Consider a matrix $A\in SL_2(\mathbb R)$, and let $\Gamma\begin{pmatrix}+&-\\+&-\end{pmatrix}\subset M_{2\times 2}(\mathbb R)$ denote the set of matrices whose coefficients have the indicated signs. Under what conditions on $A$ do we have the following: $$ \{A^n\}_{n\geq 1}\subset \Gamma\begin{pmatrix}+&-\\+&-\end{pmatrix} $$