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Erschienen in:
2020 | OriginalPaper | Buchkapitel
12. Boundary Fixed Points and Infinitesimal Generators
verfasst von : Filippo Bracci, Manuel D. Contreras, Santiago Díaz-Madrigal
Erschienen in: Continuous Semigroups of Holomorphic Self-maps of the Unit Disc
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Abstract
In this chapter, we continue the analysis of the boundary behavior of semigroups, and we concentrate on boundary fixed points. We show that if a positive iterate of a semigroup has a boundary fixed point (in the sense of non-tangential limit), then such a point is indeed fixed for all the iterates of the semigroup. Moreover, the boundary dilation coefficients of the semigroup at a boundary fixed point are either identically infinity or they are of the form \(e^{-\lambda t}\) for some \(\lambda \in \mathbb R\). The latter is the case of boundary regular fixed points. We prove that there is a one-to-one correspondence between boundary regular fixed points and boundary regular critical points of infinitesimal generators.