Pleasant extensions retaining algebraic structure, II

In this paper we combine the general tools developed in (arXiv:0905.0518) with several ideas taken from earlier work on one-dimensional nonconventional ergodic averages by Furstenberg and Weiss, Host and Kra and Ziegler to study the averages $\frac{1}{N}\sum_{n=1}^N(f_1\circ T^{n\bf{p}_1})(f_2\circ T^{n\bf{p}_2})(f_3\circ T^{n\bf{p}_3})$for$f_1,f_2,f_3 \in L^\infty(\mu)$associatedtoatripleofdirections$\bf{p}_1,\bf{p}_2,\bf{p}_3 \in $\mathbb{Z}$^2$thatlieingeneralpositionalongwith$0 \in $\mathbb{Z}$^2$. We will show how to construct a `pleasant’ extension of an initially-given ${\mathbb{Z}}^2$-system for which these averages admit characteristic factors with a very concrete description, involving one-dimensional isotropy factors and two-step pro-nilsystems. We also use this analysis to construct pleasant extensions and then prove norm convergence for the polynomial nonconventional ergodic averages $\frac{1}{N}{\sum }_{n=1}^N(f_1\circ T_1^{n^2})(f_2\circ T_1^{n^2}{T}_2^n)$ associated to two commuting transformations $T_1$, $T_2$.