Mathematical Sciences: Recurrence, Convergence and Entropy in Ergodic Theory

小行星9401093 本计画的目的是：1）利用组合与结构的方法，研究动力系统的多重递归类型，包括Bergelson与Liebman的多项式Szemeredi定理的各个方面与改进; 2）研究遍历理论中平均值稳定性的各种测度的界的存在性，包括各种过程差异的特殊平方函数，通过测度论和傅立叶变换技术，以及3）考虑熵，同构，和多重混合的问题，为不同的群体作用在测度空间中，使用这两种方法介绍了研究动力系统中的熵与群体作用，并适用于此类系统的适当数论估计。 这个项目的目的是从各种角度更好地理解动力系统的长期行为。 首先，通过研究动力系统的基本结构的性质，将研究经常性行为的多次重复的类型。其次，利用调和分析方法，讨论了动力系统长期平均的稳定性和收敛速度。第三，通过积分和数论的方法，将更好地理解具有多种变换模式的动力系统的混沌行为的性质和程度。 ***

9401093 Bergelson The purposes of this project are to 1) to study types of multiple recurrence for dynamical systems, including various aspects and refinements of the polynomial Szemeredi theorem of Bergelson and Liebman, by combinatorial and structural methods, 2) to examine the existence of bounds for various measures of the stability of averages in ergodic theory, including particular square functions for differences of various processes, through measure-theoretic and Fourier transform techniques, and also 3) to consider problems in entropy, isomorphism, and multiple mixing for different group actions in measure spaces, using both the methods introduced for studying entropy in dynamical systems with group actions, and the appropriate number-theoretic estimates that apply to such systems. The purpose of this project is to understand better the long-term behavior of dynamical systems from a variety of viewpoints. First, by examining the nature of the underlying structure of the dynamical system, types of many-fold repetition of recurrent behavior will be studied. Second, using harmonic analysis methods, the stability and rate of convergence of long term averages in the dynamical system will be considered. Third, by methods of integration and number theory, the nature and extent of chaotic behavior of dynamical systems with many modes of transformation will be better understood. ***