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

Abstract

Let $T_n(x)=nx\pmod 1$ on $\T=\mathbb R/\mathbb Z$, and let $\Sigma\subset\mathbb N$ 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 $\int f\,d\mu$ over the $\Sigma$-invariant probability measures satisfies $h_\mu(T_r)=0$ for all $r\in\Sigma\setminus\{1\}$. For each fixed $s$, a single dense $G_\delta$ subset of $W^{1,s}(\T)$ works simultaneously for all finitely generated nonlacunary multiplicative semigroups. The proof combines Rudolph--Johnson entropy rigidity with periodic-grid perturbations that generically exclude Haar measure from the optimizing faces. In particular, without any uniqueness assumption, the result applies to simultaneous $T_p,T_q$-invariance whenever $p,q\ge2$ are multiplicatively independent.