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 $O^-(x)=\{y:x=F^ny\text {\ for\ some\ }n\ge 0\}$ 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 of PTT 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 perfect Polish spaces. The converse is false.