Streams, Graphs and Global Attractors of Dynamical Systems on Locally Compact Spaces

In a recent article, we introduced the concept of streams and graphs of a semiflow. An important related concept is the one of semiflow with { compact dynamics}, which we defined as a semiflow F with a { compact global trapping region}. In this follow-up, we restrict to the important case where the phase space X is locally compact and we move the focus on the concept of { global attractor}, a maximal compact set that attracts every compact subset of X. A semiflow F can have many global trapping regions but, if it has a global attractor, this is unique. We modify here our original definition and we say that F has compact dynamics if it has a global attractor G. We show that most of the qualitative properties of F are inherited by the restriction FG of F to G and that, in case of Conley's chains stream of F, the qualitative behavior of F and FG coincide. Moreover, if F is a continuous-time semiflow, then its graph is identical to the graph of its time-1 map. Our main result is that, for each semiflow F with compact dynamics over a locally compact space, the graphs of the prolongational relation of F and of every stream of F are connected if the global attractor is connected.