Pincaré Recurrence And Measure-theoretic Entropy In Dynamical Systems

This paper is divided into four parts, we study Poincaré recurrence and a-weighted entropy of multifractal analysis in dynamical systems. In the first part of this paper, we introduce the backgrounds of Poincaré recurrence and the ba-sic knowledge of a-weighted entropy. In the second part of this paper, we explore multifractal analysis of local entropy. In the third part of this paper, we study Ornstein-Weiss theorem for r-ball. In the fourth part, we explore Poincaré recur-rence for weighted background.This paper is organized as follows:In chapter 1, the backgrounds of Poincaré recurrence and the basic knowledge of a-weighted entropy.In chapter 2, Multifractal analysis of local entropy.In chapter 3, Ornstein-Weiss theorem for r-ball.In chapter 4, Poincaré recurrence and a-weighted entropy.