Abstract
This chapter is an introduction to symbolic dynamics, with emphasis on its relations to hyperbolic dynamics. In particular, it is sometimes easier to solve certain problems of hyperbolic dynamics, such as those concerning periodic points, after associating a symbolic dynamics (also called a coding) to a hyperbolic set. After introducing some basic notions of symbolic dynamics, we illustrate with several examples how one can associate naturally a coding to several dynamical systems considered in the former chapters. These include expanding maps, quadratic maps and the Smale horseshoe. We also consider topological Markov chains, and we study their periodic points, topological entropy, and recurrence properties. Finally, we consider briefly the notion of the zeta function of a dynamical system.
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References
Dieck, T.: Algebraic Topology. Textbooks in Mathematics. Eur. Math. Soc., Zürich (2008)
Munkres, J.: Topology: A First Course. Prentice-Hall, New York (1975)
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© 2013 Springer-Verlag London
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Barreira, L., Valls, C. (2013). Symbolic Dynamics. In: Dynamical Systems. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-4835-7_7
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DOI: https://doi.org/10.1007/978-1-4471-4835-7_7
Publisher Name: Springer, London
Print ISBN: 978-1-4471-4834-0
Online ISBN: 978-1-4471-4835-7
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