A renormalization scheme for semi-regular continued fractions

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, T̂slow T ^ slow and T̂fast T ^ fast : these maps are defined for (x,y)∈ [0,1] ( x , y ) ∈ [ 0 , 1 ] , where x is the number for which a semi- regular continued fraction representation is developed by T̂slow 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 T̂fast T ^ fast is a “sped up" version of the map T̂slow T ^ slow , and we show that T̂fast T ^ fast is ergodic with respect to a probability measure which is mutually absolutely continuous with Lebesgue measure. In contrast, T̂slow T ^ slow preserves no such measure, but does preserve an infinite, σ σ -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 under rotation by x modulo one. 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 α α -continued fractions.