This is a thorough introduction to the dynamics of one-sided and two-sided Markov shifts on a finite alphabet and to the basic properties of Markov shifts on a countable alphabet. These are the symbolic dynamical systems defined by a finite transition rule. The basic properties of these systems are established using elementary methods. The connections to other types of dynamical systems, cellular automata and information theory are illustrated with numerous examples. The book is written for graduate students and others who use symbolic dynamics as a tool to study more general systems.

Klappentext

Nearly one hundred years ago Jacques Hadamard used infinite sequences of symbols to analyze the distribution of geodesics on certain surfaces. That was the beginning of symbolic dynamics. In the 1930's and 40's Arnold Hedlund and Marston Morse again used infinite sequences to investigate geodesics on surfaces of negative curvature. They coined the term symbolic dynamics and began to study sequence spaces with the shift transformation as dynamical systems. In the 1940's Claude Shannon used sequence spaces to describe infor­ mation channels. Since that time symbolic dynamics has been used in ergodic theory, topological dynamics, hyperbolic dynamics, information theory and complex dynamics. Symbolic dynamical systems with a finite memory are stud­ ied in this book. They are the topological Markov shifts. Each can be defined by transition rules and the rules can be summarized by a transition matrix. The study naturally divides into two parts. The first part is about topological Markov shifts where the alphabet is finite. The second part is concerned with topological Markov shifts whose alphabet is count ably infinite. The techniques used in the two cases are quite different. When the alphabet is finite most of the methods are combinatorial or algebraic. When the alphabet is infinite the methods are much more analytic. This book grew from notes for a graduate course taught at Wesleyan Uni­ versity in the fall of 1994 and is intended as a graduate text and as a reference book for mathematicians working in related fields.

Inhalt

1. Background and Basics.- § 1.1 Subshifts of Finite Type.- § 1.2 Examples.- § 1.3 Perron-Frobenius Theory.- § 1.4 Basic Dynamics.- Notes.- References.- 2. Topology Conjugacy.- § 2.1 Decomposition of Topological Conjugacies.- § 2.2 Algebraic Consequences of Topological Conjugacy.- Notes.- References.- 3. Automorphisms.- § 3.1 Automorphisms.- § 3.2 Automorphisms as Conjugacies.- § 3.3 Subgroups of the Automorphism Group.- § 3.4 Actions of Automorphisms.- § 3.5 Summary.- Notes.- References.- 4. Embeddinggs and Factor Maps.- § 4.1 Factor Maps.- § 4.2 Finite-to-one Factor Maps.- §4.3 Special Constructions Involving Factor Maps.- § 4.4 Subsystems and Infinite-to-One Factor Maps.- Notes.- References.- 5. Almost-Topological Conjugacy.- § 5.1 Reducible Subshifts of Finite Type.- § 5.2 Almost-Topological Conjugacy.- Notes.- References.- 6. Further Topics.- § 6.1 Sofic Systems.- § 6.2 Markov Measures and the Maximal Measure.- § 6.3 Markov Subgroups.- § 6.4 Cellular Automata.- § 6.5 Channnel Codes.- Notes.- References.- 7. Countable State Markov Shifts.- § 7.1 Perron-Frobenius Theory.- § 7.2 Basic Symbolic Dynamics.- Notes.- References.- Name Index.