Pro-Nilfactors of the Space of Arithmetic Progressions in Topological Dynamical Systems

For a topological dynamical system (X, T), $$l\in \mathbb {N}$$ and $$x\in X$$ , let $$N_l(X)$$ $$L_x^l(X)$$ be the orbit closures of diagonal point $$(x,\ldots ,x)$$ (l times) under actions $$\mathscr {G}_{l}$$ $$\tau _l $$ respectively, where is generated by $$T\times \ldots \times T$$ _l=T\times T^l$$ . In this paper, we show that for minimal T) $$d,l\in maximal d-step pro-nilfactor $$(N_l(X),\mathscr {G}_{l})$$ $$(N_l(X_d),\mathscr $$X_d$$ pronilfactor T). Meanwhile, when nilsystem, also calculate pro-nilfactors $$(L_x^l(X),\tau _l)$$ almost every x w.r.t. Haar measure. particular, there exists 2-step nilsystem (Y, countable subset $$\Omega Y such $$y\in Y\backslash \Omega equicontinuous factor $$(L_y^2(Y),\tau _2)$$ not $$(L_{\pi _1(y)}^2(Y_{1}),\tau $$Y_1$$ $$\pi _1:Y\rightarrow Y_1$$ map.