Entropy and domination for quasi-Hitchin representations

Let $S$ be a closed oriented surface of genus $g\geq 2$. We consider an $n$-pleated representation $ρ: π_1(S) \to \mathrm{PSL}_n(\mathbb{C})$ obtained by bending a Hitchin representation $ρ_0:π_1(S) \to \mathrm{PSL}_n(\mathbb{R})$ along a maximal geodesic lamination. The space of such $n$-pleated re