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

Let f be a univalent self-map of the unit disc. We introduce technique, that we call semigroup-fication, which allows to construct continuous semigroup (ϕ t ) holomorphic self-maps disc whose time one map ϕ 1 is, in sense, very close f. The semigroup-fication is same type as (elliptic, hyperbolic, parabolic positive step or zero step) and there one-to-one correspondence between set boundary regular fixed points with given multiplier corresponding for . Moreover, case (and hence has no interior points, slope orbits converging Denjoy–Wolff point same. construction based on models, localization techniques Gromov hyperbolicity. As an application this construction, prove non-elliptic case, converge non-tangentially if only Koenigs domain “almost symmetric” respect vertical lines.