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Symbolic Dynamics

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Mathematics of Complexity and Dynamical Systems

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Article Outline

Glossary

Definition of the Subject

Introduction

Origins of Symbolic Dynamics: Modeling of Dynamical Systems

Shift Spaces and Sliding Block Codes

Shifts of Finite Type and Sofic Shifts

Entropy and Periodic Points

The Conjugacy Problem

Other Coding Problems

Coding for Data Recording Channels

Connections with Information Theory and Ergodic Theory

Higher Dimensional Shift Spaces

Future Directions

Bibliography

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Abbreviations

Almost conjugacy:

(Sect. “Other Coding Problems”) A common extension of two shift spaces given by factor codes that are one-to-one almost everywhere.

Automorphism:

(Sect. “The Conjugacy Problem”) An invertible sliding block code from a shift space to itself; equivalently, a shift‐commuting homeomorphism from a shift space to itself; equivalently, a topological conjugacy from a shift space to itself.

Dimension group:

(Sect. “The Conjugacy Problem”) A particular group associated to a shift of finite type. This group, together with a distinguished sub‐semigroup and an automorphism, captures many invariants of topological conjugacy for shifts of finite type.

Embedding:

(Sect. “Shift Spaces and Sliding Block Codes”) A one-to-one sliding block code from one shift space to another; equivalently, a one-to-one continuous shift‐commuting mapping from one shift space to another.

Factor map:

(Sect. “Shift Spaces and Sliding Block Codes”) An onto sliding block code from one shift space to another; equivalently, an onto continuous shift‐commuting mapping from one shift space to another. Sometimes called Factor Code.

Finite equivalence:

(Sect. “Other Coding Problems”) A common extension of two shift spaces given by finite-to-one factor codes.

Full shift:

(Sect. “Shift Spaces and Sliding Block Codes”) The set of all bi‐infinite sequences over an alphabet (together with the shift mapping). Typically, the alphabet is finite.

Higher dimensional shift space:

(Sect. “Higher Dimensional Shift Spaces”) A set of bi‐infinite arrays of a given dimension, determined by a collection of finite forbidden arrays. Typically, the alphabet is finite.

Markov partition:

(Sect. “Origins of Symbolic Dynamics: Modeling of Dynamical Systems”) A finite cover of the underlying phase space of a dynamical system, which allows the system to be modeled by a shift of finite type. The elements of the cover are closed sets, which are allowed to intersect only on their boundaries.

Measure of maximal entropy:

(Sect. “Connections with Information Theory and Ergodic Theory”) A shift‐invariant measure of maximal measure‐theoretic entropy on a shift space. Its measure‐theoretic entropy coincides with the topological entropy of the shift space.

Road problem:

(Sect. “Other Coding Problems”) A recently‐solved classical problem in symbolic dynamics, graph theory and automata theory.

Run‐length limited shift:

(Sect. “Coding for Data Recording Channels”) The set of all bi‐infinite binary sequences whose runs of zeros, between two successive ones, are bounded below and above by specific numbers.

Shift equivalence:

(Sect. “The Conjugacy Problem”) An equivalence relation on defining matrices for shifts of finite type. This relation characterizes the corresponding shifts of finite type, up to an eventual notion of topological conjugacy.

Shift space:

(Sect. “Shift Spaces and Sliding Block Codes”) A set of bi‐infinite sequences determined by a collection of finite forbidden words; equivalently, a closed shift‐invariant subset of a full shift.

Shift of finite type:

(Sect. “Shifts of Finite Type and Sofic Shifts”) A set of bi‐infinite sequences determined by a finite collection of finite forbidden words.

Sliding block code:

(Sect. “Shift Spaces and Sliding Block Codes”) A mapping from one shift space to another determined by a finite sliding block window; equivalently, a continuous shift‐commuting mapping from one shift space to another.

Sofic shift:

(Sect. “Shifts of Finite Type and Sofic Shifts”) A shift space which is a factor of a shift of finite type; equivalently, a set of bi‐infinite sequences determined by a finite directed labeled graph.

State splitting:

(Sect. “The Conjugacy Problem”) A splitting of states in a finite directed graph that creates a new graph, whose vertices are the split states. The operation that creates the new graph from the original graph is a basic building block for all topological conjugacies between shifts of finite type.

Strong shift equivalence:

(Sect. “The Conjugacy Problem”) An equivalence relation on defining matrices for shifts of finite type. In principle, this relation characterizes the corresponding shifts of finite type, up to topological conjugacy.

Topological conjugacy:

(Sect. “Shift Spaces and Sliding Block Codes”) A bijective sliding block code from one shift space to another; equivalently, a shift‐commuting homeomorphism from one shift space to another. Sometimes called conjugacy.

Topological entropy:

(Sect. “Entropy and Periodic Points”) The asymptotic growth rate of the number of finite sequences of given length in a shift space (as the length goes to infinity).

Zeta function:

(Sect. “Entropy and Periodic Points”) An expression for the number of periodic points of each given period in a shift space.

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  1. Department of Mathematics, University of British Columbia, Vancouver, Canada

    Brian Marcus

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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Marcus, B. (2012). Symbolic Dynamics. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_108

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