发表机构
University of Michigan; Institute for Advanced Study（密歇根大学; 高等研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究Fuchsian表示乘积的极值熵，通过计数极值拉伸闭测地线并证明其次指数增长，进而推导增长率指标在Benoist极限锥边界消失，并推广到一般Borel Anosov子群，加强了Thurston结果。
AI 中文摘要
本文中，我们计数了一对（非共轭的）闭曲面群的Fuchsian表示的（几乎）极值拉伸闭测地线的数量，并证明其具有次指数增长。我们进而推断，作为$\mathsf{PO}(2, 1) \times \mathsf{PO}(2, 1)$的离散子群，乘积表示的增长率指标在Benoist极限锥的边界上消失。我们还证明了对于$\mathsf{PO}(2, 1) \times \mathsf{PO}(2, 1)$的一般Zariski稠密Borel Anosov子群，其增长率指标在Benoist极限锥的至少一个边界分量上消失，但未必在两个分量上都消失。我们的第一个结果可视为对Thurston结果的加强，即存在唯一的测地线层$\lambda$，使得每个相对于这两个表示使长度比最大化的测度层其支撑包含在$\lambda$中。我们希望这将成为更一般的极值熵研究的起点。
英文摘要
In this paper, we count the number of (almost) extremally stretched closed geodesics for a pair of (non-conjugate) Fuchsian representations of a closed surface group, and show that it has subexponential growth. We then deduce that, as a discrete subgroup of $\mathsf{PO}(2, 1) \times \mathsf{PO}(2, 1)$, the growth indicator of the product representation vanishes on the boundary of the Benoist limit cone. We also prove that for a general Zariski dense Borel Anosov subgroup of $\mathsf{PO}(2, 1) \times \mathsf{PO}(2, 1)$, its growth indicator vanishes on at least one boundary component of the Benoist limit cone, but not necessarily on both. One may view our first result as a sharpening of Thurston's result that there is a unique geodesic lamination $λ$ such that every measured lamination maximizing the ratio of lengths with respect to the two representations has support contained in $λ$. We hope this will be a starting point for a more general study of extremal entropy.
