$α$-expansions with odd partial quotients

Florin P. Boca, Claire Merriman

Published 2018-06-16Version 1

We consider an analogue of Nakada's $\alpha$-continued fraction transformations 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 $x\in [\alpha-2,\alpha)$ can be uniquely represented as $ x=\frac{e_1}{d_1+\frac{e_2}{d_2+\frac{e_3}{d_3+\cdots}}} $, with $e_i =e_i(x;\alpha) \in \{ \pm 1\}$ and $d_i =d_i(x;\alpha) \in 2{\mathbb N} -1$ determined by the iterates of the transformation $\varphi_\alpha (x) := \frac{1}{| x|} - 2 \left[ \frac{1}{2| x|} +\frac{1-\alpha}{2} \right]-1$ of $[\alpha -2,\alpha)$. We also describe the natural extension of $\varphi_\alpha$ and prove that the endomorphism $\varphi_\alpha$ is exact.