Induced random β-transformation

We study the first return map defined on the switch region induced by the greedy map and the lazy map. In particular we study the allowable sequences of return times, and when the first return map is a generalised Lüroth series transformation. We show that there exists a countable collection (In)∞n=1 of disjoint intervals such that all sequences of return times are permissible if and only if β∈In for some n. Moreover, we show that there exists a set M⊆(1,2) of Hausdorff dimension 1 and Lebesgue measure zero for which the first return map is a generalised Lüroth series transformation if and only if β∈M.