Aleksandrov-Clark measures and semigroups of analytic functions in the unit disc

Bracci, Filippo; Contreras, Manuel D.; Diaz-Madrigal, Santiago

Aleksandrov-Clark measures and semigroups of analytic functions in the unit disc

Authors

Bracci, Filippo

Contreras, Manuel D.

Diaz-Madrigal, Santiago

Description

In this paper we prove a formula describing the infinitesimal generator of a continuous semigroup $(\v_t)$ of holomorphic self-maps of the unit disc with respect to a boundary regular fixed point. The result is based on Alexandrov-Clark measures techniques. In particular we prove that the Alexandrov-Clark measure of $(\v_t)$ at a boundary regular fixed points is differentiable (in the weak$^\ast$-topology) with respect to $t$.
10 pages