  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.
