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Let $U \subset \mathbb{R}^n$ be an open and connected set. We assume there is a vector field $F \in \mathcal{C}^1(\overline{U})$ giving rise to a DS ($\overline{U}$ denotes the closure) $$\dot{\mathbf{x}}=F(\mathbf{x})$$ We set $\Gamma_{\mathbf{x}_0}=\{\mathbf{x}(t)|t \in I_{max}\}$ where $I_{max}$ is the maximal interval of existence.

My question is the following: Is there a DS/ can somebody give an example of a DS which has a dense trajectory in $\overline{U}$, i.e. $\overline{\Gamma_{x_0}}=\overline{U}$?

I am unable to come up with an example and wonder if something like this actually exists.

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  • $\begingroup$ Cross posted: math.stackexchange.com/questions/4750810/… $\endgroup$
    – NicAG
    Commented Aug 11, 2023 at 15:12
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    $\begingroup$ For instance, the complement to the trefoil knot in $S^3$ admits an ergodic dynamical system, the geodesic flow of the modular orbifold. $\endgroup$
    – Moishe Kohan
    Commented Aug 12, 2023 at 3:32
  • $\begingroup$ @MoisheKohan Very interesting. Could you point me to some reference? $\endgroup$
    – NicAG
    Commented Aug 14, 2023 at 11:31
  • $\begingroup$ I am not sure about your level. You can start by reading E.Ghys' plenary talk at ICM-2006 in Madrid. You also need to know that ergodicity of a flow implies that almost every orbit is dense. $\endgroup$
    – Moishe Kohan
    Commented Aug 14, 2023 at 14:13

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