On the prevalence of the periodicity of maximizing measures

Abstract

For a continuous map T:X→X on a compact metric space (X,d), we say that a function f:X→R has the property PT if its time averages along forward orbits of T are maximized at a periodic orbit. In this paper, we prove that for the one-sided full shift on two symbols, the property PT is prevalent (in the sense of Hunt–Sauer–Yorke) in spaces of Lipschitz functions with respect to metrics with mildly fast decaying rate on the diameters of cylinder sets. This result is a strengthening of [3, Theorem A], confirms the prediction mentioned in the ICM proceeding contribution of J. Bochi ([1, Section 1]) suggested by experimental evidence, and is another step towards the Hunt–Ott conjectures in the area of ergodic optimization.