Horoballs and iteration of holomorphic maps on bounded symmetric domains

TL;DR

This paper generalizes Wolff's theorem to bounded symmetric domains, constructing invariant horoballs at boundary points and analyzing the boundary behavior of iterates of fixed-point free holomorphic maps.

Contribution

It introduces a family of invariant convex domains at boundary points and characterizes horoballs as invariant domains, extending classical results to infinite-dimensional settings.

Findings

Existence of invariant convex domains at boundary points.

Construction of horoballs as invariant domains in finite rank domains.

Boundary accumulation of iterates' limit functions in a single boundary component.

Abstract

Given a fixed-point free compact holomorphic self-map $f$ on a bounded symmetric domain $D$, which may be infinite dimensional, we establish the existence of a family $\{H(\xi, \lambda)\}_{\lambda >0}$ of convex $f$-invariant domains at a point $\xi$ in the boundary $\partial D$ of $D$, which generalises completely Wolff's theorem for the open unit disc in $\mathbb{C}$. Further, we construct horoballs at $\xi$ and show that they are exactly the $f$-invariant domains when $D$ is of finite rank. Consequently, we show in the latter case that the limit functions of the iterates $(f^n)$ with weakly closed range all accumulate in one single boundary component of $\partial D$.