A framework for the study of the qualitative dynamics of general Iterated Function Systems

We develop a qualitative-dynamics framework for general Iterated Function Systems (IFSs) on locally compact spaces. Our approach extends to IFSs a framework recently developed in the semiflows setting by James Yorke and the present author based on binary relations (streams) that encode which point can be regarded as “downstream” from which. This leads to a graph representation of the global qualitative dynamics that separates recurrent behavior from gradient-like behavior.