This talk is based on joint work with Vaughn Climenhaga (Maryland), in which we show that every shift space which is a factor of a beta-shift has a unique measure of maximal entropy. This provides an affirmative answer to Problem 28.2 of Mike Boyle's article 'Open problems in symbolic dynamics'.
A measure of maximal entropy is a measure which witnesses the greatest possible complexity in the orbit structure of a topological dynamical system. Establishing when a system has a unique measure of maximal entropy is a fundamental topic in ergodic theory and has been studied extensively since the 1960's. The beta-shifts are a class of symbolic spaces with an extremely rich structure and a profound connection to number theory.
Our method actually applies to a rather large general class of shift spaces, and can also be applied to non-symbolic systems. Furthermore, we have recently extended our results to establish uniqueness of equilibrium measures for a large class of potential functions.
I will give a detailed explanation of the problems described above and their motivation. I will also explain, via a detailed description of the beta-shift, the key ideas behind our method.