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Edit: According to the comment of Prof. Eremenko I revise the question.

19 years ago, I have heard the following problem from a specialist of dynamical system. During these 19 years, I was in contact with him and I asked him if there is a solution to this problem He always answered me "No, I did not find any answer yet". Now I quote the problem here:

Problem:

Is there a real analytic vector field locally defined around origin in $\mathbb{R}^3$, which has an isolated singularity at origin and satisfies the following:

There exist an orbit $\gamma(t)$ which tends to origin as $t$ goes to $+\infty$ and the closure of$\{ \frac{\gamma(t)}{|\gamma(t)|},\;t\in\mathbb{R}^+\}$ has a nonempty interior in $S^2$

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Of course not. The image of a $C^1$ function on $\mathbb R^+$ has $\sigma$-finite Hausdorff $1$-dimensional measure, and therefore has $2$-dimensional measure $0$. In fact, $C^1$ can even be weakened to differentiable: see my answer here.

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  • $\begingroup$ Thank you for your answer. May be I miss remember the precise formulation of his problem. $\endgroup$
    – Ali Taghavi
    Feb 26, 2019 at 20:10
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    $\begingroup$ He probably means that the limit set of $\gamma(t)/|\gamma(t)|$ has non-empty interior. $\endgroup$
    – Alexandre Eremenko
    Feb 26, 2019 at 20:10
  • $\begingroup$ @AlexandreEremenko Yes yes yes that is the true formulation. I revise the question. Thank you! $\endgroup$
    – Ali Taghavi
    Feb 26, 2019 at 20:13
  • $\begingroup$ @Robert I revise the question. $\endgroup$
    – Ali Taghavi
    Feb 26, 2019 at 20:16

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