Dimension gap and variational principle for Anosov representations

We consider a representation of a finitely generated CAT(–K) group $\mathrm{\Gamma }$ in SL(d, R) that is Zariski dense and k-Anosov for at least two values of k. We exhibit a gap for the Minkowski dimension of minimal sets for the action of $\mathrm{\Gamma }$ on flags spaces. The proof uses a variational principle for the action on partial flags.