$p_{i_{2}}(n+k_{j_{2}})-p_{i_{2}}(k_{j_{2}})$ is $a^{i_{2}}_{d_{i_{2}}}$. Thus
when $a_{d_{1}}^{1}\neq a_{d_{2}}^{2}$, the polynomials
$p_{i_{1}}(n+k_{j_{1}})-p_{i_{1}}(k_{j_{1}})$ and
$p_{i_{2}}(n+k_{j_{2}})-p_{i_{2}}(k_{j_{2}})$ in one variable $n$ are
essentially distinct.

When $a_{d_{1}}^{1}=a_{d_{2}}^{2}$, let $a_{d_{1}}^{1}=a_{d_{2}}^{2}=a$. Note
that the coefficient of $n^{d-1}$ in the polynomial
$p_{i_{1}}(n+k_{j_{1}})-p_{i_{1}}(k_{j_{1}})$ is
$a^{i_{1}}_{d_{i_{1}}}d_{i_{1}}k_{j_{1}}+a^{i_{1}}_{d_{i_{1}}-1}=adk_{j_{1}}+a^{i_{1}}_{d_{i_{1}}-1}$,
and the coefficient of $n^{d-1}$ in the polynomial
$p_{i_{2}}(n+k_{j_{2}})-p_{i_{2}}(k_{j_{2}})$ is
$a^{i_{2}}_{d_{i_{2}}}d_{i_{2}}k_{j_{2}}+a^{i_{2}}_{d_{i_{2}}-1}=adk_{j_{2}}+a^{i_{2}}_{d_{i_{2}}-1}$.
Thus since $|k_{1}-k_{2}|\geq L$ and $i_{1}\neq i_{2}$ , one has

$\begin{split}&\quad|(a^{i_{1}}_{d_{i_{1}}}d_{i_{1}}k_{j_{1}}+a^{i_{1}}_{d_{i_{1}}-1})-(a^{i_{2}}_{d_{i_{2}}}d_{i_{2}}k_{j_{2}}+a^{i_{2}}_{d_{i_{2}}-1})|\\\
&\geq|a|dL-(|a^{1}_{d_{1}-1}|+|a^{2}_{d_{2}-1}|)\\\
&\geq(|a^{1}_{d_{1}-1}|+|a^{2}_{d_{2}-1}|+1)-(|a^{1}_{d_{1}-1}|+|a^{2}_{d_{2}-1}|)\\\
&=1.\end{split}$

This implies that
$a^{i_{1}}_{d_{i_{1}}}d_{i_{1}}k_{j_{1}}+a^{i_{1}}_{d_{i_{1}}-1}\neq
a^{i_{2}}_{d_{i_{2}}}d_{i_{2}}k_{j_{2}}+a^{i_{2}}_{d_{i_{2}}-1}$ and so the
polynomials $p_{i_{1}}(n+k_{j_{1}})-p_{i_{1}}(k_{j_{1}})$ and
$p_{i_{2}}(n+k_{j_{2}})-p_{i_{2}}(k_{j_{2}})$ in one variable $n$ are
essentially distinct. ∎

For $n<m$, we denote by $[n,m]$ the interval $\\{n,n+1,\ldots,m\\}$ of
$\mathbb{Z}$. By the definition, it is easy to verify the following lemma.

######  Lemma 6.2.

Let $A\subseteq\mathbb{Z}^{2}$ a piecewise syndetic subset in $\mathbb{Z}^{2}$
and $B\subseteq\mathbb{Z}^{2}$. If for any $M\in\mathbb{N}$, there is some
${\bf n}_{M}\in\mathbb{Z}^{2}$ such that

${\bf n}_{M}+(A\cap[-M,M]^{2})\subseteq B,$

then $B$ is also a piecewise syndetic subset in $\mathbb{Z}^{2}$.

###  6.2. The proof of Theorem C

In last section, we have proved the first and second part of Theorem C. Now we
finish the proof of Theorem C by proving the last part of it. Let $(X,T)$ be
minimal and $\pi:X\longrightarrow X_{\infty}$ be the factor map. By the
discussion in Section 5, we know that the theorem holds for $\infty$-step pro-
nilsystems. The idea of our proof is that we lift the property from
$X_{\infty}$ to $X$ by using the saturation theorem for polynomials when $\pi$
is open. If $\pi$ is not open then we can modify it by almost one to one
extensions. Note that in the middle of the proof we will explain the idea of
the proof for some special case.

###### The proof of the last part of Theorem C.

By Theorem 3.1, if $\pi:X\rightarrow X_{\infty}$ is the factor map to
$X_{\infty}$, then there are minimal t.d.s. $X^{*}$ and $X_{\infty}^{*}$ which
are almost one to one extensions of $X$ and $X_{\infty}$ respectively, and a
commuting diagram below such that $X_{\infty}^{*}$ is a $k$-step topological
characteristic factor of $X^{*}$ for all $k\geq 2$.

