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

Filippo Bracci·Manuel D. Contreras·Santiago Diaz-Madrigal
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Abstract

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 $t$.

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Filippo Bracci, Manuel D. Contreras, Santiago Diaz-Madrigal. 2007-01-29. Aleksandrov-Clark measures and semigroups of analytic functions in the unit disc. https://arxiv.org/abs/math/0701834

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