Im currently studying differential equations and dynamical systems, specifically from the book of Gerald Teschl. I have a question about limit sets.
Def: The $w_{+}(x)$ (resp $w_{-}$) of a point $x$ is the set of those points $y$ for which there exists a sequence $t_n \to \infty $ (resp($t_n \to -\infty $)) such that $\Phi(t_n,x)\to y$, where $\Phi(t,x)$ denotes the flow of the system at time $t$ starting at $x$.
Now, the thing that confuses me is that the book continues proving lemmas/theorems about the set $w_{\sigma}(x)$, for example about its closure, invariance, compactness, connectedness and so on. Which tells me that this sets can be bigger than just a point. But, how can this set consist of more than one point? How can a solution/trajectory, have a limit on several points? In my logic, for example, $w_{+}(x)$ is either a point in the domain, or empty. Right? (I guess not..., but clarify please)
Perhaps an example will help.
Consider the flow on the circle $S^1$ generated by the unit tangent vector field $v_{x,y} = \langle -y,x\rangle$. The time $t$ map $\Phi(t,x)$ of this flow is simply rotation of the circle through an angle of $t$ radians.
Given any $x,y \in S^1$, I can find an angle $t \in (0,2\pi]$ such that $\Phi(t,x)=y$. But since rotation through angle $2\pi$ is the identity map, it follows that $\Phi(t + 2 \pi n,x)=y$ for all integers $n \ge 1$. Since $t + 2\pi n \to \infty$, it follows that $y \in w_+(x)$. This is true for every $x,y \in S^1$.
So we've shown that $w_+(x)=S^1$ for every $x \in S^1$.