Ergodic Theory and Symbolic Dynamics

摘要：Lind/Tuncel Lind将研究几个维度作用的各个方面，包括Zeta函数和方向熵。重点将放在扩展集的确定上，扩展集的一个分量的动力学如何影响另一个分量的动力学，以及有限类型的假设如何改变答案。他将进一步探讨某些行为的重合熵与相关无限图的生成树的增长率之间的关系。他还将寻求与电子网络的联系，这是因为他发现了一个新的不变量，用于研究生成树产生的马尔可夫位移。Tuncel将研究有限类型的二维移位。他观察到，当某些矩阵方程有(正)解时，它们就决定了熵，而且类似的不等式给出了熵的界限。他将继续研究方程有解的共轭类的特征，通过状态分裂来改善边界的可能性，以及与全息数据存储相关的编码算法中不等式的使用。他还将继续分析马尔可夫链及其分类，重点是开发动态不变量和处理近边界约束的编码方法。编码数据的传统问题一直集中在一维序列上，例如在计算机磁盘驱动器上或在压缩音频光盘上(以螺旋形式)找到的序列。在这种情况下，物理约束意味着，为了有效编码，必须将任意数据编码成受约束的数据。例如，光盘上的数据必须具有这样的性质：在连续的1‘S之间至少有两个，但不超过7，0’S。符号动力学为发现和研究具有实际意义的编码方法提供了数学框架。最近，工业和研究领域的数学家都在研究这些问题的更高维版本。例如，工业界对全息数据存储感兴趣，在这种存储中，信息分布在三维立方体上。研究数学家们已经在这一领域发现了令人着迷的新现象，特别是与交换代数和代数几何这两个看似无关的领域有着丰富的联系。研究人员将继续他们的高维理论工作，重点是测量信息的方法和对周期结构的研究。

ABSTRACT: Lind/Tuncel Lind will investigate various aspects of several dimensional actions, including zeta functions and directional entropies. Emphasis will be placed on the determination of the expansive sets, how the dynamics of one component of the expansive set may influence the dynamics in another, and how finite type assumptions may change the answers. He will further probe the coincidence entropies of certain actions with the growth rate for spanning trees of related infinite graphs. He will also pursue a link with electrical networks suggested by his discovery of a new invariant for Markov shifts arising from consideration of spanning trees. Tuncel will work on two-dimensional shifts of finite type. He has observed that certain matrix equations determine the entropy whenever they have (positive) solutions, and that similar inequalities give bounds on entropy. He will pursue the characterization of conjugacy classes for which the equations have solutions, the possibility of improving the bounds by state-splitting, and the use of the inequalities in coding algorithms related to holographic data storage. He will also continue his analysis of Markov chains and their classifications, with emphasis on the development of dynamical invariants and coding methods for handling near-boundary constraints. Traditional problems of encoding data have focused on one-dimensional sequences, such as those found on computer disk drives or (in spiral form) on compact audio discs. In such situations physical constraints mean that, for efficient encoding, arbitrary data must be encoded into data subject to constraints. For instance data on a compact audio disk must have the property that between consecutive 1's there are at least two, but no more that seven, 0's. Symbolic dynamics provides that mathematical framework for discovering and investigating methods of encoding that have practical implications. More recently both industrial and research mathematicians have been studying higher-dimen sional versions of these problems. Industry is interested, for example, in holographic data storage where information is spread over a three-dimensional cube. Research mathematicians have been discovering fascinating new phenomena in this area, in particular some rich connections with the seemingly unrelated fields of commutative algebra and algebraic geometry. The investigators will continue their work with the higher-dimensional theory, with emphasis on ways of measuring information and a study of periodic structures.