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 is an element of [1/2 (root 5-1), 1/2(root 5+1)], we show that every irrational number x is an element of I-alpha = [alpha - 2, alpha) can be uniquely represented as x = e(1) (x; alpha)vertical bar/vertical bar d(1) (x; alpha) + e(2) (x; alpha)vertical bar/vertical bar d(2)(x; alpha) + e(3) (x; alpha)vertical bar/vertical bar d(3) (x; alpha) + ... , with e(i) (x; alpha) is an element of {+/- 1} and d(i) (x; alpha) is an element of 2 N - 1 determined by the iterates of the transformation phi(alpha)(x) := 1/vertical bar x vertical bar - 2[1/2 vertical bar x vertical bar + 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. (C) 2018 Elsevier Inc. All rights reserved.