Let $A\in \mathcal{M}_{m\times m}(\mathbb R)$ , $det(A)=1$ , $A$ is positively definite. Which matrices $P$ satisfy the equation $$P^TAP=A$$ In fact I am interested in sequences of traces $tr P^n$ of the iterations of such solutions.

In dimension $2$ one can show that $$P^n=\left( \begin{array}{cc} \cos n\phi& -\beta \sin n\phi \\ \alpha \sin n\phi & \cos n\phi\\ \end{array}\right)$$ for some $\phi, \alpha, \beta$ satisfying$\alpha\beta=1$ hence $tr P^n= 2\cos n\phi$.