A topological dynamical system $(X,T)$ is called CF-Nil($k$) if it is strictly ergodic and the maximal measurable and maximal topological $k$-step pro-nilfactors coincide as measure preserving systems. Through constructing specific ``CF-Nil'' models, we prove that for any ergodic system $(X,\mathcal{ X},\mu,T)$, any nilsequence $\{ \psi(m,n)\}_{ m,n\in\mathbb{ Z}}$ and any $f_1,\dots,f_d\in L^{ \infty}(\mu)$, the averages \begin{ equation*} \dfrac{ 1}{ N^{ 2}} \sum_{ m,n=0}^{ N-1} \psi(m,n)\prod_{ j=1}^{ d}f_{ j}(T^{ { m+jn}}x) \end{ equation*} converge pointwise as $N$ goes to infinity. Moreover, we show the $L^2$-convergence of a certain two-dimensional averages for non-commuting transformations without zero entropy condition.
Categories: math.DS