Weak mixing of maps with bounded cutting parameter

TL;DR

This paper investigates Ornstein transformations with bounded cutting parameters, showing they are not mixing but are almost surely weakly mixing, highlighting a probabilistic distinction in their dynamical behavior.

Contribution

It demonstrates that Ornstein transformations with bounded cutting parameters are almost surely weakly mixing, despite not being strongly mixing, expanding understanding of their probabilistic properties.

Findings

Bounded cutting parameter transformations are not mixing.

They exhibit weak mixing with probability one.

The results relate to similar properties in interval exchange transformations.

Abstract

In the class of Ornstein transformations the mixing property satisfies a 0-1 law. Here we consider Ornstein's construction with bounded cutting parameter. In fact, these latter transformations are not mixing, however it is proved that the weak mixing property occurs with probability one. Our situation is similar to the case of interval exchange transformations whose link with the cutting and stacking construction relies in a dynamical process called Rauzy induction.