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</div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.01042">arXiv:2407.01042</a> <span> [<a href="https://arxiv.org/pdf/2407.01042">pdf</a>, <a href="https://arxiv.org/format/2407.01042">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> A note on weak conjugacy for homeomorphisms of surfaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Roux%2C+F+L">Fr茅d茅ric Le Roux</a>, <a href="/search/math?searchtype=author&query=Passeggi%2C+A">Alejandro Passeggi</a>, <a href="/search/math?searchtype=author&query=Sambarino%2C+M">Martin Sambarino</a>, <a href="/search/math?searchtype=author&query=Wolff%2C+M">Maxime Wolff</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="2407.01042v1-abstract-short" style="display: inline;"> We explore the relation of weak conjugacy in the group of homeomorphisms isotopic to the identity, for surfaces. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.01042v1-abstract-full" style="display: none;"> We explore the relation of weak conjugacy in the group of homeomorphisms isotopic to the identity, for surfaces. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.01042v1-abstract-full').style.display = 'none'; document.getElementById('2407.01042v1-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 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">18 pages, 2 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37E30; 57K20; 37E45; 37D15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2305.05755">arXiv:2305.05755</a> <span> [<a href="https://arxiv.org/pdf/2305.05755">pdf</a>, <a href="https://arxiv.org/ps/2305.05755">ps</a>, <a href="https://arxiv.org/format/2305.05755">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="Symplectic Geometry">math.SG</span> </div> </div> <p class="title is-5 mathjax"> Area preserving homeomorphisms of surfaces with rational rotational direction </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Guih%C3%A9neuf%2C+P">Pierre-Antoine Guih茅neuf</a>, <a href="/search/math?searchtype=author&query=Calvez%2C+P+L">Patrice Le Calvez</a>, <a href="/search/math?searchtype=author&query=Passeggi%2C+A">Alejandro Passeggi</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="2305.05755v2-abstract-short" style="display: inline;"> Let $S$ be a closed surface of genus $g\geq 2$, furnished with a Borel probability measure $位$ with total support. We show that if $f$ is a $位$-preserving homeomorphism isotopic to the identity such that the rotation vector $\mathrm{rot}_f(位)\in H_1(S,\mathbb R)$ is a multiple of an element of $H_1(S,\mathbb Z)$, then $f$ has infinitely many periodic orbits. Moreover, these periodic orbits can b… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.05755v2-abstract-full').style.display = 'inline'; document.getElementById('2305.05755v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2305.05755v2-abstract-full" style="display: none;"> Let $S$ be a closed surface of genus $g\geq 2$, furnished with a Borel probability measure $位$ with total support. We show that if $f$ is a $位$-preserving homeomorphism isotopic to the identity such that the rotation vector $\mathrm{rot}_f(位)\in H_1(S,\mathbb R)$ is a multiple of an element of $H_1(S,\mathbb Z)$, then $f$ has infinitely many periodic orbits. Moreover, these periodic orbits can be supposed to have their rotation vectors arbitrarily close to the rotation vector of any fixed ergodic Borel probability measure. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.05755v2-abstract-full').style.display = 'none'; document.getElementById('2305.05755v2-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 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 9 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 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">38 pages, 6 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37C25; 37E30; 37E45 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2305.02963">arXiv:2305.02963</a> <span> [<a href="https://arxiv.org/pdf/2305.02963">pdf</a>, <a href="https://arxiv.org/format/2305.02963">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"> Conditions implying annular chaos </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=Tal%2C+F+A">Fabio Armando Tal</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="2305.02963v2-abstract-short" style="display: inline;"> This work investigates topological chaos for homeomorphisms of the open annulus, introducing a new set of sufficient conditions based on points with distinct rotation numbers and their topological relation to invariant continua. These conditions allow us to formulate classic methods for verifying annular chaos in a finitely verifiable version supported on basic properties of the map. The results p… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.02963v2-abstract-full').style.display = 'inline'; document.getElementById('2305.02963v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2305.02963v2-abstract-full" style="display: none;"> This work investigates topological chaos for homeomorphisms of the open annulus, introducing a new set of sufficient conditions based on points with distinct rotation numbers and their topological relation to invariant continua. These conditions allow us to formulate classic methods for verifying annular chaos in a finitely verifiable version supported on basic properties of the map. The results pave the way for simple computer-assisted proofs of chaos in a wide range of annular maps, including many well known examples, and we present these proofs for some analytic families, demonstrating the effectiveness of the method. On the theoretical side, one of the consequences of the established conditions permits the proof of a folkloric conjecture about the relation between topological entropy and rotation sets. