Diabolical entropy

Milnor and Thurston's famous paper proved monotonicity of the topological entropy for the real quadratic family. Guckenheimer showed that it is H\"older continuous. We obtain a precise formula for the H\"older exponent at almost every quadratic parameter. Furthermore, the entropy of most parameters is proven to be in a set of Hausdorff dimension smaller than one, while most values of the entropy arise from a set of parameters of dimension smaller than one.