CDAM Research Report, LSE-CDAM-2008-11
Topological regular variation: I. Slow variation
N. H. Bingham and A. J. OstaszewskiMotivated by the Category Embedding Theorem, as applied to convergent automorphisms BOst11, we unify and extend the multivariate regular variation literature by a reformulation in the language of topological dynamics. Here the natural setting are metric groups, seen as normed groups (mimicking normed vector spaces). We briefly study their properties as a preliminary to establishing that the Uniform Convergence Theorem (UCT) for Baire, group-valued slowly-varying functions has two natural metric generalizations linked by the natural duality between a homogenous space and its group of homeomorphisms. Each is derivable from the other by duality. One of these explicitly extends the (topological) group version of UCT due to Bajanski and Karamata BajKar from groups to flows on a group. A multiplicative representation of the flow derived in Ost-knit demonstrates equivalence of the flow with the earlier group formulation. In companion papers we extend the theory to regularly varying functions: we establish the calculus of regular variation in BOst14 and we extend to locally compact, σ-compact groups the fundamental theorems on characterization and representation BOst15. In BOst16, working with topological flows on homogeneous spaces, we identify an index of regular variation, which in a normed-vector space context may be specified using the Riesz representation theorem, and in a locally compact group setting may be connected with Haar measure.
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