Topological entropy for induced hyperspace maps

Peña, José Salvador Cánovas; López, Gabriel Soler

doi:10.1016/j.chaos.2005.08.173

Cited by 28 publications

(5 citation statements)

References 6 publications

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“…, f X Canovas and Lopez [7] investigate the relation of topological entropy between the original system and the induced system using Bowen's notion. The result is shown in the following theorem.…”

Section: Fundamental Properties Of Topological Entropymentioning

confidence: 99%

A Note on the Notions of Topological Entropy

Sang

Salleh

2018

EJMS

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Topological entropy is used to determine the complexity of a dynamical system. This paper aims to serve as a stepping stone for the study of topological entropy. We review the notions of topological entropy, give an overview on the relation between the notions and fundamental properties of topological entropy. Besides, we cover the topological entropy of the induced hyperspaces and its connection with the original systems. We also provide a summary on the latest research topic related with topological entropy.

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Section: Fundamental Properties Of Topological Entropymentioning

confidence: 99%

A Note on the Notions of Topological Entropy

Sang

Salleh

2018

EJMS

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“…(see [5]). However, in [10,12] the author consider the extention of f to f over the class of compact an convex subsets of a real interval X.…”

Section: Theoremmentioning

confidence: 99%

Dynamics on Hyperspaces

Ayala¹,

Silva²,

Román-Flores³

2018

Preprint

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Given a compact metric space (X, ) and a continuous function f : X → X, we study the dynamics of the induced map f on the hyperspace of the compact subsets of X. We show how the chain recurrent set of f and its components are related with the one of the induced map. The main result of the paper proves that, under mild conditions, the numbers of chain components of f is greater than the ones of f . Showing the richness in the dynamics of f which cannot be perceived by f .

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“…In recent years, the research on hyperspace dynamics has been considered, for example: Román-Flores on transitivity [40]; J. Banks on Chaos [5]; Peris on mixing, weak mixing, transitivity and Auslander-Yorke chaos [39]; Liao et al [30] on transitivity, mixing and chaos; Peña and López on entropy [37]; Kwietniak and Oprocha on entropy, mixing and weak mixing [27]; Wang et al [44] on sensitivity; Lampart and Raith on entropy [28]; Li on stronger forms of sensitivity [29]; and Fernández et al…”

Section: Introductionmentioning

confidence: 99%

Conceptions of Topological Transitivity on Symmetric Products

2021

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Let X be a topological space. For any positive integer n , we consider the n -fold symmetric product of X , ℱ n ( X ), consisting of all nonempty subsets of X with at most n points; and for a given function ƒ : X → X , we consider the induced functions ℱ n ( ƒ ): ℱ n ( X ) → ℱ n ( X ). Let M be one of the following classes of functions: exact, transitive, ℤ-transitive, ℤ + -transitive, mixing, weakly mixing, chaotic, turbulent, strongly transitive, totally transitive, orbit-transitive, strictly orbit-transitive, ω-transitive, minimal, I N, T T ++ , semi-open and irreducible. In this paper we study the relationship between the following statements: ƒ ∈ M and ℱ n ( ƒ ) ∈ M .

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Topological entropy for induced hyperspace maps

Cited by 28 publications

References 6 publications

A Note on the Notions of Topological Entropy

A Note on the Notions of Topological Entropy

Dynamics on Hyperspaces

Conceptions of Topological Transitivity on Symmetric Products

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