Effective Intrinsic Ergodicity for renewal-type potentials on S-gap shifts

We establish effective intrinsic ergodicity for renewal-type potentials on one-sided $S$-gap shifts. Inducing on the one-symbol cylinder $[1]$ reduces the system to a full shift over the alphabet $S$, where the induced potential becomes a one-symbol potential and the equilibrium measure is Bernoulli. The associated renewal equation has a unique solution $P$, and under the condition $P>\phi (0^{\mathrm{\infty }})$ (automatic when $S$ is infinite), we show that $P$ is the topological pressure and that the potential admits a unique equilibrium state ${\mu }_{\phi }$. Our main result is an effective intrinsic ergodicity estimate: invariant measures whose free energy is within $\mathrm{\Delta }$ of the pressure are $O(\sqrt{\mathrm{\Delta }})$-close to ${\mu }_{\phi }$ when tested against Hölder observables. As an application, every finite-word cylinder of positive ${\mu }_{\phi }$-measure yields a uniform pressure gap for the set of orbits avoiding that cylinder, leading in the entropy case to strict entropy and Hausdorff-dimension gaps.