On the eigenvalues of the infinitesimal generator of a semigroup of composition operators

Suppose that (ϕt) is a one-parameter semigroup of holomorphic self-maps of the unit disk with associated planar domain Ω. Let (Tt) be the corresponding semigroup of composition operators on the classical Hardy space Hp. When the semigroup (ϕt) is hyperbolic, we describe the point spectrum of the infinitesimal generator of (Tt) in terms of the geometry of Ω. The proofs involve various estimates of harmonic measure. We also present various properties of the point spectrum in the parabolic case.