$\alpha$-Expansions with odd partial quotients

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

a few typos corrected in the published version