Math 635: Dynamical Systems

Catalog Information

Title

Dynamical Systems.

Credit Hours

3

Prerequisite

Math 634.

Description

This course is designed to provide students with a rigorous treatment of the theory of dynamical systems.

Desired Learning Outcomes

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

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.

Courses for which this course is prerequisite

None.