# Math 635: Dynamical Systems

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## Catalog Information

### Title

Dynamical Systems.

3

### Description

(A rigorous treatment of the theory of dynamical systems.?)

## Desired Learning Outcomes

This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include topological, symbolic, and hyperbolic dynamical systems.

### Prerequisites

Students are expected to have completed Math 634. This will provide the students with an understanding on the theory of ordinary differential equations.

### Minimal learning outcomes

Students should achieve mastery of the topics below. This means that they should know all relevant definitions, full statements of the major theorems, and examples of the various concepts. Further, students should be able to solve non-trivial problems related to these concepts, and prove theorems in analogy to proofs given by the instructor.

1. Topological dynamical systems
• Nonwandering set, chain recurrence
• Topological mixing and transitivity
• Expansive systems
• Topological entropy
• Topological and smooth conjugacy
2. Symbolic dynamical systems
• Subshifts of finite type
• Perron-Frobenius theorem
• Topological entropy for subshifts of finite type

3. Hyperbolic dynamical systems
• Hartman-Grobman theorem
• Stable manifold theorem
• Hyperbolic sets
• Anosov diffeomorphisms
• Smale Horseshoe and transverse homoclinic points
• Shadowing
• Axiom A dynamical systems and spectral decomposition
4. Low dimensional dynamical systems
• Circle homeomorphisms, circle diffeomorphisms, and rotation number
• Real quadratic maps
• Expanding endomorphisms

### Textbooks

Possible textbooks for this course include (but are not limited to):

### Additional topics

These are at the instructor's discretion as time allows examples are: Ergodic theory, Markov partitions, Sharkovsky theorem, Hamiltonian dynamics, and measure theoretic entropy.

### Recommended texts

Dynamical Systems, stability, symbolic dynamics, and chaos, by Clark Robinson; Introduction to Dynamical Systems, by Michael Brin and Garrett Stuck

None.