Non-uniform hyperbolicity in complex dynamics

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 Poincare 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 Poincare 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.