  We study the boundary behaviour of a meromorphic map $f: \mathbb C \to
\widehat{\mathbb C}$ on its invariant simply connected Fatou component $U$. To
this aim, we develop the theory of accesses to boundary points of $U$ and their
relation to the dynamics of $f$. In particular, we establish a correspondence
between invariant accesses from $U$ to infinity or weakly repelling points of
$f$ and boundary fixed points of the associated inner function on the unit
disc. We apply our results to describe the accesses to infinity from invariant
Fatou components of the Newton maps.
