Typical Weak Mixing and Exceptional Spectral Properties for Interval Translation Mappings

TL;DR

This paper studies the weak mixing properties of interval translation mappings, proving typical weak mixing behavior for Bruin-Troubetzkoy ITMs and constructing the first examples of non-weak mixing infinite type ITMs.

Contribution

It provides two proofs of typical weak mixing for Bruin-Troubetzkoy ITMs and extends the approach to other classes, also constructing non-weak mixing examples.

Findings

Typical Bruin-Troubetzkoy ITMs are weakly mixing.

The second proof approach applies to other interval translation mappings.

First examples of non-weak mixing Bruin-Troubetzkoy ITMs of infinite type.

Abstract

We investigate weak mixing for some classes of interval translation mappings. We give two distinct proofs that a typical Bruin-Troubetzkoy interval translation mapping is weakly mixing. Moreover, we show that the second approach extends to other classes of interval translation mappings. In particular, we show that Bruin interval translation mappings on any number of intervals are typically weak mixing. Finally, we construct the first examples of non weak mixing Bruin-Troubetzkoy ITM of infinite type.