Local entropy theory

Glasner, Eli; Ye, Xiangdong

doi:10.1017/s0143385708080309

Cited by 73 publications

(52 citation statements)

References 97 publications

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“…The author introduced a variant of h(G F ) in [127], the local geometric entropy h(G F , w) of G F which is a refinement of the global entropy. The local geometric entropy is analogous to the local measure-theoretic entropy for maps introduced by Brin and Katok [44,103].…”

Section: Exponential Complexitymentioning

confidence: 99%

Lectures on Foliation Dynamics

Hurder

2014

Advanced Courses in Mathematics - CRM Barcelona

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Section: Exponential Complexitymentioning

confidence: 99%

Lectures on Foliation Dynamics

Hurder

2014

Advanced Courses in Mathematics - CRM Barcelona

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“…As in the article of Mouron [20], recent developments in local entropy theory are essential to our proofs. We refer the reader to [7] for a general survey on the subject. Inspired by the concepts of entropy pairs and IE-pair, we introduce a new concept called zigzag pair (or Z-pair for short.)…”

Section: Introductionmentioning

confidence: 99%

Chaos and indecomposability

Darji

Kato

2017

Advances in Mathematics

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We use recent developments in local entropy theory to prove that chaos in dynamical systems implies the existence of complicated structure in the underlying space. Earlier Mouron proved that if X is an arc-like continuum which admits a homeomorphism f with positive topological entropy, then X contains an indecomposable subcontinuum. Barge and Diamond proved that if G is a finite graph and f : G → G is any map with positive topological entropy, then the inverse limit space lim ← − (G, f ) contains an indecomposable continuum. In this paper we show that if X is a G-like continuum for some finite graph G and f : X → X is any map with positive topological entropy, then lim ← − (X, f ) contains an indecomposable continuum. As a corollary, we obtain that in the case that f is a homeomorphism, X contains an indecomposable continuum. Moreover, if f has uniformly positive upper entropy, then X is an indecomposable continuum. Our results answer some questions raised by Mouron and generalize the above mentioned work of Mouron and also that of Barge and Diamond. We also introduce a new concept called zigzag pair which attempts to capture the complexity of a dynamical systems from the continuum theoretic perspective and facilitates the proof of the main result.

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“…Entropy pairs were introduced and studied in [3] by Blanchard, and the notion was extended to entropy tuples in [18] by Huang and Ye. Nowadays, the idea of using various tuples to obtain dynamical properties is widely accepted and developed, see for example the surveys [15,24]. In [30], Xiong extended Huang-Ye's result related to Li-Yorke chaos to the multivariant version of Li-Yorke chaos.…”

Section: Introductionmentioning

confidence: 99%

Stable sets and mean Li–Yorke chaos in positive entropy systems

Huang

2014

Journal of Functional Analysis

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It is shown that in a topological dynamical system with positive entropy, there is a measure-theoretically "rather big" set such that a multivariant version of mean Li-Yorke chaos happens on the closure of the stable or unstable set of any point from the set. It is also proved that the intersections of the sets of asymptotic tuples and mean Li-Yorke tuples with the set of topological entropy tuples are dense in the set of topological entropy tuples respectively.2000 Mathematics Subject Classification. Primary 37B05; Secondary 54H20.

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Local entropy theory

Abstract: In this survey we offer an overview of the so-called local entropy theory, which has been in development since the early 1990s. While doing so, we emphasize the connections between the topological dynamics and the ergodic theory points of view.

Cited by 73 publications

References 97 publications

Lectures on Foliation Dynamics

Lectures on Foliation Dynamics

Chaos and indecomposability

Stable sets and mean Li–Yorke chaos in positive entropy systems

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