  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 $L_x=\overline{{\mathcal{O}}}((x,x,\ldots,x), T\times
T^2\times \ldots \times T^d)$ 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 $k\ge 2$, then for any $d\in
{\mathbb N}$ and any $0\le j<k$ there is a sequence $\{n_i\}$ of $\mathbb Z$
with $n_i\equiv j\ (\text{mod}\ k)$ such that $T^{n_i}x\rightarrow x,
T^{2n_i}x\rightarrow x, \ldots, T^{dn_i}x\rightarrow x$ for $x$ in a dense
$G_\delta$ subset of $X$; (2) if $(X,T)$ is totally minimal, then
$\{T^{n^2}x:n\in {\mathbb Z}\}$ is dense in $X$ for $x$ in a dense $G_\delta$
subset of $X$; (3) for any $d\in\mathbb N$ 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.
