Spatial chaos of Wang tiles with two symbols

Jin-Yu Chen National Chiao Tung University Yu-Jie Chen National Chiao Tung University Wen-Guei Hu Sichuan University Song-Sun Lin National Chiao Tung University

Dynamical Systems mathscidoc: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
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@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.
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