Exponentially mixing, locally constant skew extensions of shift maps

TL;DR

This paper presents examples of non-abelian locally constant extensions of shift maps that exhibit exponential mixing, contrasting with the known limitations for toral extensions of hyperbolic systems.

Contribution

It introduces non-abelian locally constant compact extensions of shift maps that are exponentially mixing, utilizing Bourgain-Gamburd's result and a decoupling technique.

Findings

Examples of exponentially mixing non-abelian extensions

Contrasts with toral extension limitations

Uses Bourgain-Gamburd's theorem and decoupling argument

Abstract

It is known that locally constant toral extensions of hyperbolic systems can never mix at an exponential rate. In this note we exhibit some examples of non-abelian locally constant compact extensions of the shift map which are exponentially mixing for Holder-$L^2$ observables. The proof rests on a result of Bourgain-Gamburd and a decoupling argument.