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∈[[Formula presented](5−1),[Formula presented](5+1)], we show that every irrational number x∈I _α =[α−2,α) can be uniquely represented as with e _i (x;α)∈{±1} and d _i (x;α)∈2N−1 determined by the iterates of the transformation φ _α (x):=[Formula presented]−2[[Formula presented]+[Formula presented]]−1 of I _α . We also describe the natural extension of φ _α and prove that the endomorphism φ _α is exact.