Shadowing Properties And Bundle Dynamical Systems Of Group Actions

As an important research direction of dynamical systems,many results of Z-actions have been extended to general group actions.Firstly,shadowing properties of Zd-actions are studied in this thesis.A Zd-action is said to be non-uniform hyperbolic(partially hyperbolic)if one generator of the Zd-action is non-uniform hyperbolic(partially hyperbolic).We call that a Zd-action is non-uniform partially hyperbolic if one generator of the Zd-action has zero Lyapunov exponents.We obtain that non-uniform hyperbolic,partially hyperbolic and non-uniform partially hyperbolic Zd-actions enjoy the shadowing and quasi-shadowing properties.Secondly,some results in style number theory are generalized to amenable group actions.The existence of coverage measure determined by style number and the variational principle of style number and independent number are obtained.Finally,the relationship between the entropy of dynamical systems and that of its induced bundle dynamical systems of the amenable group actions are studied.The concent of the thesis is arranged as follows.The research background of this thesis is given in the first chapter,which mainly states the development of differential dynamical systems,and introduces the research status of pseudo-orbit shadowing theory and bundle dynamical systems.In the second chapter,we introduce some preliminaries used in this thesis:some concepts and results of differential dynamical systems and ergodic theory,style number theory,pseudo-orbit shadowing properties of hyperbolic systems,C1+α Pesin’s theory,quasi-shadowing properties of partially hyperbolic systems and bundle dynamical systems,entropy and quasi-tiling properties of amenable groups.In the third chapter,we define non-uniform hyperbolic,partial hyperbolic and non-uniform partial hyperbolic Zd-actions,and prove that they have the shadowing and quasi-shadowing properties.In Chapter four,some important results of style number theory are generalized to the action of amenable groups.In the fifth chapter,we obtain that the measure entropy and topological entropy of dynamical system and those of its induced bundle dynamical systems of amenable group actions are equal respectively.