Semigroup-fication of univalent self-maps of the unit disc

Let f be a univalent self-map of the unit disc. We introduce a technique, that we call semigroup-fication, which allows to construct a continuous semigroup (ϕt) of holomorphic self-maps of the unit disc whose time one map ϕ1 is, in a sense, very close to f. The semigroup-fication of f is of the same type as f (elliptic, hyperbolic, parabolic of positive step or parabolic of zero step) and there is a one-to-one correspondence between the set of boundary regular fixed points of f with a given multiplier and the corresponding set for ϕ1. Moreover, in case f (and hence ϕ1) has no interior fixed points, the slope of the orbits converging to the Denjoy–Wolff point is the same. The construction is based on holomorphic models, localization techniques and Gromov hyperbolicity. As an application of this construction, we prove that in the non-elliptic case, the orbits of f converge non-tangentially to the Denjoy–Wolff point if and only if the Koenigs domain of f is “almost symmetric” with respect to vertical lines