On the periodic orbits, shadowing and strong transitivity of continuous flows

Bessa, Mário; Torres, Maria Joana; Varandas, Paulo

doi:10.1016/j.na.2018.06.002

Cited by 19 publications

(23 citation statements)

References 11 publications

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“…Continuous maps with the gluing orbit property. Here we prove the genericity of the gluing orbit property on chain recurrent classes with a dense set of periodic orbits, a result of independent interest inspired by [6].…”

Section: The Set Of Points With Non-trivial Pointwise Rotation Setmentioning

confidence: 84%

On the rotation sets of generic homeomorphisms on the torus

Lima¹,

Varandas²

2020

Ergod. Th. Dynam. Sys.

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We study the rotation sets for homeomorphisms homotopic to the identity on the torus $\mathbb T^d$ , $d\ge 2$ . In the conservative setting, we prove that there exists a Baire residual subset of the set $\text {Homeo}_{0, \lambda }(\mathbb T^2)$ of conservative homeomorphisms homotopic to the identity so that the set of points with wild pointwise rotation set is a Baire residual subset in $\mathbb T^2$ , and that it carries full topological pressure and full metric mean dimension. Moreover, we prove that for every $d\ge 2$ the rotation set of $C^0$ -generic conservative homeomorphisms on $\mathbb T^d$ is convex. Related results are obtained in the case of dissipative homeomorphisms on tori. The previous results rely on the description of the topological complexity of the set of points with wild historic behavior and on the denseness of periodic measures for continuous maps with the gluing orbit property.

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Section: The Set Of Points With Non-trivial Pointwise Rotation Setmentioning

confidence: 84%

On the rotation sets of generic homeomorphisms on the torus

Lima¹,

Varandas²

2020

Ergod. Th. Dynam. Sys.

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“…In rough terms, a priori the embedding problem can also be posed by asking which homeomorphisms are generated by vector fields. In this respect, in view of Picard's theorem, it is natural to consider the space of Lipschitz vector fields endowed with the C 0 -topology (see [6]). However, since the set of bi-Lipschitz homeomorphisms form a first category subset in the space of homeomorphisms, it seldom occurs for a homeomorphisms to be the time-1 map of a flow generated by a Lipschitz vector field.…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

Continuous flows generate few homeomorphisms

Bonomo¹,

Varandas

2020

Proceedings of the Edinburgh Mathematical Society

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We describe topological obstructions (involving periodic points, topological entropy and rotation sets) for a homeomorphism on a compact manifold to embed in a continuous flow. We prove that homeomorphisms in a $C^{0}$ -open and dense set of homeomorphisms isotopic to the identity in compact manifolds of dimension at least two are not the time-1 map of a continuous flow. Such property is also true for volume-preserving homeomorphisms in compact manifolds of dimension at least five. In the case of conservative homeomorphisms of the torus $\mathbb {T}^{d} (d\ge 2)$ isotopic to identity, we describe necessary conditions for a homeomorphism to be flowable in terms of the rotation sets.

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“…Moreover, this property holds for homeomorphisms with the periodic shadowing property on each chain recurrence class of the non-wandering set. We refer the reader to [3,4,5,29] for more details.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…The characteristic polynomial is p(λ) = −λ 3 + 4λ + 1, which has three real zeros λ 1 < λ 2 < 0 < λ 3 with |λ 1 |, |λ 3 Since f A is Anosov, hence expansive, it follows that the centralizer Z 0 (f A ) is discrete [30] (hence Z 1 (f A ) is discrete). We proceed to analyze the triviality of the C 1 -centralizer.…”

Section: Anosov Diffeomorphisms With Non-trivial C 1 -Centralizermentioning

confidence: 99%

The centralizer of $C^r$-generic diffeomorphisms at hyperbolic basic sets is trivial

Rocha

Varandas

2017

Proc. Amer. Math. Soc.

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Abstract. In the late nineties, Smale proposed a list of problems for the next century and, among these, it was conjectured that for every r ≥ 1 a C r -generic diffeomorphism has trivial centralizer. Our contribution here is to prove the triviality of C r -centralizers on hyperbolic basic sets. In particular, C r -generic transitive Anosov diffeomorphisms have a trivial C 1 -centralizer. These results follow from a more general criterium for expansive homeomorphisms with the gluing orbit property. We also construct a linear Anosov diffeomorphism on T 3 with discrete, non-trivial centralizer and with elements that are not roots. Finally, we prove that all elements in the centralizer of an Anosov diffeomorphism preserve some of its maximal entropy measures, and use this to characterize the centralizer of linear Anosov diffeomorphisms on tori.

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On the periodic orbits, shadowing and strong transitivity of continuous flows

Cited by 19 publications

References 11 publications

On the rotation sets of generic homeomorphisms on the torus

On the rotation sets of generic homeomorphisms on the torus

Continuous flows generate few homeomorphisms

The centralizer of $C^r$-generic diffeomorphisms at hyperbolic basic sets is trivial

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