Dynamical property of hyperspace on uniform space

Abstract

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)\left(X,\mu ) be a uniform space, (C(X),Cμ)\left(C\left(X),{C}^{\mu }) be a hyperspace of (X,μ)\left(X,\mu ), and f:X→Xf:X\to X be uniformly continuous. By using the relationship between original space and hyperspace, we obtain the following results: (a) the map ff is equicontinous if and only if the induced map Cf{C}^{f} is equicontinous; (b) if the induced map Cf{C}^{f} is expansive, then the map ff is expansive; (c) if the induced map Cf{C}^{f} has ergodic shadowing property, then the map ff has ergodic shadowing property; (d) if the induced map Cf{C}^{f} is chain transitive, then the map ff is chain transitive. In addition, we also study the topological conjugate invariance of (G,h)\left(G,h)-shadowing property in metric GG- space and prove that the map SS has (G,h)\left(G,h)-shadowing property if and only if the map TT has (G,h)\left(G,h)-shadowing property. These results generalize the conclusions of equicontinuity, expansivity, ergodic shadowing property, and chain transitivity in hyperspace.

Keywords

• equicontinuity
• expansivity
• ergodic shadowing property
• chain transitivity
• (g, h)-shadowing property
• uniform space
• 37b65
• o189.11