Asymptotic properties of infinitesimal characters and applications

Inspired by Benoist, we study objects linked to integrable tangent vectors on the character variety of a semi-group $\Gamma$ with values in a semi-simple real-algebraic group $\mathsf G$. We prove the \emph{cone of Jordan variations} has non-empty interior and, when $\mathsf G$ is split, establish non-empty interior of the set of \emph{length-normalized variations}. We apply these techniques to pressure forms on Anosov representations and higher-rank Teichm\"uller spaces. We identify an explicit functional $\varphi\in\mathfrak a^*$ whose pressure form is compatible with Goldman's symplectic form at Fuchsian points in the Hitchin component. Finally, we show the degeneration of the Hausdorff dimension of \emph{higher}-quasi-circles is governed by a Diophantine equation.