what if an orbit is contained in its $\omega$- limit set?

Question

I guess it should be a periodic orbit, but I'm not sure whether there is an counterexample or not.

can you give me a proof or an counterexample?

Answer 1

I only really know about the 2d case. If the $\omega$ limit set has no fixed points, then Poincare-Bendixon says its a periodic orbit.

If it had a fixed point, then your orbit was just that fixed point.

So yes, if an orbit is contained in its $\omega$ limit set, it's periodic.

Answer 2

It is not true. As a counter-example, take an irrational rotation on the circle.

What it is true is that if the $\omega$ limit is equal to the orbit (say for iterations of a map acting on a complete metric space) then the orbit is finite.