxiv.org/pdf/1208.4394">pdf</a>, <a href="https://arxiv.org/ps/1208.4394">ps</a>, <a href="https://arxiv.org/format/1208.4394">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="History and Overview">math.HO</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"> On the work of Jorge Lewowicz on expansive systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="1208.4394v1-abstract-short" style="display: inline;"> We will try to give an overview of one of the landmark results of Jorge Lewowicz: his classification of expansive homeomorphisms of surfaces. The goal will be to present the main ideas with the hope of giving evidence of the deep and beautiful contributions he made to dynamical systems. We will avoid being technical and try to concentrate on the tools introduced by Lewowicz to obtain these classif… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1208.4394v1-abstract-full').style.display = 'inline'; document.getElementById('1208.4394v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1208.4394v1-abstract-full" style="display: none;"> We will try to give an overview of one of the landmark results of Jorge Lewowicz: his classification of expansive homeomorphisms of surfaces. The goal will be to present the main ideas with the hope of giving evidence of the deep and beautiful contributions he made to dynamical systems. We will avoid being technical and try to concentrate on the tools introduced by Lewowicz to obtain these classification results such as Lyapunov functions and the concept of persistence for dynamical systems. The main contribution that we will try to focus on is his conceptual framework and approach to mathematics reflected by the previously mentioned tools and fundamentally by the delicate interaction between topology and dynamics of expansive homeomorphisms of surfaces he discovered in order to establish his result. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1208.4394v1-abstract-full').style.display = 'none'; document.getElementById('1208.4394v1-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 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">Expanded version of talk given at a conference in Montevideo: http://imerl.fing.edu.uy/sdm2012/</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1207.1822">arXiv:1207.1822</a> <span> [<a href="https://arxiv.org/pdf/1207.1822">pdf</a>, <a href="https://arxiv.org/ps/1207.1822">ps</a>, <a href="https://arxiv.org/format/1207.1822">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"> Partial hyperbolicity and attracting regions in 3-dimensional manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="1207.1822v1-abstract-short" style="display: inline;"> This thesis attempts to contribute to the study of differentiable dynamics both from a semi-local and global point of view. The center of study is differentiable dynamics in manifolds of dimension 3 where we are interested in the understanding of the existence and structure of attractors as well as dynamical and topological implications of the existence of a global partially hyperbolic splitting.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1207.1822v1-abstract-full').style.display = 'inline'; document.getElementById('1207.1822v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1207.1822v1-abstract-full" style="display: none;"> This thesis attempts to contribute to the study of differentiable dynamics both from a semi-local and global point of view. The center of study is differentiable dynamics in manifolds of dimension 3 where we are interested in the understanding of the existence and structure of attractors as well as dynamical and topological implications of the existence of a global partially hyperbolic splitting. The main contributions are new examples of dynamics without attractors where we get a quite complete description of the dynamics around some wild homoclinic classes and two results on dynamical coherence of partially hyperbolic diffeomorphisms of $\TT^3$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1207.1822v1-abstract-full').style.display = 'none'; document.getElementById('1207.1822v1-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 July, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 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">PhD thesis. 294 pages, 12 figures. Advisors: Sylvain Crovisier and Mart铆n Sambarino</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1206.2860">arXiv:1206.2860</a> <span> [<a href="https://arxiv.org/pdf/1206.2860">pdf</a>, <a href="https://arxiv.org/ps/1206.2860">ps</a>, <a href="https://arxiv.org/format/1206.2860">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"> Partial hyperbolicity and foliations in $\mathbb{T}^3$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="1206.2860v2-abstract-short" style="display: inline;"> We prove that dynamical coherence is an open and closed property in the space of partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ isotopic to Anosov. Moreover, we prove that strong partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ are either dynamically coherent or have an invariant two-dimensional torus which is either contracting or repelling. We develop for this end some general resu… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1206.2860v2-abstract-full').style.display = 'inline'; document.getElementById('1206.2860v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1206.2860v2-abstract-full" style="display: none;"> We prove that dynamical coherence is an open and closed property in the space of partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ isotopic to Anosov. Moreover, we prove that strong partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ are either dynamically coherent or have an invariant two-dimensional torus which is either contracting or repelling. We develop for this end some general results on codimension one foliations which may be of independent interest. