Diabolical Entropy

  • Neil DobbsEmail author
  • Nicolae Mihalache


Milnor and Thurston’s famous paper proved monotonicity of the topological entropy for the real quadratic family. Guckenheimer showed that it is Hölder continuous. We obtain a precise formula for the Hölder exponent at almost every quadratic parameter. Furthermore, the entropy at 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.


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The authors thank Magnus Aspenberg, Viviane Baladi, Michael Benedicks, Davoud Cheraghi, Jean-Pierre Eckmann, Jacek Graczyk, Hans Koch and Masato Tsujii for helpful comments and conversations and the referees for insightful reports.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity College DublinDublin 14Ireland
  2. 2.Université Paris Est - CréteilCréteil CedexFrance

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