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 $\rho: \pi_1(S) \to \mathrm{PSL}_n(\mathbb{C})$ obtained by bending a Hitchin representation $\rho_0:\pi_1(S) \to \mathrm{PSL}_n(\mathbb{R})$ along a maximal geodesic lamination. The space of such $n$-pleated representations was recently introduced by Maloni-Martone-Mazzoli-Zhang who provided a parametrization via shear-bend cocycles. Our first result is that $\rho_0$ dominates $\rho$ in the Hilbert length spectrum and the translation-length spectrum; this generalizes our earlier result for finite laminations on punctured surfaces. Using this, we prove entropy rigidity results: namely, the Hilbert entropy of a quasi-Hitchin representation in the bending fiber is strictly greater than that of $\rho_0$, and the same for the translation-length entropy in the case that $\rho_0$ is $n$-Fuchsian. The proof involves analyzing the weighted planar networks for finite approximants of the monodromy matrix, and establishing a strict domination for \emph{most} curves using the equidistribution of closed geodesics in the unit tangent bundle of $S$.