Mathematical Sciences: Ergodic Theory of Symbolic Systems

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。Lind教授和Tuncel教授要进行的具体工作是在“符号动力学”领域。符号动力系统是来自有限字母表的双向无限符号串的集合。这些字符串满足编队规则，该规则指定当一个人沿着该字符串移动时，哪些符号被允许跟随哪个符号。近年来，符号动力系统在数据传输的建模中得到了广泛的应用。一些特殊的研究领域将涉及紧群的交换自同构的动力学，利用平面的自相似平铺构造马尔可夫划分，以及马尔可夫链的编码和不变量。这项工作强调遍历理论、代数和几何领域之间的相互作用。

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. The particular work to be undertaken by Professors Lind and Tuncel is in the area of "symbolic dynamics". A symbolic dynamical system is a collection of two-way infinite strings of symbols from a finite alphabet. These strings satisfy a formation rule which specifies which symbols are allowed to follow which as one moves along the string. Symbolic dynamical systems have recently found widespread application in the modelling of data transmission. Some of the particular areas of investigation will involve the dynamics of commuting automorphisms of compact groups, the construction of Markov partitions using self-similar tilings of the plane, and codings and invariants of Markov chains. The work stresses interactions between the fields of ergodic theory, algebra, and geometry.