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.02963v2-abstract-full').style.display = 'none'; document.getElementById('2305.02963v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 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">This new version, upload on the 17th of June of 2024 replace the article Weack Conditions Implying Annular Chaos. The content of the previous one has been significantly improved here, and some other parts which do not appear in this new version will appear in a forthcoming paper</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37E30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2006.12345">arXiv:2006.12345</a> <span> [<a href="https://arxiv.org/pdf/2006.12345">pdf</a>, <a href="https://arxiv.org/format/2006.12345">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"> Generic Rotation Sets in Hyperbolic Surfaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Alonso%2C+J">J. Alonso</a>, <a href="/search/math?searchtype=author&query=Brum%2C+J">J. Brum</a>, <a href="/search/math?searchtype=author&query=Passeggi%2C+A">A. Passeggi</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="2006.12345v1-abstract-short" style="display: inline;"> We show that for generic homeomorphisms homotopic to the identity in a closed and oriented surface of genus $g>1$, the rotation set is given by a union of at most $2^{5g-3}$ convex sets. Examples showing the sharpness for this asymptotic order are provided. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2006.12345v1-abstract-full" style="display: none;"> We show that for generic homeomorphisms homotopic to the identity in a closed and oriented surface of genus $g>1$, the rotation set is given by a union of at most $2^{5g-3}$ convex sets. Examples showing the sharpness for this asymptotic order are provided. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.12345v1-abstract-full').style.display = 'none'; document.getElementById('2006.12345v1-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, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2020. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1903.05172">arXiv:1903.05172</a> <span> [<a href="https://arxiv.org/pdf/1903.05172">pdf</a>, <a href="https://arxiv.org/format/1903.05172">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"> The bifurcation set as a topological invariant for one-dimensional dynamics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fuhrmann%2C+G">Gabriel Fuhrmann</a>, <a href="/search/math?searchtype=author&query=Gr%C3%B6ger%2C+M">Maik Gr枚ger</a>, <a href="/search/math?searchtype=author&query=Passeggi%2C+A">Alejandro Passeggi</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="1903.05172v1-abstract-short" style="display: inline;"> For a continuous map on the unit interval or circle, we define the bifurcation set to be the collection of those interval holes whose surviving set is sensitive to arbitrarily small changes of their position. By assuming a global perspective and focusing on the geometric and topological properties of this collection rather than the surviving sets of individual holes, we obtain a novel topological… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1903.05172v1-abstract-full').style.display = 'inline'; document.getElementById('1903.05172v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1903.05172v1-abstract-full" style="display: none;"> For a continuous map on the unit interval or circle, we define the bifurcation set to be the collection of those interval holes whose surviving set is sensitive to arbitrarily small changes of their position. By assuming a global perspective and focusing on the geometric and topological properties of this collection rather than the surviving sets of individual holes, we obtain a novel topological invariant for one-dimensional dynamics. We provide a detailed description of this invariant in the realm of transitive maps and observe that it carries fundamental dynamical information. In particular, for transitive non-minimal piecewise monotone maps, the bifurcation set encodes the topological entropy and strongly depends on the behavior of the critical points. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1903.05172v1-abstract-full').style.display = 'none'; document.getElementById('1903.05172v1-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 March, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 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">20 pages, 3 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1803.03294">arXiv:1803.03294</a> <span> [<a href="https://arxiv.org/pdf/1803.03294">pdf</a>, <a href="https://arxiv.org/ps/1803.03294">ps</a>, <a href="https://arxiv.org/format/1803.03294">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"> Deviations in the Franks-Misiurewicz conjecture </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=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="1803.03294v1-abstract-short" style="display: inline;"> We show that if there exists a counter example for the rational case of the Franks-Misiurewicz conjecture, then it must exhibit unbounded deviations in the complementary direction of its rotation set. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1803.03294v1-abstract-full" style="display: none;"> We show that if there exists a counter example for the rational case of the Franks-Misiurewicz conjecture, then it must exhibit unbounded deviations in the complementary direction of its rotation set. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1803.03294v1-abstract-full').style.display = 'none'; document.getElementById('1803.03294v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 March, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2018. