Research on ergodic theory and its applications

在这个项目中，许多从事遍历理论、概率论、泛函分析、解析数论、组合学和微分几何等领域的数学家进行了与遍历理论相关的多个领域的研究，并取得了许多重要的成果。在遍历理论中，深入研究了无有限不变测度的变换所特有的遍历变换的性质（即所谓的ii型和III型变换）。特别地，研究了Radon-Nikodym环的渐近性，穷举弱游荡序列的性质及其与整数Z的直接和分解的关系，得到了一些有趣的结果。进一步研究了i *型变换的多次递归性质，并建立了其与Kakutani-Parry指数的关系。考虑到遍历理论与其他领域的相互关系，确定了与三维台球相关的符号序列的复杂度，得到了双曲3流形上拉普拉斯特征函数的尖锐L^*范数估计。

In this project, researches concerning diversified areas connected with ergodic theory were carried out by a number of mathematicians working in the areas of ergodic theory, probability theory, functional analysis, analytic number theory, combinatorics and differential geometry, and many significant results were obtained.1. In ergodic theory proper, properties of ergodic transformations which are characteristic for transformations having no finite invariant measures (so-called typ II_* and type III transformations) were investigated in depth. In particular, asymptotic behavior of the Radon-Nikodym cocycles and properties of exhaustive weakly wandering sequences and their relation to the direct sum decomposition of the integers Z were studied and a number of interesting results were obtained. Furthermore, multiple recurrence properties of type II_* transformations were studied and their relation with Kakutani-Parry index was established.2. Concerning the interrelation between ergodic theory and othe areas, complexity of th esymbolic sequences associated with 3-dimensional billiard was determined, and sharp L^*-norm estimates for eigen functions for Laplacian on hyperbolic 3 manifolds were obtained.