{"title":"Periodicity of general multidimensional continued fractions using repetend matrix form","authors":"Hanka Řada , Štěpán Starosta , Vítězslav Kala","doi":"10.1016/j.exmath.2024.125571","DOIUrl":null,"url":null,"abstract":"<div><p>We consider expansions of vectors by a general class of multidimensional continued fraction algorithms. If the expansion is eventually periodic, then we describe the possible structure of a matrix corresponding to the repetend and use it to prove that a number of vectors have an eventually periodic expansion in the Algebraic Jacobi–Perron algorithm. Further, we give criteria for vectors to have purely periodic expansions; in particular, the vector cannot be totally positive.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Expositiones Mathematicae","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0723086924000380","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
We consider expansions of vectors by a general class of multidimensional continued fraction algorithms. If the expansion is eventually periodic, then we describe the possible structure of a matrix corresponding to the repetend and use it to prove that a number of vectors have an eventually periodic expansion in the Algebraic Jacobi–Perron algorithm. Further, we give criteria for vectors to have purely periodic expansions; in particular, the vector cannot be totally positive.
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