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1701.04644">arXiv:1701.04644</a> <span> [<a href="https://arxiv.org/pdf/1701.04644">pdf</a>, <a href="https://arxiv.org/format/1701.04644">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"> A Poincar茅-Bendixson theorem for translation lines and applications to prime ends </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Koropecki%2C+A">Andres Koropecki</a>, <a href="/search/math?searchtype=author&query=Passeggi%2C+A">Alejandro Passeggi</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="1701.04644v2-abstract-short" style="display: inline;"> For an orientation-preserving homeomorphism of the sphere, we prove that if a translation line does not accumulate in a fixed point, then it necessarily spirals towards a topological attractor. This is in analogy with the description of flow lines given by Poincar茅-Bendixson theorem. We then apply this result to the study of invariant continua without fixed points, in particular to circloids and b… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1701.04644v2-abstract-full').style.display = 'inline'; document.getElementById('1701.04644v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1701.04644v2-abstract-full" style="display: none;"> For an orientation-preserving homeomorphism of the sphere, we prove that if a translation line does not accumulate in a fixed point, then it necessarily spirals towards a topological attractor. This is in analogy with the description of flow lines given by Poincar茅-Bendixson theorem. We then apply this result to the study of invariant continua without fixed points, in particular to circloids and boundaries of simply connected open sets. Among the applications, we show that if the prime ends rotation number of such an open set $U$ vanishes, then either there is a fixed point in the boundary, or the boundary of $U$ is contained in the basin of a finite family of topological "rotational" attractors. This description strongly improves a previous result by Cartwright and Littlewood, by passing from the prime ends compactification to the ambient space. Moreover, the dynamics in a neighborhood of the boundary is semiconjugate to a very simple model dynamics on a planar graph. Other applications involve the decomposability of invariant continua, and realization of rotation numbers by periodic points on circloids. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1701.04644v2-abstract-full').style.display = 'none'; document.getElementById('1701.04644v2-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 June, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 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">36 pages, 12 figures. Minor corrections. To appear in Comment. Math. Helv</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37E30; 37B45; 37E45; 54H20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1611.05498">arXiv:1611.05498</a> <span> [<a href="https://arxiv.org/pdf/1611.05498">pdf</a>, <a href="https://arxiv.org/ps/1611.05498">ps</a>, <a href="https://arxiv.org/format/1611.05498">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"> The Franks-Misiurewicz conjecture for extensions of irrational rotations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Koropecki%2C+A">Andres Koropecki</a>, <a href="/search/math?searchtype=author&query=Passeggi%2C+A">Alejandro Passeggi</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="1611.05498v1-abstract-short" style="display: inline;"> We show that a toral homeomorphism which is homotopic to the identity and topologically semiconjugate to an irrational rotation of the circle is always a pseudo-rotation (i.e. its rotation set is a single point). In combination with recent results, this allows us to complete the study of the Franks-Misiurewicz conjecture in the minimal case. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1611.05498v1-abstract-full" style="display: none;"> We show that a toral homeomorphism which is homotopic to the identity and topologically semiconjugate to an irrational rotation of the circle is always a pseudo-rotation (i.e. its rotation set is a single point). In combination with recent results, this allows us to complete the study of the Franks-Misiurewicz conjecture in the minimal case. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1611.05498v1-abstract-full').style.display = 'none'; document.getElementById('1611.05498v1-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, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 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">13 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37E45; 37E30 </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/1408.2931">arXiv:1408.2931</a> <span> [<a href="https://arxiv.org/pdf/1408.2931">pdf</a>, <a href="https://arxiv.org/ps/1408.2931">ps</a>, <a href="https://arxiv.org/format/1408.2931">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"> Rotation sets and almost periodic sequences </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=J%C3%A4ger%2C+T">Tobias J盲ger</a>, <a href="/search/math?searchtype=author&query=Passeggi%2C+A">Alejandro Passeggi</a>, <a href="/search/math?searchtype=author&query=%C5%A0timac%2C+S">Sonja 艩timac</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="1408.2931v1-abstract-short" style="display: inline;"> We study the rotational behaviour on minimal sets of torus homeomorphisms and show that the associated rotation sets can be any type of line segments as well as non-convex and even plane-separating continua. This shows that restrictions holding for rotation sets on the whole torus are not valid on minimal sets. The proof uses a construction of rotational horseshoes by Kwapisz to transfer the probl… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1408.2931v1-abstract-full').style.display = 'inline'; document.getElementById('1408.2931v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1408.2931v1-abstract-full" style="display: none;"> We study the rotational behaviour on minimal sets of torus homeomorphisms and show that the associated rotation sets can be any type of line segments as well as non-convex and even plane-separating continua. This shows that restrictions holding for rotation sets on the whole torus are not valid on minimal sets. The proof uses a construction of rotational horseshoes by Kwapisz to transfer the problem to a symbolic level, where the desired rotational behaviour is implemented by means of suitable irregular Toeplitz sequences. