  In an article (which we will refer to as [BM]) of Boca and the fourth author
of this paper, 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 is
introduced. In this article, by using operations called singularizations and
insertions on the partial quotients of the odd continued fraction expansions
under consideration, the natural extensions from [BM] are obtained, and it is
shown that for each $\alpha,\alpha^*\in [g,G]$ the natural extensions from [BM]
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 [BM]. 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 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. In order to prove this we use a phenomena called matching.