$\begin{CD}X@<{\varsigma^{*}}<{}<X^{*}\\\ @V{}V{\pi}V@V{}V{\pi^{*}}V\\\
X_{\infty}@<{\varsigma}<{}<X_{\infty}^{*}\end{CD}$

If the result holds for $X^{*}$, then it also holds for $X$ since
$\varsigma^{*}$ is almost one to one. So without loss of generality, we may
assume that $X=X^{*}$. That is, we may assume the following diagram

---
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{*}}$$\textstyle{X^{*}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varsigma}$$\textstyle{X_{\infty}}$

where $\pi^{*}$ is open and $\varsigma$ is almost one to one. Let
$\Omega=\\{x\in X^{*}_{\infty}:\varsigma^{-1}(\varsigma(x))=\\{x\\}\\}.$ Then
$\Omega$ is a dense $G_{\delta}$ subset of $X^{*}_{\infty}$. Let

$X_{0}=X_{pol}\cap(\pi^{*})^{-1}(\Omega),$

where $X_{pol}$ is defined in (3.1). Then $X_{0}$ is a dense $G_{\delta}$
subset of $X$, and we will show it is what we are looking for, that is, for
essentially distinct integral polynomials $p_{1},\ldots,p_{d}$ with
$p_{i}(0)=0$ and $\deg(p_{i})\geq 2$, $1\leq i\leq d$, for each $x\in X_{0}$
and each neighbourhood $U$ of $x$

$N_{\\{p_{1},\ldots,p_{d}\\}}(x,U)=\\{n\in\mathbb{Z}:T^{p_{1}(n)}x\in
U,\ldots,T^{p_{d}(n)}x\in U\\}\in\mathcal{F}_{ps}.$

To make the proof easier to understand, we first outline the proof when $d=1$
and $p_{1}(n)=n^{2}$.

In this case let $x\in X_{0}$, $y=\pi^{*}(x)$ and $V=\pi^{*}(U)$ which is an
open subset of $X_{\infty}^{*}$, where $U$ is an open neighborhood of $x$. By
Theorem 5.2 and some discussions we get that

$\\{n\in\mathbb{Z}:T^{n^{2}}y\in V\\}=\\{\ldots<s_{-1}<s_{0}<s_{1}<\ldots\\}$

is syndetic. It is clear that
$T^{-s_{j}^{2}}U\cap(\pi^{*})^{-1}(y)\not=\emptyset$ for each
$j\in\mathbb{Z}$.

Let $k\in\mathbb{N}$. We will show that there is $m=m(k)\in\mathbb{Z}$ such
that

$m+s_{-k},\ldots,m+s_{k}\in\\{n\in\mathbb{Z}:T^{n^{2}}x\in U\\}$

which implies that $\\{n\in\mathbb{Z}:T^{n^{2}}x\in U\\}\in\mathcal{F}_{ps}.$

Let $q_{i}(n)=n^{2}+2s_{i}n$, $n\in\mathbb{Z},-k\leq i\leq k$. Then
$q_{-k},\ldots,q_{k}$ are pairwise distinct polynomials. Applying Theorem 3.6
to $q_{-k},\ldots,q_{k}$ and the open sets
$T^{-s_{-k}^{2}}U,\ldots,T^{-s_{k}^{2}}U$ we get that there is
$m\in\mathbb{Z}$ such that $T^{q_{j}(m)}x\in T^{-s_{j}^{2}}U$, i.e.
$T^{(m+s_{j})^{2}}x\in U$ for each $-k\leq j\leq k$. This finishes the proof
for the special case.

Now come back to the proof for the general case. By Lemma 6.1, there is some
$L\in\mathbb{N}$ such that if $k_{1},k_{2}\in\mathbb{Z}$ with
$|k_{1}-k_{2}|\geq L$, then for any
$(i_{1},j_{1})\neq(i_{2},j_{2})\in\\{1,2,\ldots,d\\}\times\\{1,2\\}$, the
polynomials $p_{i_{1}}(n+k_{j_{1}})-p_{i_{1}}(k_{j_{1}})$ and
$p_{i_{2}}(n+k_{j_{2}})-p_{i_{2}}(k_{j_{2}})$ in one variable $n$ are
essentially distinct.

Let $z=\pi^{*}(x)$. Since $x\in X_{0}$, $z\in\Omega$. Given a neighbourhood
$U$ of $x$, let $\epsilon>0$ such that $B_{2\epsilon}(x)\subseteq U$, and let
$\widetilde{U}=B_{\epsilon}(x)\subseteq U$. Note $\pi^{*}(\widetilde{U})$ is
an open neighbourhood of $z$ as $\pi^{*}$ is open. Let $z_{0}=\varsigma(z)$.
Since $z\in\Omega$, there exists an open neighbourhood $W$ of $z_{0}$ in
$X_{\infty}$ such that $\varsigma^{-1}(W)\subseteq\pi^{*}(\widetilde{U})$.
Thus

(6.1) $\displaystyle\begin{aligned} F&\triangleq
N_{\\{p_{1},\ldots,p_{d}\\}}(z,\pi^{*}(\widetilde{U}))=\\{n\in\mathbb{Z}:T^{p_{1}(n)}z,\ldots,T^{p_{d}(n)}z\in\pi^{*}(\widetilde{U})\\}\\\
&\supseteq
N_{\\{p_{1},\ldots,p_{d}\\}}(z_{0},W)=\\{n\in\mathbb{Z}:T^{p_{1}(n)}z_{0},\ldots,T^{p_{d}(n)}z_{0}\in
W\\}.\end{aligned}$

Note that the minimal $\infty$-step pro-nilsystem $X_{\infty}$ is inverse
limit of minimal nilsystems. Let
$(X_{\infty},T)=\displaystyle\lim_{\longleftarrow}(X_{r},T)_{r\in\mathbb{N}}$,
where each $(X_{r},T)$ is minimal nilsystem and
$\pi_{\infty,r}:(X_{\infty},T)\rightarrow(X_{r},T)$ is the corresponding
factor map. Since

