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In this paper we consider a class of continued fraction expansions: theso-called $N$-expansions with a finite digit set, where $N\geq 2$ is aninteger. 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 aparameter $\alpha\in (0,\sqrt{N}-1]$. 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 thispaper we show that for all $N\in \mathbb N$, $N\geq 2$, such plateaux exist. Inorder to show that the entropy is constant on such plateaux, we obtain theunderlying planar natural extension of the maps $T_{\alpha}$, the$T_{\alpha}$-invariant measure, ergodicity, and we show that for any two$\alpha,\alpha'$ from the same plateau, the natural extensions are metricallyisomorphic, and the isomorphism is given explicitly. The plateaux are found bya property called matching.

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