On the spectral value of Semigroups of Holomorphic Functions

Abstract

Let $({\varphi }_t{)}_{t\ge 0}$ be a semigroup of holomorphic self-maps of the unit disk $\mathbb{D}$ with Denjoy-Wolff point $\tau =1$. The angular derivative is ${\varphi }_t^{\prime }(1)=e^{-\lambda t}$, where $\lambda \ge 0$ is the spectral value of $({\varphi }_t)$. 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 ${\varphi }_t(K)$. 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.

17 pages, 4 figures