Matching of orbits of certain N-expansions with a finite set of digits

In this paper we consider a class of continued fraction expansions: the so-called N-expansions with a finite digit set, where N ≥ 2 is an integer. These N-expansions with a finite digit set were introduced in [13, 15], and further studied in [10, 23]. For N fixed they are steered by a parameter α ∈ (0, √N − 1]. In [13], for N = 2 an explicit interval [ A, B] was determined, such that for all α ∈ [ A, B] the entropy ℎ(T_α) of the underlying Gauss-map T_α is equal. In this paper we show that for all N ∈ N, N ≥ 2, such plateaux exist. In order to show that the entropy is constant on such plateaux, we obtain the underlying planar natural extension of the maps Tα, the Tα-invariant measure, ergodicity, and we show that for any two α, α^′ from the same plateau, the natural extensions are metrically isomorphic, and the isomorphism is given explicitly. The plateaux are found by a property called matching.