On multiple ergodicity of affine cocycles over irrational rotations

TL;DR

This paper investigates the ergodic properties of affine cocycles over irrational rotations, establishing regularity results for certain dimensions and providing examples of non-regular cocycles, with a focus on classification of step functions.

Contribution

It proves regularity of the cocycle \\Psi_{2} for all irrational \\alpha and offers sufficient conditions for higher dimensions, including examples of non-regular cases for d=3.

Findings

\\Psi_{2} is regular for any irrational \\alpha.

Sufficient conditions for regularity in higher dimensions.

Examples of non-regular cocycles for d=3.

Abstract

Let T_\alpha denote the rotation T_{\alpha}x=x+\alpha (mod 1) by an irrational number \alpha on the additive circle T=[0,1). Let \beta_1,..., \beta_d be d\geqslant 1 parameters in [0, 1). One of the goals of this paper is to describe the ergodic properties of the cocycle (taking values in R^(d+1)) generated over T_\alpha by the vectorial function \Psi_{d+1}(x):=(\phi(x), \phi(x+\beta_1),..., \phi(x+\beta_d)), with \phi(x)={x}-1/2. It was already proved in \cite{LeMeNa03} that \Psi_{2} is regular for \alpha with bounded partial quotients. In the present paper we show that \Psi_{2} is regular for any irrational \alpha. For higher dimensions, we give sufficient conditions for regularity. While the case d=2 remains unsolved, for d=3 we provide examples of non-regular cocycles \Psi_{4} for certain values of the parameters \beta_1,\beta_2,\beta_3. We also show that the problem of regularity for the cocycle \Psi_{d+1} reduces to the regularity of the cocycles of the form \Phi_{d} =(1_{[0, \beta_j]} - \beta_j)_{j= 1, ..., d} (taking values in R^d). Therefore, a large part of the paper is devoted to the classification problems of step functions with values in R^{d}.