Symbolic Representations of Iterated Maps
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.
Cite
@article{arxiv.nlin/0012007,
title = {Symbolic Representations of Iterated Maps},
author = {Xin-Chu Fu and Weiping Lu and Peter Ashwin and Jinqiao Duan},
journal= {arXiv preprint arXiv:nlin/0012007},
year = {2007}
}