Abstract. We introduce the notion of an ${\msbm R}$-group of which the classical groups ${\msbm R}$, ${\msbm Z}$ and ${\msbm R}_{+}^{\ast }$ are typical examples, and we study flows $( X,{\cal H}) $, where $X$ is a locally compact space and ${\cal H}$ is a continuous ${\msbm R}$-group action on $X$ with the further property that any compact set is \hbox{\it 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 ${\cal H}$. The case where $X$ is a locally compact abelian group is also considered. We are particularly interested in discussing the asymptotic properties of ${\cal H}$, which is made possible by proving a deep theorem about the existence of nontrivial ${\cal H}$-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.

AMS Subject Classification (1991): 37B05, 43A07, 46J10, 28A25, 28A50, 26E60, 54D45

Keyword(s): locally compact space, group actions

Received May 1, 2010, and in revised form February 8, 2011. (Registered under 36/2010.)