多维符号动力系统的非空问题与空间熵

Symbolic dynamics is an important topic in the area of dynamical systems, which is closely related to statistical physics. The results of higher-dimensional symbolic dynamical systems are fewer than those of one-dimensional case, and they are quite different essentially. Wang tiles are unit squares with colored edges. The tiling problems of Wang tiles are the shifts of finite type. Given a set of Wang tiles, the non-emptiness problem is to determine whether or not the set of global patterns that are generated by the set of Wang tiles is empty. To show the non-emptiness of global patterns, Wang conjectured that any set of Wang tiles that cantile a plane can tile the plane periodically. Currently, Wang’s conjecture is true for two or three colors but it fails for five colors. This project will investigate whether or not Wang’s conjecture holds for four colors. Also, we study the complexity of symbolic dynamical systems. Rectangular spatial entropy is usually used to measure the complexity of multidimensional symbolic dynamical systems, which is the limit of growth rate of admissible local patterns on finite rectangular sublattices which expands to whole space. For shift spaces, the topological entropy is equal to the rectangular spatial entropy. The spatial entropies of general expanding systems of sublattices of whole space are studied, which can be different from the rectangular spatial entropy. This project will investigate the relation between the spatial entropies and expanding systems, and finds an efficient method to compute spatial entropies.

符号动力系统是动力系统领域中的重要课题并且与统计物理密切相关。一维结果较为完整，但多维与一维有许多本质差异，且多维理论及具体计算方法较稀少。王氏瓷砖为边着色的正方形瓷砖，其拼砖问题为二维符号动力系统中的有限型移位。非空问题为能否用一组磁砖拼满全平面,为了解决此问题，王猜测如果一组王氏瓷砖可拼满全平面，则可周期性地拼满全平面。目前已知颜色数为5时，王猜测是不成立的；颜色数为2或3时，王猜测是对的。本项目第一个课题将研究颜色数为4时，王猜测是否成立。另一方面，我们将研究符号动力系统的复杂度，矩形空间熵是量测符号动力系统复杂度的一个常用度量，其为计算矩形花样个数随着矩形网格变大的增长速度，一般矩形空间熵等于拓朴熵。然而若选取不同网格逼近全空间的方式去计算空间熵，空间熵与矩形空间熵可能不相等。本项目第二个课题将研究网格逼近方式与空间熵之间的关系及研究计算空间熵的有效方法。

本项目主要研究多维符号系统的非空问题和空间熵。首先，非空问题中一个重要的问题就是王猜测，即一组王磁砖如果可以拼成全平面则可以周期性的拼成全平面，在此项目考虑王立方体的情况，王立方体是王磁砖的自然推广，我们证明了颜色数为2时，王猜测是对的。关于空间熵问题，空间熵是量测复杂度的重要指标，我们证明了多维移位空间一般空间熵等于拓朴熵的充分必要条件是其网格序列是Folner 序列；另外，乘法整数系统也是一类符号空间，其与多重遍历平均的等高集密切相关，我们也研究多维乘法整数系统的空间熵，得到其空间熵的级数表示公式。最后，拓朴性质是离散动力系统重要的概念，一般一个乘法整数系统可以由一个移位空间去生成，我们讨论该移位系统的混合性质与乘法整数系统的混合性质之间的关系，得到一些它们之间的充分必要条件，因此，可由移位系统的混合性质去判别乘法整数系统的混合性质。