Topological characteristic factors and nilsystems

We prove that the maximal infinite step pro-nilfactor $X_infty$ of a minimal dynamical system $(X,T)$ is the topological characteristic factor in a certain sense. Namely, we show that by an almost one to one modification of $pi:X rightarrow X_infty$, the induced open extension $pi^*:X^* rightarrow X^*_infty$ has the following property: for $x$ in a dense $G_delta$ set of $X^*$, the orbit closure $L_x=overline{{mathcal{O}}}((x,x,ldots,x), Ttimes T^2times ldots times T^d)$ is $(pi^*)^{(d)}$-saturated, i.e. $L_x=((pi^*)^{(d)})^{-1}(pi^*)^{(d)}(L_x)$. Using results derived from the above fact, we are able to answer several open questions: (1) if $(X,T^k)$ is minimal for some $kge 2$, then for any $din {mathbb N}$ and any $0le j<k$ there is a sequence ${n_i}$ of $mathbb Z$ with $n_iequiv j (text{mod} k)$ such that $T^{n_i}xrightarrow x, T^{2n_i}xrightarrow x, ldots, T^{dn_i}xrightarrow x$ for $x$ in a dense $G_delta$ subset of $X$; (2) if $(X,T)$ is totally minimal, then ${T^{n^2}x:nin {mathbb Z}}$ is dense in $X$ for $x$ in a dense $G_delta$ subset of $X$; (3) for any $dinmathbb N$ and any minimal system, which is an open extension of its maximal distal factor, ${bf RP}^{[d]}={bf AP}^{[d]}$, where the latter is the regionally proximal relation of order $d$ along arithmetic progressions.