一、課程背景與目的：

The aim of this summer course is to provide the knowledge of the transformations (on manifolds and on metric spaces) which expand the distance between nearby points. There are many reasons why this transformation has an important role in ergodic theory. On the one hand, expanding maps exhibit very rich dynamical behaviors, from the metric and topological point of view (e.g., topological exact property, shadowing property and growth rate of periodic points) as well as from the ergodic point of view (e.g., thermodynamics and fractal geometry). On the other hand, the study of the expanding map leads to paradigms that are useful for understanding other systems (e.g., partially and non-uniformly hyperbolic systems). After introducing the classical results of the expanding maps, we will touch some interesting research topics which include the weighted thermodynamics, multifractal analysis of the Gibbs measure and multiple ergodic theory. 

 

二、課程之大綱：

1. Reviews on the ergodic theory：

Entropy, pressures and variational principle

2. Expanding maps (I): 

Distortion lemma and existence of ergodic measures

3. Expanding maps (II): 

Shadowing, growth rate of periodic points and Bowen-Manning formula 

4. Expanding maps (III)：

Thermodynamic formalism

5. Expanding maps (IV)：

Multifractal analysis and some related topics

 

No class on August 14.