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.

中文翻译：

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逻辑图的图形是一个塔

动态系统的定性行为可以用图形编码。图中的每个节点都是一个链循环点的等价类，如果使用任意小的扰动，可以将从 $A$ 的任何点开始的轨迹引导到从节点 $A$ 到节点 $B$ 的边任何一点$B$。在本文中，我们描述了逻辑图的图形。我们的主要结果是图总是一个塔，即有一条边连接每对不同的节点。请注意，这些图从不包含循环。如果从节点$A$到节点$B$有一条边，$A$中某个周期轨道的不稳定流形包含最终映射到$B$的点。对于特殊的参数值，这个塔有无限多个节点。