$\alpha$-Expansions with odd partial quotients

TL;DR

This paper introduces an analogue of Nakada's $eta$-continued fractions using odd partial quotients, establishing unique representations, analyzing the transformation's properties, and describing its natural extension.

Contribution

It develops a new continued fraction framework with odd partial quotients, proves unique representations, and analyzes the transformation's ergodic properties.

Findings

Unique representation of irrationals in the specified interval.

Description of the natural extension of the transformation.

Proof that the transformation is exact.

Abstract

We consider an analogue of Nakada's $\alpha$-continued fraction transformation in the setting of continued fractions with odd partial quotients. More precisely, given $\alpha \in [\frac{1}{2}(\sqrt{5}-1),\frac{1}{2}(\sqrt{5}+1)]$, we show that every irrational number $x\in I_\alpha=[\alpha-2,\alpha)$ can be uniquely represented as $$ x= \cfrac{e_1 (x;\alpha)}{d_1 (x;\alpha) +\cfrac{e_2(x;\alpha)}{d_2(x;\alpha)+\cdots}} , $$ with $e_i(x;\alpha) \in \{ \pm 1\}$ and $d_i(x;\alpha) \in 2{\mathbb N} -1$ determined by the iterates of the transformation $$\varphi_\alpha (x) := \frac{1}{| x|} - 2 \bigg[ \frac{1}{2| x|} +\frac{1-\alpha}{2} \bigg]-1$$ of $I_\alpha$. We also describe the natural extension of $\varphi_\alpha$ and prove that the endomorphism $\varphi_\alpha$ is exact.