Typically periodic optimization conjecture

Let $(X,T)\in\mathfrak{D}$ be a suitably hyperbolic dynamical system, and let $V$ be a Banach space consisting of suitably regular continuous real-valued functions on $X$. Define $V_{Per}$ to be the set of those $f\in V$ such that the set $\mathcal M_{\max}(f)$ of maximizing measures contains at least one measure supported on a single periodic orbit.

Typically periodic optimization (TPO) conjecture. The set $V_{Per}$ contains an open dense subset of $V$.

This conjecture asks whether periodic maximizing measures are typical in suitable function spaces, extending the weaker fact that typical maximizing measures are not fully supported for certain expanding or Anosov systems. The source describes the formulation as purposefully imprecise; specific versions for expanding maps and Axiom A diffeomorphisms, including smooth functions and the Lipschitz case, were previously conjectured by Yuan and Hunt.