# A special oscillatory orbit in space

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 $$S^2$$

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.
• He probably means that the limit set of $\gamma(t)/|\gamma(t)|$ has non-empty interior. – Alexandre Eremenko Feb 26 at 20:10