A counterexample to parabolic dichotomies in holomorphic iteration

Abstract

We give an example of a parabolic holomorphic self-map $f$ of the unit ball $\mathbb B^2\subset \mathbb C^2$ whose canonical Kobayashi hyperbolic semi-model is given by an elliptic automorphism of the disc $\mathbb D\subset \mathbb C$, 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 $\\mathbb B^2$ is radically different from parabolic dynamics in the disc. The example is obtained via a geometric method, embedding the ball $\mathbb B^2$ as a domain $\Omega$ in the bidisc $\\mathbb D\times \mathbb{H}$ that is forward invariant and absorbing for the map $(z,w)\mapsto (e^{i\theta}z,w+1)$, where $\mathbb H\subset \mathbb C$ denotes the right half-plane. We also show that a complete Kobayashi hyperbolic domain $\Omega$ with such properties cannot be Gromov hyperbolic w.r.t. the Kobayashi distance (hence, it cannot be biholomorphic to $\\mathbb B^2$) if an additional quantitative geometric condition is satisfied.