Katok Entropy And Large Deviation Of Group Action

This thesis investigates some problems of the Katok entropy for-mula for amenable groups and Large deviations for Z~d-actions. In the first part of the thesis, we will introduce Katok entropy formula for amenable groups. In the second part, we will prove the Large deviations for Z~d-actions. The paper is organized as followsIn Chapter 1, the backgrounds of actions of amenable groups, the development and some results of the Katok entropy formula and Large deviations.In Chapter 2, we recall some classical definitions and theroms in compact metric space, dynamical systems and actions in amenable groups.In Chapter 3, we show the the Katok entropy formula for actions of amenable groups, Futhermore we will give its relation between Measure-theoretic entropy and prove it.In Chapter 4, we generalize the result of L.-S. Young's [32] Large deviations for Z~d actions on dynamical systems.