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