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I know that the alpha and omega limit sets of a flow on a compact connected invariant subset of a manifold must be connected and these limit sets are contained in the non wandering set.

My question is whether the non-wandering set also has to be connected? If not, are there any counter-examples that can disprove this?

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  • $\begingroup$ The flow $\dot x=\sin x$ on the reals has a disconnected non-wandering set. The glow $\dot x=0$ has a connected non-wandering set. $\endgroup$
    – Anthony Quas
    Commented Oct 29, 2020 at 15:54
  • $\begingroup$ Thank you for your answer. Can you be more specific by explicitly writing the non-wandering sets for both the examples? $\endgroup$
    – Atul Anurag Sharma
    Commented Oct 29, 2020 at 16:54
  • $\begingroup$ The non-wandering sets are the fixed points. $\endgroup$
    – Anthony Quas
    Commented Oct 29, 2020 at 21:53
  • $\begingroup$ The flow is defined by $$\phi_{t}(x) : \mathbb{R} \to \mathbb{R}$$ For $\dot{x} = 0$, we have $$\phi_{t}(x) = x$$ For this case: $$\phi_{t}(\mathbb{R}) \ \cap \ \mathbb{R} \neq \emptyset$$ and hence I can conclude that the entire real line(which is connected) is a non-wandering set. Is that right?@AnthonyQuas $\endgroup$
    – Atul Anurag Sharma
    Commented Oct 29, 2020 at 22:48
  • $\begingroup$ Let $x\in\mathbb R$. Let $N$ be any neighbourhood of $x$. Then for any $T>0$, there exists $t>T$ such that $\phi_t(x)\in N$(in fact this is trivially true for all $t$). Hence, by the definition, $x$ is non-wandering. $\endgroup$
    – Anthony Quas
    Commented Oct 29, 2020 at 23:07

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