拟希钦表示的熵与支配关系
Entropy and domination for quasi-Hitchin representations
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中文总结 AI 辅助
针对亏格≥2的闭定向曲面，研究拟希钦表示的熵与支配关系，证明弯曲所得拟希钦表示的希尔伯特熵严格大于原希钦表示的熵，推广了此前有限层叠的相关结果。
中文摘要 AI 辅助
设S为亏格g≥2的闭定向曲面。我们考虑n褶皱表示ρ：π₁(S)→PSLₙ(ℂ)，该表示通过沿极大测地线层叠弯曲希钦表示ρ₀：π₁(S)→PSLₙ(ℝ)得到。这类n褶皱表示的空间由Maloni-Martone-Mazzoli-Zhang近期引入，他们通过剪弯上循环提供了参数化。我们的首个结果是ρ₀在希尔伯特长度谱和平移长度谱中支配ρ，这推广了我们此前关于穿孔曲面上有限层叠的结果。利用该结论，我们证明了熵刚性结果：弯曲纤维中拟希钦表示的希尔伯特熵严格大于ρ₀的希尔伯特熵；当ρ₀为n-富克斯表示时，平移长度熵也满足该关系。证明涉及分析单值化矩阵有限逼近的加权平面网络，并利用S的单位切丛中闭测地线的等分布，为“大多数”曲线建立严格支配关系。
英文摘要
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 $S$.
发表机构
- Ashoka University（阿肖卡大学）
- Indian Institute of Science（印度科学研究所）
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