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

Let $Σ\subset\mathbb N$ be a finitely generated nonlacunary multiplicative semigroup acting on $\mathbb T=\mathbb R/\mathbb Z$ by $T_n(x)=nx\pmod 1$. We prove that, for an open and dense subset of $\operatorname{Lip}(\mathbb T)$, every $Σ$-invariant maximizing or minimizing measure $μ$ satisfies $h_μ(T_n)=0$ for every $n\inΣ\setminus{1}$. Moreover, there exists a single dense $G_δ$ subset of $\operatorname{Lip}(\mathbb T)$ on which this conclusion holds simultaneously for all finitely generated nonlacunary multiplicative semigroups $Σ\subset\mathbb N$. The proof combines a periodic-grid perturbation estimate with the Rudolph--Johnson positive-entropy rigidity theorem.