# MathSciDoc: An Archive for Mathematician ∫

#### Dynamical Systemsmathscidoc:1609.11003

Journal of Mathematical Physics, 57, 637-658, 2015
This investigation completely classifies the spatial chaos problem in plane edge coloring (Wang tiles) with two symbols. For a set of Wang tiles B, spatial chaos occurs when the spatial entropy h(B) is positive. B is called a minimal cycle generator if P(B) 6= ∅ and P(B′) = ∅ whenever B′ \$ B, where P(B) is the set of all periodic patterns on Z2 generated by B. Given a set of Wang tiles B, write B = C1 ∪ C2 ∪ · · · ∪ Ck ∪ N, where Cj, 1 ≤ j ≤ k, are minimal cycle generators and B contains no minimal cycle generator except those contained in C1 ∪ C2 ∪ · · · ∪ Ck. Then, the positivity of spatial entropy h(B) is completely determined by C1 ∪ C2 ∪ · · · ∪ Ck. 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 B, h(B) is positive if and only if B contains an MPE set, and h(B) is zero if and only if B is a subset of an SZE set.
Spatial chaos; spatial entropy; Wang tiles
```@inproceedings{jin-yu2015spatial,
title={Spatial chaos of Wang tiles with two symbols},
author={Jin-Yu Chen, Yu-Jie Chen, Wen-Guei Hu, and Song-Sun Lin},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914160430866050005},
booktitle={ Journal of Mathematical Physics},
volume={57},
pages={637-658},
year={2015},
}
```
Jin-Yu Chen, Yu-Jie Chen, Wen-Guei Hu, and Song-Sun Lin. Spatial chaos of Wang tiles with two symbols. 2015. Vol. 57. In Journal of Mathematical Physics. pp.637-658. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914160430866050005.