Résumé
Nous disons qu’une application rationnelle F satisfait la condition de sommabilité avec un exposant α si pour tout point critique c qui appartient à l’ensemble de Julia J, il y a un entier positif n c tel que \(\sum_{n=1}^{\infty} |(F^{n})^{\prime}(F^{n_{c}}(c))|^{-\alpha}<\infty\) et F n’a pas de points périodiques paraboliques. Soit μ max la multiplicité maximale des points critiques de F.
L’objectif est d’étudier les séries de Poincaré pour une large classe d’applications rationnelles et d’établir les propriétés ergodiques et la regularité des mesures conformes. Si F est sommable avec un exposant \(\alpha<\frac{\delta_{\textit{Poin}}(J)}{\delta_{\textit{Poin}}(J)+\mu_{\textit{max}}}\), où δ Poin (J) est l’exposant de Poincaré de l’ensemble de Julia, alors il existe une unique mesure conforme ν avec l’exposant δ Poin (J)=HDim(J) qui est invariante, ergodique, et non-atomique. De plus, F possède une mesure invariante absolument continue par rapport à ν pourvu que \(\sum_{n=1}^{\infty}n |(F^{n})^{\prime}(F^{n_{c}}(c))|^{-\alpha}<\infty\) (sommabilité de type polynômial) et que F n’a pas de points périodiques paraboliques. Cela aboutit à un nouveau résultat sur l’existence des mesures invariantes absolument continues pour des applications multimodales d’un intervalle.
Nous démontrons que si F est sommable avec un exposant \(\alpha<\frac{2}{2+\mu_{\textit{max}}}\), alors la dimension de Minkowski de J, si \(J\neq\hat{\mathbb{C}}\), est strictement plus petite que 2 et F est instable. Si F est un polynôme ou un produit de Blaschke, alors J est conformément effaçable. Si F est sommable avec \(\alpha<\frac{1}{1+\mu_{\textit{max}}}\), alors toute composante connexe de la frontière de chaque composante de Fatou invariante est localement connexe. Pour étudier la continuité de la dimension de Hausdorff des ensembles de Julia, nous introduisons le concept de la sommabilité uniforme.
Enfin, nous en déduisons un analogue conforme du théorème de Jakobson et Benedicks-Carleson. Nous montrons la continuité externe de la dimension de Hausdorff des ensembles de Julia pour presque tout point de l’ensemble de Mandelbrot par rapport à la mesure harmonique.
Abstract
We say that a rational function F satisfies the summability condition with exponent α if for every critical point c which belongs to the Julia set J there exists a positive integer n c so that \(\sum_{n=1}^{\infty} |(F^{n})^{\prime}(F^{n_{c}}(c))|^{-\alpha}<\infty\) and F has no parabolic periodic cycles. Let μ max be the maximal multiplicity of the critical points.
The objective is to study the Poincaré series for a large class of rational maps and establish ergodic and regularity properties of conformal measures. If F is summable with exponent \(\alpha<\frac{\delta_{\textit{Poin}}(J)}{\delta_{\textit{Poin}}(J)+\mu_{\textit{max}}}\) where δ Poin (J) is the Poincaré exponent of the Julia set then there exists a unique, ergodic, and non-atomic conformal measure ν with exponent δ Poin (J)=HDim(J). If F is polynomially summable with the exponent α, \(\sum_{n=1}^{\infty}n |(F^{n})^{\prime}(F^{n_{c}}(c))|^{-\alpha}<\infty\) and F has no parabolic periodic cycles, then F has an absolutely continuous invariant measure with respect to ν. This leads also to a new result about the existence of absolutely continuous invariant measures for multimodal maps of the interval.
