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 system (Formula Presented.) where the control functions t↦ Gt∈ 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 Uτ[F] formed by idD and all ϕ∈U[F]∖{idD} with the prescribed boundary Denjoy–Wolff point τ∈ ∂D∖ F. This completes the study launched in Gumenyuk, P. (Constr. Approx. 46 (2017), 435–458, https://doi.org/10.1007/s00365-017-9376-4), where the case of interior Denjoy–Wolff point τ∈ D was considered.
Parametric representations and boundary fixed points of univalent self-maps of the unit disk
Gumenyuk P.
2017-01-01
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 system (Formula Presented.) where the control functions t↦ Gt∈ 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 Uτ[F] formed by idD and all ϕ∈U[F]∖{idD} with the prescribed boundary Denjoy–Wolff point τ∈ ∂D∖ F. This completes the study launched in Gumenyuk, P. (Constr. Approx. 46 (2017), 435–458, https://doi.org/10.1007/s00365-017-9376-4), where the case of interior Denjoy–Wolff point τ∈ D was considered.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
