On the prevalence of the periodicity of maximizing measures

For a continuous map $T:X\to X$ on a compact metric space $(X,d)$, we say that a function $f:X\to \mathbb{R}$ has the property ${\mathcal{P}}_T$ if its time averages along forward orbits of $T$ are maximized at a periodic orbit. In this paper, we prove that for the one-side full shift of two symbols, the property ${\mathcal{P}}_T$ 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 \cite[Theorem~A]{BZ16}, confirms the prediction mentioned in the ICM proceeding contribution of J. Bochi (\cite[Seciton 1]{Boc18}) suggested by experimental evidence, and is another step towards the Hunt–Ott conjectures in the area of ergodic optimization.