COMMUTING SEMIGROUPS OF HOLOMORPHIC MAPPINGS

Let S1 = {Ft}t≥0 and S2 = {Gt}t≥0 be two continuous semigroups of holomorphic self-mappings of the unit disk Δ = {z : |z| < 1} generated by f and g, respectively. We present conditions on the behavior of f (or g) in a neighborhood of a fixed point of S1 (or S2), under which the commutativity of two elements, say, F1 and G1 of the semigroups implies that the semigroups commute, i.e., Ft ○ Gs = Gs ○ Ft for all s, t ≥ 0. As an auxiliary result, we show that the existence of the (angular or unrestricted) n-th derivative of the generator f of a semigroup {Ft}t≥0 at a boundary null point of f implies that the corresponding derivatives of Ft, t ≥ 0, also exist, and we obtain formulae connecting them for n = 2, 3.