Anosov representations with Lipschitz limit set

We study Anosov representations whose limit set has intermediate regularity, namely is a Lipschitz submanifold of a flag manifold. We introduce an explicit linear functional, the unstable Jacobian, whose orbit growth rate is integral on this class of representations. We prove that many interesting higher rank representations, including $\Theta$-positive representations, belong to this class, and establish several applications to rigidity results on the orbit growth rate in the symmetric space.