Absence of mixing for interval translation mappings and some generalizations
Abstract
We consider piecewise monotone maps, we show that an ergodic measure for which the map is invertible almost everywhere can not be mixing. It follows that every ergodic measure for an interval translation mapping is not mixing. We also show that double rotations without periodic points have an ergodic but not weakly mixing invariant measure. This article is dedicated to the memory of Anatoly Mikhailovich Stepin.
Keywords
Cite
@article{arxiv.2102.05904,
title = {Absence of mixing for interval translation mappings and some generalizations},
author = {Serge Troubetzkoy},
journal= {arXiv preprint arXiv:2102.05904},
year = {2021}
}
Comments
The proof of Lemma 6 contains a gap. Since all the results in this article are based on this lemma none of them have been established. A version of Lemma 6 with the same gap was used in (Kryzhevich, Invariant measures for interval translations and some other piecewise continuous maps, Math. Model. Nat. Phen. 15 (2020)) and thus some of the results of this article have also not been established