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1206.2860v2-abstract-full').style.display = 'none'; document.getElementById('1206.2860v2-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 July, 2014; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 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">45 pages, 4 figures. To appear in JMD. This version is more compact and includes many improvements clarifying proofs thanks to the referee report</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37C05; 37C20; 37C25; 37C29; 37D30; 57R30 </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/1106.3741">arXiv:1106.3741</a> <span> [<a href="https://arxiv.org/pdf/1106.3741">pdf</a>, <a href="https://arxiv.org/ps/1106.3741">ps</a>, <a href="https://arxiv.org/format/1106.3741">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.2012.124">10.1017/etds.2012.124 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Wild Milnor attractors accumulated by lower dimensional dynamics </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="1106.3741v2-abstract-short" style="display: inline;"> We present new examples of open sets of diffeomorphisms such that a generic diffeomorphisms in those sets have no dynamically indecomposable attractors in the topological sense and have infinitely many chain-recurrence classes. We show that except from one particular class, the other classes are contained in periodic surfaces. This study allows us to obtain existence of Milnor attractors as well a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1106.3741v2-abstract-full').style.display = 'inline'; document.getElementById('1106.3741v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1106.3741v2-abstract-full" style="display: none;"> We present new examples of open sets of diffeomorphisms such that a generic diffeomorphisms in those sets have no dynamically indecomposable attractors in the topological sense and have infinitely many chain-recurrence classes. We show that except from one particular class, the other classes are contained in periodic surfaces. This study allows us to obtain existence of Milnor attractors as well as studying ergodic properties of the diffeomorphisms in those open sets by using the ideas and results from [BV] and [BF]. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1106.3741v2-abstract-full').style.display = 'none'; document.getElementById('1106.3741v2-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 June, 2012; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 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">30 pages, 4 figures. This paper contains the results of http://arxiv.org/abs/1003.2280 and some more. It has been completely rewritten</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Ergod. Th. Dynam. Sys. 34 (2014) 236-262 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1103.4576">arXiv:1103.4576</a> <span> [<a href="https://arxiv.org/pdf/1103.4576">pdf</a>, <a href="https://arxiv.org/ps/1103.4576">ps</a>, <a href="https://arxiv.org/format/1103.4576">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"> Recurrence of non-resonant homeomorphisms on the torus </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="1103.4576v2-abstract-short" style="display: inline;"> We prove that a homeomorphism of the torus homotopic to the identity whose rotation set is reduced to a single totally irrational vector is chain-recurrent. In fact, we show that pseudo-orbits can be chosen with a small number of jumps, in particular, that the nonwandering set is weakly transitive. We give an example showing that the nonwandering set of such a homeomorphism may not be transitive. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1103.4576v2-abstract-full" style="display: none;"> We prove that a homeomorphism of the torus homotopic to the identity whose rotation set is reduced to a single totally irrational vector is chain-recurrent. In fact, we show that pseudo-orbits can be chosen with a small number of jumps, in particular, that the nonwandering set is weakly transitive. We give an example showing that the nonwandering set of such a homeomorphism may not be transitive. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1103.4576v2-abstract-full').style.display = 'none'; document.getElementById('1103.4576v2-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> 2 May, 2011; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 March, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 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">11 pages. Some corrections made</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37E45 37B20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1003.2280">arXiv:1003.2280</a> <span> [<a href="https://arxiv.org/pdf/1003.2280">pdf</a>, <a href="https://arxiv.org/ps/1003.2280">ps</a>, <a href="https://arxiv.org/format/1003.2280">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"> Non existence of attractors and dynamics around some wild homoclinic classes </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="1003.2280v2-abstract-short" style="display: inline;"> We present new examples of generic diffeomorphisms without attractors. Also, we study how these wild classes are accumulated by infinitely many other classes (obtaining that the chain recurrence classes different from the only quasi-attractor are contained in center stable manifolds). The construction relies on some derived from Anosov (DA) constructions and uses strongly the semiconjugacy obtaine… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1003.2280v2-abstract-full').style.display = 'inline'; document.getElementById('1003.2280v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1003.2280v2-abstract-full" style="display: none;"> We present new examples of generic diffeomorphisms without attractors. Also, we study how these wild classes are accumulated by infinitely many other classes (obtaining that the chain recurrence classes different from the only quasi-attractor are contained in center stable manifolds). The construction relies on some derived from Anosov (DA) constructions and uses strongly the semiconjugacy obtained by these diffeomorphisms. An interesting feature of this examples is that we can show that robustly, they present a unique attractor in the sense of Milnor. