3

I am reading parts of the book:

Hasselblatt B., Katok A. (1995) Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press.

In Exercise 1.6.2 we are asked to find a compact, smooth, Riemannian manifold $(M,g)$ and a smooth function $F:M\rightarrow \mathbb{R}$ such that there exists some $x\in M$ with the property that the $\omega$-limit set of $x$ of the corresponding gradient field $Y:=\nabla F$ contains more than one point. It was already shown in Proposition 1.6.4 that the $\omega$-limit set consists either of a single point or of infinitely many points.

I was wondering whether it is possible to take $M=S^2$ (with its standard metric). Due to the fact that the $\omega$-limit set must contain infinitely many points I was imagining that $Y$ might vanish on the equator and for instance at the north pole and that an appropriate field line $\gamma$ might 'emerge from' the north pole at $t=-\infty$ and 'spiral' down to the equator, such that the field line gets arbitrarily close to all points of the equator as time passes. However I don't know whether such a situation can be realised by a gradient vector field. I am aware of the Poincaré-Bendixson theorem, which restricts the field line behaviour of vector fields on $S^2$ altogether, that's why I am particularly curious about the case $M=S^2$. Thus my precise question is:

Is it possible to find a smooth function $f:S^2\rightarrow \mathbb{R}$ such that the corresponding gradient field (with respect to the standard metric) admits an orbit whose $\omega$-limit set contains more than one (and hence infintely many) points.

Kind regards

Dennis

Dennis
  • 347
  • 1
    The math.stackexchange website suggest this related question: https://math.stackexchange.com/questions/501007/when-does-gradient-flow-not-converge?rq=1. It seems that the example shown can be adapted to the sphere. – D. Thomine Sep 02 '19 at 17:47
  • 1
    Yes, the situation you describe can be realized, for instance by clever use of Whitney's extension theorem: if you specify along a close set (here, the reunion of your spiral, the north pole and the equator) something which looks like the restriction of the 1-jet of a smooth map, then it genuinely is. Here the "restricted 1-jet" should come from a gradient flow, which is where the challenge lies. Note that such a function would vanish along the equator. Hence it doesn't contradict the Poincaré-Bendixson theorem which holds in presence of a vector field vanishing in a finite number of points. – Jordan Payette Sep 02 '19 at 17:50
  • Of course, I met to say that the gradient of the function would vanish on the equator... – Jordan Payette Sep 02 '19 at 18:00
  • Thank you for your comments, I'll dwell into it! As for the linked stackexchange thread, Anthony constructs a specific metric on the torus such that the gradient of a given function with respect to this metric has the desired property. However this metric is not necessarily the standard metric. Nonetheless the discussion therein was insightful. Thank you! – Dennis Sep 02 '19 at 21:28

0 Answers0