Backward orbits in the unit ball

Abstract

We show that, if $f:{\mathbb{B}}^q\to {\mathbb{B}}^q$ 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 $1\le k\le q$, $\tau$ 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].

9 pages