Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Measures with positive Lyapunov exponent and conformal measures in rational dynamics

Author: Neil Dobbs
Journal: Trans. Amer. Math. Soc. 364 (2012), 2803-2824
MSC (2010): Primary 37F10, 37D25, 37D35
Published electronically: January 25, 2012
MathSciNet review: 2888229
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Ergodic properties of rational maps are studied, generalising the work of F. Ledrappier. A new construction allows for simpler proofs of stronger results. Very general conformal measures are considered. Equivalent conditions are given for an ergodic invariant probability measure with positive Lyapunov exponent to be absolutely continuous with respect to a general conformal measure. If they hold, we can construct an induced expanding Markov map with integrable return time which generates the invariant measure.

References [Enhancements On Off] (What's this?)

  • 1. Manfred Denker and Mariusz Urbański.
    Ergodic theory of equilibrium states for rational maps.
    Nonlinearity, 4(1):103-134, 1991. MR 1092887 (92a:58112)
  • 2. Manfred Denker and Mariusz Urbański.
    On Sullivan's conformal measures for rational maps of the Riemann sphere.
    Nonlinearity, 4(2):365-384, 1991. MR 1107011 (92f:58097)
  • 3. Jacek Graczyk and Stanislav Smirnov.
    Non-uniform hyperbolicity in complex dynamics.
    Invent. Math., 175(2):335-415, 2009. MR 2470110 (2010e:37056)
  • 4. François Ledrappier.
    Some properties of absolutely continuous invariant measures on an interval.
    Ergodic Theory Dynamical Systems, 1(1):77-93, 1981. MR 627788 (82k:28018)
  • 5. François Ledrappier.
    Quelques propriétés ergodiques des applications rationnelles.
    C. R. Acad. Sci. Paris Sér. I Math., 299(1):37-40, 1984. MR 756305 (86c:58091)
  • 6. N. Makarov and S. Smirnov.
    On thermodynamics of rational maps. II. Non-recurrent maps.
    J. London Math. Soc. (2), 67(2):417-432, 2003. MR 1956144 (2004e:37067)
  • 7. Ricardo Mañé.
    On the Bernoulli property for rational maps.
    Ergodic Theory Dynam. Systems, 5(1):71-88, 1985. MR 782789 (86i:58082)
  • 8. William Parry.
    Topics in ergodic theory, volume 75 of Cambridge Tracts in Mathematics.
    Cambridge University Press, Cambridge, 1981. MR 614142 (83a:28018)
  • 9. Karl Petersen.
    Ergodic theory, volume 2 of Cambridge Studies in Advanced Mathematics.
    Cambridge University Press, Cambridge, 1989.
    Corrected reprint of the 1983 original. MR 1073173 (92c:28010)
  • 10. Feliks Przytycki.
    On the Perron-Frobenius-Ruelle operator for rational maps on the Riemann sphere and for Hölder continuous functions.
    Bol. Soc. Brasil. Mat. (N.S.), 20(2):95-125, 1990. MR 1143178 (93b:58120)
  • 11. Feliks Przytycki.
    Lyapunov characteristic exponents are nonnegative.
    Proc. Amer. Math. Soc., 119(1):309-317, 1993. MR 1186141 (93k:58193)
  • 12. Feliks Przytycki.
    On the hyperbolic Hausdorff dimension of the boundary of a basin of attraction for a holomorphic map and of quasirepellers.
    Bull. Pol. Acad. Sci. Math., 54(1):41-52, 2006. MR 2270793 (2007h:37065)
  • 13. Feliks Przytycki and Juan Rivera-Letelier.
    Statistical properties of topological Collet-Eckmann maps.
    Ann. Sci. École Norm. Sup. (4), 40(1):135-178, 2007. MR 2332354 (2008j:37093)
  • 14. Feliks Przytycki, Juan Rivera-Letelier, and Stanislav Smirnov.
    Equality of pressures for rational functions.
    Ergodic Theory Dynam. Systems, 24(3):891-914, 2004. MR 2062924 (2005e:37103)
  • 15. Juan Rivera-Letelier.
    A connecting lemma for rational maps satisfying a no-growth condition.
    Ergodic Theory Dynam. Systems, 27(2):595-636, 2007. MR 2308147 (2008a:37051)
  • 16. V. A. Rohlin.
    Exact endomorphisms of a Lebesgue space.
    Amer. Math. Soc. Transl. (2), 39:1-36, 1964.
  • 17. Dennis Sullivan.
    Conformal dynamical systems.
    In Geometric dynamics (Rio de Janeiro, 1981), volume 1007 of Lecture Notes in Math., pages 725-752. Springer, Berlin, 1983. MR 730296 (85m:58112)
  • 18. Peter Walters.
    An introduction to ergodic theory, volume 79 of Graduate Texts in Mathematics.
    Springer-Verlag, New York, 1982. MR 648108 (84e:28017)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 37F10, 37D25, 37D35

Retrieve articles in all journals with MSC (2010): 37F10, 37D25, 37D35

Additional Information

Neil Dobbs
Affiliation: Department of Mathematics, KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden
Address at time of publication: IBM T. J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598

Received by editor(s): April 23, 2008
Received by editor(s) in revised form: January 20, 2010
Published electronically: January 25, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society