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, \(\hat{T}_{slow}\) ${\widehat{T}}_{\mathit{slow}}$ and \(\hat{T}_{fast}\) ${\widehat{T}}_{\mathit{fast}}$ : these maps are defined for \((x,y)\in [0,1]\) $(x,y)\in [0,1]$ , where x is the number for which a semi- regular continued fraction representation is developed by \(\hat{T}_{slow}\) ${\widehat{T}}_{\mathit{slow}}$ 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 \(\hat{T}_{fast}\) ${\widehat{T}}_{\mathit{fast}}$ is a “sped up" version of the map \(\hat{T}_{slow}\) ${\widehat{T}}_{\mathit{slow}}$ , and we show that \(\hat{T}_{fast}\) ${\widehat{T}}_{\mathit{fast}}$ is ergodic with respect to a probability measure which is mutually absolutely continuous with Lebesgue measure. In contrast, \(\hat{T}_{slow}\) ${\widehat{T}}_{\mathit{slow}}$ preserves no such measure, but does preserve an infinite, \(\sigma \) $\sigma$ -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 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 \(\alpha \) $\alpha$ -continued fractions.