CF-Nil systems and convergence of two-dimensional ergodic averages

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.