$\pi_{\infty,1}^{-1}\big{(}\pi_{\infty,1}(z_{0})\big{)}\supseteq\pi_{\infty,2}^{-1}\big{(}\pi_{\infty,2}(z_{0})\big{)}\supseteq\cdots\supseteq\pi_{\infty,r}^{-1}\big{(}\pi_{\infty,r}(z_{0})\big{)}\supseteq\cdots$

and

$\bigcap_{r=1}^{\infty}\pi_{\infty,r}^{-1}\big{(}\pi_{\infty,r}(z_{0})\big{)}=\\{z_{0}\\},$

we can find $r_{*}\in\mathbb{N}$ such that
$\pi_{\infty,r_{*}}^{-1}\big{(}\pi_{\infty,r_{*}}(z_{0})\big{)}\subseteq W$.
Let $z_{*}=\pi_{\infty,r_{*}}(z_{0})$. Note that $\pi_{\infty,r_{*}}$ is an
open map, we can find an open neighborhood $W_{*}$ of $z_{*}$ in $X_{r_{*}}$
such that $\pi_{\infty,r_{*}}^{-1}(W_{*})\subseteq W$. Hence

(6.2) $N_{\\{p_{1},\ldots,p_{d}\\}}(z_{0},W)\supseteq
N_{\\{p_{1},\ldots,p_{d}\\}}(z_{*},W_{*})=\\{n\in\mathbb{Z}:T^{p_{1}(n)}z_{*},\ldots,T^{p_{d}(n)}z_{*}\in
W_{*}\\}.$

Since $(X_{r_{*}},T)$ is a minimal nilsystem, there exist a nilpotent lie
group $G$, a discrete cocompact subgroup $\Gamma$ of $G$ and $a\in G$ such
that $X_{r_{*}}=G/\Gamma$ and $T(h\Gamma)=ah\Gamma$ for any $h\in G$. Let
$\widetilde{G}=G\times G\times\cdots\times G$ ($d$-times),
$\widetilde{\Gamma}=\Gamma\times\Gamma\times\cdots\times\Gamma$ ($d$-times)
and $g:\mathbb{Z}\rightarrow\widetilde{G}$ with
$g(n)=(a^{p_{1}(n)},\cdots,a^{p_{d}(n)})$ for any $n\in\mathbb{Z}$. Then
$g:\mathbb{Z}\rightarrow\widetilde{G}$ is a polynimal. Let

$\widetilde{z_{*}}=(z_{*},z_{*},\cdots,z_{*})\in\widetilde{G}/\widetilde{\Gamma}=G/\Gamma\times
G/\Gamma\times\cdots\times G/\Gamma.$

Thus by Theorem 5.2, we have a continuous surjective map
$\eta:\widehat{X}\rightarrow\widetilde{G}/\widetilde{\Gamma}$ of a ‘larger’
nil-manifold $\widehat{X}=\widehat{G}/\widehat{\Gamma}$, a homomorphism
$\phi:\mathbb{Z}\rightarrow\widehat{G}$ and $\hat{x}\in\widehat{X}$ such that

(6.3) $g(n)\widetilde{z_{*}}=\eta(\phi(n)\hat{x}),\ \forall n\in\mathbb{Z}.$

Take $\hat{W}=\eta^{-1}(W_{*}\times W_{*}\times\cdots\times W_{*})$,
$\hat{a}=\phi(1)\in\hat{G}$, and
$R(\hat{h}\hat{\Gamma})=\hat{a}\hat{h}\hat{\Gamma}$ for $\hat{h}\in\hat{G}$.
Then $(\widehat{X},R)$ is nilsystem, $\hat{W}$ is an open neighbourhood of
$\hat{x}$ in $\widehat{X}$, and

$\displaystyle N_{\\{p_{1},\ldots,p_{d}\\}}(z_{*},W_{*})$
$\displaystyle=\\{n\in\mathbb{Z}:(T^{p_{1}(n)}z_{*},\ldots,T^{p_{d}(n)}z_{*})\in
W_{*}\times\cdots\times W_{*}\\}$
$\displaystyle=\\{n\in\mathbb{Z}:g(n)\widetilde{z_{*}}\in W_{*}\times
W_{*}\times\cdots\times W_{*}\\}$ $\displaystyle\overset{\eqref{eeeee-
eq-1}}{=}\\{n\in\mathbb{Z}:\phi(n)\hat{x}\in\eta^{-1}(W_{*}\times
W_{*}\times\cdots\times W_{*})\\}$
$\displaystyle=\\{n\in\mathbb{Z}:R^{n}\hat{x}\in\hat{W}\\}$
$\displaystyle=N_{R}(\hat{x},\hat{W}).$

Combining this with (6.1) and (6.2), one has

$F=N_{\\{p_{1},\ldots,p_{d}\\}}(z,\pi^{*}(\widetilde{U}))\supseteq
N_{R}(\hat{x},\hat{W}).$

Hence $F\in\mathcal{F}_{s}$ as $N_{R}(\hat{x},\hat{W})$ is syndetic. Let

$F=\\{n^{\prime}_{j}\\}_{j\in\mathbb{Z}}=\\{\ldots<n^{\prime}_{-2}<n^{\prime}_{-1}<n^{\prime}_{0}<n^{\prime}_{1}<n^{\prime}_{2}<\ldots\\}.$

We take $n_{j}=n^{\prime}_{Lj}$ for $j\in\mathbb{Z}$. Then
$\\{n_{j}\\}_{j\in\mathbb{Z}}\in\mathcal{F}_{s}$ and $n_{i+1}-n_{i}\geq
L,\forall i\in\mathbb{Z}$.

