  It has been more than twenty years since Moshe Newman, based on work by Neil
Calkin and Herbert Wilf, introduced an explicit bijection between the rational
and natural numbers. Interestingly, this bijection is dynamic in nature.
Indeed, Newman's map has the property that the orbit of zero provides the
required bijection. Claudio Bonanno and Stefano Isola, using continued
fractions expansions, described the dynamics of its first return time map T.
They proved that it is topologically conjugated to the dyadic odometer. In this
article, we prove that the correct numerical system needed to analyze this map
is the backward continued fractions. Indeed, this approach has the advantage
that it provides explicitly the action of T on the expansion. As a by-product,
we naturally obtain an explicit formula for Minkowski's question mark function
in terms of backward continued fractions. The whole point of Newman was to
provide an explicit bijection, our approach shares the same taste for the
explicit.
