Mathematical Sciences: Ergodic Theory and Combinatorics

Ornstein和Katznelson将研究以下领域的问题：圆的微分同构群，“圆技术”在扭转映射中的应用；统计 稳定;熵，返回率和数据压缩；伯努利 系统;组合学中的遍历和拓扑动态方法，特别是确定给定指数多项式的遍历系统的特征因子；以及分形映射的正则性。这个项目涉及遍历理论的研究。遍历理论一般关注理解系统的平均行为，其动力学过于复杂或混乱，无法在微观细节上进行跟踪。在“动力学”的标题下，可以放置关于抽象变换如何作用于光滑空间的现代理论。通过这种方式，遍历理论在寻求对均匀空间上的流动进行分类时与几何联系在一起。

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.