Generic Zero-Entropy Optimization for Finitely Generated Nonlacunary Actions on the Circle

Let $T_n(x)=nx\pmod 1$ on $\T=\mathbb R/\mathbb Z$, and let $Σ\subset\mathbb N$ be a finitely generated nonlacunary multiplicative semigroup. We prove that, for every $1\le s\le\infty$, there is an open dense set of potentials $f\in W^{1,s}(\T)$ such that every measure that maximizes or minimizes $\