We prove the $L^{ 2}$ convergence for the linear multiple ergodic averages of commuting transformations $T_{ 1}, ..., T_{ l}$, assuming that each map $T_i$ and each pair $T_iT_j^{ -1}$ is ergodic for $i\neq j$. The limiting behavior of such averages is controlled by a particular factor, which is an inverse limit of nilsystems. As a corollary we show that the limiting behavior of linear multiple ergodic averages is the same for commuting transformations.
Categories: math.DS