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arXiv · 2008.08338

The graph of the logistic map is a tower

Roberto De Leo·James A. Yorke
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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.

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Roberto De Leo, James A. Yorke. 2020-08-19. The graph of the logistic map is a tower. https://arxiv.org/abs/2008.08338

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