  In this article we study a renormalization scheme with which we find all
semi-regular continued fractions of a number in a natural way. We define two
maps, T_slow and T_fast: these maps are defined for (x,y) in [0,1], where x is
the number for which a semi-regular continued fraction representation is
developed by T_slow according to the parameter y. The set of all possible
semi-regular continued fraction representations of x are bijectively
constructed as the parameter y varies. The map T_fast is a "sped-up" version of
the map T_slow, and we show that T_fast is ergodic with respect to a
probability measure which is mutually absolutely continuous with Lebesgue
measure. In contrast, T_slow preserves no such measure, but does preserve an
infinite, sigma-finite measure mutually absolutely continuous with Lebesgue
measure. Furthermore, we generate a sequence of substitutions which generate a
symbolic coding of the orbit of y. In the last section we highlight how our
scheme can be used to generate semi-regular continued fractions explicitly for
specific continued fraction algorithms such as Nakada's alpha continued
fractions.
