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We introduce the notion of regular (boundary) poles for infinitesimalgenerators of semigroups of holomorphic self-maps of the unit disc. Wecharacterize such regular poles in terms of $\beta$-points (i.e. pre-images ofvalues with positive Carleson-Makarov $\beta$-numbers) of the associatedsemigroup and of the associated K\"onigs intertwining function. We also definea natural duality operation in the cone of infinitesimal generators and showthat the regular poles of an infinitesimal generator correspond to the regularnull poles of the dual generator. Finally we apply such a construction to studyradial multi-slits and give an example of a non-isolated radial slit whose tiphas not a positive Carleson-Makarov $\beta$-number.

Regular poles and $\beta$-numbers in the theory of holomorphic semigroups