Contact points and fractional singularities for semigroups of holomorphic self-maps of the unit disc

We study boundary singularities which can appear for infinitesimal generators of one-parameter semigroups of holomorphic self-maps of the unit disc. We introduce “regular” fractional singularities and characterize them in terms of the behavior of the associated semigroups and Kœnigs functions. We also provide necessary and sufficient geometric criteria on the shape of the image of the Kœnigs function for having such singularities. In order to do this, we study contact points of semigroups and prove that any contact (not fixed) point of a one-parameter semigroup corresponds to a maximal arc on the boundary to which the associated infinitesimal generator extends holomorphically as a vector field tangent to this arc.