Dynamical properties of the tent map

where β = β(β−1)−1, M ∈ N∪{∞}, and l0, l1, . . . is a uniquely determined (finite or infinite) sequence of non-negative integers (cf. [2]). Analogously to other types of expansions we may ask for bases β such that the set of numbers with periodic or even finite representation is as large as possible. In particular, we say that Tβ satisfies the periodicity property (P) if the orbit (T β (x))n≥1 is eventually periodic for all x ∈ Q(β) ∩ [0, 1], and Tβ has the finiteness property (F) whenever the Tβ-orbit of each element of Q(β) ∩ [0, 1] contains 0. The principle intention of the talk is to present result concerning (P) and (F). At the beginning we summarise several well-known facts published by Lagarias et al. in [2, 3]. The focus is on a recently discovered connection between tent maps and generalised beta-transformations that allows us to transfer results concerning periodicity and finiteness properties (see [4]). Define the set