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Math."],"published-print":{"date-parts":[[2025,4]]},"abstract":"Abstract<\/jats:title>\n 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 \n \n $$f_{\\varvec{p}}$$<\/jats:tex-math>\n \n \n f<\/mml:mi>\n \n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> are parametrized by a quintuple \n \n $$\\varvec{p}$$<\/jats:tex-math>\n \n \n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> of real numbers satisfying inequations. Viewing \n \n $$f_{\\varvec{p}}$$<\/jats:tex-math>\n \n \n f<\/mml:mi>\n \n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> as a circle map, we show that it has a rotation number \n \n $$\\rho (f_{\\varvec{p}})$$<\/jats:tex-math>\n \n \n \u03c1<\/mml:mi>\n (<\/mml:mo>\n \n f<\/mml:mi>\n \n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and we compute \n \n $$\\rho (f_{\\varvec{p}})$$<\/jats:tex-math>\n \n \n \u03c1<\/mml:mi>\n (<\/mml:mo>\n \n f<\/mml:mi>\n \n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> as a function of \n \n $$\\varvec{p}$$<\/jats:tex-math>\n \n \n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> in terms of Hecke\u2013Mahler series. As a corollary, we prove that \n \n $$\\rho (f_{\\varvec{p}})$$<\/jats:tex-math>\n \n \n \u03c1<\/mml:mi>\n (<\/mml:mo>\n \n f<\/mml:mi>\n \n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is a rational number when the components of \n \n $$\\varvec{p}$$<\/jats:tex-math>\n \n \n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> are algebraic numbers.<\/jats:p>","DOI":"10.1007\/s00010-024-01064-2","type":"journal-article","created":{"date-parts":[[2024,4,22]],"date-time":"2024-04-22T13:01:31Z","timestamp":1713790891000},"page":"511-530","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Rotation number of 2-interval piecewise affine maps"],"prefix":"10.1007","volume":"99","author":[{"given":"Jos\u00e9 Pedro","family":"Gaiv\u00e3o","sequence":"first","affiliation":[]},{"given":"Michel","family":"Laurent","sequence":"additional","affiliation":[]},{"given":"Arnaldo","family":"Nogueira","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,4,22]]},"reference":[{"key":"1064_CR1","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1007\/s12346-022-00679-7","volume":"21","author":"A Antunes","year":"2022","unstructured":"Antunes, A., Bugeaud, Y., Pires, B.: Switched server systems whose parameters are normal numbers in base 4. 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