Article Contents
Article Contents

# Permutations with restricted movement

This work was supported by the Israel Science Foundation (ISF) under grant No. 270/18 and 1052/18

• A restricted permutation of a locally finite directed graph $G = (V, E)$ is a vertex permutation $\pi: V\to V$ for which $(v, \pi(v))\in E$, for any vertex $v\in V$. The set of such permutations, denoted by $\Omega(G)$, with a group action induced from a subset of graph isomorphisms form a topological dynamical system. We focus on the particular case presented by Schmidt and Strasser [18] of restricted ${\mathbb Z}^d$ permutations, in which $\Omega(G)$ is a subshift of finite type. We show a correspondence between restricted permutations and perfect matchings (also known as dimer coverings). We use this correspondence in order to investigate and compute the topological entropy in a class of cases of restricted ${\mathbb Z}^d$-permutations. We discuss the global and local admissibility of patterns, in the context of restricted ${\mathbb Z}^d$-permutations. Finally, we review the related models of injective and surjective restricted functions.

Mathematics Subject Classification: Primary: 37B50,; Secondary: 37B05, 37B40.

 Citation:

• Figure 2.1.  (a) The two-dimensional honeycomb lattice. (b) The fundamental domain

Figure 2.2.  (a) Paths configuration corresponding to an elements in $\Omega(G_{A_L})$. (b) Paths configuration corresponding to an elements in $\Omega(G_{A_+})$

Figure 2.3.  The graphs corresponding to $A_+$ and $A_L$

Figure 3.1.  The graph $G_{A_L}'$

Figure 3.2.  The quotient of $L_{H}$ by the action of $(2 {\mathbb Z})^2$

Figure 3.3.  A correspondence between a function in $B_{3, 3}(A_L)$ and a perfect cover of $\hat{V}_{3, 3}$ in $G_{A_L}$

Figure 3.4.  (a) The $T$-gadget and the weights on its edges. (b) The construction of $\hat{G}$ from $G$

Figure 3.5.  The correspondence between a restricted permutation of the honeycomb lattice, perfect matchings and permutations of ${\mathbb Z}^2$ restricted by $A_L$

Figure 3.6.  The corresponding graph for $G_{A_+}$ from Theorem 1. Black points represent vertices of the form $(I, n)$ and white points represent vertices of the form $(O, n)$

Figure 4.1.  A locally admissible pattern which is not globally admissible where the restricting set is $A_+$

Figure 4.2.  The extension of $\pi_v$ to a $\pi\in \Omega(A_\oplus)$

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