2020/02/19 by Filippo Bracci, Oliver Roth, Bracci, Filippo +1 · 2 citations
Mathematics · #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2002.08252
openalex publication_date 2020/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let f be a univalent self-map of the unit disc. We introduce a technique, that we call \sl 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 semigrup-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.