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

Abstract

In this paper we consider a class of continued fraction expansions: the so-called $N$-expansions with a finite digit set, where $N\ge 2$ is an integer. These \emph{$N$-expansions with a finite digit set} were introduced in [KL,L], and further studied in [dJKN,S]. For $N$ fixed they are steered by a parameter $\alpha \in (0,\sqrt{N}-1]$. In [KL], for $N=2$ an explicit interval $[A,B]$ was determined, such that for all $\alpha \in [A,B]$ the entropy $h(T_{\alpha })$ of the underlying Gauss-map $T_{\alpha }$ is equal. In this paper we show that for all $N\in \mathbb{N}$, $N\ge 2$, 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 $T_{\alpha }$, the $T_{\alpha }$-invariant measure, ergodicity, and we show that for any two $\alpha ,{\alpha }^{\prime }$ 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.