Paper detail

On multiple ergodicity of affine cocycles over irrational rotations

Let T_αdenote the rotation T_αx=x+α(mod 1) by an irrational number αon the additive circle T=[0,1). Let β_1,..., β_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_αby the vectorial function Ψ_{d+1}(x):=(ϕ(x), ϕ(x+β_1),..., ϕ(x+β_d)), with ϕ(x)={x}-1/2. It was already proved in \cite{LeMeNa03} that Ψ_{2} is regular for αwith bounded partial quotients. In the present paper we show that Ψ_{2} 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 Ψ_{4} 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 Φ_{d} =(1_{[0, β_j]} - β_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}.