I have seen a theorem which says that under certain hypothesis, given an elliptic fixed point $P$ for particular discrete dynamical systems ($X_{k+1}=S(X_k)$ with $S$ a conservative diffeomorphism not depending on time), there are invariant curves for every neighborhood of $P$ called Moser curves. Then it is concluded that if I have these curves, then the point is stable. I don't understand something though.
If I consider the region between two such invariant curves, why should the internal points into internal points of the image? I mean, the boundary given by these curves stays the same because of invariance, but why does the interior points get mapped to interior points? If I had a continous dynamical system I could say that every solution would be bound to stay in that set because otherwise it should pass through a Moser curve, violating the uniqueness of solution. But for discrete systems, why does this happen?
Is it true that a diffeomorhphism sends interior points to interior points? Or is it because there is also the hypothesis of the diffeomorphism being conservative?