PARRY'S TOPOLOGICAL TRANSITIVITY AND f-EXPANSIONS

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(n)y for some n >= 0} of some x is an element of [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.