Topological entropy for set valued maps

Lampart, Marek; Raith, Peter

doi:10.1016/j.na.2010.04.054

Cited by 28 publications

(16 citation statements)

References 13 publications

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“…For this reason, there still appear new considerations related to those connected with entropy in relation to various problems (e.g. [23], [27]). For the need of our considerations we can assume that a function (multifunction, map) is chaotic if corresponding dynamical system has a positive entropy.…”

Section: Generalized Entropymentioning

confidence: 99%

On Baire generalized topological spaces and some problems connected with discrete dynamical systems

Loranty¹,

Pawlak²

2013

Traditional and Present-Day Topics in Real Analysis. Dedicated to Professor Jan Stanisław Lipiński

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Section: Generalized Entropymentioning

confidence: 99%

On Baire generalized topological spaces and some problems connected with discrete dynamical systems

Loranty¹,

Pawlak²

2013

Traditional and Present-Day Topics in Real Analysis. Dedicated to Professor Jan Stanisław Lipiński

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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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“…Fedeli [18], Guirao et al [19] and Hou et al [20] studied Devaney chaos for set-valued discrete systems. Lampart and Raith [21] discussed topological entropy for set-valued maps. Liu et al [22] and Wang et al [23] studied sensitivity of set-valued discrete systems.…”

Section: Introductionmentioning

confidence: 99%

On Local Aspects of Topological Transitivity and Weak Mixing in Set-Valued Discrete Systems

Liu

2013

Discrete Dynamics in Nature and Society

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Blanchard and Huang introduced the notion of weakly mixing subset, and Oprocha and Zhang gave the concept of transitive subset and studied its basic properties. In this paper our main goal is to discuss the weakly mixing subsets and transitive subsets in set-valued discrete systems. We prove that a set-valued discrete system has a transitive subset if and only if original system has a weakly mixing subset. Moreover, we give an example showing that original system has a transitive subset, which does not imply set-valued discrete system has a transitive subset.

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Topological entropy for set valued maps

Cited by 28 publications

References 13 publications

On Baire generalized topological spaces and some problems connected with discrete dynamical systems

On Baire generalized topological spaces and some problems connected with discrete dynamical systems

Conceptions of Topological Transitivity on Symmetric Products

On Local Aspects of Topological Transitivity and Weak Mixing in Set-Valued Discrete Systems

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