A renormalization scheme for semi-regular continued fractions

Abstract

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