Commuting semigroups of holomorphic mappings

• Autores: Mark Elin, Marina Levenshtein, Simeon Reich, David Shoikhet
• Localización: Mathematica scandinavica, ISSN 0025-5521, Vol. 103, Nº 2, 2008, págs. 295-319
• Idioma: inglés
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• Resumen

  • 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.