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1408.2931v1-abstract-full').style.display = 'none'; document.getElementById('1408.2931v1-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 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">MSC Class:</span> 37E30; 37E45; 54H20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1305.0723">arXiv:1305.0723</a> <span> [<a href="https://arxiv.org/pdf/1305.0723">pdf</a>, <a href="https://arxiv.org/ps/1305.0723">ps</a>, <a href="https://arxiv.org/format/1305.0723">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> On Torus Homeomorphisms Semiconjugate to irrational Rotations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=J%C3%A4ger%2C+T">Tobias J盲ger</a>, <a href="/search/math?searchtype=author&query=Passeggi%2C+A">Alejandro Passeggi</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="1305.0723v2-abstract-short" style="display: inline;"> In the context of the Franks-Misiurewicz Conjecture, we study homeomorphisms of the two-torus semiconjugate to an irrational rotation of the circle. As a special case, this conjecture asserts uniqueness of the rotation vector in this class of systems. We first characterise these maps by the existence of an invariant foliation by essential annular continua (essential subcontinua of the torus whose… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1305.0723v2-abstract-full').style.display = 'inline'; document.getElementById('1305.0723v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1305.0723v2-abstract-full" style="display: none;"> In the context of the Franks-Misiurewicz Conjecture, we study homeomorphisms of the two-torus semiconjugate to an irrational rotation of the circle. As a special case, this conjecture asserts uniqueness of the rotation vector in this class of systems. We first characterise these maps by the existence of an invariant foliation by essential annular continua (essential subcontinua of the torus whose complement is an open annulus) which are permuted with irrational combinatorics. This result places the considered class close to skew products over irrational rotations. Generalising a well-known result of M. Herman on forced circle homeomorphisms, we provide a criterion, in terms of topological properties of the annular continua, for the uniqueness of the rotation vector. As a byproduct, we obtain a simple proof for the uniqueness of the rotation vector on decomposable invariant annular continua with empty interior. In addition, we collect a number of observations on the topology and rotation intervals of invariant annular continua with empty interior. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1305.0723v2-abstract-full').style.display = 'none'; document.getElementById('1305.0723v2-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 May, 2013; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 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">20 pages, 5 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary 54H20; Secondary 37E30; 37E45 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1208.2614">arXiv:1208.2614</a> <span> [<a href="https://arxiv.org/pdf/1208.2614">pdf</a>, <a href="https://arxiv.org/ps/1208.2614">ps</a>, <a href="https://arxiv.org/format/1208.2614">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1112/jlms/jdt040">10.1112/jlms/jdt040 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Rational Polygons as Rotation Sets of Generic Homeomorphisms of the Two-Torus </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Passeggi%2C+A">Alejandro Passeggi</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="1208.2614v2-abstract-short" style="display: inline;"> We prove the existence of an open and dense set D\subset? Homeo0(T2) (set of toral homeomorphisms homotopic to the identity) such that the rotation set of any element in D is a rational polygon. We also extend this result to the set of axiom A dif- feomorphisms in Homeo0(T2). Further we observe the existence of minimal sets whose rotation set is a non-trivial segment, for an open set in Homeo0(T2)… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1208.2614v2-abstract-full').style.display = 'inline'; document.getElementById('1208.2614v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1208.2614v2-abstract-full" style="display: none;"> We prove the existence of an open and dense set D\subset? Homeo0(T2) (set of toral homeomorphisms homotopic to the identity) such that the rotation set of any element in D is a rational polygon. We also extend this result to the set of axiom A dif- feomorphisms in Homeo0(T2). Further we observe the existence of minimal sets whose rotation set is a non-trivial segment, for an open set in Homeo0(T2). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1208.2614v2-abstract-full').style.display = 'none'; document.getElementById('1208.2614v2-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 August, 2012; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 August, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2012. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1208.1650">arXiv:1208.1650</a> <span> [<a href="https://arxiv.org/pdf/1208.1650">pdf</a>, <a href="https://arxiv.org/ps/1208.1650">ps</a>, <a href="https://arxiv.org/format/1208.1650">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 classification of minimal sets for surface homeomorphisms </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=Xavier%2C+J">Juliana Xavier</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="1208.1650v3-abstract-short" style="display: inline;"> We classify minimal sets of (closed and oriented) hyperbolic surface homeomorphisms by studying the connected components of their complement. This extends the classification given by F. Kwakkel, T.J盲ger and A. Passeggi in the torus. The given classification is studied in the non-wandering setting and in light of the Nielsen-Thurston Theory. