On the spectral value of Semigroups of Holomorphic Functions

Abstract

Let (φt) t≥0 be a semigroup of holomorphic self-maps of the unit disk D with Denjoy-Wolff point τ = 1. The angular derivative is φ t (1) = e −λt , where λ ≥ 0 is the spectral value of (φt). If λ > 0 the semigroup is hyperbolic, otherwise it is parabolic. Suppose K is a compact non-polar subset of D with positive logarithmic capacity. We specify the type of the semigroup by examining the asymptotic behavior of φt(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.