The graph of the logistic map is a tower

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ A $\end{document}</tex-math></inline-formula> to node <inline-formula><tex-math id="M2">\begin{document}$ B $\end{document}</tex-math></inline-formula> if, using arbitrary small perturbations, a trajectory starting from any point of <inline-formula><tex-math id="M3">\begin{document}$ A $\end{document}</tex-math></inline-formula> can be steered to any point of <inline-formula><tex-math id="M4">\begin{document}$ B $\end{document}</tex-math></inline-formula>. In this article we describe the graph of the logistic map. Our main result is that the graph is always a tower, namely there is an edge connecting each pair of distinct nodes. Notice that these graphs never contain cycles. If there is an edge from node <inline-formula><tex-math id="M5">\begin{document}$ A $\end{document}</tex-math></inline-formula> to node <inline-formula><tex-math id="M6">\begin{document}$ B $\end{document}</tex-math></inline-formula>, the unstable manifold of some periodic orbit in <inline-formula><tex-math id="M7">\begin{document}$ A $\end{document}</tex-math></inline-formula> contains points that eventually map onto <inline-formula><tex-math id="M8">\begin{document}$ B $\end{document}</tex-math></inline-formula>. For special parameter values, this tower has infinitely many nodes.</p>