课程大纲
COURSE SYLLABUS
1.
课程代码/名称
Course Code/Title
MAT7062 双曲动力学 Hyperbolic Dynamics
2.
课程性质
Compulsory/Elective
Elective
3.
课程学分/学时
Course Credit/Hours
Course credits 3 – 48 hours
4.
授课语言
Teaching Language
English
5.
授课教师
Instructor(s)
Jana Rodriguez Hertz, Professor; Raul Ures, Professor
6.
是否面向本科生开放
Open to undergraduates
or not
Yes
7.
先修要求
Pre-requisites
（ 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.）
MA401 动力系统
Any basic course of Dynamical Systems. No differences between
undergraduate and graduate students.
8.
教学目标
Course Objectives
（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.）
This course introduces the basic concepts of hyperbolic dynamics, both uniform and non-uniform. It
focusses on some classical results that are important for the following development of the theory as well on
some recent developments, preparing the students for the study of more advanced topics and research.
9.
教学方法
Teaching Methods
（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.）
In presence class. No differences between undergraduate and graduate students.
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
1. Hyperbolic dynamics
1.1 The Smale horseshoe
1.2 The Arnold’s cat map
1.3 Hyperbolic sets
1.4 The Stable Manifold Theorem
1.5 Smale’s spectral decomposition Theorem
Section 2
1. Non-uniformly hyperbolic dynamics
2.1 Lyapunov exponents. Examples.
2.2. The Oseledets Theorem
2.3 Pesin Stable Manifold Theorem
2.4 Elements of Pesin Theory: Pesin Blocks, Pesin Region,
etc.
2.5 Absolute continuity
2.6 Ergodicity criterion
Section 3
Partial hyperbolicity
3.1 Partial hyperbolicity, definition and examples
3.2 Accessibility
3.3 Ergodic theorems for partially hyperbolic
diffeomorphisms
3.4 Dominated splitting
3.5 Open problems and conjectures
Section 4
Section 5
Section 6
Section 7
Section 8
Section 9
Section 10
…………
11.
课程考核
Course Assessment
（
○
1 考核形式 Form of examination；
○
2 .分数构成 grading policy；
○
3 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.）
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
Pollicott, Mark, Lectures on ergodic theory and Pesin Theory on compact manifolds,
Cambridge University Press
Katok, A., Hasselblatt, B., Introduction to the Modern Theory of Dynamical Systems,
Cambridge University Press
Rodriguez Hertz, F.; Rodriguez Hertz, J.; Ures, R., Partially Hyperbolic Dynamics.
Publicações Matemáticas do IMPA, 2011.
