Regular variation, tological dynamics, and the uniform boundedness theorem

摘要: In the metrizable topological groups context, a semi-direct product construction provides canonical multiplicative representation for arbitrary continuous flows. This implies, modulo metric differences, equivalence of natural flow formalization regular variation N. H. Bingham and A. J. Ostaszewski in [Topological variation: I. Slow variation, [to appear Topology its Applications]with B. Bajsanski Karamata group formulation [Regularly varying functions principle egui-continuity, Publ. Ramanujan Inst. 1 (1968/1969), 235-246]. consequence, theorems concerning subgroup actions may be lifted to setting. Thus, Bajsanski-Karamata Uniform Boundedness Theorem (UBT), as it applies cocycles Baire cases, reformulated refined hold under Baire-style Caratheodory conditions. Its connection classical UBT, due Stefan Banach Hugo Steinhaus, is clarified. An application algebras given.