Well-Approximable Points for Julia Sets with Parabolic and Critical Points
Résumé
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 $f$.