On the spectral value of Semigroups of Holomorphic Functions

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.