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تسجيل مستخدم جديدSpatial chaos of Wang tiles with two symbols
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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 $mathcal{P}(mathcal{B}) eqemptyset$ 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 $mathcal{B}=C_{1}cup C_{2} cupcdots cup C_{k} cup N$, where $C_{j}$, $1leq jleq k$, are minimal cycle generators and $mathcal{B}$ contains no minimal cycle generator except those contained in $C_{1}cup C_{2} cupcdots cup C_{k}$. Then, the positivity of spatial entropy $h(mathcal{B})$ is completely determined by $C_{1}cup C_{2} cupcdots cup C_{k}$. 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.
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