Chaos and the shadowing property

Kościelniak, Piotr; Mazur, Marcin

doi:10.1016/j.topol.2006.06.010

Cited by 18 publications

(6 citation statements)

References 22 publications

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“…It is known that on many manifolds, shadowing homeomorphisms with positive entropy are generic [21]. Hence the following question is natural to ask: Question 4 Is there a compact topological manifold M and a transitive homeomorphism f : M → M such that ∅ = E p ( f ) = M?…”

Section: Theorem 65mentioning

confidence: 99%

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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: Theorem 65mentioning

confidence: 99%

“…with respect to the topology of uniform convergence) in various cases (e.g. see [8,21] and the references therein). This gives a motivation for further research on non-hyperbolic systems with shadowing, but also generates many problems, since non-expansive systems with shadowing are much harder to study.…”

Section: Introductionmentioning

confidence: 99%

Shadowing, entropy and minimal subsystems

Moothathu

Oprocha

2013

Monatsh Math

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“…However, it is nowadays well known that C 0 -generic maps have a dense set of periodic points (see e.g. [26]) and, in particular, C 0 -generic homeomorphisms f are not uniquely ergodic. In conclusion, there is no open set of homeomorphisms f so that f ♯ has a unique hyperbolic fixed point of saddle type.…”

Section: Specification and Hyperbolicitymentioning

confidence: 99%

Specification and thermodynamical properties of semigroup actions

Rodrigues

Varandas

2016

Journal of Mathematical Physics

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Abstract. In the present paper we study the thermodynamical properties of finitely generated continuous subgroup actions. We propose a notion of topological entropy and pressure functions that does not depend on the growth rate of the semigroup and introduce strong and orbital specification properties, under which, the semigroup actions have positive topological entropy and all points are entropy points. Moreover, we study the convergence and Lipschitz regularity of the pressure function and obtain relations between topological entropy and exponential growth rate of periodic points in the context of semigroups of expanding maps, obtaining a partial extension of the results obtained by Ruelle for Z d -actions [33] . The specification properties for semigroup actions and the corresponding one for its generators and the action of push-forward maps is also discussed.

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“…Shadowing and ergodic properties in discrete dynamical systems have received increasing attention in recent years [4][5][6][7]. Many authors investigated the relation between shadowing properties and other ergodic properties such as mixing and transitivity [10,12,14]. In [2] Blank introduced the notion of average-shadowing property and Gu [9] followed the same scheme to introduce the notion of the asymptotic average shadowing property.…”

Section: Introductionmentioning

confidence: 99%

Chaos and Shadowing in General Systems

Nia¹,

Bahabadi²

2022

KgJMat

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In this paper we describe some basic notions of topological dynamical systems for maps of type f : X × X → X named general systems. This is proved that every uniformly expansive general system has the shadowing property and every uniformly contractive general system has the (asymptotic) average shadowing and shadowing properties. In the rest, Devaney chaos for general systems is considered. Also, we show that topological transitivity and density of periodic points of a general systems imply topological ergodicity. We also obtain some results on the topological mixing and sensitivity for general systems.

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Chaos and the shadowing property

Cited by 18 publications

References 22 publications

Shadowing, entropy and minimal subsystems

Shadowing, entropy and minimal subsystems

Specification and thermodynamical properties of semigroup actions

Chaos and Shadowing in General Systems

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