Parametric Representations and Boundary Fixed Points of Univalent Self-Maps of the Unit Disk

Abstract

A classical result in the theory of Loewner's parametric representation states that the semigroup U * of all conformal self-maps φ of the unit disk D normalized by φ(0) = 0 and φ ′ (0) > 0 can be obtained as the reachable set of the Loewner -Kufarev control systemwhere the control functions t → G t ∈ Hol(D, C) form a certain convex cone. Here we extend this result to the semigroup U[F ] consisting of all conformal φ : D → D whose set of boundary regular fixed points contains a given finite set F ⊂ ∂D and to its subsemigroup U τ [F ] formed by id D and all φ ∈ U[F ]\ {id D } with the prescribed boundary Denjoy -Wolff point τ ∈ ∂D \ F . This completes the study launched in [29], where the case of interior Denjoy -Wolff point τ ∈ D was considered. 2010 Mathematics Subject Classification. Primary: 30C35, 30C75; Secondary: 30D05, 30C80, 34H05, 37C25.

Infinitesimal generators of semigroups with prescribed boundary fixed points

“…In this section we combine our results with the theory developed in [10,27,28] in order to develop a parametric representation of univalent self-maps ϕ ∈ Hol(D, D) with given boundary regular fixed points based on a Loewner-Kufarev-type ODE. Note that in this case, in contrast to the previous sections, we do not suppose that ϕ is an element of a one-parameter semigroup.…”

“…According to [28,Theorem 2] there exists an evolution family (ϕ s,t ) ⊂ U τ [F] such that ϕ = ϕ 0,1 . Using [10, Theorem 1.1], we see that f (t) := log n k=1 ϕ 0,t (σ k ) is locally absolutely continuous on [0, +∞).…”

Infinitesimal generators of semigroups with prescribed boundary fixed points

“…In this section we combine our results with the theory developed in [12,30,31] in order to develop a parametric representation of univalent self-maps ϕ ∈ Hol(D, D) with given boundary regular fixed points based on a Loewner -Kufarev-type ODE. Note that in this case, in contrast to the previous sections, we do not suppose that ϕ is an element of a one-parameter semigroup.…”

“…According to [31,Theorem 2] there exists an evolution family (ϕ s,t ) ⊂ U τ [F ] such that ϕ = ϕ 0,1 . Using [12, Theorem 1.1], we see that f (t) := log n k=1 ϕ ′ 0,t (σ k ) is locally absolutely continuous on [0, +∞).…”