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arXiv · 2606.17031

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

Farhod Haydarov·Shirali Kadyrov·Mirmukhsin Makhmudov·Khudoyor Mamayusupov
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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 \(P>\phi(0^\infty)\) (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 \(\Delta\) of the pressure are \(O(\sqrt{\Delta})\)-close to \(\mu_\phi\) when tested against H\"older 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.

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Farhod Haydarov, Shirali Kadyrov, Mirmukhsin Makhmudov, Khudoyor Mamayusupov. 2026-06-15. Effective Intrinsic Ergodicity for renewal-type potentials on S-gap shifts. https://arxiv.org/abs/2606.17031

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