Sets of nondifferentiability for conjugacies between expanding interval maps

Tom 206 / 2009

Thomas Jordan, Marc Kesseböhmer, Mark Pollicott, Bernd O. Stratmann Fundamenta Mathematicae 206 (2009), 161-183 MSC: Primary 37C45; Secondary 28A80, 37A10. DOI: 10.4064/fm206-0-10


We study differentiability of topological conjugacies between expanding piecewise $C^{1+\epsilon }$ interval maps. If these conjugacies are not $C^1$, then their derivative vanishes Lebesgue almost everywhere. We show that in this case the Hausdorff dimension of the set of points for which the derivative of the conjugacy does not exist lies strictly between zero and one. Moreover, by employing the thermodynamic formalism, we show that this Hausdorff dimension can be determined explicitly in terms of the Lyapunov spectrum. These results then give rise to a “rigidity dichotomy” for the type of conjugacies under consideration.


  • Thomas JordanDepartment of Mathematics
    University of Bristol
    Bristol, BS8 1TW, UK
  • Marc KesseböhmerFachbereich 3–Mathematik und Informatik
    Universität Bremen
    D-28359 Bremen, Germany
  • Mark PollicottMathematics Institute
    University of Warwick
    Coventry, CV4 7AL, UK
  • Bernd O. StratmannMathematics Institute
    University of St Andrews
    St Andrews, KY16 9SS, Scotland

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