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

In this paper, we introduce the definitions of - shadowing property, - shadowing property, topological ergodicity, and strong ergodicity of iterated function systems . Then, we show the following: if has the - shadowing property (respectively, - shadowing property), then has the - shadowing property (respectively, - shadowing property) for any ; if has the - shadowing property (respectively, - shadowing property) for some , then has the - shadowing property (respectively, - shadowing property); if has the - shadowing property or - shadowing property, and or is surjective, then is chain mixing; let be open maps. For with the - shadowing property (respectively, - shadowing property), if is dense in X , and s is a minimal point of or for any , then is strongly ergodic, and hence, is strongly ergodic; and for with the average shadowing property, if is dense in X , and s is a quasi-weakly almost periodic point of or for any , then is ergodic.