Deformations of Anosov subgroups: Limit cones and Growth indicators

Let $G$ be a connected semisimple real algebraic group. We prove the continuity of limit cones along deformations of Anosov subgroups of $G$ under a certain convexity assumption. This convexity assumption turns out to be necessary. We discuss an application to the notion of sharpness of an action of a discrete subgroup on a non-Riemannian homogeneous space. We also obtain that growth indicators, certain critical exponents and the Hausdorff dimension of limit sets vary continuously in the space of Anosov representations.