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 representations was recently introduced by Maloni-Martone-Mazzoli-Zhang who provided a parametrization via shear-bend cocycles. Our main result is that $ρ_0$ dominates $ρ$ in the Hilbert length spectrum and the translation-length spectrum, with a strict domination for $\textit{most}$ curves, that we call $\textit{statistical}$ domination. Using this, we prove some entropy rigidity results: namely, the Hilbert entropy of any quasi-Hitchin representation in the bending fiber is strictly greater than that of $ρ_0$, the same for the translation length entropy when $ρ_0$ is $n$-Fuchsian, and in the latter case a new proof that for hyperconvex representations the Hausdorff dimension of the full limit set increases. The proof involves analyzing the weighted planar networks for finite approximants of the monodromy matrix, and establishing a strict matrix domination for generic monodromy using the equidistribution of closed geodesics in the unit tangent bundle of $S$.