Parry's topological transitivity and f-expansions

Abstract

In his 1964 paper on f-expansions, Parry studied piecewise-continuous, piecewise-monotonic maps F of the interval [0,1), and introduced a notion of topological transitivity different from any of the modern definitions. This notion, which we call Parry topological transitivity, (PTT) is that the backward orbit O^-(x)={y:x=F^ny for some n\ge 0} of some x\in[0,1) is dense. We take topological transitivity (TT) to mean that some $x$ has a dense forward orbit. Parry's application to f-expansions is that PTT implies the partition of [0,1) into the "fibers" of F is a generating partition (i.e., f-expansions are "valid"). We prove the same result for TT, and use this to show that for interval maps F, TT implies PTT. A separate proof is provided for continuous maps F of compact metric spaces. The converse is false.