Parry's topological transitivity and \(f\)-expansions (Q2790925)

A map \(F: X \rightarrow X\) is topologically transitive (TT) if some \(x \in X\) has a dense forward orbit. \(F\) may be called Parry topologically transitive (PTT) if some \(x \in X\) has a dense backward orbit. It was proved for piecewise continuous, piecewise monotone interval maps that PTT implies \(F\)-representations are valid by \textit{W. Parry} [Acta Math. Acad. Sci. Hung. 15, 95--105 (1964; Zbl 0136.35104)]. In the paper reviewed here, the author proves for the same class of interval maps that TT implies \(F\)-representations are valid. He also shows that TT implies PTT for piecewise interval maps and for continuous maps on perfect Polish spaces. The author points out that \textit{A. Nagar} et al. [Real Anal. Exch. 27, No. 1, 325--334 (2002; Zbl 1015.37030)] proved for a continuous map \(F: \mathbb{R} \rightarrow \mathbb{R}\) that TT implies PTT.