TY - JOUR

T1 - A renormalization scheme for semi-regular continued fractions

AU - Langeveld, Niels

AU - Ralston, David

N1 - Publisher Copyright: © The Author(s) 2024.

PY - 2024/10/30

Y1 - 2024/10/30

N2 - 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)∈[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, 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-continued fractions.

AB - 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)∈[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, 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-continued fractions.

KW - 11A55

KW - 11J70

KW - 37A44

KW - 37B10

KW - 37E05

KW - Approximating sequences

KW - Circle rotations

KW - Renormalization

KW - Semi-regular continued fractions

KW - Symbolic encodings

UR - http://www.scopus.com/inward/record.url?scp=85207931702&partnerID=8YFLogxK

U2 - 10.1007/s00605-024-02031-4

DO - 10.1007/s00605-024-02031-4

M3 - Article

VL - ??? Stand: 6. Dezember 2024

JO - Monatshefte für Mathematik

JF - Monatshefte für Mathematik

SN - 0026-9255

IS - ??? Stand: 6. Dezember 2024

ER -