While working on the proof of the stable manifold theorem, I came across a problem that I'm not able to really grasp. Given some Anosov map $f: M \to M$ on a compact Riemann manifold $M$, one can prove that for $\epsilon > 0$ small enough, for every $x \in M$, then $W^s_\epsilon(x) :\,= \{ y \in M : d(f^n(x), f^n(y)) < \epsilon \}$ is a $C^1$ embedded submanifold of $M$. If one defines the 'global stable manifold' $W^s(x) :\,= \{ y \in M : d(f^n(x), f^n(y)) \rightarrow_n 0 \}$, then one has that: \begin{align*} W^s(x) = \bigcup_n f^{-n}(W^s_\epsilon(f^n(x)) \end{align*} From that claim, I've seen several references (for instance Shub's Global stability of dynamical systems) which conclude that $W^s(x)$ is an immersed submanifold of $M$. My questions are:

  • What does immersed means from a dynamics standpoint? As far as I'm concerned, $N \subset M$ is an immersed submanifold of $M$ if $N$ can be endowed with a smooth structure such that the inclusion map $\iota : N \to M$ is a smooth immersion, which is the same as a local embedding.

  • By looking at the problem here, we see that $W^s(x)$ is actually given by an increasing union of embedded submanifolds. It is true that an increasing union of embedded submanifolds is an immersed submanifold? If so, do we have some reference for this claim?



The map $\theta \mapsto (\sin \theta,\sin 2 \theta)$ define on $0 < \theta < 2\pi$ has image a figure 8, and is an immersion, but not a local embedding, i.e. not embedded in any neighborhood of the origin (the point where the 8 crosses itself).

The submanifold structure on each embedded $W^s(x)$ shows us that it is also immersed. Near each point of the union, there is a neighborhood in which that union consists precisely of the points of some one particular $W^s(x)$, i.e. the union is immersed.


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