Topological regular variation: III. Regular variation

This paper extends the topological theory of regular variation of the slowly varying case of Bingham and Ostaszewski (2010) to the regularly varying functions between metric groups, viewed as normed groups (see also Bingham and Ostaszewski (2010). This employs the language of topological dynamics, especially flows and cocycles. In particular we show that regularly varying functions obey the chain rule and in the non-commutative context we characterize pairs of regularly varying functions whose product is regularly varying. The latter requires the use of a ‘differential modulus’ akin to the modulus of Haar integration.