Absorptive continuous ℝ-group actions on locally compact spaces

We introduce the notion of an ℝ-group of which the classical groups ℝ, ℤ and ℝ+* are typical examples, and we study flows (X, ℋ), where X is a locally compact space and ℋ is a continuous ℝ-group action on X with the further property that any compact set is absorbed (in the ordinary meaning in use in the theory of topological vector spaces) by any neighbourhood of some characteristic point in X called the center of ℋ. The case where X is a locally compact abelian group is also considered. We are particularly interested in discussing the asymptotic properties of ℋ, which is made possible by proving a deep theorem about the existence of nontrivial ℋ-homogeneous positive measures on X. Also, a close connection with homogenization theory is pointed out. It appears that the present paper lays the foundation of the mathematical framework that is needed to undertake a systematic study of homogenization problems on manifolds, Lie groups included.