Mathematical Sciences: Symbolic Dynamics and Related Topics

摘要 波义耳 波义耳将研究涉及符号动力学的各种主题，包括满射细胞自动机的周期点的密度;在整数和其他子环的实数的非负矩阵的逆谱问题;这个问题的代数推广;有限类型的可约移位的分类在一个 本文讨论了这些系统的一般代数分析的函子设置及其与C ~* 代数的关系、拓扑轨道等价、一般符号动力学框架和剩余熵。 为了对符号动力学有一个粗略的初步概念，想想研究涉及很长的0和1带的问题。适当的理想化是考虑无限磁带。 当考虑数据传输和存储问题时，这样的研究是有意义的。 它也是研究物理系统如何随时间演化的关键之一：想象一下，将一个系统切成（比如说）两部分，并根据它在两部分中的行程将一个0和1的序列与一个点联系起来。通过相关的行程来研究一个系统实际上是有用的。 正如数学中所发生的那样， 已经被证明与其他领域表面上无关的问题有关。两个例子是这个建议的研究对象：理解非负矩阵的代数结构，这是数学的许多应用中的基础， 一个框架，用于理解与数学主题C*-代数的某些关系。该提案的另一个主题涉及理解细胞自动机（计算机游戏生命是一个例子），它可以被视为物理模型， 进化最后一个主题，轨道等价，涉及研究的性质的动力系统，不依赖于行程的顺序。

Abstract Boyle Boyle will study various topics involving symbolic dynamics, including the density of periodic points for surjective cellular automata; the inverse spectral problem for nonnegative matrices over the integers and other subrings of the reals; algebraic generalizations of this problem; the classification of reducible shifts of finite type in a functorial setting suitable to the general algebraic analysis of these systems and their relations with C* algebras; topological orbit equivalence; general symbolic dynamical frameworks and residual entropy. To have a crude initial idea of symbolic dynamics, think of studying problems involving very long tapes of zeros and ones. An appropriate idealization is to consider infinite tapes. Such a study is significant when one considers problems of data transmission and storage. It also is one key to the study of how physical systems evolve over time: imagine cutting up a system into (say) two pieces, and associating to a point a sequence of zeros and ones according to its itinerary through the two pieces. It is actually useful to study a system by way of the associated itineraries ... As happens in math, the theory developed for this has turned out to be relevant to superficially unrelated problems in other areas. Two examples are objects of study of this proposal: understanding the algebraic structure of nonnegative matrices, which are basic in many applications of mathematics, and improving a framework for understanding certain relations with the mathematical subject of C*-algebras. Another topic of this proposal involves understanding cellular automata (the computer game of Life is an example) which can be viewed as models for physics and evolution. The final topic, orbit equivalence, involves the study of properties of a dynamical system which do not depend on the way itineraries are ordered.