On the connectedness of attractors for dynamical systems (Q678029)

The paper addresses the question of when the global attractor of a continuous semigroup of operators acting on a connected metric space is connected (assuming that the global attractor exists). The two main results are that the attractor is connected if the semigroup is actually a semiflow, or for arbitrary semigroups, if the space is also locally connected. In establishing this, they also show that if a global attractor is not connected, then it has infinitely many components, yet it is `invariantly connected' (meaning that it is not the disjoint union of two nonempty, compact, positively invariant sets). The paper concludes with an example of a map defined on a subset of \(\mathbb{R}^{2}\) that generates a semigroup whose global attractor is disconnected.