To finish the proof we now show that for each $l\in\mathbb{N}$, there is some
$m\in\mathbb{Z}$ such that $m+\\{n_{j}\\}_{-l\leq j\leq l}\subseteq
N_{\\{p_{1},\ldots,p_{d}\\}}(x,U)$ and hence
$N_{\\{p_{1},\ldots,p_{d}\\}}(x,U)\in\mathcal{F}_{ps}$.

Fix $l\in\mathbb{N}$. Let $\mathcal{A}^{\prime}=\\{p_{i}(n+n_{j}):-l\leq j\leq
l,1\leq i\leq d\\}$. Then $\mathcal{A}^{\prime}$ is a set of essentially
distinct integral polynomials, since $n_{i+1}-n_{i}\geq L$. By Theorem 3.6,
$\overline{\mathcal{O}}_{\mathcal{A}^{\prime}}(x^{\otimes d^{\prime}})$ is
${\pi^{*}}^{(d^{\prime})}$-saturated, that is,

$\overline{\mathcal{O}}_{\mathcal{A}^{\prime}}(x^{\otimes
d^{\prime}})=({\pi^{*}}^{(d^{\prime})})^{-1}\Big{(}{\pi^{*}}^{(d^{\prime})}(\overline{\mathcal{O}}_{\mathcal{A}^{\prime}}(x^{\otimes
d^{\prime}}))\Big{)},$

where $d^{\prime}=d(2l+1)$. One has that

$F_{\mathcal{A}^{\prime}}(x)\cap U^{d(2l+1)}\neq\emptyset.$

In particular, there is some $m\in\mathbb{Z}$ such that

$\Big{(}T^{p_{i}(m+n_{j})}(x)\Big{)}_{1\leq i\leq d,-l\leq j\leq
l}\in{U}^{d(2l+1)}.$

Thus

$m+\\{n_{j}\\}_{-l\leq j\leq l}=\\{m+n_{j}\\}_{-l\leq j\leq l}\subseteq
N_{\\{p_{1},\ldots,p_{d}\\}}(x,U).$

The proof is complete. ∎

By Theorem 4.11, we have the following corollary of Theorem C:

######  Corollary 6.3.

Let $(X,T)$ be a minimal t.d.s. There is a dense $G_{\delta}$ subset $X_{0}$
of $X$ such that for each $x\in X_{0}$ and for any family
$\mathcal{A}=\\{p_{1},p_{2},\cdots,p_{d}\\}$ of integral polynomials
satisfying $(\spadesuit)$ and $\deg p_{i}\geq 2,1\leq i\leq d$,
$(W_{x}^{\mathcal{A}},\sigma)$ is an M-system and $\xi_{x}^{\mathcal{A}}$ is a
transitive point.

###  6.3. The proof of Theorem D

In this subsection we prove Theorem D.

####  6.3.1. A lemma

######  Lemma 6.4.

Let $E$ be a syndetic subset of $\mathbb{Z}^{2}$. Then there is some
$m_{*}\in\mathbb{Z}$ such that

$E(m_{*}):=\\{n\in\mathbb{Z}:(m_{*},n)\in E\\}\in\mathcal{F}_{ps}.$

###### Proof.

Since $E$ is $\mathbb{Z}^{2}$-syndetic, there is some $L\in\mathbb{N}$ such
that

(6.4) $E+[-L,L]^{2}=\mathbb{Z}^{2},$

where $[-L,L]=\\{-L,-L+1,\ldots,L\\}$ is an interval of $\mathbb{Z}$. That is,
for each $(m,n)\in\mathbb{Z}^{2}$, there exits $(i,j)\in[-L,L]^{2}$ such that
$(m+i,n+j)=(m,n)+(i,j)\in E.$

For each $i\in[-L,L]$, let

$E(i)\triangleq\\{n\in\mathbb{Z}:(i,n)\in E\\}.$

By (6.4), for each $n\in\mathbb{Z}$, there is some $(i,j)\in[-L,L]^{2}$ such
that

$(0,n)+(i,j)\in E.$

That is, $n+j\in E(i)$. It follows that

$\mathbb{Z}=\bigcup_{-L\leq i,j\leq L}(E(i)-j).$

Since $\mathcal{F}_{ps}$ has the Ramsey property (see [13, Theorem 1.24]),
there are some $m_{*},j\in[-L,L]$ such that $E(m_{*})-j\in\mathcal{F}_{ps}$.
And hence

$E(m_{*})\in\mathcal{F}_{ps},$

as $\mathcal{F}_{ps}$ is translation invariant. ∎

####  6.3.2. The proof of Theorem D

Now we are ready to give the proof of Theorem D. Note that the idea of the
proof of D is similar to the one of the last part of Theorem C. Since linear
terms may appear in $\mathcal{A}$, this makes the argument longthy.

###### Proof of Theorem D.

By Theorem 3.6, if $\pi:X\rightarrow X_{\infty}$ is the factor map to
$X_{\infty}$, then there are minimal t.d.s. $X^{*}$ and $X_{\infty}^{*}$ which
are almost one to one extensions of $X$ and $X_{\infty}$ respectively, and a
commuting diagram below.

