Critical exponent and Hausdorff dimension for Anosov representations

Whenever G is a convex cocompact subgroup of the group of
isometries of the hyperbolic space, Patterson-Sullivan theory allows to
relate the asymptotic growth rate of orbit points for the action of G on
H^n and the Hausdorff dimension of the limit set of G on the boundary.

Anosov representations form a robust generalization of convex
cocompactness for discrete subgroups of higher rank Lie groups. However
the relation between the Hausdorff dimension of their limit set and a
suitable orbit growth rate is much more elusive since, on the one hand,
the action of G on the boundary is not conformal, and, on the other,
many different orbit growth functions can be considered. In my talk I'll
report on joint work with A. Sambarino and A. Wienhard in which we find
large classes of Anosov representations for which we can obtain such a
relation.