In addition to a chapter in the First Edition on advanced topics and a comprehensive bibliography, the Second Edition includes a detailed Addendum, with companion bibliography, describing major developments and new research directions since publication of the First Edition. There are more than 500 exercises to test the reader's understanding. Topics are carefully developed and motivated with many illustrative examples. It is written at an elementary level and aimed at students, well-established researchers, and experts in mathematics, electrical engineering, and computer science. This Second Edition maintains the introductory character of the original 1995 edition as a general textbook on symbolic dynamics and its applications to coding. It has established strong connections with many areas, including linear algebra, graph theory, probability, group theory, and the theory of computation, as well as data storage, statistical mechanics, and $C^*$-algebras. Symbolic dynamics is a mature yet rapidly developing area of dynamical systems. recoverable measure from a corresponding deterministic system. We also suggest a procedure of constructing an Drawing on tools from ergodic theory, we prove some properties of entropy-maximizing measures. We study properties of measures on infinite sequences that maximize the metric entropy under the recoverability condition. In the second part of the paper we consider a modification of the problem wherein the entries in the sequence are viewed as random variables over a finite alphabet that follow some joint distribution, and the recovery condition requires that the Shannon entropy of the The techniques that we employ rely on the connection of this problem with constrained systems. We address the problem of finding the maximum growth rate of the set, which we term capacity, as well as constructions of explicit families that approach the optimal rate. tuple of consecutive entries is uniquely recoverable from its ![]() Motivated by the established notion of storage codes, we consider sets of infinite sequences over a finite alphabet such that every On the expository side, we present a short proof of Chung’s variational principle for sofic topological pressure. We also prove that for any group-shift over a sofic group, the Haar measure is the unique measure of maximal sofic entropy for every sofic approximation sequence, as long as the homoclinic group is dense. ![]() This extends a classical theorem of Lanford and Ruelle, as well as previous generalizations of Moulin Ollagnier, Pinchon, Tempelman and others, to the case where the group is sofic.Īs applications of our main result we present a criterion for uniqueness of an equilibrium measure, as well as sufficient conditions for having that the equilibrium states do not depend upon the chosen sofic approximation sequence. ![]() We show that for any sufficiently regular potential f : X → R f \colon X \to \mathbb, any translation-invariant Borel probability measure on X X which maximizes the measure-theoretic sofic pressure of f f with respect to Σ \Sigma is a Gibbs state with respect to f f. Further assume that X X is a shift of finite type, or more generally, that X X satisfies the topological Markov property. Let Γ \Gamma be a sofic group, Σ \Sigma be a sofic approximation sequence of Γ \Gamma and X X be a Γ \Gamma -subshift with non-negative sofic topological entropy with respect to Σ \Sigma.
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