Generalized one-sided shifts and maps of the interval

Abstract

The author considers a generalization of the one-sided shift, suitable for describing a certain class of maps in the interval that preserve a Cantor set. The author shows that if such a map is single-valued, it has a finite Markov partition (i.e. that its symbolic dynamics is regular), but if it is multiple-valued, its symbolic dynamics can be an arbitrary context-free language. The scaling properties of sets corresponding to such languages are discussed, and an example is given where the semigroup of scaling operations is infinite-dimensional but finitely describable. The author also discusses the problems of embedding a computationally complex process in one dimension.