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:


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 $S^2$


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.

  • $\begingroup$ Thank you for your answer. May be I miss remember the precise formulation of his problem. $\endgroup$ Feb 26 '19 at 20:10
  • 2
    $\begingroup$ He probably means that the limit set of $\gamma(t)/|\gamma(t)|$ has non-empty interior. $\endgroup$ Feb 26 '19 at 20:10
  • $\begingroup$ @AlexandreEremenko Yes yes yes that is the true formulation. I revise the question. Thank you! $\endgroup$ Feb 26 '19 at 20:13
  • $\begingroup$ @Robert I revise the question. $\endgroup$ Feb 26 '19 at 20:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.