Chain Mixing, Shadowing Properties and Multi-transitivity

We prove that a map f is chain mixing if and only if f r × f s is chain transitive for some positive integers r , s . We prove that a map which has the average shadowing property with dense 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 Δ -mixing, the completely uniform positive entropy and the average shadowing property are equivalent mutually for a surjective map which has the shadowing property.