On periodic shadowing, transitivity, chain mixing and expansivity in uniform dynamical systems

Abstract

In this paper we extend some results on the notions such as Expansive, Pseudo Orbit Tracing Property (P.O.T.P.), Chain Transitive, Periodic Shadowing, Chain Recurrent. We prove that if an expansive dynamical system (X,f) on compact uniform space has P.O.T.P., then it has periodic shadowing. If a continuous self map f on a compact uniform space has finite shadowing, then f has P.O.T.P. We find that a dynamical system (X,f) on compact uniform space has shadowing property if (X,f) has periodic shadowing provided f is expansive. If f is chain mixing on a compact uniform space (X,U), then fn is chain transitive for each n ≥ 1. If f has the periodic shadowing then fn has periodic shadowing for all n > 1. The periodic shadowing is invariant of topological conjugacy provided that the conjugacy and its inverse are Lipschitz.