- Symbolic dynamics
In

mathematics ,**symbolic dynamics**is the practice of modelling a topological or smoothdynamical system by a discrete space consisting of infinitesequence s of abstract symbols, each of which corresponds to a state of the system, with the dynamics (evolution) given by theshift operator .**History**The idea goes back to

Jacques Hadamard 's 1898 paper on thegeodesic s onsurface s of negativecurvature . It was applied byMarston Morse in 1921 to the construction of a nonperiodic recurrent geodesic. Related work was done byEmil Artin in 1924 (for the system now calledArtin billiard ), P. J. 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 M. L. Cartwright–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 toinformation theory .The theory was further advanced in the 1960s and 1970s, notably, in the works of

Steve Smale and his school, and ofYakov Sinai and the Soviet school ofergodic theory . A spectacular application of the methods of symbolic dynamics isSharkovskii's theorem aboutperiodic orbit s of a continuous map of an interval into itself (1964).**Applications**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 infinitesequence of symbols — represented effectively as strings. If the system states are not inherently discrete, then thestate vector must be discretized, so as to get acoarse-grained description of the system.**ee also***

Measure-preserving dynamical system

*Shift space

*Shift of finite type

*Markov partition **Further reading*** Bruce Kitchens, "Symbolic dynamics. One-sided, two-sided and countable state Markov shifts". Universitext,

Springer-Verlag , Berlin, 1998. x+252 pp. ISBN 3-540-62738-3 MathSciNet|id=1484730

* Douglas Lind and Brian Marcus, " [*http://www.math.washington.edu/SymbolicDynamics/ An Introduction to Symbolic Dynamics and Coding*] ".Cambridge University Press , Cambridge, 1995. xvi+495 pp. ISBN 0-521-55124-2 MathSciNet|id=1369092

* M. Morse andG. A. Hedlund , "Symbolic Dynamics", American Journal of Mathematics, 60 (1938) 815–866

* G. A. Hedlund, " [*http://www.springerlink.com/content/k62915l862l30377/ Endomorphisms and automorphisms of the shift dynamical system*] ". Math. Systems Theory, Vol. 3, No. 4 (1969) 320–3751

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