A Renormalization Scheme for Semi-Regular Continued Fractions

TL;DR

This paper introduces a renormalization scheme with two maps, T_slow and T_fast, to systematically generate and analyze semi-regular continued fractions, revealing ergodic properties and symbolic codings.

Contribution

It develops a novel renormalization framework with ergodic analysis for semi-regular continued fractions, including explicit generation methods for specific algorithms.

Findings

T_fast is ergodic with respect to an absolutely continuous measure.

T_slow preserves an infinite, sigma-finite measure.

The scheme enables explicit generation of semi-regular continued fractions for specific algorithms.

Abstract

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.