Denjoy–Wolff theory for finite-dimensional bounded symmetric domains

Let B be a finite-dimensional bounded symmetric domain and f: B→ B be a holomorphic map having no fixed point in B. For subsequential limits, g, of (fⁿ) , we establish conditions, in terms of the Wolff point, ξ, of f, on which boundary components of B can contain g(B). We extend Hervé’s 1954 theorem on the bidisc to any finite product of bounded symmetric domains, namely if B= <inf>B1</inf>× ⋯ × <inf>Bn</inf> and ξ= (<inf>ξ1</inf>, … , <inf>ξn</inf>) then there exists d= (<inf>d1</inf>, … , <inf>dn</inf>) ∈ ∂B, satisfying <inf>K<inf>di</inf></inf>¯ ∩ <inf>K<inf>ξi</inf></inf>¯ ≠ ∅ , such that πi(g(B))⊆di+B0(di),where <inf>Kx</inf> denotes the affine boundary component of x, <inf>πi</inf> is projection on the ith coordinate and <inf>B0</inf>(<inf>di</inf>) is a bounded symmetric subdomain of <inf>Bi</inf>. This simplifies if <inf>ξi</inf> is extreme, and even more so if <inf>Bi</inf> is a Hilbert ball.