Commuting holomorphic maps in strongly convex domains

Let D be a bounded strongly convex C^3 domain of ℂ^n. We prove that if f,g ∊ Hol (D,D) are commuting holomorphic self-maps of D, then they have a common fixed point in D (if it belongs to ∂D, we mean fixed in the sense of A'-limits). Furthermore, if f and g have no fixed points in D and f o g = g o f then f and g have the same Wolff point, unless their restrictions to the complex geodesic whose closure contains the Wolff points of f and g, are two commuting (hyperbolic) automorphisms of such geodesic.