A renormalization scheme for semi-regular continued fractions
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In: Monatshefte für Mathematik, Vol. ??? Stand: 12. November 2024, 30.10.2024.
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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: 12. November 2024
JO - Monatshefte für Mathematik
JF - Monatshefte für Mathematik
SN - 0026-9255
ER -