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A topological dynamical system with two different positive sofic entropies. (English) Zbl 1497.37013

The question of whether a mixing action might have two different non-negative values of sofic entropy remained unsolved prior to the publication of this comprehensive article. The mixing subshift of finite type with two different positive sofic entropies is explicitly illustrated by the authors. The main result of this article is contained in the following theorem.
Theorem. There exist a countable group \(\Gamma\), a mixing action \(\Gamma\curvearrowright X\) by homeomorphisms on a compact metrizable space \(X\) and two sofic approximations \(\Sigma_1,\Sigma_2\) to \(\Gamma\) such that \[ 0 < h_{\Sigma_1}(\Gamma\curvearrowright X) < h_{\Sigma_2}(\Gamma\curvearrowright X) < \infty. \]
Let \(\sigma: \Gamma \to\mathrm{Sym}(V )\) be a homomorphism and \(G_{\sigma} = (V,E_{\sigma})\) be the hypergraph with vertices \(V\) and edges equal to the orbits of the generator subgroups, where \(\mathrm{Sym}(V)\) is the group of permutations of \(V\). It is proved that if \(\Sigma = \{\sigma_n\}_{n\geq 1}\) is a sofic approximation to \(\Gamma\) by uniform homomorphisms then the \(\Gamma\)-entropy of \(\Gamma\curvearrowright X\) is: \[ h_{\Sigma}(\Gamma\curvearrowright X):=\inf\limits_{\epsilon>0}\lim\sup\limits_{i\to \infty}\log\sharp\{\epsilon-\text{proper 2-coloring of} \ G_{\sigma_i}\}. \] The authors prove that \(\{\mathbb{P}_{u}^{n}\}_{n\geq 1}\) is a random sofic approximation, where \(\mathbb{P}_{u}^{n}\) is a uniform probability measure on the set of all uniform homomorphisms from \(\Gamma\) to Sym\((n)\). Additionally they prove numerous lemmas and propositions. In particular, by using a natural Markov model on the space of proper colorings, the authors study a \(\Gamma\)-invariant measure \(\mu\). Finally, two appendices are given.

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B40 Topological entropy
37B10 Symbolic dynamics
37B51 Multidimensional shifts of finite type
37E25 Dynamical systems involving maps of trees and graphs
05C65 Hypergraphs
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