Iterates of a compact holomorphic map on a finite rank homogeneous ball

We study iterates, fn, of a fixed-point free compact holomorphic map f : B → B where B is the open unit ball of any JB∗-triple of finite rank. These spaces include L(H,K), H,K Hilbert, dim(H) arbitrary, dim(K) < 1, or any classical Cartan factor or C∗-algebra of finite rank. Apart from the Hilbert ball, the sequence of iterates (fn)n does not generally converge (locally uniformly on B) and little is known of accumulation points. We present a short proof of a Wolff theorem for B and establish key properties of the resulting f-invariant subdomains. We define a concept of closed convex holomorphic hull, Ch(x), for x ϵ ∂B and prove the following. There is a unique tripotent u in ∂B such that all constant subsequential limits of (fn)n lie in Ch(u). As a consequence we also get a short proof of the classical Hilbert ball results.