Quasi-shadowing Property For Flows

The pseudo-orbit shadowing theory is an important basic tool in the qualitative theory of dynamical systems.In differential dynamical systems,uniform hyperbolic and non-uniform hyperbolic properties imply pseudo-orbit shadowing property,but partially hyperbolic systems generally no longer have pseudo-orbit shadowing property.This thesis studies and proves the quasi-shadowing properties and limit quasi-shadowing properties of partially hyperbolic flows and quasi-partially hyperbolic strings of flows.We first introduced the research background of this thesis in the first chapter,gave a brief overview of dynamical systems theory and pseudo-orbit shadowing theory,and introduced the research status of quasi-shadowing properties of partially hyperbolic systems.Then in the second chapter,some basic knowledge of continuous dynamical system is introduced,and some pseudo-orbit shadowing properties(for diffeomorphisms and flows)and quasi-shadowing properties of partially hyperbolic diffeomorphisms are reviewed.We proved in Chapter 3 that,without any additional conditions,the C~1partially hyperbolic flows have the quasi-shadowing property and the limit quasi-shadowing properties.In Chapter 4,we prove that the quasi-partially hyperbolic strings of flows have the quasi-shadowing property and limit quasi-shadowing properties.Finally,using the quasi-shadowing properties of the partial hyperbolic flows,we prove that the C~1partial hyperbolic flows with a C~1central foliation satisfies the central closing lemma.