  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 iterates 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.

Dynamical Systems