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arXiv · 2607.05602

Measures of maximal entropy for Markovian dynamics on the Gehman dendrite

Piotr Oprocha·Jakub Tomaszewski
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Abstract

We study transitive dynamical systems on the Gehman dendrite $\mathcal{G}$ for which the endpoint set $\mathrm{End}(\mathcal{G})$ is invariant. Our goal is to approximate such systems by maps whose measure-theoretic behaviour at maximal entropy is governed by an explicit countable Markov structure. We introduce a class of Markovian maps, encode their dynamics by countable Markov graphs, and use the criteria of Vere-Jones, Gurevich, Salama and Ruette to control the existence of measures of maximal entropy. The main theorem gives two arbitrarily close mixing Markovian perturbations of any given system in the considered class: one has a unique measure of maximal entropy, while the other has none.

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Piotr Oprocha, Jakub Tomaszewski. 2026-07-06. Measures of maximal entropy for Markovian dynamics on the Gehman dendrite. https://arxiv.org/abs/2607.05602

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