scholarly journals Topologically Transitive and Mixing Properties of Set-Valued Dynamical Systems

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Koon Sang Wong ◽  
Zabidin Salleh
Keyword(s):  
Dynamical Systems ◽  
Mixing Properties ◽  
Topologically Transitive ◽  
Topologically Mixing

We introduce and study two properties of dynamical systems: topologically transitive and topologically mixing under the set-valued setting. We prove some implications of these two properties for set-valued functions and generalize some results from a single-valued case to a set-valued case. We also show that both properties of set-valued dynamical systems are equivalence for any compact intervals.

scholarly journals ON ERGODIC BEHAVIOR OF p-ADIC DYNAMICAL SYSTEMS

2001 ◽  
Vol 04 (04) ◽  
pp. 569-577 ◽  
Author(s):  
MATTHIAS GUNDLACH ◽  
ANDREI KHRENNIKOV ◽  
KARL-OLOF LINDAHL
Keyword(s):  
Dynamical Systems ◽  
Natural Number ◽  
Unit Circle ◽  
Complex Plane ◽  
Haar Measure ◽  
Analogous Result ◽  
Topologically Transitive ◽  
Ergodic Behavior ◽  
Adic Case

Monomial mappings, x ↦ xn, are topologically transitive and ergodic with respect to Haar measure on the unit circle in the complex plane. In this paper we obtain an analogous result for monomial dynamical systems over p-adic numbers. The process is, however, not straightforward. The result will depend on the natural number n. Moreover, in the p-adic case we will not have ergodicity on the unit circle, but on the circles around the point 1.

On Devaney chaotic generalized shift dynamical systems

2013 ◽  
Vol 50 (4) ◽  
pp. 509-522 ◽  
Author(s):  
Fatemah Shirazi ◽  
Javad Sarkooh ◽  
Bahman Taherkhani
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Point ◽  
Periodic Points ◽  
List Type ◽  
Topologically Transitive ◽  
Generalized Shift ◽  
One To One ◽  
Shift Dynamical System ◽  
Infinite Sequences

In the following text we prove that in a generalized shift dynamical system (XГ, σφ) for infinite countable Г and discrete X with at least two elements the following statements are equivalent: the dynamical system (XГ, σφ) is chaotic in the sense of Devaneythe dynamical system (XГ, σφ) is topologically transitivethe map φ: Г → Г is one to one without any periodic point.Also for infinite countable Г and finite discrete X with at least two elements (XГ, σφ) is exact Devaney chaotic, if and only if φ: Г → Г is one to one and φ: Г → Г has niether periodic points nor φ-backwarding infinite sequences.

Sequence Entropy and Mild Mixing

1992 ◽  
Vol 44 (1) ◽  
pp. 215-224 ◽  
Author(s):  
Qing Zhang
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Spectrum ◽  
Infinite Subset ◽  
Mixing Properties ◽  
Sequence Entropy ◽  
Number Of Authors

Entropy characterizations of different spectral and mixing properties of dynamical systems were dealt with by a number of authors (see [5], [6] and [8]).Given an infinite subset Γ = {tn}of N and a dynamical system (X, β,μ, T) one can define sequence entropy along for any finite Petition ξ, and hΓ(T) —supξ hΓ(T,ξ). In [6] Kushnirenko used sequence entropy to give a characterization of systems with discrete spectrum.

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