$α$-Expansions with odd partial quotients

Abstract

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