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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

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    Does $tr(A) = tr(B)$?2012-10-08
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    No! $tr(A)$ can be diferent of $tr(B)$2012-10-08
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    Could you clarify your notation? $A,B$ are matrices and $F$ is a function. What do $A\circ F$ and $F\circ B$ mean? Are you first applying $F$ then multiplying by $A$ and for the latter are you multiplying by $B$ and then applying $F$?2012-10-08
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    I tried to clarify my question. if it has not been clear can ask more questions2012-10-08
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    What have you done so far?2012-10-08
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    Do you want integer entries? Otherwise there are lots of easy counterexamples.2012-10-08
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    yes integer entries2012-10-08

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