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Finitary measures for subshifts of finite type and sofic systems. (English) Zbl 0594.28024

Mem. Am. Math. Soc. 338, 68 p. (1985).
Let G be a finite semigroup with zero element \(0\in G\) satisfying \(g\cdot 0=0\cdot g=0\) for all \(g\in G\). Let \(A\subseteq G\) be a subset which generates \(G\setminus \{0\}\). The pair (G,A) is called a sofic pair. It defines a subshift S of \(A^{{\mathbb{Z}}}\) by declaring that \(x=(x_ n)\in S\) if and only if for any \(\ell,n\in {\mathbb{Z}}\), \(\ell \geq 1\), we have \(x_ nx_{n+1}...x_{n+\ell}\neq 0\) in G. If in addition the subshift S is transitive (there exists a point \(x\in S\) whose orbit \(\{\sigma^ n(x):\quad n\in {\mathbb{N}}\}\) is dense in S, where \(\sigma\) is the shift map) then S is called a sofic system. A well-known subclass of sofic systems consists of the subshifts of finite type which can be considered as being defined by 0-1 matrices.
The authors study a class of measures which are natural for sofic systems in the same way that Markov measures are natural for subshifts of finite type. These are the finitary measures (or semigroup measures) on sofic systems, defined in terms of finite semigroups using stochastic transition matrices.
Different characterizations of semigroup measures are given, in particular generalizing the work of the authors [Isr. J. Math., to appear] and of W. Krieger [preprint]. Semigroup measures are shown to display strong thermodynamic properties and structure, and behave well under continuous maps. It is shown that the unique equilibrium state of a locally constant function is a semigroup measure for any pair (G,A) describing the sofic system S. A number of other results are given describing semigroup measures and examples are given illustrating various points. Finally the authors remark that the semigroup measures they consider seem to be of considerable interest in communication and computing.
Reviewer: G.Goodson

MSC:

28D20 Entropy and other invariants
54H20 Topological dynamics (MSC2010)
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