Planar discrete dynamical systems' omega limit sets

Question

This is just curiosity. Given a discrete time dynamical system $x_{n+1}=f(x_n)$ on the plane, $f:X\rightarrow X$ for $X$ a (relatively) compact region of $\mathbb{R}^2$, $f$ continuous on $X$. Suppose we know that an orbit has the property that its omega-limit set is compact, non-empty and contains no fixed point. Is it possible to conclude anything along the lines of Poincare'-Bendixon theorem (that is something like: then the omega limit set must be an orbit itself etc...)? I have searched a while but it seems that Poincare'-Bendixon theory in general really does not apply to discrete dynamical systems. Are there any close/weaker results to a situation as the above anyone happens to know about? Note: I have not assumed that the omega-limit set has finitely many points. Otherwise many things could be said about such omega-limit set (at least in the compact case).