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 = {Gabriela I. Sebe}, title = {A {Gauss-Kuzmin} theorem for the {Rosen} fractions}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {667--682}, publisher = {Universit\'e Bordeaux I}, volume = {14}, number = {2}, year = {2002}, zbl = {1067.11044}, mrnumber = {2040700}, language = {en}, url = {https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_667_0/} }
TY - JOUR AU - Gabriela I. Sebe TI - A Gauss-Kuzmin theorem for the Rosen fractions JO - Journal de théorie des nombres de Bordeaux PY - 2002 SP - 667 EP - 682 VL - 14 IS - 2 PB - Université Bordeaux I UR - https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_667_0/ LA - en ID - JTNB_2002__14_2_667_0 ER -
Gabriela I. Sebe. A Gauss-Kuzmin theorem for the Rosen fractions. Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, pp. 667-682. https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_667_0/
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