Affine Transformations On Nilmanifolds And Katok's Conjecture

In this paper,we mainly study the affine transformations on nilpotent manifolds.The details are as follows:In the first chapter,we briefly review the history of topological dynamical system and ergodic theory.Also some backgroud and main results of our study are presented.In Chapter 2,we briefly introduce some basic definitions and important properties of topological dynamical systems and ergodic theory.In Chapter 3,we will focus on the affine transformations on nilpotent manifolds.Firstly,we study affine transformations on 1-step nilpotent manifolds(that is,torus),and show that the Katok's conjecture holds for such systems;Then we review some basic properties of nilpotent manifolds and give some important results;Finally,we show that if the affine transformation on a nilpotent manifold has a fixed point,the Katok's conjecture holds for such systems.Especially,the Katok's conjecture holds for quasi-hyperbolic affine transformations on nilpotent manifolds.