INSTRUCTOR: Steven Hurder
OFFICE: 505 SEO
CONTACT: (312) 413-2154, hurder@uic.edu
OFFICE HOURS: 10-12, MWF in 340 SEO, or by appointment

Dynamical systems is a very rich and still growing area of mathematical research, which studies a wide variety of examples and phenomenon that arise from nature, physics, computer science and many areas of pure mathematics.

This course will give a broad introduction to dynamical systems, illustrated by applying the theory developed to a selection of some classical examples and some much more recent fundamental ones.

It is not a course in ergodic theory, although some of the main ideas of ergodic theory will be presented.

The course starts by introducing a mixture of key examples. We will then discuss basic concepts of topological dynamics - topological transitivity and topologically mixing systems, expansive systems, and topological entropy.

Next we will introduce symbolic dynamical systems, which provide many great examples and key ideas for the study of all dynamical systems.

Then we discuss the ideas of ergodic theory, and some of the relations and applications to topological dynamics and symbolic dynamics.

Finally, we delve into the world of differentiable dynamical systems, and study the class of uniformly hyperbolic systems. This includes the work of Anosov and general Anosov systems, Smale's Horseshoe Examples, and the ideas of stable manifolds and their applications.

The book by Brin and Stuck for the course is chosen to cover this broad approach to the subject, and while the reviews call it "terse" we will illustrate the ideas with may examples and discussions in class. The book includes several "real life" examples. For example, the first author's son is one of the founders of Google, and there is a section describing the application of the theory to search engine theory. The chapter and examples on symbolic dynamical systems owes a strong debt to Adler, Kitchens and Marcus who were senior researchers at IBM for several decades.

Prerequisites for the course are multivariable calculus, a good grasp of linear algebra, and a good understanding of point-set topology would be very useful. The course does not require the graduate course in Real Analaysis Math 533, nor the graduate course in Manifold Theory, Math 549. However, in the last part of the course on hyperbolic dynamical systems, ideas of manifold theory will be introduced (even if just for Euclidean space and the study of tori) and the deeper study of these systems does require a solid understanding of Riemannian manifolds.

The main text is a relatively recent book which covers the topics listed above:

• Introduction to Dynamical Systems (Hardcover, 2002) by Michael Brin and Garrett Stuck
  The book sells for $43.63 on amazon.com

There are three additional "recommended texts". The first by Devaney is one of the standard texts for an undergraduate course in dynamical systems. It includes somewhat more discussion of many of the same topics, although it is not as thorough.

• An Introduction to Chaotic Dynamical Systems, 2nd Edition (Paperback, 2003) by Robert L. Devaney
  The book sells for $45.00 on amazon.com

The other two books are by Hasselblatt and Katok. Their 2003 book "A First Course" is at approximately the same level as Brin and Stuck book, but with more emphasis on smooth dynamical systems. Their 1996 book is a modern classic, and has become the standard reference for the study of smooth dynamical systems.

• A First Course in Dynamics: with a Panorama of Recent Developments (Paperback, 2003) by Boris Hasselblatt and Anatole Katok
  The book sells for $37.84 on amazon.com
  

• Introduction to the Modern Theory of Dynamical Systems (Paperback, 1996) by Anatole Katok and Boris Hasselblatt
  The book sells for $57.27 on amazon.com
  

This book is a very nice, accessible treatment of the theory of fractal sets, which are closely related to dynamical systems -- they are often closed invariant sets defined by a dynamical system.

• Measure, Topology, and Fractal Geometry (Undergraduate Texts in Mathematics) by Gerald Edgar
  The book sells for $46.45 on amazon.com