We prove that if F is summable with an exponent \(\alpha< \frac{2}{2+\mu_{\textit{max}}}\) then the Minkowski dimension of J is strictly less than 2 if \(J\neq\hat{\mathbb{C}}\) and F is unstable. If F is a polynomial or Blaschke product then J is conformally removable. If F is summable with \(\alpha<\frac{1}{1+\mu_{\textit{max}}}\) then connected components of the boundary of every invariant Fatou component are locally connected. To study continuity of Hausdorff dimension of Julia sets, we introduce the concept of the uniform summability.
Finally, we derive a conformal analogue of Jakobson’s (Benedicks–Carleson’s) theorem and prove the external continuity of the Hausdorff dimension of Julia sets for almost all points c from the Mandelbrot set with respect to the harmonic measure.
References
Aspenberg, M.: The Collet–Eckmann condition for rational maps on the Riemann sphere. Ph.D. Thesis, KTH Sweden (2004)
Beardon, A.F.: Iteration of Rational Functions. Springer, New York (1991)
Benedicks, M., Carleson, L.: On iterations of 1-ax2 on (-1,1). Ann. Math. (2) 122(1), 1–25 (1985)
Benedicks, M., Carleson, L.: The dynamics of the Hénon map. Ann. Math. (2) 133(1), 73–169 (1991)
Bishop, C.J.: Minkowski dimension and the Poincaré exponent. Mich. Math. J. 43(2), 231–246 (1996)
Bishop, C.J., Jones, P.W.: Hausdorff dimension and Kleinian groups. Acta Math. 179(1), 1–39 (1997)
Bruin, H., van Strien, S.: Expansion of derivatives in one-dimensional dynamics. Isr. J. Math. 137, 223–263 (2003)
Bruin, H., Luzzatto, S., van Strien, S.: Decay of correlations in one-dimensional dynamics. Ann. Sci. Éc. Norm. Supér., IV. Sér. 36(4), 621–646 (2003)
Bruin, H., Shen, W., van Strien, S.: Invariant measures exist without a growth condition. Commun. Math. Phys. 241(2–3), 287–306 (2003)
Carleson, L., Gamelin, T.W.: Complex Dynamics. Springer, New York (1993)
De Melo, W., van Strien, S.: One-Dimensional Dynamics. Springer, Berlin (1993)
Denker, M., Urbański, M.: On Sullivan’s conformal measures for rational maps of the Riemann sphere. Nonlinearity 4(2), 365–384 (1991)
Denker, M., Urbański, M.: On the existence of conformal measures. Trans. Am. Math. Soc. 328(2), 563–587 (1991)
Douady, A., Sentenac, P., Zinsmeister, M.: Implosion parabolique et dimension de Hausdorff. C.R. Acad. Sci. Paris, Sér. I 325(7), 765–772 (1997)
Federer, H.: Geometric Measure Theory. Grundl. Math. Wissensch., vol. 153. Springer, New York (1969)
Gehring, F.W., Martio, O.: Lipschitz classes and quasiconformal mappings. Ann. Acad. Sci. Fenn., Ser. A I, Math. 10, 203–219 (1985)
Graczyk, J., Sands, D., Świątek, G.: Metric attractors for smooth unimodal maps. Ann. Math. (2) 159(2), 725–740 (2004)
Graczyk, J., Smirnov, S.: Collet, Eckmann and Hölder. Invent. Math. 133(1), 69–96 (1998)
Graczyk, J., Świątek, G.: The Real Fatou Conjecture. Princeton University Press, Princeton, NJ (1998)
Graczyk, J., Świątek, G.: Harmonic measure and expansion on the boundary of the connectedness locus. Invent. Math. 142(3), 605–629 (2000)
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1988). Reprint of the 1952 edition
Heinonen, J., Koskela, P.: Definitions of quasiconformality. Invent. Math. 120(1), 61–79 (1995)
Jakobson, M.V.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys. 81(1), 39–88 (1981)
Jones, P.W., Smirnov, S.K.: Removability theorems for Sobolev functions and quasiconformal maps. Ark. Mat. 38(2), 263–279 (2000)
