Backward orbits in the unit ball

Abstract

We show that if f : B q → B q f\colon \mathbb {B}^q\to \mathbb {B}^q is a holomorphic self-map of the unit ball in C q \mathbb {C}^q and ζ ∈ ∂ B q \zeta \in \partial \mathbb {B}^q is a boundary repelling fixed point with dilation λ > 1 \lambda >1 , then there exists a backward orbit converging to ζ \zeta with step log ⁡ λ \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 ( B k , ℓ , τ ) (\mathbb {B}^k,\ell , \tau ) associated with ζ \zeta where 1 ≤ k ≤ q 1\leq k\leq q , τ \tau is a hyperbolic automorphism of B k \mathbb {B}^k , and whose image ℓ ( B k ) \ell (\mathbb {B}^k) is precisely the set of starting points of backward orbits with bounded step converging to ζ \zeta . This answers questions of Ostapyuk (2011) and the first author (2015, 2017).