It's not clear to me exactly what "$1$-dimensional dynamical system" means, so I'll stick with continuous (but not necessarily invertible) transformations of $S^1$, and with the $\omega$-limit taken in forward time.
- The map $f \colon [-1,1] \to [-1,1]$ given by $f(x):=4x^3-3x$ has a dense orbit and fixes $1$ and $-1$. Take the disjoint union of two copies of $[-1,1]$ with this map defined on each. Now identify $1$ on the first interval with $1$ on the other, and identify $-1$ with the copy of $-1$ too. This creates a map of the circle with two invariant closed intervals. The "dense orbit" in each interval has $\omega$-limit set equal to the whole interval (i.e. a closed semicircle). I think that with a little care one could come up with an analytic example.