Boundary of Anosov coaffine representations

Let $\mathrm{\Gamma }\le \mathrm{S}\mathrm{L}(d+1,\mathbb{R})$ be a torsion-free hyperbolic group dividing a properly and strictly convex domain $\mathrm{\Omega }\subset \mathbb{R}{\mathrm{P}}^d$, and let $\rho :\mathrm{\Gamma }\to \mathrm{S}\mathrm{L}(d+1,\mathbb{R})$ be the inclusion. We study representations of $\mathrm{\Gamma }$ into the coaffine group that are deformations of $\rho \oplus 1\in \mathrm{H}\mathrm{o}\mathrm{m}(\mathrm{\Gamma },\mathrm{G}\mathrm{L}(d+2,\mathbb{R}))$ and that lie on the boundary of the locus of projective Anosov representations. In this setting, we identify a canonical limit set $\mathrm{\Lambda }$ for the action and study its fine geometric features which are, in a suitable sense, stable. We prove the existence of an ending lamination that encodes the failure of the Anosov condition and the non-existence of a Cannon–Thurston map. Under certain conditions, we compute the Hausdorff dimension and show that the corresponding measure is positive.