ABSTRACT

This talk mainly reviews and introduces recent progress on the typical optimization of dynamical systems: the symbolic systems case. The contents include: the theory of maximizing sets in dynamical systems, for the study of ergodic optimization in systems with weak hyperbolicity, but where the Mañé cohomology lemma does not hold. The theory yields a structural theorem that isolates the part of the system responsible for any robust non-periodic optimization. The structural theorem is developed further in the setting of symbolic dynamics: given any shift space, for typical Lipschitz functons, the maximizing measure is shown to be either periodic or supported on the Markov boundary of the shift space. It follows that Contreras' Typical Periodic Optimization theorem for shifts of finite type can be extended to a wide class of shift spaces, including every sofic shift. This is joint work with Oliver Jenkinson, Leiye Xu, and Yiwei Zhang.