I've got a linear transformation on $\mathbb{R}^2$ with complex eigenvalues and determinant equal to $1$, so it induces some kind of rotation on the plane. If I look at the orbit of a non-zero point under this transformation, I get points on an ellipse. Is it known that these points are either finite in number or else dense on the ellipse, according to whether the transformation is periodic or not?
Is this a standard result? I need it for something, and I'm trying to avoid re-inventing very many wheels along the way.
The way I see it most easily is that the transformation is just a change-of-coordinates away from being rotation on a circle through an angle $\theta$, and $\theta$ is either a rational multiple of $\pi$, or else you can use the pigeonhole principle to show that its multiples are densely distributed on $(0,2\pi)$. At least, I think that works.