The Characteristic Factor Of Dynamical Systems And Its Applications

In this thesis,we study the characteristic factor of dynamical systems and its ap-plications in independent pair along arithmetic progressions and ?-transitivity.We also give a new characterization of measure-theoretic rigidity.The thesis is organized as follows:In Introduction,we briefly recall the history and developments of topological dy-namics and ergodic theory.We also introduce the backgrounds of the study and the main results in this thesis.In Chapter 1,we recall the basic concepts in topological dynamics and ergodic theory.Some related knowledge and useful tools which will be used in this thesis are also introduced.In Chapter 2.we study the topological characteristic factor.In 1977,Fursten-berg gave a new proof of Szemeredi theorem via ergodic theory.introduced the idea of characteristic factor.In 1994,Glasner introduced the analogous concept of topological characteristic factor in topological dynamics.Glasner proved that for a distal minimal system,its largest distal factor of order d-1 is its d-step topological characteristic factor.He also gave the generalized result for a general minimal system.Glasner only dealt with totally minimal system.In this thesis,we deal with general minimal system and generalize Glasner' s work to the product system of finitely many minimal systems.In the proof of L2 convergence of the multiple ergodic averages,Host and Kra used the idea of characteristic factor.In this thesis,we give an analogous result of Host and Kra's work in topological dynamics.We introduce the concept of topological characteristic factor along cubes and prove that for a distal minimal system.its maximal(d-1)-step pro-nilfactor is a d-step topological characteristic factor along cubes.We also give a generalized result for a general minimal system.In Chapter 3,we study the application of topological characteristic factor in inde-pendent pair along arithmetic progressions and ?-transitivity.Using the generalization of Glasner's result in this thesis,we prove that for a distal minimal system,if the set of independent pair along arithmetic progressions order d is equal to its diagonal,then it is its largest distal factor of order d.We also give a generalized result for a gen-eral minimal system.We study the relation between independent pair along arithmetic progressions.?-transitivity and topological characteristic factor.We prove that for a dynamical system(X,T).the following three conditions are equivalent:(1)(X,T)is?-transitive:(2)the set of independent pair along arithmetic progressions of(X,T)is the whole space X×X:(3)the trivial system is a topological characteristic factor of(X,T).In the last chapter,we give a new characterization of measure-theoretic rigidity.We prove that measure-theoretic equicontinuity along an IP-set(or a sequence of N)is equal to measure-theoretic rigidity;measure-theoretic mean equicontinuity along a sequence of N with positive upper density implies measure-theoretic rigidity.