The d-Shadowing Property and Average Shadowing Property for Iterated Function Systems. (29th April 2020) - Jiang, Jie; Wang, Lidong; Zhao, Yingcui
Abstract : In this paper, we introduce the definitions of d ¯ -shadowing property, d ¯ -shadowing property, topological ergodicity, and strong ergodicity of iterated function systems IFS f 0, f 1 . Then, we show the following: 1 if IFS f 0, f 1 has the d ¯ -shadowing property (respectively, d ¯ -shadowing property), then ℱ k has the d ¯ -shadowing property (respectively, d ¯ -shadowing property) for any k ∈ Z + ; 2 if ℱ k has the d ¯ -shadowing property (respectively, d ¯ -shadowing property) for some k ∈ Z +, then IFS f 0, f 1 has the d ¯ -shadowing property (respectively, d ¯ -shadowing property); 3 if IFS f 0, f 1 has the d ¯ -shadowing property or d ¯ -shadowing property, and f 0 or f 1 is surjective, then IFS f 0, f 1 is chain mixing; 4 let f 0, f 1 be open maps. For IFS f 0, f 1 with the d ¯ -shadowing property (respectively, d ¯ -shadowing property), if A ⊂ X is dense in X, and s is a minimal point of f 0 or f 1 for any s ∈ A, then IFS f 0, f 1 is strongly ergodic, and hence, ℱ k is strongly ergodic; and 5 for IFS f 0, f 1 with the average shadowing property, if S ⊂ X is dense in X, and s is a quasi-weakly almost periodic point of f 0 or f 1 for any s ∈ S, then IFS f 0, f 1 is ergodic.