From the Publisher:
Although it originated as a method to study general dynamical systems, symbolic dynamics is useful in coding for data storage and transmission as well as in linear algebra. Requiring only a undergraduate knowledge of linear algebra, this first general textbook includes over 500 exercises.

An Introduction to Symbolic Dynamics and Coding

In this paper we present a class of C*-algebras and point out its close relationship to topological Markov chains, whose theory is part of symbolic dynamics. The C*-algebra construction starts from a matrix A =(A (i,j))i,~ z, Z a finite set, A(i,j)c{0, l}, and where every row and every column of A is non-zero. (That A(i,j)e{O, 1} is assumed for convenience only. All constructions and results extend to matrices with entries in 2~+. We comment on this in Remark 2.18.) A C*-algebra 6~ A is then generated by partial isometries Si~O(i~X ) that act on a Hilbert space in such a way that their support projections Qi=S*S~ and their range projections P~ =SIS* satisfy the relations

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A Class of C*-Algebras and Topological Markov Chains.

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Ergodic equivalence relations, cohomology, and von Neumann algebras. II

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Ergodic equivalence relations, cohomology, and von Neumann algebras

A branch of ordered algebraic structures has grown, motivated by $K$-theoretic applications and mainly concerned with partially ordered abelian groups satisfying the Riesz interpolation property This monograph is the first source in which the algebraic and analytic aspects of these interpolation groups have been integrated into a coherent framework for general reference The author provides a solid foundation in the structure theory of interpolation groups and dimension groups (directed unperforated interpolation groups), with applications to ordered $K$-theory particularly in mind Although interpolation groups are defined as purely algebraic structures, their development has been strongly influenced by functional analysis This cross-cultural development has left interpolation groups somewhat estranged from both the algebraists, who may feel intimidated by compact convex sets, and the functional analysts, who may feel handicapped by the lack of scalars This book, requiring only standard first-year graduate courses in algebra and functional analysis, aims to make the subject accessible to readers from both disciplines High points of the development include the following: characterization of dimension groups as direct limits of finite products of copies of the integers; the double-dual representation of an interpolation group with order-unit via affine continuous real-valued functions on its state space; the structure of dimension groups complete with respect to the order-unit norm, as well as monotone sigma-complete dimension groups and dimension groups with countably infinite interpolation; and an introduction to the problem of classifying extensions of one dimension group by another The book also includes a development of portions of the theory of compact convex sets and Choquet simplices, and an expository discussion of various applications of interpolation group theory to rings and $C^*$-algebras via ordered $K_0$ A discussion of some open problems in interpolation groups and dimension groups concludes the book Of interest, of course, to researchers in ordered algebraic structures, the book will also be a valuable source for researchers seeking a background in interpolation groups and dimension groups for applications to such subjects as rings, operator algebras, topological Markov chains, positive polynomials, compact group actions, or other areas where ordered Grothendieck groups might be useful This is a reprint of the 1986 original (SURV/20S)

Partially ordered abelian groups with interpolation

Let X be a second countable zero-dimensional a-compact space. Denote by ~ x the Boolean ring of compact open subsets of X and consider a countable locally finite group N of homeomorphisms of X. Assume that every element of G has an open fixed point set, and, for the moment, assume also, that the fixed point set of every element of ff has a compact complement. If we are given a finite family A~, ia~, of disjoint compact open non-empty subsets of X together with a family g~(-f, i e J , such that the g~A~, i ~ J , are also disjoint, and such that

On dimension functions and topological Markov chains

More generally, we consider the subshifts of F z. These are the closed subsets of F z that are invariant under S. We denote for such a subshift X by C(X) the set of continuous real-valued functions on X, and by d//(X) the set of shiftinvariant Borel probability measures on X. h(/z) will stand for the entropy of a tz E rig(X). (For the entropy and its properties see, e.g., [11].) We set for a subshift X,

On the uniqueness of the equilibrium state

The results ofOn sofic systems I on topological Markov chains extending sofic systems are completed. To homomorphisms of sofic systems are canonically associated homomorphisms of Markov extensions. Also considered is a class of finitary codes for sofic systems.

Wolfgang Krieger

Papers

On dimension functions and topological Markov chains

On the uniqueness of the equilibrium state

On sofic systems II

On a Dimension for a Class of Homeomorphism Groups

On the Araki-Woods asymptotic ratio set and non-singular transformations of a measure space