no. 9
Mathematical analysis/Dynamical systems
A note on singularity of a recently introduced family of Minkowski's question-mark functions
[Note sur la singularité d'une famille de fonctions « Minkowski's question-mark » récemment introduite]

Présenté par : the Editorial Board

Fernández Sánchez, Juan1 ; Trutschnig, Wolfgang2
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
Comptes Rendus. Mathématique, Tome 355 (2017) no. 9, pp. 956-959.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.09.009
Fernández Sánchez, Juan 1 ; Trutschnig, Wolfgang 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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Fernández Sánchez, Juan; Trutschnig, Wolfgang. A note on singularity of a recently introduced family of Minkowski's question-mark functions. Comptes Rendus. Mathématique, Tome 355 (2017) no. 9, pp. 956-959. doi : 10.1016/j.crma.2017.09.009. http://www.numdam.org/articles/10.1016/j.crma.2017.09.009/

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

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

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

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