Symbolic dynamics arises naturally in areas as diverse as applied mathematics, number theory, combinatorics, probability, ergodic theory, coding and information theory. A plethora of motivating examples and numerous connections to other fields make symbolic dynamics an especially attractive and suitable medium for involving students at both the undergraduate and graduate level, thereby providing an excellent opportunity for achieving vertical integration and broadening education.

The term symbolic dynamics refers to dynamical systems on spaces of sequences where the dynamics is generated by a particular map, namely, the shift. These systems are readily introduced (any shift-invariant set of sequences from a fixed alphabet is a symbolical dynamical system!) and can be used to motivate, illustrate and study various key concepts in dynamical systems such as ergodicity, entropy, invariant measures and many others. They also contribute new tools to other areas. The following, by far not exhaustive, list of applications of symbolic dynamics amply demonstrates its ubiquity and potential:

• Hyperbolic diffeomorphisms, geodesic flows and billiards [2,9]
• Data storage and transmission in coding and information theory [6]
• Perron-Frobenius theory of nonnegative matrices [1]
• Ramsey theory and additive number theory via ergodic theory [3]
• Markov chains and probability [5]
• Mathematical linguistics [7]
• Chaos, Smale's horseshoe, Sharkovskii's theorem [8]
• Homoclinic and heteroclinic cycles in ODEs [4]
• Applications to rotating convection rolls in Rayleigh-Benard convection and forced oscillations of elastic beams in magnetic fields [4].

References:

[1] M. Boyle and D. Handelman.
    The spectra of nonnegative matrices via symbolic dynamics.
    Ann. Math. 133 (1991) 249-316.
[2] L.A. Bunimovich and Y.G. Sinai.
    Markov partitions for dispersed billiards. 
    Comm. Math. Phys. 73 (1980) 247-280.
[3] H. Furstenberg.
    Recurrence in ergodic theory and combinatorial number theory.
    Princeton University Press, 1981.
[4] J. Guckenheimer and P. Holmes. 
    Nonlinear oscillations, dynamical systems, and bifurcations of vector fields.
    Springer, New York, 1983.
[5] B. Kitchens.
    Symbolic Dynamics. Springer, 1998.
[6] D. Lind and B. Marcus.
    Symbolic Dynamics and Coding. Cambridge University Press, 1995.
[7] M. Lothaire.
    Combinatorics on Words. Cambridge University Press, 1983.
[8] J. Palis and F. Takens.
    Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations.
    Cambridge University Press, 1993.
[9] C. Series.
    Symbolic dynamics for geodesic flows.
    Acta Math. 146 (1981) 103-128.