论文标题
层压和叶子的奇异定理：最新结果和观点
Ergodic theorems for laminations and foliations: recent results and perspectives
论文作者
论文摘要
该报告讨论了riemann表面层压层面的千古理论中的最新结果以及新观点，并强调了曲线的奇异霍顿叶子。这些事态发展的中心概念是叶片库纳尔度量标准，定向谐波电流，乘法性循环和莱普诺诺夫指数。我们处理了这种层压层的各种千古定理：随机和操作者的ergodic定理，（几何）birkhoff ergodic定理，oseledec乘法性千古定理和独特的ergodicition定理。还提供了这些定理的应用。特别是，我们为紧凑的投影表面上的大型奇异群体叶子定义并研究了规范的Lyapunov指数。这些特征数字的拓扑和代数几何解释也得到了处理。 这些结果突出了地图和黎曼表面层压层的千古理论之间的强烈相似性以及基本差异。这里报告的大多数结果都是已知的。但是，抽象热扩散与叶轮的传播相吻合（第5.2小节）的足够条件是新的。
This report discusses recent results as well as new perspectives in the ergodic theory for Riemann surface laminations, with an emphasis on singular holomorphic foliations by curves. The central notions of these developments are leafwise Poincaré metric, directed positive harmonic currents, multiplicative cocycles and Lyapunov exponents. We deal with various ergodic theorems for such laminations: Random and Operator Ergodic Theorems, (Geometric) Birkhoff Ergodic Theorems, Oseledec Multiplicative Ergodic Theorem and Unique Ergodicity Theorems. Applications of these theorems are also given. In particular, we define and study the canonical Lyapunov exponents for a large family of singular holomorphic foliations on compact projective surfaces. Topological and algebro-geometric interpretations of these characteristic numbers are also treated. These results highlight the strong similarity as well as the fundamental differences between the ergodic theory of maps and that of Riemann surface laminations. Most of the results reported here are known. However, sufficient conditions for abstract heat diffusions to coincide with the leafwise heat diffusions (Subsection 5.2) are new ones.
