Exact ${\cal C}^{\infty }$ covering maps of the circle without (weak) limit measure

Volume 93 / 2002

Roland Zweimüller Colloquium Mathematicum 93 (2002), 295-302 MSC: 28D05, 37A40, 37E10. DOI: 10.4064/cm93-2-9


We construct ${\cal C}^{\infty }$ maps $T$ on the interval and on the circle which are Lebesgue exact preserving an absolutely continuous infinite measure $\mu \ll \lambda $, such that for any probability measure $\nu \ll \lambda $ the sequence $(n^{-1}\sum _{k=0}^{n-1}\nu \circ T^{-k})_{n\geq 1}$ of arithmetical averages of image measures does not converge weakly.


  • Roland ZweimüllerInstitut für Mathematik
    Universität Salzburg
    Hellbrunnerstraße 34
    A-5020 Salzburg, Austria

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