Joint typical periodic optimization

We prove a generalised Yuan--Hunt--Ma\~n\'e Conjecture: if $\mathcal{F}$ is the Banach space of $\alpha$-H\"older 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.