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Dynamical property of hyperspace on uniform space

Ji Zhanjiang

Demonstratio Mathematica

2023

Vol. 56, nr 1

art. no. 20230264

First, we introduce the concepts of equicontinuity, expansivity , ergodic shadowing property, and chain transitivity in uniform space. Second, we study the dynamical properties of equicontinuity, expansivity , ergodic shadowing property, and chain transitivity in the hyperspace of uniform space. Let (X,μ) be a uniform space, (C(X),Cμ) be a hyperspace of (X,μ) , and ƒ:X→X be uniformly continuous. By using the relationship between original space and hyperspace, we obtain the following results: (a) the map ƒ is equicontinous if and only if the induced map Cƒ is equicontinous; (b) if the induced map Cƒ is expansive, then the map f is expansive; (c) if the induced map Cƒ has ergodic shadowing property, then the map f has ergodic shadowing property; (d) if the induced map Cƒ is chain transitive, then the map ƒ is chain transitive. In addition, we also study the topological conjugate invariance of (G,h) -shadowing property in metric G - space and prove that the map S has (G,h) -shadowing property if and only if the map T has (G,h) -shadowing property. These results generalize the conclusions of equicontinuity, expansivity, ergodic shadowing property, and chain transitivity in hyperspace.