CDAM: Computational, Discrete and Applicable Mathematics@LSE

 CDAM Research Report, LSE-CDAM-2008-15

August 2008

Regular variation, topological dynamics, and the Uniform Boundedness Theorem

A. J. Ostaszewski

In the metrizable topological groups context, a direct product construction (mimicking the `action groupoid') provides a multiplicative representation canonical for arbitrary continuous flows. This implies, modulo metric differences, the topological equivalence of the natural, flow setting of regular variation of BOst13 with the Bajšanski and Karamata BajKar group formulation. In consequence topological theorems concerning subgroup actions may be lifted to the flow setting. Thus the Bajšanski-Karamata Uniform Boundedness Theorem (UBT), as it applies to cocycles in the continuous and Baire cases, may be reformulated and refined to hold under Baire-style Carath$eacute;odory conditions. Its connection to the Banach-Steinhaus UBT is clarified. An application to Banach algebras is given.

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