Backward orbits in the unit ball

TL;DR

This paper investigates backward orbits in the unit ball for holomorphic self-maps, establishing their convergence properties near boundary fixed points and introducing a canonical pre-model that characterizes these orbits.

Contribution

It proves the existence and uniqueness of backward orbits with specific convergence behavior and constructs a canonical pre-model for boundary repelling fixed points.

Findings

Backward orbits converge to boundary repelling fixed points with step log λ.

Any two backward orbits to the same fixed point remain at finite distance.

A unique canonical pre-model is associated with each boundary repelling fixed point.

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 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].