Trajectories of semigroups of holomorphic functions and harmonic measure

Abstract

We give an explicit relation between the slope of the trajectory of a semigroup of holomorphic functions and the harmonic measure of the associated planar domain Ω. We use this to construct a semigroup whose slope is an arbitrary interval in [−π/2, π/2]. The same method is used for the slope of a backward trajectory approaching a super-repulsive fixed point.

Characterizations of convergence by a given set of angles in simply connected domains

“…However, in general, the convegence can be tangential or non-tangential and the cluster set of arg(1 − τ φ t (z)), as t → +∞, can be either a singleton {θ} or a set [θ 1 , θ 2 ], but it is always a compact connected subset of [− π 2 , π 2 ]. In [3,11] examples are constructed where the slope is the whole interval [− π 2 , π 2 ], while in [6,16] examples are constructed where the slope is a closed subinterval of (− π 2 , π 2 ). Furthermore, in [5], the authors find geometric conditions that guarantee that the slope reduces to {0}.…”

Angular Extents and Trajectory Slopes in the Theory of Holomorphic Semigroups in the Unit Disk

“…In contrast, for parabolic semigroups of zero hyperbolic step, the arrival slope set does not have to reduce to a unique point (see [3,6,10,14]). However, according to the following result by the first two authors, it does not depend on the initial point.…”

On the spectral value of Semigroups of Holomorphic Functions

“…Later, one-parameter semigroups have been an issue of scientific interest, especially due to their application in other mathematical areas such as dynamical systems, Markov stochastic processes, etc. Recent advances in this area, as well as, the classical theory of semigroups can be found in [5,7,8,10,13,19].…”