Study On Several Problems Of Sensitivity And Disjointness In Dynamical Systems

In this thesis,we mainly study the properties of sensitivity via Furstenberg families in topological dynamical system and measurable dynamical system,the properties of disjointness are also involved.The thesis is organized as follows:In Chapter 1,we briefly recall the development course and main objectives of the topological dynamical system and ergodic theory.We also introduce the research background and main results of our study.In Chapter 2,we briefly introduce some basic definitions and properties of topo-logical dynamical system and ergodic theory.We also discuss some concepts and propositions of the thesis.In Chapter 3,we will discuss topological sensitivity via Furstenberg families.We introduce the notion of block thick sensitivity,block IP sensitivity,strong thick sensi-tivity and strong IP sensitivity.By using structure theorem for minimal systems,we obtain that a minimal system is either strongly thick sensitive or a proximal extension of its maximal distal factor.By using the methods of ergodic theory and the properties of maximal ?-step nilfactor,we prove the following results:(1)a minimal system is either block IP sensitive or an almost one-to-one extension of its maximal ?-step nil-factor;(2)a minimal system is either block thick sensitive or a proximal extension of its maximal equicontinous factor;(3)a minimal system is either strongly IP sensitive or an almost one-to-one extension of its maximal distal factor.These results connect sen-sitivity via Furstenberg families with the structure of minimal systems,giving another way to characterize the structure of minimal systems.In Chapter 4,we will investigate measurable sensitivity via Furstenberg families.For a topological dynamical system(X,T),there exists invariant Borel probability mea-sure ?.Thus(X,Bx,?,T)can be viewed as a measurable dynamical system,where BX is the Borel ?-algebra of X.We introduce the notion of thick sensitivity for ?,IP sen-sitivity for ?,block thick sensitivity for ? and block IP sensitivity for ?.We show that for minimal system:(1)thick sensitivity for ? is equivalent to thick sensitivity;(2)block thick sensitivity for ? is equivalent to block thick sensitivity;(3)block IP sensitivity for ? is equivalent to block IP sensitivity.In Chapter 5,we introduce the notion of vector sensitivity,and mainly investigate two special cases of vector sensitivity:l-sensitivity and ?-l-sensitivity.We show that even the system is l-sensitive for any positive integer l,the system need not to be multi-sensitive.We also construct a minimal system which is l-sensitive but not(l+1)-sensitive.To distinguish between ?-1-sensitivity and ?-(l + 1)-sensitivity,we construct a minimal system(weakly mixing system)which is ?-l-sensitive but not ?-(l + 1)-sensitive.In Chapter 6,we study the properties of systems which are disjoint from all mini-mal systems under group actions.We show that if(X,G)is weakly mixing with dense distal points and with G being abelian,then(X,G)is disjoint from all minimal systems,extending Dong,Shao and Ye's work[20]to abelian group actions.We also prove that if(X,Zd)is disjoint from all minimal systems and transitive,then(X,Zd)is a weakly mixing M-system without nontrivial minimal factor,extending Huang and Ye's work[58]to Zd-actions.