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I've been drawn to a problem that requires ascertaining the existence of fixed points in the following recurrence relation, any ideas would be much appreciated. I seek neccessary conditions on $A,B$ such that a fixed point exists.

For $A,B \in \{0,1\}^{mn}$ and $U,L,R,D$, the permutation matrices that shift rows up, left, right, and down respectively (rows and columns both cycling), i.e. $[UA]_{1j} = [A]_{mj}$ etc;

and $\circ_{i=0}^{k}A = A\circ A\circ A ...$, where $\circ$ denotes the hadamard product, excuse the awful notation.

$A^{n+1} A_{n+1} = UA^n UA_n + (I-U)A^n\circ\sum_{i=1}^{m}\circ_{k=1}^{i}(D^kB^n)$I-U)A_n\circ\sum_{i=1}^{m}\circ_{k=1}^{i}(D^kB_n)B^{n+1} B_{n+1} = RB^n RB_n + (I-R)B^n\circ\sum_{i=1}^{m}\circ_{k=1}^{i}(L^kA^n)$I-R)B_n\circ\sum_{i=1}^{m}\circ_{k=1}^{i}(L^kA_n)$

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I've been drawn to a problem that requires ascertaining the existence of fixed points in the following recurrence relation, any ideas would be much appreciated. I seek neccessary conditions on $A,B$ such that a fixed point exists.

For $A,B \in {0,1}^{mn}$ ; \{0,1\}^{mn}$ and$U,L,R,D$, the permutation matrices that shift rows up, left, right, and down respectively (rows and columns both cyclingcycling), i.e. $[UA]{1j} [UA]_{1j} = [A]{mj}$A]_{mj}$ etc;

and $\circ_{i=0}^{k}A = A\circ A\circ A ...$ ..$, where$\circ$denotes the hadamard product, excuse the awful notation.$A^{n+1} = UA^n + (I-U)A^n\circ\sum_{i=1}^{m}\circ_{k=1}^{i}(D^kB^n)B^{n+1} = RB^n + (I-R)B^n\circ\sum_{i=1}^{m}\circ_{k=1}^{i}(L^kA^n)$2 added 4 characters in body; deleted 6 characters in body; added 14 characters in body I've been drawn to a problem that requires ascertaining the existence of fixed points in the following recurrence relation, any ideas would be much appreciated. I seek neccessary conditions on$A,B$such that a fixed point exists. For$A,B \in {0,1}^{mn}$;$U,L,R,D$, the permutation matrices that shift rows up left right and down respectively (rows and columns both cycling, i.e.$(UA)[UA]{1j} = (A)[A]{mj}$; mj}$ etc;

and $\circ_{i=0}^{k}A = A\circ A\circ A ...$ where $\circ$ denotes the hadamard product, excuse the awful notation.

$A^{n+1} = UA^n + (I-U)A^n\circ\sum_{i=1}^{m}\circ_{k=1}^{i}(D^kB)$I-U)A^n\circ\sum_{i=1}^{m}\circ_{k=1}^{i}(D^kB^n)B^{n+1} = RB^n + (I-R)B^n\circ\sum_{i=1}^{m}\circ_{k=1}^{i}(L^kA)$I-R)B^n\circ\sum_{i=1}^{m}\circ_{k=1}^{i}(L^kA^n)$

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