Dynamical property of hyperspace on uniform space

Abstract

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