Intuition for almost periodic solution and Poincaré recurrence theorem

Question

I would like to ask a question that I had asked yesterday on the site math.stackexchange and I still have not received an answer.

Suppose that we have a PDE that admit a solution $u$ that can be expressed in a certain system of coordinates (angle-action variables) as advection with constant velocity on tori. And suppose also that the solution is almost periodic i.e. $$\forall \epsilon >0,\ \exists \ell>0,\ \forall a \in \mathbb R,\ \exists \tau \in [a,a+\ell],\ \forall t\in \mathbb R,\ \|u(t+\tau)-u(t)\|<\epsilon.$$ I'm trying to understood the behavior of the solution intuitively. Does that mean that if we start from an initial data $u_0$ and if we follow the evolution which is given by the winding of a straight line on the torus with constant speed, then we don't necessarily have that the curve (the wound straight line) closes on itself, but rather we have that the wound straight line is going to pass an infinity of times "close" to the starting point, which makes us deduce that the evolution of the solution verifies the criterion of Poincaré's recurrence, which says roughly that

Given a $u_0$, there exist a sequence $t_n\to\infty$ such that the solution $u_0(t_n)\to u_0$.

Answer 1

The definition which you wrote can be translated in words like this.

You evaluate the solution at time $t$. Then you fix a small distance $\epsilon>0$. Then you can find a time length $\ell(\epsilon)$ (independent of $t$) having the following property: within $\ell$ time units from whatever instant $a$, you will find an instant $\tau$ such that the solution at $\tau$ is $\epsilon$-close to the solution at $t$.

So yes, the integral curves do not need to close (although that would happen in the particular case of a genuinely periodic solution). But in fact you have more than you claim. Indeed, not only you will pass infinitely many times "close" to the solution at $t$ (in topological dynamics this would be called recurrence), but you can control from above, through the duration $\ell$, the gaps between successive "close" passages of the solution, that is your recurrence is uniform.

If, in addition, you can control $\ell$ also from below, then your solution would be called quasi-periodic, and you could say that $\ell$ is the $\epsilon$- period. Finally, if the period is independent of $\epsilon$ you can see easily that the solution is periodic.