The d-Shadowing Property and Average Shadowing Property for Iterated Function Systems

Abstract

In this paper, we introduce the definitions of d¯-shadowing property, d¯-shadowing property, topological ergodicity, and strong ergodicity of iterated function systems IFSf0,f1. Then, we show the following: 1 if IFSf0,f1 has the d¯-shadowing property (respectively, d¯-shadowing property), then ℱk has the d¯-shadowing property (respectively, d¯-shadowing property) for any k∈Z+; 2 if ℱk has the d¯-shadowing property (respectively, d¯-shadowing property) for some k∈Z+, then IFSf0,f1 has the d¯-shadowing property (respectively, d¯-shadowing property); 3 if IFSf0,f1 has the d¯-shadowing property or d¯-shadowing property, and f0 or f1 is surjective, then IFSf0,f1 is chain mixing; 4 let f0,f1 be open maps. For IFSf0,f1 with the d¯-shadowing property (respectively, d¯-shadowing property), if A⊂X is dense in X, and s is a minimal point of f0 or f1 for any s∈A, then IFSf0,f1 is strongly ergodic, and hence, ℱk is strongly ergodic; and 5 for IFSf0,f1 with the average shadowing property, if S⊂X is dense in X, and s is a quasi-weakly almost periodic point of f0 or f1 for any s∈S, then IFSf0,f1 is ergodic.