Let $\pi: (X,T)\rightarrow (Y,T)$ be a factor map of topological dynamics and
$d\in {\mathbb {N}}$. $(Y,T)$ is said to be a $d$-step topological
characteristic factor if there exists a dense $G_\delta$ set $X_0$ of $X$ such
that for each $x\in X_0$ the orbit closure $\overline{\mathcal O}((x,
\ldots,x), T\times T^2\times \ldots \times T^d)$ is $\pi\times \ldots \times
\pi$ ($d$ times) saturated...
In 1994 Eli Glasner studied the topological
characteristic factor for minimal systems. For example, it is shown that for a
distal minimal system, its largest distal factor of order $d-1$ is its $d$-step
topological characteristic factor. In this paper, we generalize Glasner's work
to the product system of finitely many minimal systems and give its relative
version. To prove these results, we need to deal with $(X,T^m)$ for $m\in
{\mathbb {N}}$. We will study the structure theorem of $(X,T^m)$. We show that
though for a minimal system $(X,T)$ and $m\in {\mathbb {N}}$, $(X,T^m)$ may not
be minimal, but we still can have PI-tower for $(X,T^m)$ and in fact it looks
the same as the PI tower of $(X,T)$. We give some applications of the results
developed. For example, we show that if a minimal system has no nontrivial
independent pair along arithmetic progressions of order $d$, then up to a
canonically defined proximal extension, it is PI of order $d$; if a minimal
system $(X,T)$ has a nontrivial $d$-step topological characteristic factor,
then there exist ``many'' $\Delta$-transitive sets of order $d$.
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Abstract