We propose an analytic framework for multi-agent learning that, unlike standard approaches, is not connected to convergence to an equilibrium concept nor to payoff guarantees for the agents. We view multi-agent systems as reservoirs that allow for the long term survival of rich spatiotemporal correlations (i.e., patterns) amongst the agents' behaviors. Our aim is to develop abstractions that allow us to capture details about the possible limit behaviors of such systems. Our approach is based on the contrast between weakly and strongly persistent properties. Informally, a property is weakly persistent if for each starting point there exist limit points that satisfy it. A property is strongly persistent if it is satisfied by all limit points. In the case of non-converging dynamics the set of weakly persistent properties can be significantly richer than that of the strongly persistent properties reflecting topological properties of the system limit sets in a concise and algorithmically tractable manner.