Absence of mixing for interval translation mappings and some generalizations

TL;DR

This paper proves that certain interval translation maps and double rotations lack mixing properties for their ergodic measures, highlighting limitations in their statistical behavior.

Contribution

It establishes that ergodic measures for piecewise monotone maps and interval translation mappings cannot be mixing, extending to double rotations without periodic points.

Findings

Ergodic measures for invertible piecewise monotone maps are not mixing.

Interval translation mappings have ergodic measures that are not mixing.

Double rotations without periodic points have ergodic but not weakly mixing measures.

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.