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  • Cited by 2
  • Print publication year: 2003
  • Online publication date: August 2009

Symbolic dynamics and topological models in dimensions 1 and 2



Symbolic dynamics has a long and distinguished history, going back to Hadamard's work on the geodesic flow on negatively curved surfaces. Because of its success in describing the dynamics of systems with Markov partitions, it is most closely associated with the study of such systems. However, in the 1970's, beginning with the work of Metropolis, Stein and Stein and culminating with the kneading theory of Milnor and Thurston, symbolic dynamics was used in a quite different way in the study of 1-dimensional discrete dynamics: the partition was fixed (given by the critical points) and the maps were allowed to vary in families, as long as the critical points remained the same. This is fundamentally different from the use of symbolic dynamics in the presence of Markov partitions. Not only does it not require the maps being studied to have such partitions, but also, by fixing the partition, it permits the description of all maps in a family in terms of the same symbols, thus allowing the comparison of different maps.

One of the conclusions of kneading theory is that every family of unimodal maps (i.e. continuous piecewise monotone self-maps of the unit interval with exactly two monotone pieces) presents essentially the same dynamical behaviour as it passes from trivial to chaotic dynamics.

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