Matching of orbits of certain N-expansions with a finite set of digits
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Abstract
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(Tα) of the underlying Gauss-map Tα 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.