Points with maximal Birkhoff average oscillation

Li, Jinjun; Wu, Min

doi:10.1007/s10587-016-0252-3

Cited by 4 publications

(13 citation statements)

References 35 publications

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“…Several extensions of these results have been considered later on, building mainly on the concept of 'specification' introduced by Bowen in the early 1970s and the concept of shadowing (see e.g. [4,12,16,29,37,50,51] and references therein).…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…It is clear from the construction that R(q, ε) ⊂ B(q, ε) for every q ∈ D and ε > 0, and thatR ⊂ R. The following lemma, identical to Propositions 2.2 and 2.3 in [29], ensures that R is a Baire generic subset of X.…”

Section: The Set Of Points With Historic Behaviormentioning

confidence: 89%

“…Baire genericity of historic behavior. This subsection is devoted to the proof of Theorem D, whose strategy is strongly inspired by [3,29]. The differences lie in the fact that, due to the higher-dimensional features of observables, we need to restrict to connected subsets in the set of all accumulation vectors, and that we have transition time functions instead of a determined time to shadow pieces of orbits (see Remark 6.1 below).…”

Section: The Set Of Points With Historic Behaviormentioning

confidence: 99%

“…In contrast with the case when f has the specification property, the previous functions need not be constant and, consequently, the collection of points of the form y = y(x k,i ) need not be 3ε-separated by a suitable iterate of the dynamics. For that reason, not only an argument to select a 'large set' of distinguishable orbits would require us to compare points with the same transition times, which strongly differs from [3,29].…”

Section: The Set Of Points With Historic Behaviormentioning

confidence: 99%

“…If, on the one hand, the proofs of Theorems D and E are inspired by [5,29,51], the argument in the proof of Theorem E is much more challenging and presents novelties on how to construct a 'large amount' of points whose finite pieces of orbits up to time n have a controlled behavior and that are separated by the dynamics. This is crucial to estimate topological pressure and metric mean dimension.…”

mentioning

confidence: 99%

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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: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Section: The Set Of Points With Historic Behaviormentioning

confidence: 89%

Section: The Set Of Points With Historic Behaviormentioning

confidence: 99%

Section: The Set Of Points With Historic Behaviormentioning

confidence: 99%

mentioning

confidence: 99%

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On the rotation sets of generic homeomorphisms on the torus

Lima¹,

Varandas²

2020

Ergod. Th. Dynam. Sys.

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Metric mean dimension of irregular sets for maps with shadowing

Foryś-Krawiec,

Oprocha

2024

Journal of Difference Equations and Applications

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On the rotation sets of generic homeomorphisms on the torus $\mathbb T^d$

Lima,

Varandas

2019

Preprint

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We study the rotation sets for homeomorphisms homotopic to the identity on the torus T d , d ≥ 2. In the conservative setting, we prove that there exists a Baire residual subset of the set Homeo 0,λ (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 T 2 , and that it carries full topological pressure and full metric mean dimension. Moreover, we prove that for every d ≥ 2 the rotation set of C 0 -generic conservative homeomorphisms on 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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Points with maximal Birkhoff average oscillation

Cited by 4 publications

References 35 publications

On the rotation sets of generic homeomorphisms on the torus

On the rotation sets of generic homeomorphisms on the torus

Metric mean dimension of irregular sets for maps with shadowing

On the rotation sets of generic homeomorphisms on the torus $\mathbb T^d$

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