AI 中文总结
本文研究双曲曲面的熵谱，证明其为具有限重数的逆良序多重集，在数值1附近存在可量化间隙，该间隙源于带边界曲面上曲线数量的几何估计。
AI 中文摘要
本文引入并研究双曲曲面的熵谱，即其子曲面的熵构成的集合。主要结果为：该熵谱是具有有限重数的逆良序多重集，且在数值1附近存在可量化的间隙。该间隙源于非填充测地线数量的计数结果，而计数结果又基于用几何数据对带边界曲面上曲线数量的显式估计，这些几何数据包括边界测地线长度及所谓的边界宽度，边界宽度用于衡量到边界的最大距离，也可通过曲面拓扑和收缩（systole）来解释。
英文摘要
This article introduces and studies the entropy spectrum of a hyperbolic surface, that is the set of entropies of its subsurfaces. The main results are that the entropy spectrum is a reverse well-ordered multiset, with finite multiplicities, and that there is a quantifiable gap around the value $1$. This gap comes from a counting result on the number of non-filling geodesics which in turn comes from explicit estimates on the number of curves on surfaces with boundary in terms of geometric data. The geometric data includes lengths of boundary geodesics and so-called boundary width, which measures maximal distance to the boundary but can also be interpreted in terms of the topology of the surface and the systole.
Comments62 pages, 14 figures
