Topologies on the set of iterates of a holomorphic function in infinite dimensions

Let f : B 7 → B be a compact holomorphic map on the open unit ball B of a complex Banach space Z in possibly infinite dimensions, where f compact means f (B) is relatively compact. The sequence of iterates (f n)n of f (where f n := f ◦ f n−1, f 1 := f ) is of much interest and, since it generally does not converge, the set of all its subsequential limits for a particular topology have been studied instead. We prove that the pointwise limit of any subsequence of (f n)n is itself a holomorphic function. We show, in fact, that on the set of iterates {f n : n ∈ N} the topology of pointwise convergence on B coincides with any finer topology on the space, H(B, Z), of holomorphic functions from B to Z. In particular, it coincides with both the compact open topology and the topology of local uniform convergence on B. Despite the fact that these topologies are not first countable, we prove that the set of accumulation points of (f n)n coincides with the set of all its subsequential limits.