Mathematical Sciences: Flow Equivalence of Subshifts

[401079]黄提出将有限型的所有二分量位移都归为流动等价。当1不是定义矩阵的特征值时，他最近分类了有限型的n分量移位。该项目是通过证明一个重要的“双面”实现定理来完成一般有限型n分量位移的分类，该定理也可能对C*-代数有新的含义。此外，在已有的研究结果的基础上，作者打算研究sofic位移、随机Markov位移和更一般的Laurent多项式矩阵的设置的（几乎）流等效性。特别是，他计划探索流动等价和C*-代数之间更有趣的关系，包括他为与有限型2分量位移相关的非简单Cuntz-Krieger代数的分类所做的一个猜想。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，遍历理论在对流和代数结构进行分类的过程中与几何和代数接触。＊＊＊

9401079 Huang The proposer has classified up to flow equivalence all 2- component shifts of finite type. He has recently classified n-component shifts of finite type when 1 is not an eigenvalue of the defining matrix. The project is to complete the classification for general n-component shifts of finite type by proving a crucial "two-sided" realization theorem which may also have a new implication to C*-algebras. Moreover, based on the existing results, the proposer intends to study (almost) flow equivalence of sofic shifts, of stochastic Markov shifts and of the more general setting of Laurent polynomial matrices. Especially, he plans to explore more interesting relations between flow equivalence and C*-algebras, including a conjecture he made for the classification of non-simple Cuntz-Krieger algebras associated to 2-component shifts of finite type. 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 of "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 and algebra in their quests to clasify flows and algebraic structures. ***