On the spectral value of Semigroups of Holomorphic Functions

TL;DR

This paper investigates the spectral value of semigroups of holomorphic functions on the unit disk, linking it to potential theory and the asymptotic behavior of the semigroup on compact sets.

Contribution

It provides a novel characterization of the spectral value using potential theoretic quantities and relates it to the asymptotic behavior of the semigroup.

Findings

Spectral value determines if the semigroup is hyperbolic or parabolic.

Asymptotic behavior of the semigroup on compact sets reveals spectral properties.

Representation of spectral value via harmonic measure, Green function, and capacity.

Abstract

Let $(\phi_t)_{t \geq 0}$ be a semigroup of holomorphic self-maps of the unit disk $\mathbb{D}$ with Denjoy-Wolff point $\tau=1$. The angular derivative is $\phi_t^{\prime}(1)= e^{-\lambda t}$, where $\lambda \geq 0$ is the spectral value of $(\phi_t)$. If $\lambda>0$ the semigroup is hyperbolic, otherwise it is parabolic. Suppose $K$ is a compact non-polar subset of $\mathbb{D}$ with positive logarithmic capacity. We specify the type of the semigroup by examining the asymptotic behavior of $\phi_t(K)$. We provide a representation of the spectral value of the semigroup with the use of several potential theoretic quantities e.g. harmonic measure, Green function, extremal length, condenser capacity.