Academic Journal

Extremal values of semi‐regular continuants and codings of interval exchange transformations

التفاصيل البيبلوغرافية
العنوان: Extremal values of semi‐regular continuants and codings of interval exchange transformations
المؤلفون: de Luca, Alessandro, Edson, Marcia, Zamboni, Luca, Q
المساهمون: Combinatoire, théorie des nombres (CTN), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)
المصدر: ISSN: 0025-5793.
بيانات النشر: HAL CCSD
University College London
سنة النشر: 2023
المجموعة: Université Jean Monnet – Saint-Etienne: HAL
مصطلحات موضوعية: Continued fractions extremal values of continuants Markoff property Sturmian words and interval exchange transformations. MSC[2010]: Primary 11J70, 37B10 68R15, Continued fractions, extremal values of continuants, Markoff property, Sturmian words and interval exchange transformations. MSC[2010]: Primary 11J70, 37B10, 68R15, [MATH]Mathematics [math]
الوصف: International audience ; Abstract Given a set consisting of positive integers and a k ‐term partition , find the extremal denominators of the regular and semi‐regular continued fraction with partial quotients and where each occurs precisely times in the sequence . In 1983, G. Ramharter gave an explicit description of the extremal arrangements of the regular continued fraction and the minimizing arrangement for the semi‐regular continued fraction and showed that in each case the arrangement is unique up to reversal and independent of the actual values of the positive integers . However, an explicit determination of a maximizing arrangement for the semi‐regular continuant turned out to be substantially more difficult. Ramharter conjectured that as in the other three cases, the maximizing arrangement is unique (up to reversal) and depends only on the partition P and not on the actual values of the . He further verified the conjecture in the special case of a binary alphabet. In this paper, we confirm Ramharter's conjecture for sets with and give an algorithmic procedure for constructing the unique maximizing arrangement. We also show that Ramharter's conjecture fails for sets with in that the maximizing arrangement is in general neither unique nor independent of the values of the digits in . The central idea is that the extremal arrangements satisfy a strong combinatorial condition. This combinatorial condition may also be stated more or less verbatum in the context of infinite sequences on an ordered set. We show that in the context of bi‐infinite binary words, this condition coincides with the Markoff property, discovered by A. A. Markoff in 1879 in his study of minima of binary quadratic forms. We further show that this same combinatorial condition is the fundamental property which describes the orbit structure of the natural codings of points under a symmetric k ‐interval exchange transformation.
نوع الوثيقة: article in journal/newspaper
اللغة: English
DOI: 10.1112/mtk.12185
الاتاحة: https://hal.science/hal-04764024
https://hal.science/hal-04764024v1/document
https://hal.science/hal-04764024v1/file/Extremal%20values%20of%20semi-regular%20continuants%20%28revised%2006%3A12%3A22%29.pdf
https://doi.org/10.1112/mtk.12185
Rights: info:eu-repo/semantics/OpenAccess
رقم الانضمام: edsbas.3AE1BB15
قاعدة البيانات: BASE