Symbolic dynamics

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In mathematics, symbolic dynamics is the practice of modelling a dynamical system by a space consisting of infinite sequences of abstract symbols, each symbol corresponding to a state of the system, and a shift operator corresponding to the dynamics. Symbolic dynamics were first introduced by Emil Artin in 1924, in the study of Artin billiards.

Symbolic dynamics originated as a method to study general dynamical systems; now its techniques and ideas have found significant applications in data storage and transmission, linear algebra, 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 sequence 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.

A formal definition of the symbolic dynamics of a dynamical system is given in the article measurdynamical system. See also shift of finite type.

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