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 \begin{document}$ A $\end{document} to node \begin{document}$ B $\end{document} if, using arbitrary small perturbations, a trajectory starting from any point of \begin{document}$ A $\end{document} can be steered to any point of \begin{document}$ B $\end{document} . In this article we describe the graph of the logistic map. Our main result is that the graph is always a tower, namely there is an edge connecting each pair of distinct nodes. Notice that these graphs never contain cycles. If there is an edge from node \begin{document}$ A $\end{document} to node \begin{document}$ B $\end{document} , the unstable manifold of some periodic orbit in \begin{document}$ A $\end{document} contains points that eventually map onto \begin{document}$ B $\end{document} . For special parameter values, this tower has infinitely many nodes.