  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.
