The question that I have in mind is the following:

*Which kind of closed sets can arise as the $\omega$-limit of a point for a $1$-dimensional dynamical system?*

It is probably somewhat naive, but nowhere I looked did I find a hint of what kind of answer I was to expect. Let me specify a bit what got me to formulate my question this way.

Some of the simplest examples of such systems are rotations of the circle. For those, the $\omega$-limits are either a finite orbit or the whole circle and are from the topological point of view very simple. The Denjoy construction takes it to another level, it builds an homeomorphism of the circle for which the orbit of any point accumulates to a Cantor set. The price to pay though is that such a construction cannot be made $\mathcal{C }^2$.

Do those examples cover every possible accumulation sets: periodic orbits, Cantor sets and the whole circle?

Is it possible that the $\omega$-limit has non-empty interior, but is not the whole circle?

Is it possible to build examples for which an $\omega$-limit can be somehow 'hybrid'?

Can a $\omega$-limit be the union of several periodic orbits?

Are there countable $\omega$-limits which are not finite?

My last question is motivated by the existence of 'very complicated' countable closed subsets of $\mathbb{R}$: there exists countable closed subsets of $\mathbb{R}$ whose sequence of derived sets is non empty until any fixed ordinal. Can those appear as the $\omega$-limit of a homeomorphism of $\mathbb{R}$ or $S^1$? If so is there any restriction on the regularity of such an homeomorphism?

All those questions extend to other $1$-dimensional systems such as codimension $1$ foliations on manifolds, interval exchange transformation, groups acting on the circle...

I'd be very pleased if you could share any interesting example/theorem going some way to answering any of the question I asked. I understand some of those might be very stupid, I apologize in advance :)