A flower structure of backward flow invariant domains for semigroups

Abstract

In this paper, we study conditions which ensure the existence of backward flow invariant domains for semigroups of holomorphic selfmappings 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 Ω ⊂ D such that each element of S is an automorphism of Ω, in other words, such that S forms a one-parameter group on Ω.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.

Asymptotic Behavior of One-Parameter Semigroups and Rigidity of Holomorphic Generators

“…For semigroups of parabolic type this inequality just shows that the disks (horocycles) D(τ, k) = {z ∈ Δ : d(z, τ) < k}, k > 0, internally tangent to ∂Δ at τ , are F t -invariant for each t ≥ 0. Also, it can be shown (see [14] and Section 5 below) that if τ ∈ ∂Δ is a boundary regular null point of f ∈ G(Δ), different from the Denjoy-Wolff point of the semigroup S = {F t (z)} t≥0 generated by f (i.e., f (τ ) < 0), then there is a nonempty, simply connected domain Ω in Δ such that S can be extended to a one-parameter group of automorphisms of Ω. So, for z ∈ Ω, inequality (1.10) can be useful for understanding the asymptotic behavior of the trajectory {F t (z)} t∈R , as t tends to −∞.…”

“…First we mention the following notion introduced in [14]. The following result [14] asserts that the existence of a (backward) flowinvariant domain is equivalent to the existence of a boundary regular null point of the generator (or, what is one and the same, the existence of a repelling fixed point of the generated semigroup).…”

“…First, we suppose that the point τ = 1 is the Denjoy-Wolff point of S. Denote β = f (1). It was shown in [14] that the mapping ϕ satisfies…”

“…Using Lemma 3 in [14], one can show that there is a positive measure σ on ∂Δ such that the function h admits the representation…”

“…We answer this question in Section 5 by employing the so-called backward flow-invariant domains introduced in [14].…”