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Topological regular variation: I. Slow variation

机译：拓扑规则变化：I.缓慢变化

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Motivated by the Category Embedding Theorem, as applied to convergent automorphisms (Bingham and Ostaszewski (in press) [11]), 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 Bajsanski and Karamata (1969) [4] from groups to flows on a group. A multiplicative representation of the flow derived in Ostaszewski (2010) [45] 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 Bingham and Ostaszewski (2010) [13] and we extend to locally compact, σ-compact groups the fundamental theorems on characterization and representation (Bingham and Ostaszewski (2010) [14]). In Bingham and Ostaszewski (2009) [15], 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.
机译：受类别嵌入定理（适用于会聚自同构（Bingham和Ostaszewski（印刷中）[11]）的启发），我们通过拓扑动力学语言的重新表述统一并扩展了多元正则变异文献。这里的自然环境是度量标准组，被视为规范组（模仿规范化向量空间）。我们简要地研究了它们的性质，以此作为建立Baire的统一收敛定理（UCT），群值缓慢变化函数具有通过同构空间及其同胚同族之间的自然对偶联系在一起的两个自然度量的概括。每个都可以通过二元性相互推导。其中之一是由于Bajsanski和Karamata（1969）[4]将UCT的（拓扑）组版本从组扩展到组中的流。 Ostaszewski（2010）[45]中得到的流量的乘积表示证明了该流量与较早的组公式等效。在伴随论文中，我们将理论扩展到有规律的变化的函数：我们在Bingham和Ostaszewski（2010）[13]中建立了有规律变化的演算，并且将关于特征和表示的基本定理扩展到局部紧凑的σ-compact组（Bingham和Ostaszewski（2010）[14]）。在Bingham和Ostaszewski（2009）[15]中，我们研究了齐次空间上的拓扑流，确定了正则变化的索引，在规范向量空间的上下文中，可以使用Riesz表示定理指定局部变量，设置可能与Haar测量有关。

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