Convergence of multiple ergodic averages for some commuting transformations

We prove the $L^2$ convergence for the linear multiple ergodic averages of commuting transformations $T_1,\mathrm{.}\mathrm{.}\mathrm{.},T_l$, assuming that each map $T_i$ and each pair $T_i{T}_j^{-1}$ is ergodic for $i\mathrm{\ne }j$. 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.