$\begin{CD}X@<{\varsigma^{*}}<{}<X^{*}\\\ @V{}V{\pi}V@V{}V{\pi^{*}}V\\\
X_{\infty}@<{\varsigma}<{}<X_{\infty}^{*}\end{CD}$

If the result in Theorem D holds for $X^{*}$, then it also holds for $X$. So
without loss of generality, we may assume that $X=X^{*}$. That is, we may
assume

---
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{*}}$$\textstyle{X^{*}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varsigma}$$\textstyle{X_{\infty}}$

where $\pi^{*}$ is open and $\varsigma$ is almost one to one. Let
$\Omega=\\{x\in X^{*}_{\infty}:\varsigma^{-1}(\varsigma(x))=\\{x\\}\\}.$ Then
$\Omega$ is a dense $G_{\delta}$ subset $X^{*}_{\infty}$. Let

$X_{0}=X_{pol}\cap(\pi^{*})^{-1}(\Omega),$

where $X_{pol}$ is defined in (3.1). Then $X_{0}$ is a dense $G_{\delta}$
subset of $X$, $T(X_{0})=X_{0}$ and for all $x\in X_{0}$ statements in Theorem
3.6 still hold.

We will show that for integral polynomials $p_{1},\ldots,p_{d}$ with
$p_{i}(0)=0$, $1\leq i\leq d$, for any non-empty open set $U$, we can find
some $x\in X_{0}\cap U$ such that

$N_{\\{p_{1},\ldots,p_{d}\\}}(x,U)=\\{n\in\mathbb{Z}:T^{p_{1}(n)}x\in
U,\ldots,T^{p_{d}(n)}x\in U\\}\in\mathcal{F}_{ps}.$

Without loss of generality, we may require that $p_{1},\ldots,p_{d}$ are
essentially distinct non-constant integral polynomials such that $p_{i}(0)=0$,
$1\leq i\leq d$.

If $\deg(p_{i})\geq 2,\forall 1\leq i\leq d$, then by the third part of
Theorem C it is done. So we need to deal with the case when there are linear
polynomials in $p_{1},\ldots,p_{d}$. Let

(6.5)
$\displaystyle\mathcal{A}=\\{p_{1},\ldots,p_{s},p_{s+1}(n),\ldots,p_{d}(n)\\},$

where $p_{i}(n)=a_{i}n$, $1\leq i\leq s$ and $\deg(p_{i})\geq 2,s+1\leq i\leq
d$. Since $p_{1},\ldots,p_{d}$ are essentially distinct non-constant integral
polynomials, $a_{1},\ldots,a_{s}\in\mathbb{Z}\setminus\\{0\\}$ are distinct.

Let $U$ be a non-empty open set of $X$. Choose an $\epsilon>0$ and an non-
empty open set $\widetilde{U}$ of $X$ such that
$B_{2\epsilon}(\widetilde{U})\subseteq U$. First we need the following claim
whose proof will be given in the next subsection.

Claim: There is some $x\in\widetilde{U}\cap X_{0}$ such that

$N_{\\{p_{1},\ldots,p_{s}\\}}(x,\widetilde{U})\cap
N_{\\{p_{s+1},\ldots,p_{d}\\}}(\pi^{*}(x),\pi^{*}(\widetilde{U}))\in\mathcal{F}_{ps}.$

By Lemma 6.1, there is some $L\in\mathbb{N}$ such that if
$k_{1},k_{2}\in\mathbb{Z}$ with $|k_{1}-k_{2}|\geq L$, then for any
$(i_{1},j_{1}),(i_{2},j_{2})\in\\{s+1,\ldots,d\\}\times\\{1,2\\}$ with
$(i_{1},j_{1})\neq(i_{2},j_{2})$, the polynomials
$p_{i_{1}}(n+k_{j_{1}})-p_{i_{1}}(k_{j_{1}})$ and
$p_{i_{2}}(n+k_{j_{2}})-p_{i_{2}}(k_{j_{2}})$ are essentially distinct.

Let $z=\pi^{*}(x)$, where $x\in\widetilde{U}\cap X_{0}$ is as in the Claim
above. Note $\pi^{*}(\widetilde{U})$ is an open neighbourhood of $z$ as
$\pi^{*}$ is open. Let

$F=N_{\\{p_{1},\ldots,p_{s}\\}}(x,\widetilde{U})\cap
N_{\\{p_{s+1},\ldots,p_{d}\\}}(\pi^{*}(x),\pi^{*}(\widetilde{U})).$

By Claim,
$F=\\{n_{j}\\}_{j\in\mathbb{Z}}=\\{\ldots<n_{-2}<n_{-1}<n_{0}<n_{1}<n_{2}<\ldots\\}$
is a piecewise syndetic set of $\mathbb{Z}$. By taking a piecewise syndetic
subset, we may assume that $|n_{i}-n_{j}|\geq L,\forall i\neq j\in\mathbb{Z}$.

We show that for each $l\in\mathbb{N}$, there is some $m\in\mathbb{Z}$ such
that $m+\\{n_{j}\\}_{-l\leq j\leq l}\subseteq N_{\mathcal{A}}(x,U)$ and hence
$N_{\mathcal{A}}(x,U)\in\mathcal{F}_{ps}$ by Lemma 6.2, where $\mathcal{A}$ is
defined in (6.5).

