On the Hausdorff dimension of the Sierpiński Julia sets

Abstract

We estimate the Hausdorff dimension of hyperbolic Julia sets of maps from the well-known family $F_{\lambda,n}(z) = z^n + \lambda/z^n$, $n \ge 2$, $\lambda \in \mathbb{C} \setminus \{0\}$. In particular, we show that $\dim_H J(F_{\lambda,n}) = \mathcal O (1/\ln |\lambda|)$ for large $|\lambda|$, and $\dim_H J(F_{\lambda,n}) = 1 + \mathcal O (1/\ln n)$ for large $n$ in the three cases: when $J(F_{\lambda,n})$ is a Cantor set, a Cantor set of quasicircles and a Sierpinski curve.