mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, symbolic dynamics is the practice of modeling a topological or smooth

dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...

by a discrete space consisting of infinite

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...

s of abstract symbols, each of which corresponds to a

state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States * ''Our S ...

of the system, with the dynamics (evolution) given by the

shift operator In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the lag operator. Shift ...

. Formally, a Markov partition is used to provide a finite cover for the smooth system; each set of the cover is associated with a single symbol, and the sequences of symbols result as a trajectory of the system moves from one covering set to another.

History

The idea goes back to

Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. Biography The son of a teac ...

's 1898 paper on the

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connecti ...

s on

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

s of negative

curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...

. It was applied by Marston Morse in 1921 to the construction of a nonperiodic recurrent geodesic. Related work was done by

Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing l ...

in 1924 (for the system now called Artin billiard), Pekka Myrberg, Paul Koebe,

Jakob Nielsen Jacob or Jakob Nielsen may refer to: * Jacob Nielsen, Count of Halland (died c. 1309), great grandson of Valdemar II of Denmark * , Norway (1768-1822) * Jakob Nielsen (mathematician) (1890–1959), Danish mathematician known for work on automorphis ...

, G. A. Hedlund. The first formal treatment was developed by Morse and Hedlund in their 1938 paper. George Birkhoff, Norman Levinson and the pair Mary Cartwright and J. E. Littlewood have applied similar methods to qualitative analysis of nonautonomous second order

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...

Claude Shannon Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American mathematician, electrical engineer, and cryptographer known as a "father of information theory". As a 21-year-old master's degree student at the Massachusetts I ...

used symbolic sequences and shifts of finite type in his 1948 paper ''

A mathematical theory of communication "A Mathematical Theory of Communication" is an article by mathematician Claude E. Shannon published in ''Bell System Technical Journal'' in 1948. It was renamed ''The Mathematical Theory of Communication'' in the 1949 book of the same name, a sma ...

'' that gave birth to

information theory Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. ...

. During the late 1960s the method of symbolic dynamics was developed to hyperbolic toral automorphisms by

Roy Adler Roy Lee Adler (February 22, 1931 – July 26, 2016) was an American mathematician. Adler earned his Ph.D. in 1961 from Yale University under the supervision of Shizuo Kakutani (''On some algebraic aspects of measure preserving transformations'') ...

and

Benjamin Weiss Benjamin Weiss ( he, בנימין ווייס; born 1941) is an American-Israeli mathematician known for his contributions to ergodic theory, topological dynamics, probability theory, game theory, and descriptive set theory. Biography Benjamin ...

, and to

Anosov diffeomorphism In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "cont ...

s by Yakov Sinai who used the symbolic model to construct Gibbs measures. In the early 1970s the theory was extended to Anosov flows by

Marina Ratner Marina Evseevna Ratner (russian: Мари́на Евсе́евна Ра́тнер; October 30, 1938 – July 7, 2017) was a professor of mathematics at the University of California, Berkeley who worked in ergodic theory. Around 1990, she proved ...

, and to

Axiom A In mathematics, Smale's axiom A defines a class of dynamical systems which have been extensively studied and whose dynamics is relatively well understood. A prominent example is the Smale horseshoe map. The term "axiom A" originates with Stephen Sm ...

diffeomorphisms and flows by

Rufus Bowen Robert Edward "Rufus" Bowen (23 February 1947 – 30 July 1978) was an internationally known professor in the Department of Mathematics at the University of California, Berkeley, who specialized in dynamical systems theory. Bowen's work dealt p ...

. A spectacular application of the methods of symbolic dynamics is Sharkovskii's theorem about

periodic orbit In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Iterated functions Given a ...

s of a

continuous map In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...

of an interval into itself (1964).

Examples

Concepts such as heteroclinic orbits and

homoclinic orbit In mathematics, a homoclinic orbit is a trajectory of a flow of a dynamical system which joins a saddle equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of the stable manifold and the unstable manifold of ...

s have a particularly simple representation in symbolic dynamics.

Itinerary

Itinerary of point with respect to the partition is a sequence of symbols. It describes dynamic of the point. Mathematics of Complexity and Dynamical Systems by Robert A. Meyers. Springer Science & Business Media, 2011, , 9781461418054

Applications

Symbolic dynamics originated as a method to study general dynamical systems; now its techniques and ideas have found significant applications in

data storage Data storage is the recording (storing) of information (data) in a storage medium. Handwriting, phonographic recording, magnetic tape, and optical discs are all examples of storage media. Biological molecules such as RNA and DNA are consi ...

and

transmission Transmission may refer to: Medicine, science and technology * Power transmission ** Electric power transmission ** Propulsion transmission, technology allowing controlled application of power *** Automatic transmission *** Manual transmission ** ...

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...

, the motions of the planets and many other areas. The distinct feature in symbolic dynamics is that time is measured in '' discrete'' intervals. So at each time interval the system is in a particular ''state''. Each state is associated with a symbol and the evolution of the system is described by an infinite

of symbols—represented effectively as strings. If the system states are not inherently discrete, then the state vector must be discretized, so as to get a

coarse-grained Granularity (also called graininess), the condition of existing in granules or grains, refers to the extent to which a material or system is composed of distinguishable pieces. It can either refer to the extent to which a larger entity is sub ...

description of the system.

References

External links

ChaosBook.org
Chapter "Transition graphs" Dynamical systems Combinatorics on words