Computational Methods and Function Theory 1 (2001), No. 1, 89--97
Copyright Heldermann Verlag 2001
Bernd O. Stratmann
bos@maths.st-and.ac.uk, Mathematical Institute, University of St Andrews, St Andrews KY16 9SS, Scotland.
Mariusz Urbanski
urbanski@unt.edu, Department of Mathematics, University of North Texas, Denton, TX 76203-1430, U.S.A.
Michel Zinsmeister
Michel.Zinsmeister@labomath.univ-orleans.fr, MAPMO, Matématiques, Universite d'Orleans, BP 6759 45067 Orléans Cedex, France.
In this paper we consider rational functions $f\colon \oc \to \oc$ with parabolic and critical points contained in their Julia sets $J(f)$ such that $$ \sum_{n=1}^\infty|(f^n)'(f(c))|^{-1}<\infty $$ for each critical point $c \in J(f)$. We calculate the Hausdorff dimensions of subsets of $J(f)$ consisting of elements $z$ for which $$ \inf\{\dist(f^n(z),\Crit(f)):\, n\ge 0\}>0 $$ and which are well-approximable by backward iterates of the parabolic periodic points of $f$.
Keywords: Rational functions, critical phenomena, ergodic theory, fractal geometry, Diophantine analysis.
MSC 2000: 37A45, 30C99, 30D40, 11K36.
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