Topics in Symbolic Dynamics

在当前的项目中，PI将探索符号动力学领域的几个问题和公开问题。动力学是对封闭系统的研究，其元素根据确定性规则随着时间的推移而演变；一些例子包括行星运动，或粒子在盒子内的运动。符号动力学是对动力学的研究，其中所讨论的元素是无限的符号序列，随着时间的演变来自于水平移动序列。符号动力学最初是作为研究更复杂动力系统的工具引入的，但它本身已成为一个基本领域，其应用和联系范围从计算机科学(例如，二进制数据存储)到统计物理(例如，磁性的伊辛模型)。目前一个高度活跃的领域是多维符号动力学，其中考虑的不是序列，而是可以向任何方向移动的符号的多维阵列。虽然核心的定义和想法保持不变，但多个维度发生了显著的变化；许多在一个维度上容易回答的问题变得相当困难，在一个维度上行为相对简单的系统类可能会表现出奇怪和复杂的行为。这项拨款支持的研究将处理符号动力学中的各种问题，从更经典的一维情况到较少被理解的多维情况。贯穿始终的一个共同主线是使用来自其他领域的想法、技术和观点，包括概率论、统计物理学和渗流理论，以解决基本问题。这两个关于一维符号动力学的项目分别提出了关于从动集的经典概念和规范性质的新的研究方向。第一个多维项目提出了一种有效逼近有限类型多维位移熵的新方法，与现有技术不同，该方法可用于具有多个最大熵的系统。最后一个项目将扩展PI和Kevin McGoff最近在随机多维SFT上的工作，在该工作中定义了一个概率框架，其中“通常”选择的多维SFT不会表现出该类最糟糕的病理行为；我们现在计划使用这个框架来证明一些经典的一维结果的多维概率版本。所有工作将通过同行评议期刊上的出版物、会议报告以及作者与年轻研究人员(包括他的两名博士生)的持续合作来传播。

In the current project, the PI will explore several questions and open problems in the area of symbolic dynamics. Dynamics is the study of closed systems whose elements evolve over time according to deterministic rules; some examples include planetary motion, or a particle's movement inside a box. Symbolic dynamics is the study of dynamics where the elements in question are infinite sequences of symbols, and the evolution over time comes from horizontally shifting the sequence. Symbolic dynamics was introduced as a tool for studying more complicated dynamical systems, but has become a fundamental area in its own right, with applications and connections ranging from computer science (e.g. data storage in binary) to statistical physics (e.g. the Ising model for magnetism). A current area of high activity is multidimensional symbolic dynamics, in which one considers not sequences, but multidimensional arrays of symbols, which can be shifted in any direction. Though the core definitions and ideas remain the same, there are remarkable changes that occur for multiple dimensions; many questions that were easy to answer in one dimension become quite difficult, and classes of systems whose behavior was relatively simple in one dimension can exhibit strange and complex behaviors.The research supported by this grant will treat a variety of problems in symbolic dynamics, ranging from the more classical one-dimensional case to the less well-understood multidimensional case. A common thread throughout is the use of ideas, techniques, and viewpoints from other areas, including probability theory, statistical physics, and percolation theory, to attack fundamental problems. The two projects on one-dimensional symbolic dynamics propose new directions of research about the classical notions of follower sets and the specification property respectively. The first multidimensional project proposes a new method for the efficient approximation of entropies of multidimensional shifts of finite type (SFT) which, unlike existing techniques, can be used on systems with multiple measures of maximal entropy. The final project will extend recent work by the PI and Kevin McGoff on random multidimensional SFTs, in which a probabilistic framework was defined where a "typically" chosen multidimensional SFT does not exhibit the worst pathological behaviors of the class; we now plan to use this framework to prove multidimensional probabilistic versions of some classical one-dimensional results. All work will be disseminated through publications in peer-reviewed journals, conference presentations, and the author's continued collaboration with young researchers, including his two Ph.D. students.