A Counterexample to Parabolic Dichotomies in Holomorphic Iteration

We give an example of a parabolic holomorphic self-map f of the unit ball 𝔹^2⊂ℂ^2 whose canonical Kobayashi hyperbolic semi-model is given by an elliptic automorphism of the disc 𝔻⊂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 𝔹^2 is radically different from parabolic dynamics in the disc. The example is obtained via a geometric method, embedding the ball 𝔹^2 as a domain Ω in the bidisc 𝔻×ℍ that is forward invariant and absorbing for the map (z,w)↦ (e^iθz,w+1) , where ℍ⊂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 𝔹^2 ) if an additional quantitative geometric condition is satisfied.