A Gauss-Kuzmin theorem for the Rosen fractions

Gabriela I. Sebe

Journal de théorie des nombres de Bordeaux (2002)

  • Volume: 14, Issue: 2, page 667-682
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Sebe, Gabriela I.. "A Gauss-Kuzmin theorem for the Rosen fractions." Journal de théorie des nombres de Bordeaux 14.2 (2002): 667-682. <http://eudml.org/doc/248914>.

@article{Sebe2002,
abstract = {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.},
author = {Sebe, Gabriela I.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Rosen fractions; Gauss-Kuzmin theorem},
language = {eng},
number = {2},
pages = {667-682},
publisher = {Université Bordeaux I},
title = {A Gauss-Kuzmin theorem for the Rosen fractions},
url = {http://eudml.org/doc/248914},
volume = {14},
year = {2002},
}

TY - JOUR
AU - Sebe, Gabriela I.
TI - A Gauss-Kuzmin theorem for the Rosen fractions
JO - Journal de théorie des nombres de Bordeaux
PY - 2002
PB - Université Bordeaux I
VL - 14
IS - 2
SP - 667
EP - 682
AB - 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.
LA - eng
KW - Rosen fractions; Gauss-Kuzmin theorem
UR - http://eudml.org/doc/248914
ER -

References

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  1. [1] R.M. Burton, C. Kraaikamp, T.A. Schmidt, Natural extensions for the Rosen fractions. Trans. Amer. Math. Soc.352 (2000), 1277-1298. Zbl0938.11036MR1650073
  2. [2] K. Gröchenig, A. Haas, Backward continued fractions and their invariant measures. Canad. Math. Bull.39 (1996), 186-198. Zbl0863.11045MR1390354
  3. [3] A. Haas, C. Series, The Hurwitz constant and Diophantine approximation on Hecke groups. J. London Math. Soc.34 (1986), 219-234. Zbl0605.10018MR856507
  4. [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. Zbl0801.47003MR1229957
  5. [5] M. Iosifescu, On the Gauss-Kuzmin-Lévy theorem, III. Rev. Roumaine Math. Pures Appl.42 (1997), 71-88. Zbl1013.11045MR1650087
  6. [6] M. Iosifescu S.GRIGORESCU, Dependence with complete connections and its applications. Cambridge Univ. Press, Cambrigde, 1990. Zbl0749.60067MR1070097
  7. [7] J. Lehner, Diophantine approximation on Hecke groups, Glasgow Math. J.27 (1985), 117-127. Zbl0576.10023MR819833
  8. [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. Zbl0790.11004MR1210534
  9. [9] H. Nakada, Metrical theory for a class of continued fraction transformations and their natural extensions. Tokyo J. Math.7 (1981), 399-426. Zbl0479.10029MR646050
  10. [10] H. Nakada, Continued fractions, geodesic flows and Ford circles. In: Algorithms, fractals, and dynamics (Okayama/Kyoto, 1992), 179-191, Plenum, New York, 1995. Zbl0868.30005MR1402490
  11. [11] D. Rosen, A class of continued fractions associated with certain properly discontinuous groups. Duke Math. J.21 (1954), 549-563. Zbl0056.30703MR65632
  12. [12] D. Rosen, T.A. Schmidt, Hecke groups and continued fractions. Bull. Austral. Math. Soc.46 (1992), 459-474. Zbl0754.11012MR1190349
  13. [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. Zbl0790.11043

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