Suppose $(M, g)$ is a compact Riemannian manifold, and $N subset M$ a closed Riemannian submanifold. Assume that there’s a smooth endomorphism of $M$, $f : M to M$, such that $f^*g = Ccdot g$, with $C > 0$ a constant.

Now let’s consider the subset $X$ of $M$ defined as the union of all the images $f^k(N)$ of $N$ under iterates of $f$, for all $kge 0$ ($f^0(N)=N$) and let’s endow $X$ with the topology induced by $M$.

Is there any meaningful way (and if so, under what additional assumptions on $f$?) to view $X$ as a foliation on $M$?

I’m guessing we’d have to assume something about the fixed point set of $f$ and perhaps we’d have to assume $f$ is submersive. I’d be greatful for any literature reference relevant to this question.