Fix $l\in\mathbb{N}$. Since $\\{n_{j}\\}_{-l\leq j\leq l}\subseteq
N_{\\{p_{1},\ldots,p_{s}\\}}(x,\widetilde{U})$, we have

$x\in\widetilde{U}\cap\bigcap_{j=-l}^{l}\left(T^{-a_{1}n_{j}}\widetilde{U}\cap
T^{-a_{2}n_{j}}\widetilde{U}\cap\ldots\cap
T^{-a_{s}n_{j}}\widetilde{U}\right).$

Let $0<\epsilon^{\prime}<\epsilon$ such that

$x\in
B_{2\epsilon^{\prime}}(x)\subseteq\widetilde{U}\cap\bigcap_{j=-l}^{l}\left(T^{-a_{1}n_{j}}\widetilde{U}\cap
T^{-a_{2}n_{j}}\widetilde{U}\cap\ldots\cap
T^{-a_{s}n_{j}}\widetilde{U}\right).$

Thus we have

(6.6) $T^{a_{t}n_{j}}B_{2\epsilon^{\prime}}(x)\subseteq\widetilde{U},\quad
1\leq t\leq s,-l\leq j\leq l.$

Let

$\mathcal{A}^{\prime}=\\{p_{1},\ldots,p_{s}\\}\cup\\{p_{i}^{j}:-l\leq j\leq
l,s+1\leq i\leq d\\},$

where $p_{i}^{j}(n)=p_{i}(n+n_{j})$. Then $\mathcal{A}^{\prime}$ is a set of
essentially distinct integral polynomials. As $x\in X_{0}$, by Theorem 3.6
$\overline{\mathcal{O}}_{\mathcal{A}^{\prime}}(x^{\otimes d^{\prime}})$ is
${\pi^{*}}^{(d^{\prime})}$-saturated, that is,

$\overline{\mathcal{O}}_{\mathcal{A}^{\prime}}(x^{\otimes
d^{\prime}})=({\pi^{*}}^{(d^{\prime})})^{-1}\Big{(}{\pi^{*}}^{(d^{\prime})}(\overline{\mathcal{O}}_{\mathcal{A}^{\prime}}(x^{\otimes
d^{\prime}}))\Big{)},$

where $d^{\prime}=(d-s)(2l+1)+s$. One has that

$F_{\mathcal{A}^{\prime}}(x)\cap\Big{(}B_{2\epsilon^{\prime}}(x)^{s}\times
U^{(d-s)(2l+1)}\Big{)}\neq\emptyset,$

where $F_{*}$ is defined in Lemma 3.5. In particular, there is some
$m\in\mathbb{Z}$ such that

$\left(T^{a_{1}m}x,\ldots,T^{a_{s}m}x,\big{(}T^{p_{i}(m+n_{j})}(x)\big{)}_{s+1\leq
i\leq d,-l\leq j\leq l}\right)\in B_{2\epsilon^{\prime}}(x)^{s}\times
U^{(d-s)(2l+1)}.$

Thus

$T^{p_{i}(m+n_{j})}x\in U,\quad s+1\leq i\leq d,-l\leq j\leq l,$

and by (6.6),

$T^{a_{t}(m+n_{j})}x\in
T^{a_{t}n_{j}}B_{2\epsilon^{\prime}}(x)\subseteq\widetilde{U}\subseteq U,\quad
1\leq t\leq s,-l\leq j\leq l.$

Thus

$m+\\{n_{j}\\}_{-l\leq j\leq l}=\\{m+n_{j}\\}_{-l\leq j\leq l}\subseteq
N_{\\{p_{1},\ldots,p_{d}\\}}(x,U).$

The proof is complete. ∎

####  6.3.3. The proof of the Claim

###### Proof of Claim.

Let $x_{0}\in\widetilde{U}\cap X_{0}$ and $z_{0}=\pi^{*}(x_{0})$. Since
$\pi^{*}$ is open, $\pi^{*}(\widetilde{U})$ is an open neighbourhood of
$z_{0}$. By the same arguing in the proof of Theorem C (the third part), we
can find a nil-manifold $\widehat{X}=\widehat{G}/\widehat{\Gamma}$, a
homomorphism $\phi:\mathbb{Z}^{2}\rightarrow\widehat{G}$,
$\hat{x}\in\widehat{X}$ and an open neighbourhood $\hat{W}$ of $\hat{x}$ in
$\widehat{X}$ such that

(6.7) $\displaystyle\begin{aligned} &\hskip
14.22636pt\\{(m,n)\in\mathbb{Z}^{2}:T^{m+p_{s+1}(n)}z_{0}\in\pi^{*}(\widetilde{U}),\ldots,T^{m+p_{d}(n)}z_{0}\in\pi^{*}(\widetilde{U})\\}\\\
&\supseteq\\{(m,m)\in\mathbb{Z}^{2}:\phi(m,n)\hat{x}\in\hat{W}\\}.\end{aligned}$

We consider a $\mathbb{Z}^{2}$-action on
$\widehat{X}=\widehat{G}/\widehat{\Gamma}$ by

$g^{(m,n)}(\hat{h}\widehat{\Gamma})=\phi(m,n)\hat{h}\widehat{\Gamma}$

for any $\hat{h}\in\widehat{G}$ and $(n,m)\in\mathbb{Z}^{2}$. Then
$(\widehat{X},\\{g^{(m,n)}\\}_{(m,n)\in\mathbb{Z}^{2}})$ is a distal
$\mathbb{Z}^{2}$-t.d.s.

