Rational approximation for Hitchin representations

Abstract: A consequence of Rapinchuk et al. is that for $S$ a closed surface of genus $g\geq 2$, the set of Hitchin representations of $\pi_1(S)$ with image in $\mathrm{SL}(n,\mathbb{Q})$ is dense in the Hitchin component. We give a dynamical proof of this fact provided that $g\geq 3$. Moreover, we extend it to some other $\mathbb{Q}$-groups such as $\mathrm{Sp}(2k,\mathbb{Q})$ and $\mathrm{G}_2(\mathbb{Q})$, where the results are new.