  Suppose that $f$ is a transcendental entire function, $V \subsetneq
\mathbb{C}$ is a simply connected domain, and $U$ is a connected component of
$f^{-1}(V)$. Using Riemann maps, we associate the map $f \colon U \to V$ to an
inner function $g \colon \mathbb{D} \to \mathbb{D}$. It is straightforward to
see that $g$ is either a finite Blaschke product, or, with an appropriate
normalisation, can be taken to be an infinite Blaschke product.
  We show that when the singular values of $f$ in $V$ lie away from the
boundary, there is a strong relationship between singularities of $g$ and
accesses to infinity in $U$. In the case where $U$ is a forward-invariant Fatou
component of $f$, this leads to a very significant generalisation of earlier
results on the number of singularities of the map $g$.
  If $U$ is a forward-invariant Fatou component of $f$ there are currently very
few examples where the relationship between the pair $(f, U)$ and the function
$g$ have been calculated. We study this relationship for several well-known
families of transcendental entire functions.
  It is also natural to ask which finite Blaschke products can arise in this
way, and we show the following: For every finite Blaschke product $g$ whose
Julia set coincides with the unit circle, there exists a transcendental entire
function $f$ with an invariant Fatou component such that $g$ is associated to
$f$ in the above sense. Furthermore, there exists a single transcendental
entire function $f$ with the property that any finite Blaschke product can be
arbitrarily closely approximated by an inner function associated to the
restriction of $f$ to a wandering domain.
