Horoballs and iteration of holomorphic maps on bounded symmetric domains

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 $\mathrm{\partial }D$ 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 $\mathrm{\partial }D$.