Furstenberg transformations on Cartesian products of infinite-dimensional tori

TL;DR

This paper studies Furstenberg transformations on infinite-dimensional tori, proving unique ergodicity and countable Lebesgue spectrum under certain conditions, extending classical results to an infinite-dimensional context.

Contribution

It generalizes known one-dimensional results of Furstenberg and others to infinite-dimensional tori, using commutator methods and Bruhat functions.

Findings

Transformations are uniquely ergodic with respect to Haar measure.

Transformations have countable Lebesgue spectrum in a suitable subspace.

Results extend classical ergodic theory to infinite-dimensional settings.

Abstract

We consider in this note Furstenberg transformations on Cartesian products of infinite-dimensional tori. Under some appropriate assumptions, we show that these transformations are uniquely ergodic with respect to the Haar measure and have countable Lebesgue spectrum in a suitable subspace. These results generalise to the infinite-dimensional setting previous results of H. Furstenberg, A. Iwanik, M. Lemanzyk, D. Rudolph and the second author in the one-dimensional setting. Our proofs rely on the use of commutator methods for unitary operators and Bruhat functions on the infinite-dimensional torus.