Entropy Rigidity for Maximal Representations

Let $\mathrm{\Gamma }\subset \mathsf{P}\mathsf{S}\mathsf{L}(2,\mathbb{R})$ be a lattice and $\rho :\mathrm{\Gamma }\to \mathsf{S}\mathsf{p}(2n,\mathbb{R})$ be a maximal representation. We show that $\rho$ satisfies a measurable $(1,1,2)-$hypertransversality condition. With this we define a measurable Gromov product and the Bowen-Margulis-Sullivan measure associated to the unstable Jacobian introduced by Pozzetti, Sambarino and Wienhard. As a main application, we prove a strong entropy rigidity result for $\rho$.