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(ξ,λ)}λ>0 of convex f-invariant domains at a point ξ in the boundary ∂D of D, which generalises completely Wolff's theorem for the open unit disc in C. Further, we construct horoballs at ξ 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 (fn) with weakly closed range all accumulate in one single boundary component of ∂D.