Joint typical periodic optimization

We prove a generalised Yuan–Hunt–Mañé Conjecture: if $\mathcal{F}$ is the Banach space of $\alpha$-Hölder functions, and $\mathcal{T}$ is either a space of Lipschitz expanding maps, or of Anosov diffeomorphisms, or the family of beta-transformations on the interval, there is an open dense subset of $\mathcal{T}\times \mathcal{F}$ consisting of map-function pairs whose maximizing invariant measure is unique and supported on a periodic orbit.