The Chain Graph of a general Iterated Function System

Abstract

Classical fractal geometry describes the metric and topological structure of attractors generated by contractive Iterated Function Systems (IFSs). Much less is known about their global qualitative dynamics once the contractive hypothesis is abandoned. In this article we focus on IFSs with "compact dynamics", namely those IFSs for which there exists a non-empty compact set invariant under the Hutchinson map that attracts every compact subset of the phase space. We call such a set the "global attractor" of the IFS. We introduce the "chain graph" of a general IFS as a directed graph encoding the qualitative dynamics features of the IFS. For an IFS with compact dynamics, the chain graph contains at least one node. Our main results are that the chain graph of an IFS with compact dynamics coincides with the chain graph of its restriction to its global attractor and that the chain graph of an IFS has at most as many nodes as the graph of its Hutchinson map.