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

• Published in: Transactions of the American Mathematical Society 1998-02, Vol.350 (2), p.717-742
• Main Author: Przytycki, Feliks
• Format: Article
• Language: English
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-98-01890-X

We prove that for every rational map on the Riemann sphere f:\overline{\C} \to \overline{\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)=\mya _{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 \mya _{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 \mya -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}