On multiple ergodicity of affine cocycles over irrational rotations

Abstract

Let ${T_\alpha }$ denote the rotation ${T_\alpha }x = x + \alpha$ (mod 1) by an irrational number α on the additive circle T = [0, 1). Let β ₁, …, β _d be d ≥ 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 Ψ _d+1(x):= (φ(x), φ(x+β ₁), …, φ(x+β _d )), with φ(x) = {x}−½.

It was already proved in [LeMeNa03] that Ψ₂ is regular for α with bounded partial quotients. In the present paper we show that Ψ₂ is regular for any irrational α. 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 Ψ₄ for certain values of the parameters β ₁, β ₂, β ₃.

We also show that the problem of regularity for the cocycle Ψ _d+1 reduces to the regularity of the cocycles of the form ${\Phi _d} = {({1_{[0,{\beta _j}]}} - {\beta _j})_{j = 1, \ldots ,d}}$ (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.