Entropy points and applications

Ye, Xiangdong; Zhang, Guo Hua

doi:10.1090/s0002-9947-07-04357-7

Cited by 51 publications

(35 citation statements)

References 21 publications

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“…We say that x ∈ X is an entropy point if h( f, U ) > 0 for every open neighborhood U of x. Entropy points were introduced in [33] as natural elements of systems with positive topological entropy. We will denote by E p ( f ) the set of all entropy points of f .…”

Section: Entropymentioning

confidence: 99%

Shadowing, entropy and minimal subsystems

Moothathu

Oprocha

2013

Monatsh Math

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We consider non-wandering dynamical systems having the shadowing property, mainly in the presence of sensitivity or transitivity, and investigate how closely such systems resemble the shift dynamical system in the richness of various types of minimal subsystems. In our excavation, we do discover regularly recurrent points, sensitive almost 1-1 extensions of odometers, minimal systems with positive topological entropy, etc. We also show that transitive semi-distal systems with shadowing are in fact minimal equicontinuous systems (hence with zero entropy) and, in contrast to systems with shadowing, the entropy points do not have to be densely distributed in transitive systems.

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Section: Entropymentioning

confidence: 99%

Shadowing, entropy and minimal subsystems

Moothathu

Oprocha

2013

Monatsh Math

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“…The entropy concept was localized by defining entropy pairs [3,6] or tuples [22,33] both in topological and in measure-theoretical situations. At the same time the theory of entropy sets [16,8], entropy points [64], sequence entropy pairs and null systems [30] and sequence entropy tuples and tame systems [29] were developed. To study the relationship between the two kinds of entropy pairs or tuples, a notable amount of attention has been paid to the study of local version of the variational principle, see [5,23,31,33,57] for related results.…”

Section: Motivationsmentioning

confidence: 99%

Conditional entropy and fiber entropy for amenable group actions

Yan

2015

Journal of Differential Equations

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In this paper, we introduce the notions of topological conditional entropy and fiber entropy for a given factor map between two amenable group actions, and prove three variational principles for conditional entropy and fiber entropy. Moreover, as an application of our variational principle we prove that the countable-to-one extension and the distal extension have zero conditional topological entropy. In particular, the topological entropy of a distal system is zero.

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“…In spite of the development of such significant literature on global behaviour of a dynamical system, mathematicians started working in finding those local behaviours which provides information about the global behaviour. In [20], authors have introduced the concept of entropy points and proved that existence of such a point implies positive entropy. Very recently, Kawaguchi has introduced [9] a weaker version of shadowable points [15] called e-shadowable points and in a subsequent paper [10] has shown that existence of certain kind of e-shadowable points implies positive entropy.…”

Section: Introductionmentioning

confidence: 99%

Measure Expansivity and Specification for Pointwise Dynamics

Das

Khan

Das

2019

Bull Braz Math Soc, New Series

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We introduce pointwise measure expansivity for bi-measurable maps. We show through examples that this notion is weaker than measure expansivity. In spite of this fact, we show that many results for measure expansive systems hold true for pointwise systems as well. Then, we study the concept of mixing, specification and chaos at a point in the phase space of a continuous map. We show that mixing at a shadowable point is not sufficient for it to be a specification point, but mixing of the map force a shadowable point to be a specification point. We prove that periodic specification points are Devaney chaotic point. Finally, we show that existence of two distinct specification points is sufficient for a map to have positive Bowen entropy. (2010): 54H20, 37C50, 37B40 Mathematics Subject Classifications

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Entropy points and applications

Cited by 51 publications

References 21 publications

Shadowing, entropy and minimal subsystems

Shadowing, entropy and minimal subsystems

Conditional entropy and fiber entropy for amenable group actions

Measure Expansivity and Specification for Pointwise Dynamics

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