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 subset of C-2 whose canonical Kobayashi hyperbolic semi-model is given by an elliptic automorphism of the disc D subset of 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 Omega in the bidisc D x H that is forward invariant and absorbing for the map (z, w) bar right arrow (e(i0) z, w + 1), where H subset of 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 B-2) if an additional quantitative geometric condition is satisfied.