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Birkhoff spectrum for piecewise monotone interval maps

Volume 252 / 2021

Thomas Jordan, Michał Rams Fundamenta Mathematicae 252 (2021), 203-223 MSC: Primary 37C45; Secondary 37D35. DOI: 10.4064/fm824-1-2020 Published online: 7 August 2020

Abstract

For piecewise monotone, piecewise continuous interval maps we look at Birkhoff spectra for regular potential functions. This means considering the Hausdorff dimension of the set of points for which the Birkhoff average of the potential takes a fixed value. In the uniformly hyperbolic case we obtain complete results, in the case with parabolic behaviour we are able to describe the part of the sets where the lower Lyapunov exponent is positive. In addition we give some lower bounds on the full spectrum in this case. This is an extension of work of Hofbauer on entropy and Lyapunov spectra.

Authors

  • Thomas JordanSchool of Mathematics
    University of Bristol
    University Walk
    Bristol, BS8 1TW, UK
    e-mail
  • Michał RamsInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-656 Warszawa, Poland
    e-mail

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