On the rate of convergence of hyperbolic semigroups of holomorphic functions

Abstract

Let { ϕ t } t ⩾ 0 be a semigroup of holomorphic self-maps of the unit disk. We assume that the Denjoy–Wolff point of the semigroup is the point 1; so 1 is the unique attractive boundary fixed point of the semigroup. We further assume that for all t > 0 , ϕ t ' ( 1 ) 1 (angular derivative), namely the semigroup is hyperbolic. We prove then that the rate of convergence of the semigroup to the point 1, as t → + ∞ , is exponential with exponent arbitrarily close to - π t / ν ( Ω ) , where ν ( Ω ) is the width of the smallest strip containing the associated planar domain Ω. We also prove that the trajectories of the semigroup approach the point 1 in a monotonic way.