Then (6.7) can be rewritten as

(6.8) $\displaystyle\begin{aligned} &\hskip
14.22636pt\\{(m,n)\in\mathbb{Z}^{2}:T^{m+p_{s+1}(n)}z_{0}\in\pi^{*}(\widetilde{U}),\ldots,T^{m+p_{d}(n)}z_{0}\in\pi^{*}(\widetilde{U})\\}\\\
&\supseteq\\{(m,n)\in\mathbb{Z}^{2}:g^{(m,n)}\hat{x}\in\hat{W}\\}.\end{aligned}$

Take $a=\max_{1\leq j\leq s}|a_{j}|$ and $t=2a+1$. Let

$\sigma_{t}=T\times\ldots\times T\ (t\ \text{times}),\ \text{and}\
\tau=T^{-a}\times\cdots\times T^{-1}\times{\rm id}\times T\times\ldots\times
T^{a}.$

By Theorem 2.7, $N_{t}(X)=\overline{\mathcal{O}}(x_{0}^{\otimes
t},\langle\sigma_{t},\tau\rangle)$ is a
$\langle\sigma_{t},\tau\rangle$-minimal t.d.s., where $x_{0}^{\otimes
t}=(x_{0},\ldots,x_{0})\in X^{t}$. We define a $\mathbb{Z}^{2}$-action on
$X^{t}\times\widehat{X}$ by

$\big{(}(x_{1},\ldots,x_{t}),\hat{h}\widehat{\Gamma}\big{)}\mapsto(\sigma_{t}^{m}\tau^{n}\times
g^{(m,n)})\big{(}(x_{1},\ldots,x_{t}),\hat{h}\widehat{\Gamma}\big{)},\forall(m,n)\in\mathbb{Z}^{2}.$

We denote the corresponding $\mathbb{Z}^{2}$-t.d.s. by
$(X^{t}\times\widehat{X},\\{\sigma_{t}^{m}\tau^{n}\times
g^{(m,n)}\\}_{(m,n)\in\mathbb{Z}^{2}})$.

Since $\hat{x}$ is a $\\{g^{(m,n)}\\}_{(m,n)\in\mathbb{Z}^{2}}$-distal point
and $x_{0}^{\otimes t}$ is a
$\\{\sigma_{t}^{m}\tau^{n}\\}_{(m,n)\in\mathbb{Z}^{2}}$-minimal point, one has
$(x_{0}^{\otimes t},\hat{x})$ is a minimal point of
$(X^{t}\times\widehat{X},\\{\sigma_{t}^{m}\tau^{n}\times
g^{(m,n)}\\}_{(m,n)\in\mathbb{Z}^{2}})$. 333Let $(X,T),(Y,S)$ be t.d.s. If
$x\in X$ is distal and $y\in Y$ is minimal, then $(x,y)$ is a $T\times
S$-minimal point [13, Theorem 9.11.]. This also holds for
$\mathbb{Z}^{2}$-t.d.s. and more general group actions [5].

Thus for the open neighborhood
$\widetilde{U}\times\cdots\times\widetilde{U}\times\hat{W}$ of
$(x_{0}^{\otimes t},\hat{x})$,

$E\triangleq\\{(m,n)\in\mathbb{Z}^{2}:\sigma_{t}^{m}\tau^{n}\times
g^{(m,n)}\big{(}x_{0}^{\otimes
t},\hat{x}\big{)}\in\widetilde{U}\times\cdots\times\widetilde{U}\times\hat{W}\\}$

is a syndetic subset of $\mathbb{Z}^{2}$. Moreover, by Lemma 6.4 we can find
$m_{*}\in\mathbb{Z}$ such that

$E(m_{*}):=\\{n\in\mathbb{Z}:(m_{*},n)\in E\\}\in\mathcal{F}_{ps}.$

For any $n\in E(m_{*})$, one has

$T^{m_{*}+jn}(x_{0})\in\widetilde{U}\text{ for }j=-a,\cdots,-1,0,1,\cdots,a$

and $g^{(m_{*},n)}\hat{x}\in\widehat{W}$, which implies
$T^{m_{*}+p_{i}(n)}z_{0}\in\pi^{*}(\widetilde{U})$ for $i=s+1,s+2,\cdots,d$ by
(6.8). In particular, $T^{m_{*}}x_{0}\in\widetilde{U}$.

Now put $x=T^{m_{*}}(x_{0})$. Then $x\in\widetilde{U}\cap X_{0}$ as $X_{0}$ is
$T$-invariant. Note that $\pi^{*}(x)=T^{m_{*}}(z_{0})$. Then by the above
discussion for any $n\in E(m_{*})$, one has that

$T^{a_{j}n}x=T^{m_{*}+a_{j}n}x_{0}\in\widetilde{U}\text{ for }j=1,\cdots,s,$

(that is, $n\in N_{\\{p_{1},\ldots,p_{s}\\}}(x,\widetilde{U})$) and

$T^{p_{i}(n)}\pi^{*}(x)=T^{m_{*}+p_{i}(n)}z_{0}\in\pi^{*}(\widetilde{U})\text{
for }i=s+1,s+2,\cdots,d,$

(that is, $n\in
N_{\\{p_{1},\ldots,p_{d}\\}}(\pi^{*}(x),\pi^{*}(\widetilde{U}))$). Hence

$E(m_{*})\subseteq N_{\\{p_{1},\ldots,p_{s}\\}}(x,\widetilde{U})\cap
N_{\\{p_{s+1},\ldots,p_{d}\\}}(\pi^{*}(x),\pi^{*}(\widetilde{U}))$

and so

$N_{\\{p_{1},\ldots,p_{s}\\}}(x,\widetilde{U})\cap
N_{\\{p_{s+1},\ldots,p_{d}\\}}(\pi^{*}(x),\pi^{*}(\widetilde{U}))\in\mathcal{F}_{ps}.$

The proof of the Claim is complete. ∎

##  7\. The proof of Theorem E

In this section we will show that for any minimal t.d.s. $(X,T)$,
$(M_{\infty}({X},\mathcal{A}),\langle T^{\infty},\sigma\rangle)$ is an
$M$-system. And as a corollary, we give a proof of Theorem E.

