2023/02/02 by Niels Langeveld, David K. Ralston, Langeveld, Niels +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #Advanced Differential Equations and Dynamical Systems #Caveolin-1 and cellular processes #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2302.00945
openalex publication_date 2023/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, Tslow and Tfast: 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 Tslow 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 Tfast is a "sped-up" version of the map Tslow, and we show that Tfast is ergodic with respect to a probability measure which is mutually absolutely continuous with Lebesgue measure. In contrast, Tslow 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.