  We introduce the notion of regular (boundary) poles for infinitesimal
generators of semigroups of holomorphic self-maps of the unit disc. We
characterize such regular poles in terms of $\beta$-points (i.e. pre-images of
values with positive Carleson-Makarov $\beta$-numbers) of the associated
semigroup and of the associated K\"onigs intertwining 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
null poles 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 whose tip
has not a positive Carleson-Makarov $\beta$-number.
