  In this paper, we study conditions which ensure the existence of backward
flow invariant domains for semigroups of holomorphic self-mappings of a simply
connected domain $D$. More precisely, the problem is the following. Given a
one-parameter semigroup $\mathcal S$ on $D$, find a simply connected subset
$\Omega\subset D$ such that each element of $\mathcal S$ is an automorphism of
$\Omega$, in other words, such that $\mathcal S$ forms a one-parameter group on
$\Omega$.
  On the way to solving this problem, we prove an angle distortion theorem for
starlike and spirallike functions with respect to interior and boundary points.
