Transitivity properties of dynamical systems on uniform hyperspaces

Subject Areas : هندسه

Farzaneh Pirfalak maloomeh ¹ , Nader Kouhestani ² , Seyyed Alireza Ahmadi ³

1 - Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran
2 - Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran
3 - Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran

Received: 2022-02-05 Accepted : 2024-04-30 Published : 2025-04-08

Keywords: Shadowing, Ergodic shadowing, Chain transitive map, Topologically ergodic map, Ergodically sensitive map,

Abstract :

In this paper we study the dynamical properties of actions on the space of compact subsets of the phase space. More precisely, if X is a uniform space, let K(X) denote the space of non-empty closed subsets of X provided with the Hausdorff topology. If f is a continuous few naturally self-map on X, then there are induced continuous self-map f ̃ on K(X) such that the dynamic of (K(X),f ̃) are richer than the dynamic of (X,f) . Our main theme is the interrelation between the dynamic of f and these induced maps. For such a study, we introduce and study the topological concepts of chain transitivity, mixing and chain mixing properties for dynamical systems induced by uniform hyperspaces. Finally, we investigate the transitivity properties of the induced maps and we prove the equivalence forms of topological chain transitivity for induced maps in uniform hyperspaces, also we prove equivalence forms of some transitivity properties for dynamical system (X,f).

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