The graph of the logistic map is a tower

TL;DR

This paper characterizes the graph structure of the logistic map's dynamical behavior, showing it forms a tower with a complete hierarchy of connections between equivalence classes of chain-recurrent points.

Contribution

It provides a detailed description of the graph of the logistic map, revealing it as a tower with a complete set of directed edges and no cycles, including cases with infinitely many nodes.

Findings

The graph of the logistic map is always a tower with edges between every pair of nodes.

The graph contains no cycles, reflecting the unidirectional nature of the dynamics.

For certain parameters, the graph has infinitely many nodes.

Abstract

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.