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In mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (evolution) given by the shift operator. 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's 1898 paper on the geodesics on surfaces of negative curvature. It was applied by Marston Morse in 1921 to the construction of a nonperiodic recurrent geodesic. Related work was done by Emil Artin in 1924 (for the system now called Artin billiard), Pekka Myrberg, Paul Koebe, Jakob Nielsen, 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 equations. Claude Shannon used symbolic sequences and shifts of finite type in his 1948 paper ''A mathematical theory of communication'' that gave birth to information theory. The theory was further advanced in the 1960s and 1970s, notably, in the works of Steve Smale and his school, and of Yakov Sinai and the Soviet school of ergodic theory. A spectacular application of the methods of symbolic dynamics is Sharkovskii's theorem about periodic orbits of a continuous map of an interval into itself (1964). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Symbolic dynamics」の詳細全文を読む スポンサード リンク
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