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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: On non-renormalizable quadratic polynomials
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by Feliks Przytycki PDF
Trans. Amer. Math. Soc. 350 (1998), 717-742 Request permission

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.
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Additional Information
  • Feliks Przytycki
  • Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8 00 950 Warszawa, Poland
  • MR Author ID: 142490
  • Email: feliksp@impan.impan.gov.pl
  • Received by editor(s): January 18, 1995
  • Received by editor(s) in revised form: July 28, 1995, and June 13, 1996
  • Additional Notes: The author acknowledges support by Polish KBN Grants 210469101 and 2 P301 01307 “Iteracje i Fraktale". He expresses also his gratitude to the Universities at Orleans and at Dijon in France, where parts of this paper were written
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 717-742
  • MSC (1991): Primary 58F23
  • DOI: https://doi.org/10.1090/S0002-9947-98-01890-X
  • MathSciNet review: 1407501