Brief Course Description: This is the second semester of a two semester sequence of courses on Ergodic Theory and Dynamical Systems. Dynamical Systems forms the mathematical foundations of what is popularly known as "Chaos Theory". The first semester studied topological and differential properties of evolving systems, or more precisely, of the orbits of differential equations and iterated functions. The second semester, which will be independent of the first, will focus on Ergodic Theory which studies the statistical properties of the evolution of complex systems.

Ergodic theory originated in Boltzmann's work on statistical mechanics regarding the question of when is is valid to assume that the probability of an evolving system being in a certain state is the same as the relative size of that state in the space of all possible states, or more succinctly, when can one interchange time and space averages. The mathematical origins of ergodic theory are due to von Neumann, Birkhoff, and Koopman in the 1930s and it has grown into one of the fundamental areas of modern mathematics with connections to fields as diverse as probability and statistics, real and functional analysis, combinatorics, image processing, number theory and cryptography. The course will cover the basics of Ergodic Theory including recurrence, the Ergodic Theorems, and entropy.

Prerequisites: A knowledge of basic metric space topology, measure theory and convergence of functions (eg. from Rudin's Principles of Mathematical Analysis) and a solid course in multi-variable (advanced) calculus. Otherwise the course will be self-sufficient.

Text: The course will be based on lectures. Some standard measure theory books will be recommended, but not required.

Homework: Homework will be assigned each Friday and is due at the start of class the next Friday. Homework may be turned in late for 2/3 credit until the start of class the following Monday. After that, no late homework will be accepted. Homework must be in hard copy form, no electronic submissions will be accepted. Please see the section below for the policy on homework assistance.

Grades The course grade will be based on the homework and possibly a Final Exam.

Honor Code: On all work submitted for credit by students at the university, the following pledge is either required or implied:

On my honor, I have neither given nor received unauthorized aid in doing this assignment.

For more information on the student honor code see the Dean of Students Website.

Homework Assistance Policy: You may discuss any homework problems with other students in the course. You may not discuss the homework with students not in the course or with faculty members. You cannot look at the written solutions of your fellow students or at the solution to the problem in any books or on-line resources.

Accommodations for Students with Disabilities: The University Policy: "Students requesting classroom accommodation must first register with the Dean of Students Office. The Dean of Students Office will provide documentation to the student who must then provide this documentation to the Instructor when requesting accommodation."