课程大纲

COURSE SYLLABUS

1.

课程代码/名称

Course Code/Title

MAT7061 Ergodic Theory 光滑遍历论

2.

课程性质

Compulsory/Elective

Elective

3.

课程学分/学时

Course Credit/Hours

3/48

4.

授课语言

Teaching Language

英语

English

5.

授课教师

Instructor(s)

Raul Ures, Professor；Jana Hertz， Professor

6.

是否面向本科生开放

Open to undergraduates

or not

是 Yes

7.

先修要求

Pre-requisites

MA301 实变函数

MA301Theory of Functions of a Real Variable, Real analysis, Topology

8.

教学目标

Course Objectives

This course introduces the basic concepts of ergodic theory mainly focused on differentiable ergodic theory. It

focusses on the classical results that are important for the following development of the theory, preparing the

students for the study of more advanced topics and research.

9.

教学方法

Teaching Methods

The course will be taught in the standard way ( “ chalk and board”, in-class discussion, homework, office

hours, closed-book tests).

10.

教学内容

Course Contents

（如面向本科生开放，请注明区分内容。 If the course is open to undergraduates, please indicate the

difference.）

Section 1

(6 hours)

Measure preserving transformations. Existence of invariant measures.

Poincaré recurrence theorem.

Section 2

(8 hours)

Ergodicity. Ergodic theorems of Von Neumann and Birkhoff.

Section 3

(6 hours)

Ergodic hierarchy. Mixing, K-automorphisms. Unique ergodicity

Section 4

(8 hours)

Examples: shifts, subshifts of finite type, toral automorphisms, toral

translations. Hopf argument.

Section 5

(6 hours)

Entropy: metric and topological entropy. Variational principle.

Section 6

(6 hours)

Ergodicity of Anosov diffeomorphisms. Hopf argument revisited.

Section 7

(8 hours)

Introduction to additional topics of differentiable ergodic theory: SRB

measures for hyperbolic attractors, Osedelec’s theorem, Pesin theory, Ruelle

inequality, Pesin equality, etc.

11.

课程考核

Course Assessment

Homework 30%+ Mid-term Exam (closed-book) 30%+Final Exam (closed book) 40%

12.

教材及其它参考资料

Textbook and Supplementary Readings

1. Mañé, Ricardo - Ergodic Theory and Differentiable Dynamics. Springer-Verlag, New York-

Berlin, 1987.

2. Brin, Michael, Stuck, Garret – Introduction to Dynamical Systems. Cambridge University Press,

2004.

3. Walters, Peter - An introduction to ergodic theory. Springer-Verlag, New York-

Berlin, 1982.

4. Bowen, Rufus - Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture

Notes in Math., Springer-Verlag, 1975.

5. Viana, Marcelo; Oliveira, Krerley – Foundations of Ergodic Theory. Cambridge studies in

advanced mathematics 151, Cambridge University Press, 2016.