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Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: On non-renormalizable quadratic polynomials. (English) Zbl 0892.58063
Summary: 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 $$\text{HD}_{\text{ess}} (J)=\alpha_{0}=\text{HD} (J)$$.
Here $$\text{HD}_{\text{ess}} (J)$$ is the so-called essential, dynamical or hyperbolic dimension, $$\text{HD} (J)$$ is the 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 $$\text{HD}(J)$$.
We prove ergodicity of every $$\alpha$$-conformal measure on $$J$$ assuming $$f$$ has one critical point $$c\in J$$, not 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 $$\text{HD}_{\text{ess}} (J)=\text{HD} (J)$$ also for every non-renormalizable quadratic polynomial $$z\mapsto z^{2}+c$$ with $$c$$ not in the main cardioid in the Mandelbrot set.

##### MSC:
 37F99 Dynamical systems over complex numbers 37A99 Ergodic theory 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) 28A78 Hausdorff and packing measures
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