Topological regular variation. I. Slow variation

Motivated 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 brie‡y study their properties as a preliminary to establishing that the Uniform Convergence Theorem (UCT) for Baire, group-valued slowlyvarying 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 Bajšanski and Karamata [BajKar] from groups to ‡ows on a group. A multiplicative representation of the ‡ow derived in [Ost-knit] demonstrates equivalence of the ‡ow with the earlier group formulation. In