Mathematics 797DE - ST-Dynamical Sys & ErgodicThry

Dynamical systems is an exciting and very active field in pure and applied mathematics, that involves tools and techniques from many areas such as analyses, probability and number theory. A dynamical system can be obtained by iterating a function or letting evolve in time the solution of equation. Even if the rule of evolution is deterministic, the long term behavior of the system is often chaotic. Different branches of dynamical systems, in particular ergodic theory, provide tools to quantify this chaotic behavior and predict it in average. At the beginning of this lecture course we will give a strong emphasis on presenting many fundamental examples of dynamical systems, such as circle rotations, the baker map on the square and the continued fraction map. Driven by the examples, we will introduce some of the phenomena and main concepts which one is interested in studying. In the second part of the course, we will formalize these concepts and cover the basic definitions and some fundamental theorems and results in topological dynamics, in symbolic dynamics and in particular in ergodic theory. During the course we will also mention some applications to other areas of mathematics, such as number theory, and to Monte Carlo method, etc.

Open to Graduate students only. MATH 623 and 645 Graduate Students who have taken the pre-requisite courses outside of UMass may contact the instructor for permission to enroll.