Pro-nilfactors of the space of arithmetic progressions in topological dynamical systems

Abstract

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 $(x,x,\dots ,x)$ ($l$ times) under the actions ${\mathcal{G}}_l$ and ${\tau }_l$ respectively, where ${\mathcal{G}}_l$ is generated by $T\times T\times \dots \times T$ ($l$ times) and ${\tau }_l=T\times T^2\times \dots \times T^l$. 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 $(L_{{\pi }_1(y)}^2(Y_1),{\tau }_2)$.