Deformations of Anosov subgroups: Limit cones and Growth indicators

Let $G$ be a connected semisimple real algebraic group. We prove that limit cones vary continuously under deformations of Anosov subgroups of $G$ under a certain convexity assumption, which turns out to be necessary. We apply this result to the notion of sharpness for the action of a discrete subgroup on a non-Riemannian homogeneous space. Finally, we show that, within the space of Anosov representations, the growth indicator, the critical exponents, and the Hausdorff dimension of limit sets (with respect to an appropriate non-Riemannian metric) all vary continuously.