Some Shadowing Properties Research For Iterated Function Systems

Suppose that?X,d?is a compact metric space,f₀f₁ be continuous self-maps on X.iterated function systems is the action of the semi-group generated by {f₀,f₁} on X,short-hand for IFS?f₀,f₁?.The study of the average shadowing property and the asymptotic average shadowing property of the iterative function system began with the work of Bahabadi and Nia.This paper introduced the definitions of ergodicity and strong ergodicity of the iteration func=tion system,and made a research on the basis of this.Especially under similar conditions,?X,f?is transitive,but this article proved that IFS?f₀,f₁?is ergodic.The better results are obtained.In addition,we obtain the sufficient condition that the iterative function system IFS?f₀,f₁?does not have the asymptotic average shadowing property.Secondly,by introducing the definitions of ???-shadowing,???-shadowing and Lyapunov stable,the following results are obtained:IFS?f₀,f₁?has the ???-shadowing property????-shadowing property????F^k has the ???-shadowing property?d-shadowing property?for any k?Z⁺;Finally,we discussed the relationship among the d-shadowing property,chain transitive,chain mixing,transitivity,strong ergodicity.