A flower structure of backward flow invariant domains for semigroups

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.