Symbolic dynamics

Symbolic dynamics

In mathematics, symbolic dynamics is the practice of modelling 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.


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), 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 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).


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.

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, " [ 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 and G. A. Hedlund, "Symbolic Dynamics", American Journal of Mathematics, 60 (1938) 815–866
* G. A. Hedlund, " [ Endomorphisms and automorphisms of the shift dynamical system] ". Math. Systems Theory, Vol. 3, No. 4 (1969) 320–3751

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