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Symbolic dynamics.

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Symbolic dynamics is a mature yet rapidly developing area of dynamical systems. 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. 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 is written at an elementary level and aimed at students, well-established researchers, and experts in mathematics, electrical engineering, and computer science. Topics are carefully developed and motivated with many illustrative examples. There are more than 500 exercises to test the reader's understanding. 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.

Symbolic dynamics. --- Coding theory.

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Ergodic theory. Information theory --- Symbolic dynamics --- Geometry, Hyperbolic

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Topological dynamics --- Symbolic dynamics --- Dynamique topologique --- Dynamique symbolique --- Dynamics. --- Interpolation.

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"In topological dynamics, the Symbolic Extension Entropy Theorem (SEET) (Boyle and Downarowicz, 2004) describes the possibility of a lossless digitalization of a dynamical system by extending it to a subshift on finitely many symbols. The theorem gives a precise estimate on the entropy of such a symbolic extension (and hence on the necessary number of symbols). Unlike in the measure-theoretic case, where Kolmogorov-Sinai entropy serves as an estimate in an analogous problem, in the topological setup the task reaches beyond the classical theory of measure-theoretic and topological entropy. Necessary are tools from an extended theory of entropy, the theory of entropy structures developed in Downarowicz (2005). The main goal of this paper is to prove the analog of the SEET for actions of (discrete infinite) countable amenable groups: Let a countable amenable group G act by homeomorphisms on a compact metric space X and let MG(X) denote the simplex of all G-invariant Borel probability measures on X. A function EA on MG(X) equals the extension entropy function h[pi] of a symbolic extension [pi] : (Y,G) [arrow] (X,G), where h[pi]([mu]) [equals] sup{hV(Y,G) : V [element of] [pi]-1([mu])} ([mu] [element of] MG(X)), if and only if EA is a finite affine superenvelope of the entropy structure of (X,G). Of course, the statement is preceded by the presentation of the concepts of an entropy structure and its superenvelopes, adapted from the case of Z-actions. In full generality we are able to prove a slightly weaker version of SEET, in which symbolic extensions are replaced by quasi-symbolic extensions, i.e., extensions in form of a joining of a subshift with a zero-entropy tiling system. The notion of a tiling system is a subject of several earlier works (e.g. Downarowicz and Huczek (2018) and Downarowicz, Huczek, and Zhang (2019)) and in this paper we review and complement the theory developed there. The full version of the SEET (with genuine symbolic extensions) is proved for groups which are either residually finite or enjoy the so-called comparison property. In order to describe the range of our theorem more clearly, we devote a large portion of the paper to studying the comparison property. Our most important result in this aspect is showing that all subexponential groups have the comparison property (and thus satisfy the SEET). To summarize, the heart of the paper is the presentation of the following four major topics and the interplay between them: Symbolic extensions, Entropy structures, Tiling systems (and their encodability), The comparison property"--

Group actions (Mathematics) --- Symbolic dynamics. --- Topological entropy. --- Tiling (Mathematics)