Chain Mixing, Shadowing Properties and Multi-transitivity

Abstract

We prove that a map f is chain mixing if and only if $$f^r\times f^s$$ is chain transitive for some positive integers r, s. We prove that a map which has the average shadowing property with dense $$\underline{0}$$ -recurrent points is transitive, and by this result we point out that a map is multi-transitive if it has the average shadowing property and an invariant Borel probability measure with full support. Moreover, we show that $$\Delta $$ -mixing, the completely uniform positive entropy and the average shadowing property are equivalent mutually for a surjective map which has the shadowing property.