Typical periodic optimization for dynamical systems: symbolic dynamics

Abstract

We develop a new theory of maximizing sets in dynamical systems, for the study of ergodic optimization in systems with weak hyperbolicity but where the Ma\~n\'e cohomology lemma does not hold. This leads to new solutions of the Typical Periodic Optimization problem in the Lipschitz category: existence of an open dense set of Lipschitz functions such that each member has a unique maximizing measure and this measure is periodic (an equi-distribution on a single periodic orbit). 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 functions 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. The structural theorem is used to provide the first known example of a shift space where Typical Periodic Optimization fails despite periodic measures being dense in the set of all invariant measures.