α-Expansions with odd partial quotients

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∈[12(5−1),12(5+1)], we show that every irrational number x∈Iα=[α−2,α) can be uniquely represented as with ei(x;α)∈{±1} and di(x;α)∈2N−1 determined by the iterates of the transformationφα(x):=1|x|−2[12|x|+1−α2]−1 of Iα. We also describe the natural extension of φα and prove that the endomorphism φα is exact.