Isomorphism classes for certain expanding maps and their group extensions

TL;DR

This paper demonstrates that expanding toral endomorphisms and their perturbations, along with certain group extensions, are isomorphic to 1-sided Bernoulli shifts, revealing their strong stochastic properties.

Contribution

It extends the isomorphism result from expanding toral endomorphisms to their smooth perturbations and specific group extensions, broadening the class of systems known to be Bernoulli.

Findings

Expanding toral endomorphisms are isomorphic to 1-sided Bernoulli shifts.

Smooth perturbations of these endomorphisms retain Bernoulli properties.

Certain group extensions of these systems are also Bernoulli under mild conditions.

Abstract

We show that expanding toral endomorphisms, together with their respective Lebesgue measure are isomorphic to 1-sided Bernoulli shifts. This result is then extended to smooth perturbations of expanding toral endomorphisms, together with their respective measures of maximal entropy. Also we study group extensions of expanding toral endomorphisms and show that under certain, not too restrictive conditions on the extension cocycle, these skew products are 1-sided Bernoulli as well.