How to get asymptotic solution for vector iteration process as $f^{n+1}=B G B^T f^{n}$?

Question

Where $B$ is $M\times N$ matrix, $G$ is $N \times N$ orthonormal matrix, and $B^T$ is the transpose of $B$. For $N=1$, it is easy to see the asymptotic solution is the eigenvector of $B B^T$. But for $N>1$, I would like to know any simple relation to the matrix $B B^T$. Thanks.