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. 
Mark Elin, David Shoikhet and Lawrence Zalcman