  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.
