Strong Shift Equivalence Theory

[9971501] wagoner Shift动力系统和Markov链是研究从离散时间光滑动力学、统计力学到信息和编码理论等领域现象的有用模型。在实践中，通常有许多描述或构造具有本质上相同动力学行为的移位系统的方法。即使在一维情况下，有效地识别两个具体模型何时是等效的也是一个基本的开放性问题。另一个与自然有关的问题是研究两个系统之间不同的等价或对称性。给定系统的对称性通常称为可逆元胞自动机。研究一维位移系统的分类和对称群问题的一种卓有成效的方法涉及强位移等价理论。近年来，这导致了漂移等效性问题的进展。这是符号动力学中一个诱人的问题，可以用矩阵和图的初等形式提出，自1974年以来一直开放。它通过对矩阵进行通常的行、列运算，但施加了一定的不等式，得到了一种新的有序环的正代数k理论。强位移等价理论是代数和拓扑学的结合，它产生于被称为位移系统及其对称群的系统动力学。事实证明，它与动力学以外的数学领域有着意想不到的、有趣的联系，比如代数k理论、循环同调和拓扑量子场论。瓦格纳将继续探索这些新的发展

9971501Wagoner Shift dynamical systems and Markov chains are useful models forphenomena in areas ranging from discrete time smooth dynamics, tostatistical mechanics, to information and coding theory. In practice,there are generally many ways of describing or constructing shiftsystems that have essentially the same dynamical behaviour. Even inthe one-dimensional case, effectively recognizing when two concretemodels are equivalent is a fundamental open problem. Another naturaland related problem is to study the different equivalencies orsymmetries between two systems. Symmetries of a given system areoften called reversible cellular automata. One approach to theclassification and symmetry group problems for one-dimensional shiftsystems that has been fruitful involves strong shift equivalencetheory. In recent years, this has led to progress on the ShiftEquivalence Problem. This is a tantalizing question in symbolicdynamics that can be posed in elementary terms using matrices andgraphs and that has been open since since 1974. It has led to a newtype of positive algebraic K-theory for ordered rings by way of theusual row and column operations on matrices but with certaininequalities imposed. Strong shift equivalence theory is a combination of algebra andtopology that grew out of the dynamics of systems known as shiftsystems and their symmetry groups. It turns out to have unexpectedand interesting connections with areas of mathematics outsidedynamics, such as algebraic K-theory, cyclic homology, and topologicalquantum field theory. Wagoner will continue to explore these newdevelopments.***