Well-Approximable Points for Julia Sets with Parabo and Critical Points

Abstract

In this paper we consider rational functions \(f: \overline {\rm C} \rightarrow \overline {\rm C}\) with parabolic and critical points contained in their Julia sets J(f) such that $$\sum^{\infty}_{n = 1}|(f^{n})^\prime(f(c))|^{-1}< \infty$$ for each critical point c ∈ J(f). We calculate the Hausdorff dimensions of subsets of J(f) consisting of elements z for which $$\inf \left\{ {dist\left( {f^n \left( z \right), Crit\left( f \right)} \right):n \geqslant 0} \right\} > 0$$ and which are well-approximable by backward iterates of the parabolic periodic points of f.