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

Abstract

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 \(P>ϕ(0^\infty)\) (automatic when \(S\) is infinite), we show that \(P\) is the topological pressure and that the potential admits a unique equilibrium state \(μ_ϕ\). Our main result is an effective intrinsic ergodicity estimate: invariant measures whose free energy is within \(Δ\) of the pressure are \(O(\sqrtΔ)\)-close to \(μ_ϕ\) when tested against Hölder observables. As an application, every finite-word cylinder of positive \(μ_ϕ\)-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.