# at.algebraic topology – A question on foliations by images

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.