Anosov group action on flag varieties

Given a self-map of a projective variety, we may ask how subvarieties of the projective variety behave under iterations of the self-map. Brolin and Lyubich discovered an equidistribution phenomenon for self-maps of the projective line. People expect the same for higher dimensional varieties, i.e. different subvarieties converge in the same way under iterations. We can also ask the same question for iterations under a group action instead of a single self-map. In dimension one Patterson-Sullivan theory can be thought of as an anologue of Brolin-Lyubich's work in the group action setting. In this ongoing joint work with Bac Dang, Michael Kapovich and Mikhail Lyubich, we try to generalize some aspects of the one dimensional theory to higher dimension.