Math 147a: Dynamical Systems (Fall 2006)

MW 12:30 – 2:00 // 257 Sloan

Instructor: Anton Gorodetski
Office: 282 Sloan
Phone: 395-4350
E-mail: asgor@caltech.edu

Office Hours: after the lectures or by appointment

Dynamical systems is the study of the long-term behavior of evolving systems. The modern theory of dynamical systems originated at the end of the 19th century with fundamental question concerning the stability and evolution of the solar system. Attempts to answer those questions led to the development of a rich and powerful field with applications to physics, biology, meteorology, astronomy, economics, and other areas. The mathematical core of the theory is the study of the global orbit structure of maps and flows with emphasis on properties invariant under coordinate changes.

The course is the first of three courses in the sequence, which includes also Complex Dynamics and Hamiltonian Dynamics. This introductory part is aimed at advanced undergraduates, graduate students, physicists and other non-experts who may want to gain a basic understanding of the subject.

The following topics will be covered:

  • Basic notions and examples of topological and smooth dynamics: equivalence, classification, invariants, conjugacy, structural stability.
  • Symbolic dynamics, coding, horseshoes, attractors.
  • Chaos and fractals.
  • Basics of ergodic theory: ergodicity, mixing, examples. Ergodic Theorems.
  • Topological entropy, measure-theoretic entropy, the variational principle.
  • Hyperbolicity. The Hadamard-Perron Theorem. The Hartman-Grobman Theorem. Hyperbolic sets and attractors.
  • Applications to Number Theory.
  • Last one or two lectures will be devoted to an overview of some recent developments at a theory of dynamical systems, including the discussion of several most challenging open problems. 

Prerequisites: Ma 108ab, Ma 109a, or equivalent.

Main Texts:

  • A.Katok, B.Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, any edition.
  • M.Brin, G.Stuck,  Introduction to Dynamical Systems, Cambridge University Press, 2002.

Additional references will be given for a few topics not covered by these books.

Collaboration Policy: You may discuss homework problems with other students, but solutions should be written up individually in your own words. Take-home exams must be your own work, with outside references properly attributed.

Homework and exams

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