From hyperbolic geometry to nonlinear Perron-Frobenius theory

经典的Perron-Frobenius理论涉及非负矩阵的谱性质，被认为是矩阵分析中最美丽的主题之一，在概率论，动力系统理论和离散数学中有重要应用。非线性Perron-Frobenius理论将这一经典理论扩展到非线性正算子，并处理了这样的问题：当一个非线性正算子在对应于谱半径的锥中有一个特征向量时？什么时候本征向量位于圆锥的内部？这类运算符的迭代如何进行？这些问题自然出现在广泛的数学学科，如博弈论，分形分析和热带数学。伯克霍夫表明，人们可以使用希尔伯特几何来分析这些问题。伯克霍夫的发现之间的协同作用的非线性佩龙-弗罗贝纽斯理论和度量几何最近才开始充分结晶，是该项目的主题。我们将集中讨论几个关于希尔伯特几何的中心开放问题。希尔伯特几何是双曲几何的自然非黎曼推广。度量几何的最新发展引发了人们对希尔伯特几何的新兴趣，并为解决其中一些问题提供了令人兴奋的机会。我们的第一个目标是证明希尔伯特几何的Denjoy-Wolff型定理，它提供了关于锥内部没有本征向量的非线性正算子的动力学的详细信息。Denjoy-Wolff定理是复分析中关于单位圆盘上无不动点解析自映射动力学的经典结果。Beardon发现了一个惊人的概括这一结果不动点自由非扩张映射度量空间具有温和的双曲性质。他的工作留下了许多令人着迷的问题，其中一些我们希望在这个项目中解决。我们的第二个目标是证明几个命题的de拉哈尔佩关于等距群的希尔伯特几何。在最近的一项工作中，我们发现了一种全新的方法来处理这些20年前的问题，它将非线性Perron-Frobenius理论的思想与度量几何中的新概念（如时函数边界中的Busemann点和迂回度量）相结合。在德拉哈尔佩的解和对称锥理论之间似乎有一种有趣的联系，我们希望能解开这个联系。

The classical Perron-Frobenius theory concerns the spectral properties of nonnegative matrices, and is considered one of the most beautiful topics in matrix analysis with important applications in probability theory, dynamical systems theory, and discrete mathematics. Nonlinear Perron-Frobenius theory extends this classical theory to nonlinear positive operators, and deals with questions like: When does a nonlinear positive operator have an eigenvector in the cone corresponding to the spectral radius? When does the eigenvector lie in the interior of the cone? How do the iterates of such operators behave? These questions arise naturally in a wide range of mathematical disciplines such as game theory, analysis on fractals, and tropical mathematics. Birkhoff showed that one can use Hilbert geometries to analyse these questions. Birkhoff's discovery of the synergy between nonlinear Perron-Frobenius theory and metric geometry has only recently started to fully crystallise, and is the main theme of the project. We will focus on several central open problems concerning Hilbert geometries. Hilbert geometries are a natural non-Riemannian generalisation of hyperbolic geometry. Recent developments in metric geometry have triggered a renewed interest in Hilbert geometries, and opened up exciting opportunities to solve some of these problems. Our first goal is to prove Denjoy-Wolff type theorems for Hilbert geometries, which provide detailed information about the dynamics of nonlinear positive operators without eigenvectors in the interior of the cone. The Denjoy-Wolff theorem is a classical result in complex analysis about the dynamics of fixed point free analytic self-maps of the unit disc. Beardon discovered a striking generalisation of this result to fixed point free non-expansive maps on metric spaces that possess mild hyperbolic properties. His work left open a number a fascinating problems some of which we hope to resolve in this project. Our second goal is to prove several conjectures by de la Harpe about the isometry group of Hilbert geometries. In a recent work we found a completely novel approach to these twenty-year old conjectures, which combines ideas from nonlinear Perron-Frobenius theory with new concepts in metric geometry such as the Busemann points in the horofunction boundary and the detour metric. There appears to be an intriguing connection between the solution of de la Harpe's conjectures and the theory of symmetric cones, which we hope to unravel.