Natural extensions and entropy of $ \alpha $-continued fraction expansion maps with odd partial quotients

In [1], Boca and the fourth author of this paper introduced a new class of continued fraction expansions with odd partial quotients, parameterized by a parameter $ \alpha\in [g, G] $, where $ g = \tfrac{1}{2}(\sqrt{5}-1) $ and $ G = g+1 = 1/g $ are the two golden mean numbers. Using operations called singularizations and insertions on the partial quotients of the odd continued fraction expansions under consideration, the natural extensions from [1] are obtained, and it is shown that for each $ \alpha, \alpha^*\in [g, G] $ the natural extensions from [1] are metrically isomorphic. An immediate consequence of this is, that the entropy of all these natural extensions is equal for $ \alpha\in [g, G] $, a fact already observed in [1]. Furthermore, it is shown that this approach can be extended to values of $ \alpha $ smaller than $ g $, and that for values of $ \alpha \in [\tfrac{1}{6}(\sqrt{13}-1), g] $ all natural extensions are still isomorphic. In the final section of this paper further attention is given to the entropy, as function of $ \alpha\in [0, G] $. It is shown that if there exists an ergodic, absolutely continuous $ T_{\alpha} $-invariant measure, in any neighborhood of $ 0 $ we can find intervals on which the entropy is decreasing, intervals on which the entropy is increasing and intervals on which the entropy is constant. Moreover, we identify the largest interval on which the entropy is constant. In order to prove this we use a phenomenon called matching.