双符王氏瓦当的空间混沌
Spatial chaos of Wang tiles with two symbols
摘要：
对平面边双符号着色（王图）中的空间混沌问题进行了完全分类。对于一组Wang瓦片$\mathcal{B}$，当空间熵$h(\mathcal{B})$为正时，空间混沌发生。如果$\mathcal{P}(\mathcal{B})\neq\emptyset$和$\mathcal{P}(\mathcal{B}')=\emptyset$在任何时候都是$\mathcal{B}'\subsetneqq \mathcal{B}$，则$\mathcal{B}$称为最小循环生成器，其中$\mathcal{P}(\mathcal{B})$是由$\mathcal{B}$生成的$\mathbb{Z}^{2}$上的所有周期模式的集合。给定一组Wang图块$\mathcal{B}$，写为$\mathcal{B}=C_{1}\cup C_{2} \cup\cdots \cup C_{k} \cup N$，其中$C_{j}$、$1\leq j\leq k$是最小循环生成器，并且$\mathcal{B}$不包含除$C_{1}\cup C_{2} \cup\cdots \cup C_{k}$中所包含的最小循环生成器之外的最小循环生成器。然后，空间熵$h(\mathcal{B})$的正性完全由$C_{1}\cup C_{2} \cup\cdots \cup C_{k}$决定。此外，有39个等价类的边际正熵（MPE）的王瓦片和18个等价类的饱和零熵（SZE）的王瓦片集。对于Wang瓦片$\mathcal{B}$的集合，当且仅当$\mathcal{B}$包含MPE集合时，$h(\mathcal{B})$为正，并且当且仅当$\mathcal{B}$是SZE集合的子集时，$h(\mathcal{B})$为零。
英文摘要：
This investigation completely classifies the spatial chaos problem in plane edge coloring (Wang tiles) with two symbols. For a set of Wang tiles $\mathcal{B}$, spatial chaos occurs when the spatial entropy $h(\mathcal{B})$ is positive. $\mathcal{B}$ is called a minimal cycle generator if $\mathcal{P}(\mathcal{B})\neq\emptyset$ and $\mathcal{P}(\mathcal{B}')=\emptyset$ whenever $\mathcal{B}'\subsetneqq \mathcal{B}$, where $\mathcal{P}(\mathcal{B})$ is the set of all periodic patterns on $\mathbb{Z}^{2}$ generated by $\mathcal{B}$. Given a set of Wang tiles $\mathcal{B}$, write $\mathcal{B}=C_{1}\cup C_{2} \cup\cdots \cup C_{k} \cup N$, where $C_{j}$, $1\leq j\leq k$, are minimal cycle generators and $\mathcal{B}$ contains no minimal cycle generator except those contained in $C_{1}\cup C_{2} \cup\cdots \cup C_{k}$. Then, the positivity of spatial entropy $h(\mathcal{B})$ is completely determined by $C_{1}\cup C_{2} \cup\cdots \cup C_{k}$. Furthermore, there are 39 equivalent classes of marginal positive-entropy (MPE) sets of Wang tiles and 18 equivalent classes of saturated zero-entropy (SZE) sets of Wang tiles. For a set of Wang tiles $\mathcal{B}$, $h(\mathcal{B})$ is positive if and only if $\mathcal{B}$ contains an MPE set, and $h(\mathcal{B})$ is zero if and only if $\mathcal{B}$ is a subset of an SZE set.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
