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The purpose of this article is to describe and characterize the limitdistributions of translates of a bounded open "piece of orbit" of a reductivesubgroup on a space of S-arithmetic lattices. This is accomplished under a mildassumption of "analytic stability" on the sequence of translates. It isimportant to note however that it is not necessary to assume that the reductivesubgroup or its centralizer are algebraic. Moreover, it is also not necessaryto assume that the initial orbit is of finite measure or even closed. Thisarticle thus provides important generalizations on previously known results andopens ways to new applications in number theory, two of which are brieflymentioned.

Limit distribution of Translated pieces of possibly irrational leaves in S-arithmetic homogeneous spaces