Inventiones mathematicae

, 175:335 | Cite as

Non-uniform hyperbolicity in complex dynamics

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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.

Keywords

Hausdorff Dimension Summability Condition Harmonic Measure Blaschke Product Kleinian Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.MathématiqueUniversité de Paris-SudOrsayFrance
  2. 2.Section de MathématiquesUniversité de GenèveGenève 4Switzerland

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