Chain Transitive And Sub-shadowings In Nonautonomous Dynamical System

In this paper,we mainly study topological dynamics properties of chain tran-sitive and sub-shadowings for nonautonomous discrete dynamical systems.The details are as follows:At first Chapter,we simply introduced the development and origin of nonau-tonomous dynamical systems and some results of related dynamic properties.In chapter 2,we introduced some related concepts of the nonautonomous dynamic system and the definition of various sub-shadowings properties and the related concepts of cournot mapping are given in this paper.In chapter 3,we mainly study chain transitive property and topology tran-sitive property in nonautonomous dynamical systems.In section 3.1,At first,we pointed that if F = {fn}n=0 ?n is a continuous mapping family with chain mixing property,then Fk has chain mixing property for positive integer k.In addi-tion,we pointed a nonautonomous dynamical system that T has not chain mix-ing property but T2 has chain mixing property.In section 3.2,we prove that chain transitive and chain mixing are topological uniform conjugation invariants in nonautonomous dynamical systems.In section 3.3,We prove that:in nonau-tonomous dynamic system F = {fn}n=0 ?,topological transitive contains chain transitive,topological mixing contains chain mixing;if F = {fn}n=0 ? has pseudo orbit shadowing property,topological transitive is equivalent to chain transitive,topological mixing is equivalent to chain mixing.In chapter 4,we study several sub-shadowings property in nonautonomous dynamical systems.In section 4.1,we point out that if every mapping of F={fn}n=0 ? are full mapping,and system F has 0-average shadowing property,then it is chain transitive.In section 4.2,we point out that a system F has average shadowing property if and only if T has 0-average shadowing property when q ?[0,1).In chapter 5,we study several dynamic property of cournot mapping in nonau-tonomous dynamic system,Let ?(x,y)=(f(y),g(x))is a cournot mapping,if f:Y ? X and g:X ? Y are continuous map,(x,y)? X × Y.we proved that ?(x,y)has pseudo-orbit shadowing property if and only f?g and g?f have pseudo-orbit shadowing property;?(x,y)has average shadowing property if and only f?g and g?f have average shadowing property;?(x,y)is chain mixing if and only f?g and g?f are chain mixing.