Ergodicity of cocycles over 2-dimensional rotations

We study recurrence and ergodicity of cocycles with values in $\mathbb {R}^d$, $d \geq 1$, over a rotation on $\mathbb {T}^\rho $, $\rho \gt 1$, under Diophantine conditions for the rotation and the discontinuities of the functions generating the cocycles. For simplicity of notation we mainly consider the cases $\rho = 2$, $d=1$ and 2.