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1999 | 160 | 1 | 39-61
Tytuł artykułu

Mesures invariantes pour les fractions rationnelles géométriquement finies

Treść / Zawartość
Warianty tytułu
Języki publikacji
FR
Abstrakty
EN
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 $\frac{p(T)+1}{p(T)}δ>2$. 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.
Słowa kluczowe
Kategorie tematyczne
Rocznik
Tom
160
Numer
1
Strony
39-61
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-04-20
poprawiono
1998-08-05
poprawiono
1999-02-23
Twórcy
  • MAPMO-UMR 6628 Université d'Orléans, B.P. 6759, 45067 Orléans Cedex 2, France
Bibliografia
  • [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.
  • [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.
  • [Bow] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math. 470, Springer, 1975.
  • [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.
  • [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.
  • [Fo] S. Foguel, The Ergodic Theory of Markov Processes, Van Nostrand, 1969.
  • [Mc,Mu1] C. McMullen, Hausdorff dimension and conformal dynamics 2: Geometrically finite rational maps, preprint, 1997.
  • [Mc,Mu2] C. McMullen, Hausdorff dimension and conformal dynamics 3: Computation of dimension, preprint, 1997.
  • [Mi] J. Milnor, Dynamics in One Complex Variable: Introductory Lectures, Sto-ny Brook IMS preprint, 1990.
  • [Po] C. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, 1992.
  • [Pr,Ur] F. Przytycki and M. Urbański, Fractals in the complex plane-ergodic theory methods, to appear.
  • [Ru1] D. Ruelle, Thermodynamic Formalism, Addison-Wesley, 1978.
  • [Ru2] D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems 2 (1982), 99-107.
  • [Sm] S. Smirnov, Spectral analysis of Julia sets, thesis, California Institute of Technology, 1996.
  • [Su] D. Sullivan, Conformal dynamical systems, in: Geometric Dynamics, Lecture Notes in Math. 1007, Springer, 1983, 725-752.
  • [Ur1] M. Urbański, Rational functions with no recurrent critical points, Ergodic Theory Dynam. Systems 14 (1994), 391-414.
  • [Ur2] M. Urbański, Geometry and ergodic theory of conformal non-recurrent dynamics, ibid. 17 (1997), 1449-1476.
  • [Wa1] P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math. 97 (1976), 937-971.
  • [Wa2] P. Walters, An Introduction to Ergodic Theory, Springer, 1982.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-fmv160i1p39bwm
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