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I have already finished understanding the Poincaré-Bendixson theorem as a consequence of Schwartz's theorem, but I also want to analyze the example that Denjoy gave in $C^1$ that is not within the case taken in Schwartz's theorem. Does anyone know where I can find that detailed example? or maybe some generalization of this example? I would like to review it in detail to finish closing the idea of why Schwartz's theorem cannot be valid for flows of class $C^1$.

Schwartz Theorem.- Let $M$ be a compact, connected, two-dimensional manifold of class $C^2$. Let $\alpha: \mathbb R \times M \to M$ be a $C^2$ flow on $M$. Let $\Omega \subset M$ be an $\alpha$-minimal set. Then $\Omega$ must be one of the following:
$a)$ a singleton consisting of a fixed point
$b)$ a single, closed orbit homeomorphic to $S^1$
$c)$ all of $M$ which is homeomorphic to a torus $T^2$

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  • $\begingroup$ Suggestion: State explicitly the content of the Schwartz theorem you refer to. You might also specify which person named Schwartz proved it. $\endgroup$ Jul 22 at 21:36
  • $\begingroup$ Done I edited it. $\endgroup$
    – Zaragosa
    Jul 22 at 23:51
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    $\begingroup$ See p. 37 of these excellent notes, for a careful description of the Denjoy counterexamples: math.ens.psl.eu/~sghazoua/cdln.pdf $\endgroup$ Jul 23 at 1:30
  • $\begingroup$ There are good notes on the Denjoy counterexample by John Milnor, here: math.stonybrook.edu/~jack/DYNOTES/dn15.pdf. $\endgroup$ Jul 24 at 0:10

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