This talk is based on a series of papers by Vaughn Climenhaga (Houston) and myself where we develop a new approach to prove uniqueness of equilibrium measures for dynamical systems with various non-uniform structures.
One class of model examples that motivated our results is the family of S-gap shifts. The S-gap shifts are a natural family of symbolic spaces which are easy to define but can be challenging to study! For a fixed subset S of the natural numbers, the corresponding S-gap shift is the collection of binary sequences which satisfy the condition that the length of every run of consecutive 0’s is a member of S. Examples are the even shift (we let S be the even numbers) and the prime gap shift (we let S be the prime numbers).
The key difficulty in the study of S-gap shifts is a spectacular failure of the Markov property, and we develop techniques to deal with this. These techniques can be adapted to apply to many other interesting dynamical systems beyond the well understood uniformly hyperbolic case (e.g. beta-shifts, interval maps with parabolic fixed points, non-uniformly expanding maps in higher dimensions, some partially hyperbolic examples).
I will explain our approach and motivations focusing on the example of S-gap shifts, building up to our result that ‘Every subshift factor of an S-gap shift (or a beta-shift) is intrinsically ergodic’. If time permits, I will also describe a brand new result by Climenhaga, myself and Kenichiro Yamamoto (Tokyo Denki University) which establishes a large deviations principle for S-gap shifts.