2023/10/24 by Leandro Arosio, Filippo Bracci, Arosio, Leandro +3
Mathematics · #32F45 #32H50 #53C23 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Holomorphic and Operator Theory #Mathematics and Applications #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2310.15739
openalex publication_date 2023/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give an example of a parabolic holomorphic self-map f of the unit ball \mathbb B2⊂ \mathbb C2 whose canonical Kobayashi hyperbolic semi-model is given by an elliptic automorphism of the disc \mathbb D⊂ \mathbb C, which can be chosen to be different from the identity. As a consequence, in contrast to the one dimensional case, this provides a first example of a holomorphic self-map of the unit ball which has points with zero hyperbolic step and points with nonzero hyperbolic step, solving an open question and showing that parabolic dynamics in the ball mathbb B2 is radically different from parabolic dynamics in the disc. The example is obtained via a geometric method, embedding the ball \mathbb B2 as a domain Ω in the bidisc mathbb D× ℍ that is forward invariant and absorbing for the map (z,w)↦ (eiθz,w+1), where \mathbb H⊂ \mathbb C denotes the right half-plane. We also show that a complete Kobayashi hyperbolic domain Ω with such properties cannot be Gromov hyperbolic w.r.t. the Kobayashi distance (hence, it cannot be biholomorphic to mathbb B2) if an additional quantitative geometric condition is satisfied.