Geodesic
Invariant Measures and Natural Extensions
Canadian mathematical bulletin, Tome 45 (2002) no. 1, pp. 97-108

Voir la notice de l'article provenant de la source Cambridge University Press

We study ergodic properties of a family of interval maps that are given as the fractional parts of certain real Möbius transformations. Included are the maps that are exactly $n$ -to-1, the classical Gauss map and the Renyi or backward continued fraction map. A new approach is presented for deriving explicit realizations of natural automorphic extensions and their invariant measures.
DOI : 10.4153/CMB-2002-011-4
Mots-clés : 11J70, 58F11, 58F03, Continued fractions, interval maps, invariant measures 
Haas, Andrew. Invariant Measures and Natural Extensions. Canadian mathematical bulletin, Tome 45 (2002) no. 1, pp. 97-108. doi: 10.4153/CMB-2002-011-4
@article{10_4153_CMB_2002_011_4,
     author = {Haas, Andrew},
     title = {Invariant {Measures} and {Natural} {Extensions}},
     journal = {Canadian mathematical bulletin},
     pages = {97--108},
     year = {2002},
     volume = {45},
     number = {1},
     doi = {10.4153/CMB-2002-011-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2002-011-4/}
}
TY  - JOUR
AU  - Haas, Andrew
TI  - Invariant Measures and Natural Extensions
JO  - Canadian mathematical bulletin
PY  - 2002
SP  - 97
EP  - 108
VL  - 45
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2002-011-4/
DO  - 10.4153/CMB-2002-011-4
ID  - 10_4153_CMB_2002_011_4
ER  - 
%0 Journal Article
%A Haas, Andrew
%T Invariant Measures and Natural Extensions
%J Canadian mathematical bulletin
%D 2002
%P 97-108
%V 45
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2002-011-4/
%R 10.4153/CMB-2002-011-4
%F 10_4153_CMB_2002_011_4

[1] [1] Adler, R. L. and Flatto, L., The backward continued fraction map and the geodesic flow. Ergodic Theory Dynamical Systems 4 (1984), 487–492. Google Scholar

[2] [2] Bosma, W., Jager, H. and Wiedijk, F., Some metrical observations on the approximation of continued factions. Indag.Math. 45 (1983), 281–299. Google Scholar

[3] [3] Cornfeld, I. P., Fomin, S. V. and Sinai, Ya. G., Ergodic Theory. Springer-Verlag, Berlin, Heidelberg, New York, 1982. Google Scholar

[4] [4] de Melo, W. and van Strien, S., One-Dimensional Dynamics. ErgebnisseMath. 25, Springer-Verlag, Berlin, Heidelberg, New York, 1993. Google Scholar

[5] [5] Gröchenig, K. and Haas, A., Backward continued fractions, invariant measures, and mappings of the interval. Ergodic Theory Dynamical Systems 16 (1996), 1241–1274. Google Scholar

[6] [6] Gröchenig, K. and Haas, A., Backward continued fractions and their invariant measures. Canad. Math. Bull. 39 (1996), 186–198. Google Scholar

[7] [7] Gröchenig, K. and Haas, A., Invariant Measures for Certain Linear Fractional Transformations Mod 1. Proc. Amer. Math. Soc. 127 (1999), 3439–3444. Google Scholar

[8] [8] Nakada, H., Metrical theory for a class of continued fraction transformations and their natural extensions. Tokyo J.Math. 4 (1981), 399–426. Google Scholar

[9] [9] Ornstein, D. S., Ergodic Theory, Randomness, and Dymanical Systems. Yale Univ. Press, New Haven, 1974. Google Scholar

[10] [10] Parry, W., Entropy and generators in ergodic theory. Benjamin Inc., New York, Amsterdam, 1969. Google Scholar

[11] [11] Rudolfer, S. M., Ergodic properties of linear fractional transformations mod one. Proc. London Math. Soc. 23 (1971), 515–531. Google Scholar

[12] [12] Rohlin, V. A., New Progress in the Theory of Transformations with Invariant Measure. Russian Math. Surveys (4) 15 (1960), 1–22. Google Scholar

[13] [13] Rohlin, V. A., Exact endomorphisms of a Lebesgue space. Trans. Amer.Math. Soc. 39 (1964), 1–37. Google Scholar

[14] [14] Rudolfer, S. M. and Wilkinson, K. M., A number-theoretic class of weak Bernoulli transformations. Math. Systems Theory 7 (1973), 14–24 Google Scholar

[15] [15] Rychlik, M., Bounded variation and invariant measures. Studia Math. 76 (1983), 69–80. Google Scholar

[16] [16] Series, C., The modular group and continued fractions. J. London Math. Soc. 31 (1985), 69–80. Google Scholar

[17] [17] Thaler, M., Transformations on [0, 1] with infinite invariant measure. Israel J. Math. 46 (1983), 67–96. Google Scholar

Cité par Sources :