José Pedro Gaivão, Michel Laurent, Arnaldo Nogueira
{"title":"Rotation number of 2-interval piecewise affine maps","authors":"José Pedro Gaivão, Michel Laurent, Arnaldo Nogueira","doi":"10.1007/s00010-024-01064-2","DOIUrl":null,"url":null,"abstract":"<p>We study maps of the unit interval whose graph is made up of two increasing segments and which are injective in an extended sense. Such maps <span>\\(f_{\\varvec{p}}\\)</span> are parametrized by a quintuple <span>\\(\\varvec{p}\\)</span> of real numbers satisfying inequations. Viewing <span>\\(f_{\\varvec{p}}\\)</span> as a circle map, we show that it has a rotation number <span>\\(\\rho (f_{\\varvec{p}})\\)</span> and we compute <span>\\(\\rho (f_{\\varvec{p}})\\)</span> as a function of <span>\\(\\varvec{p}\\)</span> in terms of Hecke–Mahler series. As a corollary, we prove that <span>\\(\\rho (f_{\\varvec{p}})\\)</span> is a rational number when the components of <span>\\(\\varvec{p}\\)</span> are algebraic numbers.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00010-024-01064-2","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We study maps of the unit interval whose graph is made up of two increasing segments and which are injective in an extended sense. Such maps \(f_{\varvec{p}}\) are parametrized by a quintuple \(\varvec{p}\) of real numbers satisfying inequations. Viewing \(f_{\varvec{p}}\) as a circle map, we show that it has a rotation number \(\rho (f_{\varvec{p}})\) and we compute \(\rho (f_{\varvec{p}})\) as a function of \(\varvec{p}\) in terms of Hecke–Mahler series. As a corollary, we prove that \(\rho (f_{\varvec{p}})\) is a rational number when the components of \(\varvec{p}\) are algebraic numbers.