Generalizations of Sturmian sequences associated with N-continued fraction algorithms

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Generalizations of Sturmian sequences associated with N-continued fraction algorithms. / Langeveld, Niels; Rossi, Lucia; Thuswaldner, Jörg.
in: Journal of number theory, Jahrgang 250.2023, Nr. September, 09.2023, S. 49-83.

Publikationen: Beitrag in FachzeitschriftArtikelForschung(peer-reviewed)

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@article{7429d444a3e64ab7abac5e1cb410bfee,
title = "Generalizations of Sturmian sequences associated with N-continued fraction algorithms",
abstract = "Given a positive integer N and x ∈ [0, 1] \ Q, an N-continuedfraction expansion of x is defined analogously to the classicalcontinued fraction expansion, but with the numerators beingall equal to N. Inspired by Sturmian sequences, we introducethe N-continued fraction sequences ω(x, N) and ω(x, N),which are related to the N-continued fraction expansion ofx. They are infinite words over a two letter alphabet obtainedas the limit of a directive sequence of certain substitutions,hence they are S-adic sequences. When N = 1, we are in thecase of the classical continued fraction algorithm, and obtainthe well-known Sturmian sequences. We show that ω(x, N)and ω(x, N) are C-balanced for some explicit values of Cand compute their factor complexity function. We also obtainuniform word frequencies and deduce unique ergodicity of theassociated subshifts. Finally, we provide a Farey-like map forN-continued fraction expansions, which provides an additiveversion of N-continued fractions, for which we prove ergodicityand give the invariant measure explicitly",
author = "Niels Langeveld and Lucia Rossi and J{\"o}rg Thuswaldner",
year = "2023",
month = sep,
doi = "10.1016/j.jnt.2023.03.008",
language = "English",
volume = "250.2023",
pages = "49--83",
journal = "Journal of number theory",
issn = "0022-314X",
publisher = "Academic Press Inc.",
number = "September",

}

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TY - JOUR

T1 - Generalizations of Sturmian sequences associated with N-continued fraction algorithms

AU - Langeveld, Niels

AU - Rossi, Lucia

AU - Thuswaldner, Jörg

PY - 2023/9

Y1 - 2023/9

N2 - Given a positive integer N and x ∈ [0, 1] \ Q, an N-continuedfraction expansion of x is defined analogously to the classicalcontinued fraction expansion, but with the numerators beingall equal to N. Inspired by Sturmian sequences, we introducethe N-continued fraction sequences ω(x, N) and ω(x, N),which are related to the N-continued fraction expansion ofx. They are infinite words over a two letter alphabet obtainedas the limit of a directive sequence of certain substitutions,hence they are S-adic sequences. When N = 1, we are in thecase of the classical continued fraction algorithm, and obtainthe well-known Sturmian sequences. We show that ω(x, N)and ω(x, N) are C-balanced for some explicit values of Cand compute their factor complexity function. We also obtainuniform word frequencies and deduce unique ergodicity of theassociated subshifts. Finally, we provide a Farey-like map forN-continued fraction expansions, which provides an additiveversion of N-continued fractions, for which we prove ergodicityand give the invariant measure explicitly

AB - Given a positive integer N and x ∈ [0, 1] \ Q, an N-continuedfraction expansion of x is defined analogously to the classicalcontinued fraction expansion, but with the numerators beingall equal to N. Inspired by Sturmian sequences, we introducethe N-continued fraction sequences ω(x, N) and ω(x, N),which are related to the N-continued fraction expansion ofx. They are infinite words over a two letter alphabet obtainedas the limit of a directive sequence of certain substitutions,hence they are S-adic sequences. When N = 1, we are in thecase of the classical continued fraction algorithm, and obtainthe well-known Sturmian sequences. We show that ω(x, N)and ω(x, N) are C-balanced for some explicit values of Cand compute their factor complexity function. We also obtainuniform word frequencies and deduce unique ergodicity of theassociated subshifts. Finally, we provide a Farey-like map forN-continued fraction expansions, which provides an additiveversion of N-continued fractions, for which we prove ergodicityand give the invariant measure explicitly

U2 - 10.1016/j.jnt.2023.03.008

DO - 10.1016/j.jnt.2023.03.008

M3 - Article

VL - 250.2023

SP - 49

EP - 83

JO - Journal of number theory

JF - Journal of number theory

SN - 0022-314X

IS - September

ER -