α-Expansions with odd partial quotients

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.