The main result of this paper is a proof using real analysis of the monotonicity of the topological entropy for the family of quadratic maps, sometimes called Milnor’s Monotonicity Conjecture. In contrast, the existing proofs rely in one way or another on complex analysis. Our proof is based on tools and algorithms previously developed by the authors and collaborators to compute the topological entropy of multimodal maps. Specifically, we use the number of transverse intersections of the map iterations with the so-called critical line. The approach is technically simple and geometrical. The same approach is also used to briefly revisit the superstable cycles of the quadratic maps, since both topics are closely related.

中文翻译：

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重新审视二次族的熵单调性和超稳定循环


本文的主要结果是使用二次映射族拓扑熵单调性的实分析证明，有时称为米尔诺单调性猜想。相比之下，现有的证明在某种程度上依赖于复杂的分析。我们的证明基于作者和合作者之前开发的用于计算多模态图的拓扑熵的工具和算法。具体来说，我们使用地图迭代的横向交点数量与所谓的临界线。该方法在技术上简单且几何。同样的方法也用于简要回顾二次映射的超稳定循环，因为这两个主题密切相关。