Annales Academię Scientiarum Fennicę

Mathematica

Volumen 33, 2008, 3-34

# A FLOWER STRUCTURE OF BACKWARD FLOW
INVARIANT DOMAINS FOR SEMIGROUPS

## Mark Elin, David Shoikhet and Lawrence Zalcman

ORT Braude College, Department of Mathematics

P.O. Box 78, Karmiel 21982, Israel; mark.elin 'at' gmail.com

ORT Braude College, Department of Mathematics

P.O. Box 78, Karmiel 21982, Israel; davs27 'at' netvision.net.il

Bar-Ilan University, Department of Mathematics

52900 Ramat-Gan, Israel; zalcman 'at' macs.biu.ac.il

**Abstract.**
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
*S* on *D*, find a simply connected subset \Omega\subset
*D* such that each element of *S* is an automorphism of
\Omega, in other words, such that *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.

**2000 Mathematics Subject Classification:**
Primary 37C10, 30C45.

**Key words:**
Semigroups, holomorphic mappings,
generators, fixed points.

**Reference to this article:** M. Elin, D. Shoikhet and L. Zalcman:
A flower structure of backward flow invariant domains for
semigroups. Ann. Acad. Sci. Fenn. Math. 33 (2008), 3-34.

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