Topological regular variation: I. Slow variation

Abstract

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 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,-compact groups the fundamental theorems on characterization and representation [BOst15]. In [BOst16], working with topological ‡ows on homogeneous spaces, we identify an index of regular variation, which in a normed-vector space context may be speci…ed using the Riesz representation theorem, and in a locally compact group setting may be connected with Haar measure.

Effros, Baire, Steinhaus and non-separability

“…e G with g n (x) = x n : (This and a variant occurs in [Ban,Ch. III;Th.4]; and [ChCh]; for the term see [BinO2].) For a subgroup G H(X), say that X has the crimping property w.r.t.…”

Beyond Lebesgue and Baire II: Bitopology and measure-category duality

“…1.1.1 for background) in [BOst-SteinOstr]. For applications beyond the real line including the theory of topological regular variation see [BOst12], [BOst13] and [Ost2]. Applications of Th.…”

The Index Theorem of topological regular variation and its applications

“…The Uniform Convergence Theorem (UCT) in a topological dynamics setting is established in [10] so provides the foundations for a topological theory of regular variation. Let X be a phase space, a homogeneous metric space, specifically a group with identity e X .…”

“…The theory established in our first paper [10] is concerned with slowly varying functions. This is further developed in two companion papers.…”

“…However, it emerges that it is the topological, rather than the measurable, case that is the more important (see e.g. [10][11][12]), and so we call the resulting theory that of topological regular variation.…”