The dynamics of hyperbolic rational maps with Cantor Julia sets

Abstract

Let $f:\hat{\mathbb C}\to\hat{\mathbb C}$ be a hyperbolic rational map of degree $d\ge2$ on the Riemann sphere. We give several conditions which are equivalent to the condition for the Julia set $J_f$ to be a Cantor set. It has been known that $J_f$ is a Cantor sets if and only if there exists a positive integer $n>0$ such that $\bar{f^{-n}(U)}\subset U$ for some open topological disc $U$ containing no critical values. Let $n_f$ denote the minimal positive integer satisfying the above. The problem is whether $n_f=1$ or not. Let $S_d$ denote the shift locus of rational maps of degree $d$. We show that $n_f=1$ for generic $f\in S_d$ and that there is a rational map $\bar f\in S_4$ with $n_{\bar f}=2$. We also prove that $S_d$ is connected using the generic case result. In particular, generic hyperbolic rational maps of degree $d$ with Cantor Julia sets are qc-conjugate to each other.