Let $A$ and $B$ be $2\times 2$ matrices with $\mathrm{tr}(A)>2$, $\mathrm{tr}(B)>2$ and $\det(A)=1$, $\det(B)=1$.
My Question : there exists an bijetive application $F:\mathbb{R}^{2}\to\mathbb{R}^{2}$ such that $F(\mathbb{Z}^2)=\mathbb{Z}^2$ and $A\circ F=F\circ B$??
EDIT 1: Think $A$, $B$ and $C$ as applications of $\mathbb{R}^2\to \mathbb{R}^2$
EDIT2: $A$ and $B$ has integer entries.
I apologize for the careless drafting the question