Exact ${\cal C}^{\infty }$ covering maps of the circle without (weak) limit measure
Volume 93 / 2002
Colloquium Mathematicum 93 (2002), 295-302
MSC: 28D05, 37A40, 37E10.
DOI: 10.4064/cm93-2-9
Abstract
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.
