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From hyperbolic geometry to nonlinear Perron-Frobenius theory

从双曲几何到非线性佩伦-弗罗贝尼乌斯理论

基本信息

批准号：
EP/J008508/1
负责人：
Bas Lemmens
金额：
$ 12.59万
依托单位：
University of Kent
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2012
资助国家：
英国
起止时间：
2012 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FJ008508%2F1
关键词：
hyperbolic geometry nonlinear Perron Frobenius

项目摘要

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.

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着重要的应用。非线性Perron-Frobenius理论将这一经典理论推广到非线性正算子，并讨论了如下问题：一个非线性正算子什么时候在锥中有对应于谱半径的特征向量？特征向量什么时候位于锥体内部？这样的运算符的迭代是如何行为的？这些问题自然而然地出现在广泛的数学学科中，如博弈论、分形学分析和热带数学。伯克霍夫指出，人们可以使用希尔伯特几何来分析这些问题。伯克霍夫发现了非线性Perron-Frobenius理论和度规几何之间的协同作用，直到最近才开始完全结晶，这是该项目的主要主题。我们将集中讨论与Hilbert几何有关的几个中心公开问题。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激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Hilbert几何的Denjoy-Wolff型定理，它提供了关于锥内部没有特征向量的非线性正算子的动力学的详细信息。Denjoy-Wolff定理是单位圆盘上不动点自由解析自映射动力学复数分析中的经典结果。Beardon发现了这一结果在具有温和双曲性质的度量空间上的不动点自由非扩张映射上的显著推广。他的工作留下了一些引人入胜的问题，我们希望在这个项目中解决其中的一些问题。我们的第二个目标是证明de la Harpe关于Hilbert几何的等距群的几个猜想。在最近的一项工作中，我们发现了一种全新的方法来解释这些20年前的猜想，它结合了非线性Perron-Frobenius理论的思想和度量几何中的新概念，如角函数边界上的Busemann点和迂回度量。德拉哈普猜想的解和对称圆锥理论之间似乎有一种有趣的联系，我们希望解开这一联系。