On multiple ergodicity of affine cocycles over irrational rotations
Résumé
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 $d\geq 1$ 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 $\Psi_{d+1}(x):=(\varphi(x), \varphi(x+\beta_1),..., \varphi(x+\beta_{d})), {\rm \ with \} \varphi(x)=\{x\}-\frac12.$ 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 $\R^{d}$.