Commuting semigroups of holomorphic mappings - Elin, Mark; Levenshtein, Marina; Reich, Simeon; Shoikhet, David (MATHEMATICA SCANDINAVICA, 2008)
Let S_1={F_t}_t≥ 0 and S_2={G_t}_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 S_1 (or S_2), under which the commutativity of two elements, say, F_1 and G_1 of the semigroups implies that the semigroups commute, i.e., F_t° G_s=G_s° F_t 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 {F_t}_t≥ 0 at a boundary null point of f implies that the corresponding derivatives of F_t, t≥ 0, also exist, and we obtain formulae connecting them for n=2,3.