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。重点将放在膨胀集的确定上，膨胀集的一个组成部分的动态如何影响另一个组成部分的动态，以及有限类型假设如何改变答案。他将进一步探讨相关无限图的生成树的某些作用的符合熵与增长率的关系。他还将研究与电网络的联系，这是由于他发现了一个新的不变量，用于考虑生成树引起的马尔可夫位移。Tuncel将处理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，并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，通过状态分裂改善边界的可能性，以及在与全息数据存储相关的编码算法中使用不等式。他还将继续他对马尔可夫链及其分类的分析，重点是动态不变量的发展和处理近边界约束的编码方法。传统的数据编码问题集中在一维序列上，例如在计算机磁盘驱动器或（螺旋形式）压缩音频光盘上发现的序列。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为受约束的数据。例如，压缩音频磁盘上的数据必须具有这样的属性：在连续的1之间至少有两个0，但不能超过7个0。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，工业数学家和科研数学家都在研究这些问题的高维版本。例如，工业界对全息数据存储感兴趣，其中信息分布在三维立方体上。数学家们在这一领域不断发现令人着迷的新现象，特别是与交换代数和代数几何这两个看似无关的领域之间的一些丰富联系。研究人员将继续研究高维理论，重点是测量信息的方法和周期结构的研究。

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.