Suppose that $f \colon \hat{\mathbb{C}} \to \hat{\mathbb{C}}$ is a rational map of degree $d \ge 2$. Let $z_0$ be a point in the Fatou set $F(f)$. I'm interested in finding an estimate for the distance of $z_0$ to the julia set $J(f)$.
I assume that $z_0$ is attracted to either a super-attracting or attracting fixed point located at $0$. Then we can use either Koenigs or Böttcher coordinates to find a mapping $\varphi \colon \mathcal{A} \to \mathbb{\widehat{C}}$ such that $\varphi(f(z)) = f'(0) \varphi(z)$. If $z_0$ belonged to the immediate attracting basin $\mathcal{A}_0$ of $0$, and if there were no critical points (except possibly $0$) of $f$ inside $\mathcal{A}_0$, we could find an inverse function $\varphi^{-1} \colon \mathbb{D}_r \to \mathcal{A}_0$ from some disc of radius $r$ to the basin. The hyperbolic metric in $\mathbb{D}_r$ would then let us calculate an approximation of the distance to the boundary.
Can we do anything in the precense of critical points or when $z$ is not in the immediate basin of attraction?
PS. My main motivation is in drawing Julia sets on computer, but I'm also interested in the theory.