Semigroups of holomorphic functions and condenser capacity

Abstract

Suppose $$\left( \phi _t \right) _{t \ge 0}$$ is a semigroup of holomorphic functions in the unit disk $$\mathbb {D}$$ with Denjoy–Wolff point $$\tau =1$$. Suppose K is a compact subset of $$\mathbb {D}$$. We prove that the capacity of the condenser $$(\mathbb {D}, \phi _t(K) )$$ is a decreasing function of t. Moreover, we study its asymptotic behavior as $$t \rightarrow + \infty $$ in relation with the type of the semigroup.