Classification and Invariants of Subshifts

【摘要】长期以来，Williams关于位移等价等同于强位移等价的猜想是对有限类型子位移进行分类的主要方法。有了这个猜想的否定解，我们就可以确定证明中涉及的不变量是一个完备集，或者找到新的不变量和算法来有效地对有限类型的子位移进行分类，直到拓扑共轭。我们将考虑群论、上同调和其他方法。我们还将尝试回答Nasu提出的关于他的纺织系统的一些问题（Boyle和Maass已经部分解决了）。非技术描述：符号动力学是一种数学理论，可以处理在许多连接中出现的符号串。它已被用于制定有效的计算机数据存储和传输代码。它是动力系统理论的重要基础部分，研究系统（可能是太阳系、地球大气或人体）如何随时间变化。符号动力学已应用于拓扑量子场论，它建立了与现代物理学的联系，也与其他重要的数学领域，如节理论。有限型子移是目前研究最广泛的一类符号动力系统，本文主要研究有限型子移的分类问题。

AbstractKim/RoushFor a long time the Williams conjecture that shift equivalence is the same as strong shift equivalence was the dominant approach to classifying subshifts of finite type. With our negative resolution of this conjecture, the way is open to either determine that the invariants involved in that proof are a complete set, or to find new invariants and algorithms to effectively classify subshifts of finite type up to topological conjugacy. We will consider group-theoretic, cohomological, and other methods. We will also try to answer some questions raised by Nasu on his textile systems (already partly solved by Boyle and Maass).Nontechnical description: Symbolic dynamics is a mathematical theory that can handle strings of symbols that can arise in many connections. It has been used to formulate codes for efficient computer data storage and transmission. It is an important and fundamental part of dynamical systems theory, which studies how systems (which might be something like the solar system, the earth's atmosphere, or the human body) change under time. Symbolic dynamics has been applied to topological quantum field theory, which establishes a link to modern physics, and also to other important areas of mathematics such as the theory of knots. The present proposal is primarily a project to classify subshifts of finite type, which are the most extensively studied kind of symbolic dynamical system.