Mathematical Sciences: Ergodic Theory and Combinatorics

Ornstein和Katznelson将研究下列领域的问题：圆的微分同胚群，“圆技巧”在扭曲映射中的应用；统计稳定性；熵、回报率和数据压缩；Bernoulli系统；组合学中的遍历和拓扑动力学方法，特别是确定给定指数多项式的遍历系统的特征因子；以及分形图的正则性。这个项目涉及对遍历理论的研究。一般说来，遍历理论涉及理解系统的平均行为，这些系统的动力学太复杂或太混乱，以至于无法在微观细节上进行跟踪。在标题“动力学可以被放在现代理论如何抽象变换如何作用于光滑空间。这样，遍历理论与几何联系起来，寻求在均匀空间上对流动进行分类。

Ornstein and Katznelson will investigate problems in the following areas: Groups of diffeomorphisms of the circle, application of "circle techniques" to twist maps; statistical stability; entropy, returns rates and data compression; Bernoulli systems; ergodic and topological dynamic methods in combinatorics and in particular identifying the characteristic factors of ergodic systems for given exponent polynomials; and regularity properties for maps of fractals. This project involves research in ergodic theory. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In this way ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces.