Common fixed points of commuting holomorphic maps in the unit ball of C^n

Let B^n be the unit ball of ℂ^n (n > 1). We prove that if f, g ∈ Hol(B^n,B^n) are holomorphic self-maps of B^n such that f o g = g o f, then f and g have a common fixed point (possibly at the boundary, in the sense of K-limits). Furthermore, if f and g have no fixed points in B^n, then they have the same Wolff point, unless the restrictions of f and g to the one-dimensional complex affine subset of B^n determined by the Wolff points of f and g are commuting hyperbolic automorphisms of that subset.