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Résumé |
Let $X_{k,d}$ denote the space of rank-k lattices in $R^d$. Topological and statistical properties of the dynamics of discrete subgroups of G=SL(d,R) on $X_{d,d}$ were described in the seminal works of Benoist-Quint. A key step/result in this study is the classification of stationary measures on $X_{d,d}$. Later, Sargent-Shapira initiated the study of dynamics on the spaces $X_{k,d}. When $k \ne d$, the space $X_{k,d}$ is of a different nature and a clear description of dynamics on these spaces is far from being established. Given a probability measure $\mu$ which is Zariski-dense in a copy of SL(2,R) in G, we give a classification of $\mu$-stationary measures on $X_{k,d}$ and prove corresponding equidistribution results. In contrast to the results of Benoist-Quint, the type of stationary measures that $\mu$ admits depends strongly on the position of SL(2,R) relative to parabolic subgroups of G. I will start by reviewing preceding major works and ideas. The talk will be accessible to a broad audience. Joint work with Alexander Gorodnik and Jialun Li. |
