On Chain Mixing,Shadowing Properties And Multi-transitivity

In this thesis, we explored the relations between chain mixing, shadowing prop-erties and multi-transitivity.At first Chapter, we simply introduced the development of the dynamical system and the origin of the chain mixing, shadowing properties and multi-transitivity and some related results .In Chapter 2, some concepts of the topological dynamical system involved in this paper.In Chapter 3, We proved that a map f is chain mixing if and only if fT ×fs is chain transitive for some positive integers r, s. Moreover, we gave a example that a dynamical system (X, f) such that f x f2 is chain mixing but not transitive.In Chapter 4, we discussed relation of average shadowing property and multi-transitivity. We proved that a map which has the average shadowing property with dense 0-recurrent points is transitive, and by this result we pointed out a map which has the average shadowing property and an invariant Borel probability measure with fully support is multi-transitive. Moreover, we proved that a map f is multi-transitive for a map f which has d-shadowing property and with dense periodic points.In Chapter 5, we described relation of the average shadowing property and tran-sitivity. We proved that a map f has 0.5-average shadowing property and has dense minimal points, then it is syndetic transitive.In finally Chapter, we described relation of the average shadowing property, the completely positive entropy and △-mixing. We gave an example that a dynamical system (X, f) such that the map f has the completely positive entropy, but f has not the d-shadowing property. We proved that if the map f is chain mixing and has the family Fd1-shadowing property, then f has the q-average shadowing property for any q E [0, 1). Furthermore, f has the average shadowing property. Moreover, we showed that △-mixing, the completely uniform positive entropy and the average shadowing property are all equivalent, for a surjective map which has the shadowing property.