# Looking to prove that there are infinitely many of these closed orbits

Prescribe a map:

$$\Psi:\zeta^2 \to \Bbb T^2$$

which gives a transformation of $$\zeta-$$space to the flat torus.

Let $$\zeta^2$$ consist of flow lines in which the sources correspond to $$(0,0)$$ and $$(1,0)$$ and the sinks correspond to $$(0,1)$$ and $$(1,1)$$:

$$\tau:= \{ (x, y) \in \Bbb R^2 | \ln(x)\ln(y)=\ln^2(s) \}$$ $$\mu:= \{ (x, y) \in \Bbb R^2 | \ln(1-x)\ln(y)=\ln^2(s) \}$$

$$x,y\in \Re(0,1)$$ and $$0< \Re(s)<1.$$

Combine the two sets of flow lines by adding their tangent spaces at each point.

I want to prove that for a stable, or rigid flow, there are infinitely many closed orbits on the flat torus after the map $$\Psi:\zeta^2 \to \Bbb T^2,$$ which sends vertical flow lines in $$\zeta^2$$ to closed orbits on $$\Bbb T^2.$$

A more natural site: Lorentzian geometry.

Notice that $$\tau$$ and $$\mu$$ individually are Killing flow lines preserving the same flat Lorentzian metric, $$g=\frac{dxdt}{xt},$$ so technically when I wrote $$\Bbb R^2$$ it didn't align with the natural setting for the problem, because that's a Riemannian manifold.

When these flows transversely intersect we can add up all pairs of vectors and generate a resultant Killing flow.

This suggests that the $$\Bbb T^2$$ we are considering must admit a Lorentzian metric and Killing field, as opposed to considering the Riemannian flat torus.

I think this restructuring of the problem is good and I didn't see it before. It remains to be seen how to show that the Killing field on the Lorentzian torus has infinitely many closed orbits.

Note: I think the construction above is sort of clunky. I think there is a more natural and elegant construction than the above, via conformal compactification of Lorentz-Minkowski, next adding pairs of vectors (space-like $$+$$ time-like) to get a resultant Killing flow and then stretching into the torus. This would be an alternative approach.

• I guess you’ll have more chances to obtain an answer if your state your question in a form allowing it to be understood by a mathematician with a general background, but not only to a specialist, especially, familiar with the used notations. Apr 10, 2020 at 4:02
• I'm working on the translation but it will take time Apr 15, 2020 at 17:29
• OK. Meantime, if you wish, I can try to save your bounty points by providing a fake answer. If you’ll award it with the bounty then I’ll return your points by awarding any your answer at MSE. Apr 17, 2020 at 12:07
• Now you can choose your answer which you want to be awarded. Apr 17, 2020 at 15:17
• I started the bounty. According to the rules, I'll may award it in 23 hours. Apr 17, 2020 at 15:30