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

The purpose of this article is to describe and characterize the limit distributions of translates of a bounded open "piece of orbit" of a reductive subgroup on a space of S-arithmetic lattices. This is accomplished under a mild assumption of "analytic stability" on the sequence of translates. It is important to note however that it is not necessary to assume that the reductive subgroup or its centralizer are algebraic. Moreover, it is also not necessary to assume that the initial orbit is of finite measure or even closed. This article thus provides important generalizations on previously known results and opens ways to new applications in number theory, two of which are briefly mentioned.