Empty omega limit set

Question

I understand what is meant by a limit set but I don't understand what it would mean for this set to be empty. Could someone provide an example?

Answer 1

Well, in order for the limit set to be empty, you want $\{f^n(x): n\in\mathbb{N}\}$ to have no cluster points for any $x\in X$. So basically, you want $f$ to keep pushing points further and further away. Can you think of any nice continuous function $f$ on $\mathbb{R}$ which "keeps moving to the right?"

Answer 2

Let us remember the definition of flow:

Let $E \subseteq \mathbb{R}^{n}$ open and $f \in C^{1}\left(E\right)$. Let >$D\left(x\right)$ maximal interval corresponding to $x$. We define

$\Omega=\left\{\left(t,x\right)\in \mathbb{R}\times E \: : \: t \in D\left(x\right) \right\}.$

Then, we define the flow associated to $f$ as the funtion $\Phi:\Omega\rightarrow E$ such that

$\Phi\left(0,x\right)=x$

$\frac{d\Phi\left(t,x\right)}{dt}=f\left(\Phi\left(t,x\right)\right)$

In the case of continuous dynamical systems, especially for flows, the $\omega$**-limit set** for a given $x$ is defined by:

$\omega \left(\Phi\right)=\left\{y \in E \: : \: \mbox{There is } \left(t_{n}\right)_{n \in \mathbb{N}} \mbox{ such that } \lim_{n\rightarrow\infty}t_{n}=\infty \mbox{ and } \lim_{n\rightarrow\infty}\Phi\left(t_{n},x\right)=y\right\}$

This set can be empty, for example, we consider the flow defined by $\Phi\left(t,x\right)=x+t$ associated to the constant function $f\left(x\right)=1$.