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Characterization for entropy of shifts of finite type on Cayley trees. (English) Zbl 1459.37015

Summary: The notion of tree-shifts constitutes an intermediate class between one-sided shift spaces and multidimensional ones. This paper proposes an algorithm for computing the entropy of a tree-shift of finite type. Meanwhile, the entropy of a tree-shift of finite type is \(\frac{1}{p} \text{ln} \lambda\) for some \(p\in \mathbb{N} \), where \(\lambda\) is a Perron number. This extends D. A. Lind’s work [Ergodic Theory Dyn. Syst. 4, 283-300 (1984; Zbl 0546.58035)] on one-dimensional shifts of finite type. As an application, the entropy minimality problem is investigated, and we obtain a necessary and sufficient condition for a tree-shift of finite type to be entropy-minimal with some additional conditions.

MSC:

37B51 Multidimensional shifts of finite type
37B10 Symbolic dynamics
37B40 Topological entropy
94A17 Measures of information, entropy

Citations:

Zbl 0546.58035
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References:

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