Comptes Rendus
Mathematical analysis/Dynamical systems
A note on singularity of a recently introduced family of Minkowski's question-mark functions
Comptes Rendus. Mathématique, Volume 355 (2017) no. 9, pp. 956-959.

We point out a mistake in the proof of the main theorem in a recent article on a family of generalized Minkowski's question-mark functions, saying that each member of the family is a singular homeomorphism, and provide two alternative proofs, one based on the ergodicity of the Gauss map G and the α-Lüroth map Lα, and another one focusing more on classical properties of continued fraction expansions.

Nous mettons en évidence une erreur dans la démonstration du théroème principal dans un article récent traitant d'une famille de fonctions « Minkowski's question-mark » généralisées, stipulant que chaque membre de la famille est un homéomorphisme singulier, et nous produisons deux preuves alternatives, l'une basée sur l'ergodicité de l'application de Gauss G et de l'application α-Lüroth Lα, l'autre se focalisant davantage sur des propriétés classiques des décompositions de fractions continues.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2017.09.009

Juan Fernández Sánchez 1; Wolfgang Trutschnig 2

1 Grupo de Investigación de Análisis Matemático, Universidad de Almería, La Cañada de San Urbano, Almería, Spain
2 Department for Mathematics, University Salzburg, Hellbrunnerstrasse 34, 5020 Salzburg, Austria
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Juan Fernández Sánchez; Wolfgang Trutschnig. A note on singularity of a recently introduced family of Minkowski's question-mark functions. Comptes Rendus. Mathématique, Volume 355 (2017) no. 9, pp. 956-959. doi : 10.1016/j.crma.2017.09.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.09.009/

[1] A. Arroyo Generalised Lüroth expansions and a family of Minkowski's question-mark functions, C. R. Acad. Sci. Paris, Ser. I, Volume 353 (2015), pp. 943-946

[2] K. Dajani; C.C. Kraaikamp Ergodic Theory of Numbers, Carus Mathematical Monographs, vol. 29, The Mathematical Association of America, 2002

[3] M. Iosifescu; C. Kraaikamp Metrical Theory of Continued Fractions, Springer Science+Business Media, Dordrecht, The Netherlands, 2002

[4] P. Walters An Introduction to Ergodic Theory, Springer, New York, 1982

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