Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: On non-renormalizable quadratic polynomials

Abstract:

We prove that for every rational map on the Riemann sphere $f:\overline {\mathbb {C}} \to \overline {\mathbb {C}}$, if for every $f$-critical point $c\in J$ whose forward trajectory does not contain any other critical point, the growth of $|(f^{n})’(f(c))|$ is at least of order $\exp Q \sqrt n$ for an appropriate constant $Q$ as $n\to \infty$, then $\operatorname {HD}_{\operatorname {ess}} (J)=\alpha _{0}=\operatorname {HD} (J)$. Here $\operatorname {HD}_{\operatorname {ess}} (J)$ is the so-called essential, dynamical or hyperbolic dimension, $\operatorname {HD} (J)$ is Hausdorff dimension of $J$ and $\alpha _{0}$ is the minimal exponent for conformal measures on $J$. If it is assumed additionally that there are no periodic parabolic points then the Minkowski dimension (other names: box dimension, limit capacity) of $J$ also coincides with $\operatorname {HD}(J)$. We prove ergodicity of every $\alpha$-conformal measure on $J$ assuming $f$ has one critical point $c\in J$, no parabolic, and $\sum _{n=0}^{\infty }|(f^{n})’(f(c))|^{-1} <\infty$. Finally for every $\alpha$-conformal measure $\mu$ on $J$ (satisfying an additional assumption), assuming an exponential growth of $|(f^{n})’(f(c))|$, we prove the existence of a probability absolutely continuous with respect to $\mu$, $f$-invariant measure. In the Appendix we prove $\operatorname {HD}_{\operatorname {ess}} (J)=\operatorname {HD} (J)$ also for every non-renormalizable quadratic polynomial $z\mapsto z^{2}+c$ with $c$ not in the main cardioid in the Mandelbrot set.