The Index Theorem of topological regular variation and its applications

Abstract

We develop further the topological theory of regular variation of [N.H. Bingham, A.J. Ostaszewski, Topological regular variation: I. Slow variation, LSE-CDAM-2008-11]. There we established the uniform convergence theorem (UCT) in the setting of topological dynamics (i.e. with a group T acting on a homogenous space X), thereby unifying and extending the multivariate regular variation literature. Here, working with real-time 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.

Beurling slow and regular variation

“…We begin with a lemma that yields simplifications later; it implies a Beurling analogue of the Bounded Equivalence Principle in the Karamata theory, first noted in [10]. As it shifts attention to the origin, we call it the Shift Lemma of BMA.…”

“…So T ϕ : G × X → X is not a group action, as associativity fails. However, just as in a proper flow context, here too one has a well-defined flow rate, or infinitesimal generator, at x, for which see [6,10,65,13.34] (cf. [2]),…”

“…This auxiliary group plays its part through allowing the absorption of a small 'time' variation t + s of t into a small 'space' variation in x. This involves a concatenation formula, earlier identified in [10] as a component in the abstract theory of the index of regular variation.…”

Homomorphisms from functional equations: the Goldie equation

“…For ϕ ∈ SE the limit η = η ϕ is necessarily in GS [Ost2]. Only (CF E) visibly identifies its solution K as a homomorphism -of the additive group (R, +) -whereas homomorphy is a central feature in the recent topological development of the theory of regular variation [BinO1,2], [Ost1]. The role of homomorphy is new in this context, and is one of our principal contributions here.…”

Beurling regular variation, Bloom dichotomy, and the Gołąb–Schinzel functional equation

“…0) thus call for attention. The latter is linked to the Cauchy functional equation for additive functions (for which see [Kucz], [AczD]), which already plays a key role in determining the index theory of Karamata regular variation -see [BinO1]. Here, for Beurling regular variation, there is an analogous functional equation satis…ed by the limit functions ' , namely the Go÷¾ ab-Schinzel equation…”