Parry's topological transitivity and f-expansions

TL;DR

This paper explores Parry's unique notion of topological transitivity for interval maps, demonstrating its implications for f-expansions and establishing relationships between different forms of transitivity.

Contribution

The paper proves that topological transitivity implies Parry topological transitivity for interval maps, extending Parry's original results and clarifying their relationship.

Findings

TT implies PTT for interval maps

F-expansions are valid when PTT holds

The converse implication is false

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.