On the Spectral Value of Semigroups of Holomorphic Functions

Let ( ϕ t ) 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 - λ t , where λ ≥ 0 is the spectral value of ( ϕ t ) . If λ > 0 the semigroup is hyperbolic, otherwise it is parabolic. Suppose K is a compact non-polar subset of D . We specify the type of the semigroup by examining the asymptotic behavior of ϕ 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.