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: $$ \{A^n\}_{n\geq 1}\subset \Gamma\begin{pmatrix}+&-\\+&-\end{pmatrix} $$

What techniques are there for ensuring nonnegativity of various entries of matrix powers?

Specific Question: Consider a matrix $A\in SL_2(\mathbb R)$. Let $(A^n)_{i,j}$ denote the $(i,j)$ entry of the matrix power $A^n$. Under what conditions on $A$ does the following hold: $$ \text{sgn}\ (A^n)_{i,j}=(-1)^{j+1} $$