Length functions of Hitchin representations

Given a Hitchin representation $\rho \colon \pi_1(S) \to \PSL_n(\mathbb{R})$, we construct $n$ continuous functions $\ell_i^\rho \colon \mathcal \CH(S) \to \mathbb{R}$ defined on the space of H\"older geodesic currents $\CH(S)$ such that, for a closed, oriented curve $\gamma$ in $S$, the $i$--th eigenvalue of the matrix $\rho(\gamma)\in \PSL_n(\mathbb{R})$ is of the form $\pm \mathrm{exp}\, \ell_i^\rho(\gamma)$: such functions generalize to higher rank Thurston's length function of Fuchsian re\presentations. Identities, diffe\rentiability properties of these lengths $\ell_i^\rho$, as well as applications to eigenvalue estimates, are also considered.