Equirepartition dans les espaces homogènes

Abstract : In this work, we study some properties of repartition of sets in homogeneous spaces. We use two different techniques : - first we apply adelic mixing in order to study repartition of sets of rational matrices in a compact real group. We get equirepartition results in an unitary group for sets of rational matrices defined by conditions on denominators of coefficients. This yields a local-global principle for the existence of these matrices. - Second we use some properties of polynomial dynamic, such as Ratner rigidity theorem. This yields equidistribution results for orbits of a lattice of the special linear group on a local field of characteristic 0 in an homogeneous space under this group. We also get an S-arithmetic analogue of a theorem due to Shah in the real case. Third, we focus on a different problem : given a local field k of characteristic 0, and H a finite index subgroup of k * , can we find a Zariski-dense subgroup in SL(n, k) such that all the elements have their whole spectrum inside H ? We are able to answer this question : a sufficient and necessary condition is that either -1 belong to H or n is not congruent to 2 modulo 4.
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Antonin Guilloux. Equirepartition dans les espaces homogènes. Mathématiques [math]. Université Paris Sud - Paris XI, 2007. Français. ⟨tel-00372220⟩

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