BACKWARD ORBITS IN THE UNIT BALL

We show that if f : B-q -> B-q is a holomorphic self-map of the unit ball in C-q and zeta is an element of partial derivative 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 (B-k, l, t) associated with zeta where 1 <= k <= q, t is a hyperbolic automorphism of B-k, and whose image l(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).