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

TL;DR

This paper investigates the entropy behavior of a class of N-expansions with finite digit sets, demonstrating the existence of entropy plateaux across all N, and explicitly constructing the natural extensions and isomorphisms for parameters within these plateaux.

Contribution

It generalizes the existence of entropy plateaux for N-expansions to all N ≥ 2 and explicitly constructs natural extensions and isomorphisms for parameters in these plateaux.

Findings

Existence of entropy plateaux for all N ≥ 2.

Explicit construction of natural extensions and invariant measures.

Isomorphism between natural extensions within the same plateau.

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 \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 $N\in \mathbb N$, $N\geq 2$, 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.