Xin-Chu Fu, Weiping Lu, Peter Ashwin and Jinqiao Duan
Symbolic Representations of Iterated Maps
(76K, Latex)
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.