In 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 ...

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 tea ...

'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 connection. ...

s on surfaces 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 ...

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 automorphi ...

, 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, a ...

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 In ...

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 and Benjamin Weiss, and to Anosov diffeomorphisms 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, 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. A spectacular application of the methods of symbolic dynamics is Sharkovskii's theorem about periodic orbits 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 va ...

of an interval into itself (1964).

Examples

Concepts such as

heteroclinic orbit In mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit (sometimes called a heteroclinic connection) is a path in phase space which joins two different equilibrium points. If the equilibrium points at the start and end ...

s 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 o ...

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 cons ...

and 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 matric ...

, 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 description of the system.

References

External links

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