Tohoku Mathematical Journal
2025
September
SECOND SERIES VOL. 77, NO. 3

Tohoku Math. J.
77 (2025), 319-343

Title MATCHING OF ORBITS OF CERTAIN $N$-EXPANSIONS WITH A FINITE SET OF DIGITS

Author Yufei Chen and Cor Kraaikamp

(Received January 18, 2023, revised August 1, 2023)
Abstract. In this paper we consider a class of continued fraction expansions: the so-called $N$-expansions with a finite digit set, where $N\geq 2$ is an integer. These $N$-expansions with a finite digit set were introduced in [10, 23], and further studied in [13, 15]. For $N$ fixed they are steered by a parameter $\alpha\in (0,\sqrt{N}-1]$. In [13], 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 $N\in{\mathbb N}$, $N\geq 2$, 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 $\alpha,\alpha'$ 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.

Mathematics Subject Classification. Primary 11J70; Secondary 37E05.
Key words and phrases. Continued fractions, dynamical systems, gaps, matching.

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