Backward orbits in the unit ball

Abstract:

We show that if $f\colon \mathbb {B}^q\to \mathbb {B}^q$ is a holomorphic self-map of the unit ball in $\mathbb {C}^q$ and $\zeta \in \partial \mathbb {B}^q$ is a boundary repelling fixed point with dilation $\lambda >1$, then there exists a backward orbit converging to $\zeta$ with step $\log \lambda$. Morever, any two backward orbits converging to the same boundary repelling fixed point stay at finite distance. As a consequence there exists a unique canonical premodel $(\mathbb {B}^k,\ell , \tau )$ associated with $\zeta$ where $1\leq k\leq q$, $\tau$ is a hyperbolic automorphism of $\mathbb {B}^k$, and whose image $\ell (\mathbb {B}^k)$ is precisely the set of starting points of backward orbits with bounded step converging to $\zeta$. This answers questions of Ostapyuk (2011) and the first author (2015, 2017).