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Let $X$ be a $C^1$ vector field on $\mathbb{R}^n$. Let $S$ be a compact submanifold of dimension $s(<n)$. Suppose $S$ is invariant under the flow of $X$ and that we know everything about the dynamics on $S$, in particular all the $\omega$-limit (and $\alpha$-limit) sets in $S$. Denote by $\Omega_S$ the set of all $\omega$-limit sets and $\alpha$-limit sets in $S$ and assume that $\gamma$ is a trajectory of $X$ which converges to $S$.

I'd like to know if it is true (or not) that $\omega(\gamma)\in\Omega_S$.

This may very well be a known result (although I haven't been able to find it yet), so a reference would be appreciated.

My attempt goes as follows

Since $\gamma$ converges to $S$, then $\omega(\gamma)\subseteq S$ and therefore $\omega(\gamma)$ is compact. Next, there are two cases:

  1. If $\gamma$ converges to $S$ transversally, then $\omega(\gamma)$ is an equilibrium point and surely $\omega(\gamma)\in\Omega_S$.

  2. Now we assume that $\gamma$ converges tangentially to $S$. Here I do not have much arguments, but here it goes: Since $\gamma$ converges to $S$, there exists a trajectory $\sigma\in S$, a sufficiently large $T>0$ and a sufficiently small $\delta>0$ such that $|\gamma(t)-\sigma(t)|<\delta$ for all $t>T$.

Edit: actually the previous argument is not necessarily true. At most I think we can say that ...there exists a trajectory $\sigma\in S$, a sufficiently large $T>0$,a sufficiently small $\delta>0$ and times $t_1,t_2$, $T\leq t_1<t_2$ such that $|\gamma(t)-\sigma(t)|<\delta$ for all $t\in(t_1,t_2)$.

Then my first thought would be that if Argument 2 really holds for all $t>T$, then $\omega(\gamma)=\omega(\sigma)$ and therefore $\omega(\gamma)\in\Omega_S$.

But on the other hand, I also think there might be some pathological cases in which this does not hold. For example, if $S$ is a disc, there might be the possibility that $\omega(\gamma)$ is the whole disc right? For example if $S$ has an infinite number of homoclinic orbits? I mean $\gamma$ may densely fills $S$, or some other strange things?

Any suggestion?

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You can have $X|_S = 0$ while $\omega(\gamma)$ is more than one point. For instance, something like $X(r, \theta) = f(r,\theta)(1-r, 1)$ (in polar coordinates) where $f$ is a function that is $0$ on the unit circle and positive elsewhere should have orbits in the unit disk which converge towards the whole unit circle $S$, while $\Omega_S$ consists of only single point elements. You can even make such an example with a unique singularity $p$ in $S$ by replacing $f$ with a function that is positive anywhere except on $p$, so $\Omega_S=\{\{p\}\}$.

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  • $\begingroup$ Thanks, agreed, the example you give is useful. However, perhaps I should change my question to is $\omega(\gamma)\in S\cup\Omega_S$? We know that $\omega(\gamma)\subseteq S$, where both of your examples fit. In other words, suppose $\gamma$ converges to a proper subset of $S$ .... then? $\endgroup$ – PepeToro Jun 8 '16 at 13:59
  • $\begingroup$ Add a third coordinate, and you can make a similar example where $S$ is the plane $z=0$ and $\omega(\gamma)$ is the unit circle (or other proper subsets of $S$) while $X=0$ on $S$. $\endgroup$ – Andres Koropecki Jun 27 '16 at 14:19

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