The graph of the logistic map is a tower (Q1983309)

The authors show how to construct a graph for the logistic map where each node of the graph is an equivalence class of chain-recurrent points. With each node it is assigned a numerical value. It is proved that there is an edge of the graph between each pair of nodes, from the higher-value node to the lower-value node. Furthermore, the graph has no cycles and possibly infinitely many nodes. The paper concludes with a chain-recurrent version of the spectral decomposition theorem proved for diffeomorphisms by \textit{S. Smale} [Bull. Am. Math. Soc. 73, 747--817 (1967; Zbl 0202.55202)].