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Spectrum of weighted Birkhoff average

Volume 269 / 2023

Balázs Bárány, Michał Rams, Ruxi Shi Studia Mathematica 269 (2023), 65-82 MSC: Primary 28A80; Secondary 28A78, 37D20, 40G05. DOI: 10.4064/sm210908-19-6 Published online: 3 October 2022


Let $\{s_n\}_{n\in \mathbb N}$ be a decreasing nonsummable sequence of positive reals. We investigate the weighted Birkhoff average $\frac {1}{S_n}\sum _{k=0}^{n-1}s_k\phi (T^kx)$ on an aperiodic irreducible subshift $\Sigma _{\bf A}$ of finite type where $\phi : \Sigma _{\bf A}\to \mathbb R$ is a continuous potential. Firstly, we show that the entropy spectrum of the weighted Birkhoff averages remains the same as that of the Birkhoff averages. Then we calculate the packing spectrum of the weighted Birkhoff averages. It turns out that we can have two cases: either the packing dimension of every level set equals its Hausdorff dimension or for every nonempty level set it is equal to the packing dimension of the whole space.


  • Balázs BárányDepartment of Stochastics
    Institute of Mathematics
    Budapest University of Technology and Economics
    Műegyetem rkp. 3
    H-1111 Budapest, Hungary
    MTA-BME Stochastics Research Group
    Műegyetem rkp. 3
    H-1111 Budapest, Hungary
  • Michał RamsInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-656 Warszawa, Poland
  • Ruxi ShiSorbonne Université
    75005 Paris, France

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