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.

$A_{n+1} = UA_n + (I-U)A_n\circ\sum_{i=1}^{m}\circ_{k=1}^{i}(D^kB_n)$

$B_{n+1} = RB_n + (I-R)B_n\circ\sum_{i=1}^{m}\circ_{k=1}^{i}(L^kA_n)$