The set of normalized variations and pressure forms on the Hitchin component

Given a non-zero tangent vector $v$ to the character variety $\mathfrak{X}(\mathrm{\Gamma },G)$ of a discrete semi-group $\mathrm{\Gamma }$ with values on a semi-simple Lie group $G$ we introduce, by analogy with Benoist's limit cone, the cone of Jordan variations and the set of normalized variations. The latter depends on the choice of a functional of the Cartan space of $G$ that is positive on Benoist's limit cone (of the base-point of $v$). We will explain results regarding convexity and non-empty interior for these sets and explain their implications on the non-degeneracy problem for pressure forms on the Hitchin component.