Deformation of Fuchsian representations and proper affine actions

The main goal of this article is to generalize Mess' work and using results from Labourie--Wentworth, Potrie--Sambarino and Smilga, to show that inside Hitchin representations, infinitesimal deformations of Fuchsian representations of a cocompact surface group do not act properly along the directions corresponding to the sum of a mixed odd differential and a $2m$-differential for any $1\leq m \leq \lfloor\frac{n}{2}\rfloor$. In the process, we introduce affine versions of cross ratios and triple ratios. We introduce Margulis invariants and relate them with affine crossratios and infinitesimal Jordan projections. We obtain a general equivalent criterion for existence of proper affine actions in terms of the structure of the Margulis invariant spectra. Also, using a stability argument we show the existence of proper affine actions of non-abelian free groups whose linear part is a Hitchin representation.