Math 581BA
Dynamical Systems

Boris Solomyak

Autumn 2004, Monday/Wednesday/Friday 9:30

The mathematical theory of dynamical systems studies the global orbit structure of maps and flows. The field of dynamical systems comprises several major disciplines, among them ergodic theory, symbolic dynamics, topological dynamics, and differentiable dynamics. This course aims to give a general introduction into the subject, rather than concentrating on one of these areas. We will use the encyclopedic monograph Introduction to the Modern Theory of Dynamical Systems by A. Katok and B. Hasselblatt (Cambridge University Press, 1995), although we will only be able to cover a small part of the book.

I will begin with some of the basic examples of dynamical systems, such as circle rotations and hyperbolic toral automorphisms. We will then study classification problems and methods for finding conjugacies, including the Poincaré-Siegel Theorem; as well as invariants, notably the topological entropy. This corresponds to Chapters 1--3 of the book. After that we will jump to Chapters 11 and 12, which introduce the beautiful theory of circle maps, including the classical results of Poincaré and Denjoy.

Prerequisites: Point-set topology, real analysis, advanced multivariable calculus, and linear algebra at the advanced undergraduate level should suffice, so the course should be, in principle, accessible to first-year graduate students.