2017/02/01 by Gumenyuk, Pavel
#30C35 #30C75 (Primary) #30C80 #30D05 #34H05 #37C25 (Secondary) #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1702.00344
A classical result in the theory of Loewner's parametric representation states that the semigroup \mathfrak U_* of all conformal self-maps ϕ of the unit disk \mathbbD normalized by ϕ(0) = 0 and ϕ'(0) > 0 can be obtained as the reachable set of the Loewner - Kufarev control system (d wt)/(d t)=Gt∘ wt, t\geqslant0, w0=id_\mathbbD, where the control functions t↦ Gt\inHol(\mathbbD,ℂ) form a certain convex cone. Here we extend this result to the semigroup \mathfrak U[F] consisting of all conformal ϕ:\mathbbD→\mathbbD whose set of boundary regular fixed points contains a given finite set F⊂∂\mathbbD and to its subsemigroup \mathfrak Uτ[F] formed by id_\mathbbD and all ϕ∈\mathfrak U[F]∖\id_\mathbbD\ with the prescribed boundary Denjoy - Wolff point τ∈∂\mathbbD∖ F. This completes the study launched in [P. Gumenyuk, Preprint 2016, ArXiv:1603.04043], where the case of interior Denjoy - Wolff point τ∈\mathbbD was considered.