Background and Purpose:

The aim of ergodic theory is to understand the stochastic behavior of deterministic dynamical systems by studying the ergodic invariant probability measures of the system. Given such a measure, the ergodic theorems provide quantitative information on the long term behavior of almost every orbit. Ergodic theory has been widely applied to many disciplines such as number theory, ecological systems, and complex analysis. In this course, we will introduce thermodynamic formalism on countable Markov shifts. Meanwhile, the theory of symbolic dynamics and cellular automata on amenable groups will also be discussed. Amenability, which originated from the study of the Banach-Tarski paradox, is a property of groups generalizing both commutativity and finiteness. Nowadays, it plays an important role in many areas of mathematics.