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Title

Mesures invariantes pour les fractions rationnelles géométriquement finies

Authors 1

Affiliations

  1. MAPMO-UMR 6628 Université d'Orléans, B.P. 6759, 45067 Orléans Cedex 2, France

Abstract

Let T be a geometrically finite rational map, p(T) its petal number and δ the Hausdorff dimension of its Julia set. We give a construction of the σ-finite and T-invariant measure equivalent to the δ-conformal measure. We prove that this measure is finite if and only if p(T)+1p(T)δ>2. Under this assumption and if T is parabolic, we prove that the only equilibrium states are convex combinations of the T-invariant probability and δ-masses at parabolic cycles.

Bibliography

  1. [Aa,De,Ur] J. Aaronson, M. Denker and M. Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc. 337 (1993), 495-548.
  2. [Bo,Zi] O. Bodart et M. Zinsmeister, Quelques résultats sur la dimension de Hausdorff des ensembles de Julia des polynômes quadratiques, Fund. Math. 151 (1996), 121-137.
  3. [Bow] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math. 470, Springer, 1975.
  4. [Bow] M. Denker and M. Urbański, Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point, J. London Math. Soc. 43 (1991), 107-118.
  5. [De,Ur1] M. Denker and M. Urbański, Absolutely continuous invariant measures for expansive rational maps with rationally indifferent periodic points, Forum Math. 3 (1991), 561-579.
  6. [Fo] S. Foguel, The Ergodic Theory of Markov Processes, Van Nostrand, 1969.
  7. [Mc,Mu1] C. McMullen, Hausdorff dimension and conformal dynamics 2: Geometrically finite rational maps, preprint, 1997.
  8. [Mc,Mu2] C. McMullen, Hausdorff dimension and conformal dynamics 3: Computation of dimension, preprint, 1997.
  9. [Mi] J. Milnor, Dynamics in One Complex Variable: Introductory Lectures, Sto-ny Brook IMS preprint, 1990.
  10. [Po] C. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, 1992.
  11. [Pr,Ur] F. Przytycki and M. Urbański, Fractals in the complex plane-ergodic theory methods, to appear.
  12. [Ru1] D. Ruelle, Thermodynamic Formalism, Addison-Wesley, 1978.
  13. [Ru2] D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems 2 (1982), 99-107.
  14. [Sm] S. Smirnov, Spectral analysis of Julia sets, thesis, California Institute of Technology, 1996.
  15. [Su] D. Sullivan, Conformal dynamical systems, in: Geometric Dynamics, Lecture Notes in Math. 1007, Springer, 1983, 725-752.
  16. [Ur1] M. Urbański, Rational functions with no recurrent critical points, Ergodic Theory Dynam. Systems 14 (1994), 391-414.
  17. [Ur2] M. Urbański, Geometry and ergodic theory of conformal non-recurrent dynamics, ibid. 17 (1997), 1449-1476.
  18. [Wa1] P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math. 97 (1976), 937-971.
  19. [Wa2] P. Walters, An Introduction to Ergodic Theory, Springer, 1982.
Fundamenta Mathematicae
1999/160/1
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Pages:
39-61
Main language of publication
French
Received
1998-04-20
Accepted
1998-08-05
Published
1999
Exact and natural sciences