Rotation intervals and entropy on attracting annular continua

Passeggi, Alejandro; Potrie, Rafaël; Sambarino, Martı́n

doi:10.2140/gt.2018.22.2145

Cited by 21 publications

(20 citation statements)

References 29 publications

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“…This extends a recent result of Passeggi, Potrie and Sambarino [56], who showed that a nondegenerate rotation interval of an annulus homeomorphism H on an invariant attracting cofrontier K implies positive entropy of H | K .…”

Section: Corollary 14supporting

confidence: 88%

New Exotic Minimal Sets from Pseudo-Suspensions of Cantor Systems

2021

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We develop a technique, pseudo-suspension, that applies to invariant sets of homeomorphisms of a class of annulus homeomorphisms we describe, Handel–Anosov–Katok (HAK) homeomorphisms, that generalize the homeomorphism first described by Handel. Given a HAK homeomorphism and a homeomorphism of the Cantor set, the pseudo-suspension yields a homeomorphism of a new space that combines features of both of the original homeomorphisms. This allows us to answer a well known open question by providing examples of hereditarily indecomposable continua that admit homeomorphisms with positive finite entropy. Additionally, we show that such examples occur as minimal sets of volume preserving smooth diffeomorphisms of 4-dimensional manifolds.We construct an example of a minimal, weakly mixing and uniformly rigid homeomorphism of the pseudo-circle, and by our method we are also able to extend it to other one-dimensional hereditarily indecomposable continua, thereby producing the first examples of minimal, uniformly rigid and weakly mixing homeomorphisms in dimension 1. We also show that the examples we construct can be realized as invariant sets of smooth diffeomorphisms of a 4-manifold. Until now the only known examples of connected spaces that admit minimal, uniformly rigid and weakly mixing homeomorphisms were modifications of those given by Glasner and Maon in dimension at least 2.

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Section: Corollary 14supporting

confidence: 88%

New Exotic Minimal Sets from Pseudo-Suspensions of Cantor Systems

2021

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“…Progress in this direction was recently announced by Passeggi, Potrie and Sambarino [PPS15]. We remark that, along the same lines, it is known that if a homeomorphism of the torus homotopic to the identity has a rotation set (which is a subset of R 2 ) with nonempty interior, then the homeomorphism has positive topological entropy [LM91].…”

mentioning

confidence: 73%

Realizing rotation numbers on annular continua

Koropecki

2016

Math. Z.

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An annular continuum is a compact connected set K which separates a closed annulus A into exactly two connected components, one containing each boundary component. The topology of such continua can be very intricate (for instance, non-locally connected). We adapt a result proved by Handel in the case where K = A, showing that if K is an invariant annular continuum of a homeomorphism of A isotopic to the identity, then the rotation set in K is closed. Moreover, every element of the rotation set is realized by an ergodic measure supported in K (and by a periodic orbit if the rotation number is rational) and most elements are realized by a compact invariant set. Our second result shows that if the continuum K is minimal with the property of being annular (what we call a circloid), then every rational number between the extrema of the rotation set in K is realized by a periodic orbit in K. As a consequence, the rotation set is a closed interval, and every number in this interval (rational or not) is realized by an orbit (moreover, by an ergodic measure) in K. This improves a previous result of Barge and Gillette.

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“…We formalize the notion of strange attractor for a two-parametric family of diffeomorphisms H (a,b) defined on M = [0, 1] × S 1 , endowed with the induced topology. The set M is also called by circloid in [16]. In what follows, if A ⊂ M , A denotes its topological closure.…”

Section: Preliminaries: Strange Attractors and Srb Measuresmentioning

confidence: 99%

Rank-one strange attractors versus Heteroclinic tangles

Rodrigues¹

2022

Preprint

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We present a mechanism for the emergence of strange attractors (observable chaos) in a two-parameter periodically-perturbed family of differential equations on the plane. The two parameters are independent and act on different ways in the invariant manifolds of consecutive saddles in the cycle. When both parameters are zero, the flow exhibits an attracting heteroclinic cycle associated to two equilibria. The first parameter makes the two-dimensional invariant manifolds of consecutive saddles in the cycle to pull apart; the second forces transverse intersection. These relative positions may be determined using the Melnikov method.Extending the previous theory on the field, we prove the existence of many complicated dynamical objects in the two-parameter family, ranging from "large" strange attractors supporting SRB (Sinai-Ruelle-Bowen) measures to superstable sinks and Hénon-type attractors. We draw a plausible bifurcation diagram associated to the problem under consideration and we show that the occurrence of heteroclinic tangles is a prevalent phenomenon.

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Rotation intervals and entropy on attracting annular continua

Cited by 21 publications

References 29 publications

New Exotic Minimal Sets from Pseudo-Suspensions of Cantor Systems

New Exotic Minimal Sets from Pseudo-Suspensions of Cantor Systems

Realizing rotation numbers on annular continua

Rank-one strange attractors versus Heteroclinic tangles

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