标题： 非阿基米德希尔伯特几何与实希尔伯特几何的退化 显示英文标题

标题： Non-Archimedean Hilbert geometry and degenerations of real Hilbert geometries

摘要： 我们发展了一种在一般有序赋值域上的希尔伯特几何理论，将射影空间的一个开凸子集与一个商希尔伯特度量空间相关联。 在自然的非退化假设下，我们证明了一序列缩放实希尔伯特几何的超限极限与一个罗宾逊域上的开凸射影子集的希尔伯特度量空间等距。 这个结果使我们能够证明，闭流形上凸实射影结构空间的理想点来自于没有全局固定点的非阿基米德希尔伯特几何上的作用。 我们明确描述了一个定义在赋值环子域上的非阿基米德有界多面体$P$的希尔伯特度量空间，将其作为基于 Weyl 章节的$P$的旗复形的几何实现。 作为应用，我们得到了带有缩放希尔伯特度量的实多面体的 Gromov-Hausdorff 极限的完整描述。 显示英文摘要

摘要： We develop a theory of Hilbert geometry over general ordered valued fields, associating with an open convex subset of the projective space a quotient Hilbert metric space. Under natural non-degeneracy assumptions, we prove that the ultralimit of a sequence of rescaled real Hilbert geometries is isometric to the Hilbert metric space of an open convex projective subset over a Robinson field. This result allows us to prove that ideal points of the space of convex real projective structures on a closed manifold arise from actions on non-Archimedean Hilbert geometries without global fixed point. We explicitly describe the Hilbert metric space of a non-Archimedean bounded polytope $P$ defined over a subfield of the valuation ring as the geometric realization of the flag complex of $P$ modeled on a Weyl chamber. As an application, we obtain a complete description of Gromov-Hausdorff limits of a real polytope with rescaled Hilbert metric.