Poles of the Infinitesimal Generators
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In this chapter, we introduce the notion of regular (boundary) poles for infinitesimal generators of semigroups. We characterize such regular poles in terms of \(\beta \)-points (i.e. pre-images of values with a positive (Carleson-Makarov) \(\beta \)-numbers) of the associated semigroup and of the associated Koenigs function. We also define a natural duality operation in the cone of infinitesimal generators and show that the regular poles of an infinitesimal generator correspond to the regular critical points of the dual generator. Finally we apply such a construction to study radial multi-slits and give an example of a non-isolated radial slit semigroup whose tip has not a positive (Carleson-Makarov) \(\beta \)-number.