Symbolic Representations of Iterated Maps

Xin-Chu Fu; Weiping Lu; Peter Ashwin; Jinqiao Duan

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arXiv · nlin/0012007

Symbolic Representations of Iterated Maps

Xin-Chu Fu·Weiping Lu·Peter Ashwin·Jinqiao Duan

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Abstract

This paper presents a general and systematic discussion of various symbolic representations of iterated maps through subshifts. We give a unified model for all continuous maps on a metric space, by representing a map through a general subshift over usually an uncountable alphabet. It is shown that at most the second order representation is enough for a continuous map. In particular, it is shown that the dynamics of one-dimensional continuous maps to a great extent can be transformed to the study of subshift structure of a general symbolic dynamics system. By introducing distillations, partial representations of some general continuous maps are obtained. Finally, partitions and representations of a class of discontinuous maps, piecewise continuous maps are discussed, and as examples, a representation of the Gauss map via a full shift over a countable alphabet and representations of interval exchange transformations as subshifts of infinite type are given.

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Xin-Chu Fu, Weiping Lu, Peter Ashwin, Jinqiao Duan. 2000-12-05. Symbolic Representations of Iterated Maps. https://arxiv.org/abs/nlin/0012007

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