Ergodic Theory and Symbolic Dynamics

摘要：Lind/Tuncel 林德将调查几个维度行动的各个方面，包括zeta函数和方向熵。重点将放在确定的可扩展集，如何扩展集的一个组件的动态可能会影响另一个动态，以及如何有限类型的假设可能会改变答案。他 将进一步探讨某些行动的重合熵与相关的无限图的生成树的增长率。他还将追求一个链接与电气网络建议他发现一个新的不变量马尔可夫转移所产生的考虑生成树。通塞尔将致力于有限类型的二维移位。他观察到，某些矩阵方程确定熵时，他们有（正）的解决方案，并认为类似的不平等给予界限熵。他将追求共轭类的方程有解的特征， 通过状态分裂改进边界的可能性，以及在与全息数据存储相关的编码算法中使用不等式。他还将继续他的分析马尔可夫链和他们的分类，重点是发展动态 处理近边界约束的不变量和编码方法。 编码数据的传统问题集中在一维序列上，例如在计算机磁盘驱动器上或（以螺旋形式）在压缩音频光盘上发现的那些序列。在这种情况下，物理约束意味着，为了有效编码，任意数据必须被编码成受约束的数据。例如，在一个压缩音频磁盘上的数据必须具有这样的属性，即在连续的1之间至少有两个，但不超过七个0。符号动力学提供了数学框架， 研究具有实际意义的编码方法。 最近工业和研究数学家一直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， 信息散布在三维立方体上。研究数学家已经发现了迷人的新现象在这一领域，特别是一些丰富的联系与看似无关的领域交换代数和代数几何。研究人员将继续他们的工作与高维理论， 强调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.