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1208.1650v3-abstract-full" style="display: none;"> We classify minimal sets of (closed and oriented) hyperbolic surface homeomorphisms by studying the connected components of their complement. This extends the classification given by F. Kwakkel, T.J盲ger and A. Passeggi in the torus. The given classification is studied in the non-wandering setting and in light of the Nielsen-Thurston Theory. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1208.1650v3-abstract-full').style.display = 'none'; document.getElementById('1208.1650v3-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 August, 2012; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 August, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2012. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1109.3919">arXiv:1109.3919</a> <span> [<a href="https://arxiv.org/pdf/1109.3919">pdf</a>, <a href="https://arxiv.org/ps/1109.3919">ps</a>, <a href="https://arxiv.org/format/1109.3919">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/s00209-012-1076-y">10.1007/s00209-012-1076-y <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A Classification of Minimal Sets of Torus Homeomorphisms </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jaeger%2C+T">Tobias Jaeger</a>, <a href="/search/math?searchtype=author&query=Kwakkel%2C+F">Ferry Kwakkel</a>, <a href="/search/math?searchtype=author&query=Passeggi%2C+A">Alejandro Passeggi</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="1109.3919v4-abstract-short" style="display: inline;"> We provide a classification of minimal sets of homeomorphisms of the two-torus, in terms of the structure of their complement. We show that this structure is exactly one of the following types: (1) a disjoint union of topological disks, or (2) a disjoint union of essential annuli and topological disks, or (3) a disjoint union of one doubly essential component and bounded topological disks. Periodi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1109.3919v4-abstract-full').style.display = 'inline'; document.getElementById('1109.3919v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1109.3919v4-abstract-full" style="display: none;"> We provide a classification of minimal sets of homeomorphisms of the two-torus, in terms of the structure of their complement. We show that this structure is exactly one of the following types: (1) a disjoint union of topological disks, or (2) a disjoint union of essential annuli and topological disks, or (3) a disjoint union of one doubly essential component and bounded topological disks. Periodic bounded disks can only occur in type 3. This result provides a framework for more detailed investigations, and additional information on the torus homeomorphism allows to draw further conclusions. In the non-wandering case, the classification can be significantly strengthened and we obtain that a minimal set other than the whole torus is either a periodic orbit, or the orbit of a periodic circloid, or the extension of a Cantor set. Further special cases are given by torus homeomorphisms homotopic to an Anosov, in which types 1 and 2 cannot occur, and the same holds for homeomorphisms homotopic to the identity with a rotation set which has non-empty interior. If a non-wandering torus homeomorphism has a unique and totally irrational rotation vector, then any minimal set other than the whole torus has to be the extension of a Cantor set. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1109.3919v4-abstract-full').style.display = 'none'; document.getElementById('1109.3919v4-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 May, 2014; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 September, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 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">Published in Mathematische Zeitschrift, June 2013, Volume 274, Issue 1-2, pp 405-426</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1009.0485">arXiv:1009.0485</a> <span> [<a href="https://arxiv.org/pdf/1009.0485">pdf</a>, <a href="https://arxiv.org/ps/1009.0485">ps</a>, <a href="https://arxiv.org/format/1009.0485">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"> Examples of Minimal Diffeomorphisms on $t^{2}$ Semiconjugated to an Ergodic Translation </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=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="1009.0485v4-abstract-short" style="display: inline;"> We prove that for every $蔚>0$ there exists a minimal diffeomorphism $f:\T^{2}\rightarrow\T^{2}$ of class $C^{3-蔚}$ and semiconjugate to an ergodic traslation, and have the following properties: zero entropy, sensitivity with respect to initial conditions and Li-Yorke chaos. These examples are obtained through the holonomy of the unstable foliation of Ma帽茅's example of derived from Anosov diffeomor… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1009.0485v4-abstract-full').style.display = 'inline'; document.getElementById('1009.0485v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1009.0485v4-abstract-full" style="display: none;"> We prove that for every $蔚>0$ there exists a minimal diffeomorphism $f:\T^{2}\rightarrow\T^{2}$ of class $C^{3-蔚}$ and semiconjugate to an ergodic traslation, and have the following properties: zero entropy, sensitivity with respect to initial conditions and Li-Yorke chaos. These examples are obtained through the holonomy of the unstable foliation of Ma帽茅's example of derived from Anosov diffeomorphism on $\T^3.$ <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1009.0485v4-abstract-full').style.display = 'none'; document.getElementById('1009.0485v4-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 August, 2012; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 September, 2010; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2010. </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" 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