$\alpha$-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 $$x={\displaystyle \frac{e_1(x;\alpha )}{d_1(x;\alpha )+{\displaystyle \frac{e_2(x;\alpha )}{d_2(x;\alpha )+\cdots }}}},$$ 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 $${\phi }_{\alpha }(x):=\frac{1}{\mathrm{\mid }x\mathrm{\mid }}-2[\frac{1}{2\mathrm{\mid }x\mathrm{\mid }}+\frac{1-\alpha }{2}]-1$$ of $I_{\alpha }$. We also describe the natural extension of ${\phi }_{\alpha }$ and prove that the endomorphism ${\phi }_{\alpha }$ is exact.