Unique ergodicity of asynchronous rotations, and application

TL;DR

This paper extends the concept of unique ergodicity from irrational rotations to a continuous family of tori, demonstrating that asynchronicity ensures ergodicity in certain homogeneous dynamical systems.

Contribution

It introduces an analogue of Kronecker-Weyl's unique ergodicity for continuous tori and applies it to prove ergodicity of natural lifts of invariant measures.

Findings

Asynchronicity in homogeneous systems implies unique ergodicity.

The ergodicity of natural lifts of invariant measures is established.

The work generalizes classical results to a broader class of dynamical systems.

Abstract

The main result of this paper is an analogue for a continuous family of tori of Kronecker-Weyl's unique ergodicity of irrational rotations. We show that the notion corresponding in this setup to irrationality, namely asynchronicity, is satisfied in some homogeneous dynamical systems. This is used to prove the ergodicity of naturals lifts of invariant measures.