A Gauss-Kuzmin theorem for the Rosen fractions
Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, p. 667-682

Using the natural extensions for the Rosen maps, we give an infinite-order-chain representation of the sequence of the incomplete quotients of the Rosen fractions. Together with the ergodic behaviour of a certain homogeneous random system with complete connections, this allows us to solve a variant of Gauss-Kuzmin problem for the above fraction expansion.

En utilisant les extensions naturelles des transformations de Rosen, nous obtenons une représentation de la chaîne d'ordre infini associée à la suite des quotients incomplets des fractions de Rosen. Associé au comportement ergodique d'un certain système aléatoire homogène à liaisons complètes, ce fait nous permet de résoudre une version du problème de Gauss-Kuzmin pour le développement en fraction de Rosen.

@article{JTNB_2002__14_2_667_0,
     author = {Sebe, Gabriela I.},
     title = {A Gauss-Kuzmin theorem for the Rosen fractions},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {14},
     number = {2},
     year = {2002},
     pages = {667-682},
     zbl = {1067.11044},
     mrnumber = {2040700},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2002__14_2_667_0}
}
Sebe, Gabriela I. A Gauss-Kuzmin theorem for the Rosen fractions. Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, pp. 667-682. http://www.numdam.org/item/JTNB_2002__14_2_667_0/

[1] R.M. Burton, C. Kraaikamp, T.A. Schmidt, Natural extensions for the Rosen fractions. Trans. Amer. Math. Soc. 352 (2000), 1277-1298. | MR 1650073 | Zbl 0938.11036

[2] K. Gröchenig, A. Haas, Backward continued fractions and their invariant measures. Canad. Math. Bull. 39 (1996), 186-198. | MR 1390354 | Zbl 0863.11045

[3] A. Haas, C. Series, The Hurwitz constant and Diophantine approximation on Hecke groups. J. London Math. Soc. 34 (1986), 219-234. | MR 856507 | Zbl 0605.10018

[4] M. Iosifescu, A basic tool in mathematical chaos theory: Doeblin and Fortet's ergodic theorem and Ionescu Tulcea-Marinescu's generalization. In: Doeblin and Modern Probability (Blaubeuren, 1991), 111-124. Contemp. Math. 149, Amer. Math. Soc. Providence, RI, 1993. | MR 1229957 | Zbl 0801.47003

[5] M. Iosifescu, On the Gauss-Kuzmin-Lévy theorem, III. Rev. Roumaine Math. Pures Appl. 42 (1997), 71-88. | MR 1650087 | Zbl 1013.11045

[6] M. Iosifescu S. GRIGORESCU, Dependence with complete connections and its applications. Cambridge Univ. Press, Cambrigde, 1990. | MR 1070097 | Zbl 0749.60067

[7] J. Lehner, Diophantine approximation on Hecke groups, Glasgow Math. J. 27 (1985), 117-127. | MR 819833 | Zbl 0576.10023

[8] J. Lehner, Lagrange's theorem for Hecke triangle groups. In: A tribute to Emil Grosswald: number theory and related analysis, 477-480, Contemp. Math. 143, Amer. Math. Soc., Providence, RI, 1993. | MR 1210534 | Zbl 0790.11004

[9] H. Nakada, Metrical theory for a class of continued fraction transformations and their natural extensions. Tokyo J. Math. 7 (1981), 399-426. | MR 646050 | Zbl 0479.10029

[10] H. Nakada, Continued fractions, geodesic flows and Ford circles. In: Algorithms, fractals, and dynamics (Okayama/Kyoto, 1992), 179-191, Plenum, New York, 1995. | MR 1402490 | Zbl 0868.30005

[11] D. Rosen, A class of continued fractions associated with certain properly discontinuous groups. Duke Math. J. 21 (1954), 549-563. | MR 65632 | Zbl 0056.30703

[12] D. Rosen, T.A. Schmidt, Hecke groups and continued fractions. Bull. Austral. Math. Soc. 46 (1992), 459-474. | MR 1190349 | Zbl 0754.11012

[13] T.A. Schmidt, Remarks on the Rosen λ-continued fractions. In: Number theory with an emphasis on the Markoff spectrum (Provo, UT, 1991), 227-238, Lecture Notes in Pure and Appl. Math., 147, Dekker, New York, 1993. | Zbl 0790.11043