On the rate of convergence of hyperbolic semigroups of holomorphic functions

betsakos, dimitrios;

Let {φt}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, φt(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 −πt/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.