A Denjoy-Wolff theorem for bounded symmetric domains

Abstract

Let D be a bounded symmetric domain of finite rank, realised as the open unit ball of a complex Banach space, which can be infinite dimensional. Given a fixed-point free compact holomorphic map $f:D⟶D$, with iterates$f^n=\underset{\text{n}\text{-times}}{\underbrace{f\circ \cdots \circ f}},$ such that the limit points of one orbit $\{a,f(a),f^2(a),\dots \}$ in an f-invariant horoball of unit hororadius lie in the extended Shilov boundary of D, we show that there is a holomorphic boundary component Γ of D, with closure $\bar{\Gamma }$, such that $\ell \left(D\right)\subset \bar{\Gamma }$ for all subsequential limits $\ell ={\lim }_{k\to \infty }{f}^{n_k}$ of $\left(f^n\right)$.

This generalises the Denjoy-Wolff theorem for a fixed-point free holomorphic self-map f on the disc $D=\{z\in C:|z|<1\}$.