There is a relation between the eigenvectors of a matrix $A \in \text{SL}(2,\mathbb{R})$ and its transpose $A^T$?

Question

I'm working with a function $F: \text{SL}(2,\mathbb{R}) \setminus \{ I, -I\} \to S^1$, where SL$(2, \mathbb{R})$ is the set of $2 \times 2$ real matrices with determinant $1$, $I$ is de identity matrix $2 \times 2$ and the function is defined as $$ F(A) = \begin{cases} 1, & \text{if } | \text{tr} A| \leq 2 \\ \dfrac{v^2}{w^2}, & \text{if } | \text{tr} A| > 2 \end{cases} $$ being $v, w \in S^1 \subset \mathbb{C}$ the normalized eigenvalues of $A$, with eigendirections $\pm v, \pm w$ and associated eigenvalues $\lambda, \dfrac{1}{\lambda}$, respectively, where $|\lambda| > 1$. I Have to prove the continuity of this functions and some properties like $F(A) = \overline{F(A^T)}$ and $F(R_{-\theta}AR_{\theta}) = F(A)$, being $A^T$ the transpose of $A$ and $\overline{F(A^T)}$ the complex conjugate of $F(A^T)$ and $R_{\theta} = \begin{pmatrix} \cos \theta & -\sin \theta\\ \sin \theta & \cos \theta \end{pmatrix}$ the rotation matrix by angle $\theta \in \mathbb{R}$. I've being searching for some results or some informations that can help me to find out a relation between the eigenvector $A$ and $A^T$, and I have now ideia about it, even in that special case that $A, A^T \in \text{SL}(2, \mathbb{R}) \setminus \{ I, -I\} $. Of course I know that they have same eigenvalues, but nothing about eigenvectors. Anyway, I proved the continuity (by sequences) studying case by case, and it'll work, but I'm already in trouble to prove $F(A) = \overline{F(A^T)}$. If you have some ideia of how to prove that, or a relation that can helps me, I'd be appreciated. In any case, thanks for your atenttion.