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 $u_0(t_n)\to u_0$.