A remark on the topological entropy of homeomorphisms

Yano, Koichi

doi:10.1007/bf01453235

Cited by 66 publications

(62 citation statements)

References 6 publications

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“…Thus we see that for any t > 0, C(Λ; t, σ, f ) := lim Step 4: On one hand it is known (see [23] and [21,22], respectively) that shadowing is a generic property in C(M ) and in H(M ). On the other hand, [22,45] show that for generic maps in C(M ) (resp. H(M )) we have h top (f ) = ∞.…”

Section: Proof For Theorem 17mentioning

confidence: 99%

On the irregular points for systems with the shadowing property

Dong¹,

Oprocha

Tian³

2017

Ergod. Th. Dynam. Sys.

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We prove that when f is a continuous self-map acting on a compact metric space (X, d) which satisfies the shadowing property, then the set of irregular points (i.e. points with divergent Birkhoff averages) has full entropy.Using this fact we prove that in the class of C 0 -generic maps on manifolds, we can only observe (in the sense of Lebesgue measure) points with convergent Birkhoff averages. In particular, the time average of atomic measures along orbit of such points converges to some SRB-like measure in the weak * topology. Moreover, such points carry zero entropy. In contrast, irregular points are nonobservable but carry infinite entropy.

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Section: Proof For Theorem 17mentioning

confidence: 99%

On the irregular points for systems with the shadowing property

Dong¹,

Oprocha

Tian³

2017

Ergod. Th. Dynam. Sys.

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“…ClDiff(Af) ). There are some results about generic homeomorphisms, for example [5,12], and the following theorem.…”

Section: Xçmmentioning

confidence: 99%

Generic Homeomorphisms have the Pseudo-Orbit Tracing Property

Odani¹

1990

Proceedings of the American Mathematical Society

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“…In [26], Yano studies the generic properties of continuous self-maps of manifolds with dimension at least 1. As Yano shows that the typical map in C (M) possesses a horseshoe like structure K , we can conclude that positive topological entropy, Devaney chaos on the subsystem K , and Li-Yorke chaos are all present on a dense and open set of functions in C (M).…”

Section: Introductionmentioning

confidence: 99%

“…Our work can be viewed as a continuation of that of [3,11,12,18,26]. After establishing definitions, notation and other background material in Section 2, we develop our main results in Section 3.…”

Section: Introductionmentioning

confidence: 99%

Chaotic behaviour of the map x ↦ ω(x, f)

D’Aniello

Steele

2014

Open Mathematics

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Let K(2 N ) be the class of compact subsets of the Cantor space 2 N , furnished with the Hausdorff metric. Let ∈ C (2 N ). We study the map ω : 2 N → K(2 N ) defined as ω ( ) = ω( ), the ω-limit set of under . Unlike the case of -dimensional manifolds, ≥ 1, we show that ω is continuous for the generic self-map of the Cantor space, even though the set of functions for which ω is everywhere discontinuous on a subsystem is dense in C (2 N ). The relationships between the continuity of ω and some forms of chaos are investigated. MSC:37B99, 54H20, 37B10

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A remark on the topological entropy of homeomorphisms

Cited by 66 publications

References 6 publications

On the irregular points for systems with the shadowing property

On the irregular points for systems with the shadowing property

Generic Homeomorphisms have the Pseudo-Orbit Tracing Property

Chaotic behaviour of the map x ↦ ω(x, f)

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