Ethnic Mixture Of The Pseudo Orbit Tracing Property

In this paper,we study some complexity problems of dynamical systems related to chaos,entropy and transitive properties. More precisely,In chapter l,we briefly introduce the content,methods,development and present conditions in topological dynamical system,and main results in this paper.In chapter 2,some preliminary knowledge in topological dynamical system and ergodic theory which will be used in this paper are reviewed.In chapter 3,we mainly study topological weakly mixing property in topological dynamical system under the ideas of familization and lo-calization,more precisely,extending the notion of weakly mixing pairs in [30] to families, we define F mixing pairs and discuss the relations among F mixing pairs, F totally sequence entropy pairs, F complexity pairs,F regionally proximal relation and F equicontinuity.We prove that the factor (Y, (T|⁾) of (X, T) induced by the smallest invariant equivalent relation containing the κF regionally proximal relation related to T^-1 is the maximal equicontinuous factor,the factor of (X, T) induced by the smallest invariant equivalent relation containing F mixing pairs is κF equicontinuous.In chapter 4,we study the relations between pointwise pseudo-orbit tracing property and some chaotic properties such as topological mixing. When f has pointwise pseudo-orbit tracing property, some equivalent conditions of uniform positive entropy and completely positive entropy are given.