# Denjoy example in the Poincaré–Bendixson theorem

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 $$T^2$$

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