Can Omega limit sets of dynamical systems be connected but not road connected? In the process of reading Wiggins, we have encountered the definition and properties of Omega limit sets for autonomous equations in a finite dimensional space. The connectivity of Omega limit set of a point is described in nature, but its path connectivity is not discussed. It is because the Omega limit set itself may not satisfy the road connectivity. Or is it because we can't use the Omega limit set's road connectivity in the research process?

Yes, the omega-limit set of a dynamical system can be connected but not path-connected. In fact this is the case for hyperbolic flows with an attractor, e.g. the Lorenz attractor, the Plykin attractor, some uniformly hyperbolic attractors etc.

For these systems, the omega-limit set is the attractor. This attractor contains a dense unstable leaf, hence is the closure of a path-connected set, and thus connected. On the other hand, a path in the attractor must stay in an unstable leaf, because the attractor is a Cantor set transversally to the unstable leaves. Since the attractor is not a single unstable leaf, there are points on the attractor that cannot be connected by a path. Actually, each unstable leaf is of empty interior in the attractor.