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 , 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 , l'autre se focalisant davantage sur des propriétés classiques des décompositions de fractions continues.
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Juan Fernández Sánchez 1; Wolfgang Trutschnig 2
@article{CRMATH_2017__355_9_956_0, author = {Juan Fern\'andez S\'anchez and Wolfgang Trutschnig}, title = {A note on singularity of a recently introduced family of {Minkowski's} question-mark functions}, journal = {Comptes Rendus. Math\'ematique}, pages = {956--959}, publisher = {Elsevier}, volume = {355}, number = {9}, year = {2017}, doi = {10.1016/j.crma.2017.09.009}, language = {en}, }
TY - JOUR AU - Juan Fernández Sánchez AU - Wolfgang Trutschnig TI - A note on singularity of a recently introduced family of Minkowski's question-mark functions JO - Comptes Rendus. Mathématique PY - 2017 SP - 956 EP - 959 VL - 355 IS - 9 PB - Elsevier DO - 10.1016/j.crma.2017.09.009 LA - en ID - CRMATH_2017__355_9_956_0 ER -
%0 Journal Article %A Juan Fernández Sánchez %A Wolfgang Trutschnig %T A note on singularity of a recently introduced family of Minkowski's question-mark functions %J Comptes Rendus. Mathématique %D 2017 %P 956-959 %V 355 %N 9 %I Elsevier %R 10.1016/j.crma.2017.09.009 %G en %F CRMATH_2017__355_9_956_0
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] 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] Ergodic Theory of Numbers, Carus Mathematical Monographs, vol. 29, The Mathematical Association of America, 2002
[3] Metrical Theory of Continued Fractions, Springer Science+Business Media, Dordrecht, The Netherlands, 2002
[4] An Introduction to Ergodic Theory, Springer, New York, 1982
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