The Hopf–Tsuji–Sullivan dichotomy in higher rank and applications to Anosov subgroups

We establish an extension of the Hopf–Tsuji–Sullivan dichotomy to any Zariski dense discrete subgroup of a semisimple real algebraic group G. We then apply this dichotomy to Anosov subgroups of G, which surprisingly presents a different phenomenon depending on the rank of the ambient group G.