  In this paper we investigate the Hausdorff dimension of limit sets of Anosov
representations. In this context we revisit and extend the framework of
hyperconvex representations and establish a convergence property for them,
analogue to a differentiability property. As an application of this
convergence, we prove that the Hausdorff dimension of the limit set of a
hyperconvex representation is equal to a suitably chosen critical exponent.
