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 pre-model $(\mathbb{ B}^k,\ell, \tau)$ 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 in [8] and [3,4].
Categories: math.CV math.DS