Subshifts on infinite alphabets and their entropy

Sharwin Rezagholi

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Submission date: 07. Jul. 2020 (revised version: November 2020)
Pages: 16
Bibtex
MSC-Numbers: 05C50, 05C63, 37B10, 37B40, 68R15
Keywords and phrases: Infinite graphs, Symbolic Dynamics, Topological entropy
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Abstract:
We generalize symbolic dynamics to infinite alphabets by endowing the alphabet with the cofinite topology. The topological entropy is shown to be equal to the supremum of the growth rate of the complexity function with respect to finite subalphabets. For the case of topological Markov chains induced by infinite graphs, our approach yields the same entropy as the approach of Gurevich via Alexandrov compactification [B.M. Gurevich, Topological entropy of enumerable Markov chains, 1969, Soviet Mathematics Doklady 10:4, 911-915].