  We prove that if A is the basin of immediate attraction to a periodic
attracting or parabolic point for a rational map f on the Riemann sphere, if
$A$ is completely invariant (i.e. $f^{-1}(A)=A$), and if $\mu$ is an arbitrary
$f$-invariant measure with positive Lyapunov exponents on the boundary of $A$,
then $\mu$-almost every point $q$ in the boundary of $A$ is accessible along a
curve from $A$. In fact we prove the accessability of every "good" $q$ i.e.
such $q$ for which "small neighbourhoods arrive at large scale" under iteration
of $f$. This generalizes Douady-Eremenko-Levin-Petersen theorem on the
accessability of periodic sources.