Kallunki, S., Koskela, P.: Exceptional sets for the definition of quasiconformality. Am. J. Math. 122(4), 735–743 (2000)
Kozlovski, O., Shen, W., van Strien, S.: Rigidity for real polynomials. Ann. Math. (2) 165(3), 749–841 (2007)
Koskela, P.: Old and new on the quasihyperbolic metric. In: Quasiconformal Mappings and Analysis (Ann Arbor, MI, 1995), pp. 205–219. Springer, New York (1998)
Levin, G., van Strien, S.: Local connectivity of the Julia set of real polynomials. Ann. Math. (2) 147(3), 471–541 (1998)
Mañé, R., Sad, P., Sullivan, D.: On the dynamics of rational maps. Ann. Sci. Éc. Norm. Supér., IV. Sér. 16(2), 193–217 (1983)
Mañé, R.: The Hausdorff dimension of invariant probabilities of rational maps. In: Dynamical Systems (Valparaiso, 1986). Lect. Notes Math., vol. 1331, pp. 86–117. Springer, Berlin, New York (1988)
Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge (1995)
McMullen, C.T.: Complex Dynamics and Renormalization. Princeton University Press, Princeton, NJ (1994)
McMullen, C.T.: Hausdorff dimension and conformal dynamics. II. Geometrically finite rational maps. Comment. Math. Helv. 75(4), 535–593 (2000)
Nowicki, T., van Strien, S.: Invariant measures exist under a summability condition for unimodal maps. Invent. Math. 105(1), 123–136 (1991)
Pommerenke, C.: Boundary Behavior of Conformal Maps. Springer, New York (1992)
Przytycki, F.: On measure and Hausdorff dimension of Julia sets of holomorphic Collet–Eckmann maps. In: International Conference on Dynamical Systems (Montevideo, 1995), pp. 167–181. Longman, Harlow (1996)
Przytycki, F.: Iterations of holomorphic Collet–Eckmann maps: Conformal and invariant measures. Appendix: On non-renormalizable quadratic polynomials. Trans. Am. Math. Soc. 350(2), 717–742 (1998)
Przytycki, F.: Conical limit set and Poincaré exponent for iterations of rational functions. Trans. Am. Math. Soc. 351(5), 2081–2099 (1999)
Przytycki, F., Rohde, S.: Rigidity of holomorphic Collet–Eckmann repellers. Ark. Mat. 37(2), 357–371 (1999)
Rees, M.: Positive measure sets of ergodic rational maps. Ann. Sci. Éc. Norm. Supér., IV. Sér. 19(3), 383–407 (1986)
Shishikura, M.: The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets. Ann. Math. (2) 147(2), 225–267 (1998)
Smirnov, S.: Symbolic dynamics and Collet–Eckmann conditions. Int. Math. Res. Not. 2000(7), 333–351 (2000)
Smith, W., Stegenga, D.A.: Hölder domains and Poincaré domains. Trans. Am. Math. Soc. 319(1), 67–100 (1990)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton, NJ (1970)
Sullivan, D.: Conformal dynamical systems. In: Geometric Dynamics (Rio de Janeiro, 1981), pp. 725–752. Springer, Berlin (1983)
Tsujii, M.: Positive Lyapunov exponents in families of one-dimensional dynamical systems. Invent. Math. 111(1), 113–137 (1993)
Urbański, M.: Measures and dimensions in conformal dynamics. Bull. Am. Math. Soc. 40, 281–321 (2003)
Walters, P.: Ergodic Theory – Introductory Lectures. Lect. Notes Math., vol. 458. Springer, Berlin, New York (1975)
Whyburn, G.T.: Analytic Topology. Am. Math. Soc., Providence, RI (1963)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Graczyk, J., Smirnov, S. Non-uniform hyperbolicity in complex dynamics . Invent. math. 175, 335–415 (2009). https://doi.org/10.1007/s00222-008-0152-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-008-0152-8