For a topological dynamical system $(X, T)$, $l\in\mathbb{ N}$ and $x\in X$, let $N_l(X)$ and $L_x^l(X)$ be the orbit closures of the diagonal point $(x,x,\ldots,x)$ ($l $ times) under the actions $\mathcal{ G}_{ l}$ and $\tau_l $ respectively, where $\mathcal{ G}_{ l}$ is generated by $T\times T\times \ldots \times T$ ($l $ times) and $\tau_l=T\times T^2\times \ldots \times T^l$. In this paper, we show that for a minimal system $(X,T)$ and $l\in \mathbb{ N}$, the maximal $d$-step pro-nilfactor of $(N_l(X),\mathcal{ G}_{ l})$ is $(N_l(X_d),\mathcal{ G}_{ l})$, where $\pi_d:X\to X/\mathbf{ RP}^{ [d]}=X_d,d\in \mathbb{ N}$ is the factor map and $\mathbf{ RP}^{ [d]}$ is the regionally proximal relation of order $d$. Meanwhile, when $(X,T)$ is a minimal nilsystem, we also calculate the pro-nilfactors of $(L_x^l(X),\tau_l)$ for almost every $x$ w.r.t. the Haar measure. In particular, there exists a minimal $2$-step nilsystem $(Y,T)$ and a countable set $\Omega\subset Y$ such that for $y\in Y\backslash \Omega$ the maximal equicontinuous factor of $(L_y^2(Y),\tau_2)$ is not $(L_{ \pi_1(y)}^2(Y_{ 1}),\tau_2)$.
Categories: math.DS