THE GRAPH OF THE LOGISTIC MAP IS A TOWER

The qualitative behavior of a dynamical system can be encoded in a graph. Each node of the graph is an equivalence class of chain-recurrent points and there is an edge from node A to node B if, using arbitrary small perturbations, a trajectory starting from any point of A can be steered to any point of B. In this article we describe the graph of the logistic map. Our main result is that the graph is always a tower, namely there is an edge connecting each pair of distinct nodes. Notice that these graphs never contain cycles. If there is an edge from node A to node B, the unstable manifold of some periodic orbit in A contains points that eventually map onto B. For special parameter values, this tower has infinitely many nodes.