Date and Time: Oct 1, 2021 1PM-2PM
Speaker: Daniel J Thompson (Ohio State University)
Title: Beyond Bowen’s Specification Property
Abstract: Around a decade ago, Vaughn Climenhaga and I started a project to answer a very concrete open problem from Mike Boyle’s list of Open Problems in Symbolic Dynamics - “Does every subshift factor a beta-shift have a unique measure of maximal entropy?” We were hoping an old approach of Bowen could be adapted somehow to the setting, and that we’d answer it routinely in 3 months and present the answer to Mike Boyle at the annual Maryland dynamics workshop.
So what actually happened? Our hope that the approach of Bowen could be adapted was correct. Our timeline was not correct, although we eventually solved the problem in 6 months. Our other initial misconception was that the answer would be routine - in reality, our solution required an extension of Bowen’s approach from the uniform world to the non-uniform world. The techniques we developed for that project have proved to be very flexible, and developing these ideas has kept us busy for several years. We, and other authors, have now applied this approach to a wide range of non-uniformly hyperbolic systems, including geodesic flows in non-positive curvature. My talk will tell some of this story, starting at the beginning. I will also give a brief taste of current and (expected) future applications.
This story is also told in our recent survey article with the same title as this talk, which can be found at https://arxiv.org/abs/2009.09256
The talk will be suitable for a general mathematics audience and graduate students.