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1003.2280v2-abstract-full').style.display = 'none'; document.getElementById('1003.2280v2-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, 2010; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 March, 2010; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2010. </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 remarks added</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0903.4090">arXiv:0903.4090</a> <span> [<a href="https://arxiv.org/pdf/0903.4090">pdf</a>, <a href="https://arxiv.org/ps/0903.4090">ps</a>, <a href="https://arxiv.org/format/0903.4090">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/10.1088/0951-7715/23/7/006">10.1088/0951-7715/23/7/006 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Generic bi-Lyapunov stable homoclinic classes </p> <p class="authors"> <span class="search-hit">Authors:</span> <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="0903.4090v4-abstract-short" style="display: inline;"> We study, for $C^1$ generic diffeomorphisms, homoclinic classes which are Lyapunov stable both for backward and forward iterations. We prove they must admit a dominated splitting and show that under some hypothesis they must be the whole manifold. As a consequence of our results we also prove that in dimension 2 the class must be the whole manifold and in dimension 3, these classes must have non… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0903.4090v4-abstract-full').style.display = 'inline'; document.getElementById('0903.4090v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0903.4090v4-abstract-full" style="display: none;"> We study, for $C^1$ generic diffeomorphisms, homoclinic classes which are Lyapunov stable both for backward and forward iterations. We prove they must admit a dominated splitting and show that under some hypothesis they must be the whole manifold. As a consequence of our results we also prove that in dimension 2 the class must be the whole manifold and in dimension 3, these classes must have nonempty interior. Many results on Lyapunov stable homoclinic classes for $C^1$-generic diffeomorphisms are also deduced. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0903.4090v4-abstract-full').style.display = 'none'; document.getElementById('0903.4090v4-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 May, 2010; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 March, 2009; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2009. </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. This version includes an appendix that will not appear in the published version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37C20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0806.3036">arXiv:0806.3036</a> <span> [<a href="https://arxiv.org/pdf/0806.3036">pdf</a>, <a href="https://arxiv.org/ps/0806.3036">ps</a>, <a href="https://arxiv.org/format/0806.3036">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"> Codimension one generic homoclinic classes with interior </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Potrie%2C+R">Rafael Potrie</a>, <a href="/search/math?searchtype=author&query=Sambarino%2C+M">Martin Sambarino</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="0806.3036v3-abstract-short" style="display: inline;"> We study generic diffeomorphisms with a homoclinc class with non empty interior and in particular those admitting a codimension one dominated splitting. We prove that if in the finest dominated splitting the extreme subbundles are one dimensional then the diffeomorphism is partially hyperbolic and from this we deduce that the diffeomorphism is transitive. </span> <span class="abstract-full has-text-grey-dark mathjax" id="0806.3036v3-abstract-full" style="display: none;"> We study generic diffeomorphisms with a homoclinc class with non empty interior and in particular those admitting a codimension one dominated splitting. We prove that if in the finest dominated splitting the extreme subbundles are one dimensional then the diffeomorphism is partially hyperbolic and from this we deduce that the diffeomorphism is transitive. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0806.3036v3-abstract-full').style.display = 'none'; document.getElementById('0806.3036v3-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, 2009; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 June, 2008; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2008. </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">12 pages, Some corrections</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37C20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0805.1493">arXiv:0805.1493</a> <span> [<a href="https://arxiv.org/pdf/0805.1493">pdf</a>, <a href="https://arxiv.org/ps/0805.1493">ps</a>, <a href="https://arxiv.org/format/0805.1493">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"> Local product structure for expansive homeomorphisms </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Artigue%2C+A">Alfonso Artigue</a>, <a href="/search/math?searchtype=author&query=Brum%2C+J">Joaquin Brum</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="0805.1493v2-abstract-short" style="display: inline;"> Let $f\colon M\to M$ be an expansive homeomorphism with dense topologically hyperbolic periodic points, $M$ a compact manifold. Then there is a local product structure in an open and dense subset of $M$. Moreover, if some topologically hyperbolic periodic point has codimension one, then this local product structure is uniform. In particular, we conclude that the homeomorphism is conjugated to a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0805.1493v2-abstract-full').style.display = 'inline'; document.getElementById('0805.1493v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0805.1493v2-abstract-full" style="display: none;"> Let $f\colon M\to M$ be an expansive homeomorphism with dense topologically hyperbolic periodic points, $M$ a compact manifold. Then there is a local product structure in an open and dense subset of $M$. Moreover, if some topologically hyperbolic periodic point has codimension one, then this local product structure is uniform. In particular, we conclude that the homeomorphism is conjugated to a linear Anosov diffeomorphism of a torus. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0805.1493v2-abstract-full').style.display = 'none'; document.getElementById('0805.1493v2-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, 2008; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 May, 2008; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2008. </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, Some corrections made</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37B99; 37D45; 54H20 </p> </li> </ol> <nav 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