The mathematical theory of chaotic systems describes how randomness arises from "sensitive dependence on initial conditions": a small error in measuring the current state of a system can translate into a large error in a prediction of the system's future. Systems that behave this way can be studied using the mathematical tools of probability theory. These probabilistic descriptions depend on what is called an "invariant measure", which represents the likelihood of observing different types of behavior. For example, if a die is rolled repeatedly, one invariant measure might say that all six numbers are equally likely, while another invariant measure might say that in the long run, twice as many sixes will appear as ones. If the die is fair, then the first invariant measure is the one to trust, but if it is unevenly weighted, the second could apply. One goal of the present project is to gain a better understanding of the set of all possible invariant measures for a given system. This is an important part of making valid predictions for systems with chaotic behavior. Part of this project also involves the training of graduate students and the development of a graduate textbook on "Ergodic Theory and Hyperbolic Dynamics". In more technical terms, dynamical systems with hyperbolic ("chaotic") behavior can be studied as stochastic processes by equipping them with invariant measures and using the tools of ergodic theory. There are generally many invariant measures; thermodynami