Why does the limit set contain forward orbit?

Question

I just started reading on dynamical systems and the author made a claim that I don't quite understand. If someone could elaborate it would be highly appreciated!

So here is the problem: Let $X$ be a topological space $F:X \times \mathbb{N}_0\to X$ a flow with $F(x,0) = x$ and $F(x, n+k) = F(F(x, n),k)$. For each $x\in X$, we put $x(n) = F(x,n)$.

Now, define the limit set of $x\in X$ by $$ \omega(x) = \{y\in X : \forall \text{neighbourhoods $U$ of } y, \exists n \text{ arbitrarily large such that } x(n) \in U\}. $$ The author claims that if $y\in \omega(x)$ then $y(n)\in\omega(x)$ for all $n\in \mathbb{N}$. Can someone explain why this is true?

Answer 1

The problem takes the form "If $y\in \omega(x)$ then $y(n)\in\omega(x)$". Start by restating the problem in terms of the provided definitions:

• You are given a hypothesis: for all neighborhoods $U$ of $y$, there exists $t$ arbitrarily large such that $x(t)\in U$.
• You want to prove a conclusion: for all neighborhoods $V$ of $y(n)$, there exists $s$ arbitrarily large such that $x(s)\in V$.

Hint: In order to use the hypothesis, you need to describe some neighborhood $U$ of $y$. Your $U$ will have to depend somehow on the data provided for the conclusion: the number $n$ and the neighborhood $V$ of $y(n)$. Given $n$ and $V$, how can you construct an appropriate $U$? Then, feeding $U$ into the hypothesis, you will get arbitrarily large $t$. How can you turn this into arbitrarily large $s$ that satisfies the conclusion?