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• W2963741567 abstract "Minkowski's question-mark function is a singular homeomorphism of the unit interval that maps the set of quadratic surds into the rationals. This function has deserved the attention of several authors since the beginning of the twentieth century. Using different representations of real numbers by infinite sequences of integers, called α -Lüroth expansions, we obtain different instances of the standard shift map on infinite symbols, all of them topologically conjugated to the Gauss Map. In this note we prove that each of these conjugations share properties with Minkowski's question-mark function: all of them are singular homeomorphisms of the interval, and in the “rational” cases, they map the set of quadratic surds into the set of rational numbers. In this sense, this family is a natural generalisation of Minkowski's question-mark function. La fonction point d'interrogation de Minkowski est un homéomorphisme singulier de l'intervalle unité tel que l'image des irrationnels quadratiques est contenue dans les rationnels. Cette fonction a attiré l'attention de plusieurs auteurs depuis le début du XX e siècle. En utilisant différents développements des nombres réels avec des séquences infinies de nombres entiers, appelées développements α -Lüroth, on obtient différentes instances de l'opérateur de décalage sur l'espace des mots sur une infinité de symboles, tous topologiquement conjugués à l'application de Gauss. Dans cette note, nous prouvons que toutes ces conjugaisons ont des propriétés en commun avec la fonction point d'interrogation : elles sont toutes des homéomorphismes singuliers de l'intervalle et, dans les cas « rationnels », l'image des irrationnels quadratiques est contenue dans les rationnels. En ce sens, cette famille est une généralisation naturelle de la fonction point d'interrogation de Minkowski." @default.
• W2963741567 created "2019-07-30" @default.
• W2963741567 creator A5048500481 @default.
• W2963741567 date "2015-10-01" @default.
• W2963741567 modified "2023-10-03" @default.
• W2963741567 title "Generalised Lüroth expansions and a family of Minkowski's question-mark functions" @default.
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• W2963741567 doi "https://doi.org/10.1016/j.crma.2015.08.008" @default.
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