In this section we will use the regularizer and the universal minimal t.d.s.
from the theory of minimal flows, which usually deals with group actions on
compact Hausdorff spaces. First we need some facts about minimal t.d.s. which
will be used in the proof.

###  7.1. Some facts about minimal t.d.s.

####  7.1.1. The universal minimal t.d.s.

Let $\beta\mathbb{Z}$ be the Stone-Cěch compactification of $\mathbb{Z}$,
which is a compact Hausdorff topological space where $\mathbb{Z}$ is densely
and equivariantly embedded. Moreover, the addition on $\mathbb{Z}$ can be
extended to an addition on $\beta\mathbb{Z}$ in such a way that
$\beta\mathbb{Z}$ is a compact semigroup with continuous right translations.
Let ${\bf M}$ be a minimal left ideal in $\beta\mathbb{Z}$. Then ${\bf M}$ is
a closed semigroup with continuous right translations. By the Ellis-Namakura
Theorem the set $J=J({\bf M})$ of idempotents in ${\bf M}$ is non-empty.
Moreover, $\\{v{\bf M}:v\in J\\}$ is a partition of ${\bf M}$ and every $v{\bf
M}$ is a group with unit element $v$. Let $(X,T)$ be a t.d.s. and $x\in X$. A
necessary and sufficient condition for $x$ to be minimal is that $ux=x$ for
some $u\in J$.

Let $(X,T)$ be a t.d.s. and then $\mathbb{Z}$ acts on the compact metric space
$X$ as follows: for any $m\in\mathbb{Z}$ and $x\in X$ one has $mx=T^{m}x$.
Then the sets $\beta\mathbb{Z}$ and ${\bf M}$ also act on $X$ as semigroups
and
$\beta\mathbb{Z}x=\\{px:p\in\beta\mathbb{Z}\\}=\overline{\mathcal{O}}(x,T)$.
If $(X,T)$ is minimal, then ${\bf M}x=\overline{\mathcal{O}}(x,T)=X$ for every
$x\in X$. Also, as a t.d.s., $({\bf M},T)$ is an extension of any minimal
t.d.s. $(X,T)$. We call $({\bf M},T)$ the universal minimal t.d.s. Up to
isomorphisms, the universal minimal t.d.s. is unique [4, Chapter 8].

####  7.1.2. Hyperspace t.d.s.

When $X$ is not metric, neither is $2^{X}$. And in this case $2^{X}$ can be
endowed with the Vietoris topology. A basis for this topology on $2^{X}$ is
given by

$\langle U_{1},\ldots,U_{n}\rangle=\\{A\in
2^{X}:A\subseteq\bigcup_{i=1}^{n}U_{i}\ \text{and $A\cap U_{i}\neq\emptyset$
for every $i\in\\{1,\ldots,n\\}$}\\},$

where each $U_{i}\subseteq X$ is open. Note that when $X$ is a metric space,
the Hausdorff topology consists with the topology induced by Hausdorff metric
of $2^{X}$.

Let $(X,T)$ be a t.d.s. It induces a t.d.s. on $2^{X}$, which is called the
hyperspace t.d.s. The action of $\mathbb{Z}$ on $2^{X}$ is given by
$mA=\\{ma:a\in A\\}=\\{T^{m}a:a\in A\\}$ for each $m\in\mathbb{Z}$ and $A\in
2^{X}$. This action induces another one of $\beta\mathbb{Z}$ on $2^{X}$. To
avoid ambiguities one denotes the action of $\beta\mathbb{Z}$ on $2^{X}$ by
the circle operation as follows: let $p\in\beta\mathbb{Z}$ and $A\in 2^{X}$,
and define $p\circ A={\lim\limits_{\lambda}}\ m_{\lambda}A$ for any net
$\\{m_{\lambda}\\}_{\lambda\in\Lambda}\subseteq\mathbb{Z}$ converging to $p$.
Moreover

$p\circ A=\\{x\in X:\text{for each $\lambda\in\Lambda$ there is
$d_{\lambda}\in A$ with $x=\lim_{\lambda}m_{\lambda}d_{\lambda}$}\\}$

for any fixed net $\\{m_{\lambda}\\}_{\lambda\in\Lambda}$ converging to $p$.
Observe that $pA\subseteq p\circ A$, where $pA=\\{pa:a\in A\\}$.

####  7.1.3. RIC extensions

Let $\pi:(X,T)\rightarrow(Y,T)$ be a factor map of minimal t.d.s., and
$x_{0}\in X$, $y_{0}=\pi(x_{0})$. Let $u\in J$ such that $ux_{0}=x_{0}$. We
say that $\pi$ is a RIC (relatively incontractible) extension if for every
$y=py_{0}\in Y$, $p\in{\bf M}$,

(7.1) $\pi^{-1}(y)=p\circ\big{(}u\pi^{-1}(y_{0})\big{)}.$

Note that every distal extension is RIC, and every RIC extension is open.

Every factor map between minimal systems can be lifted to a RIC extension by
proximal extensions (see [10, Theorem 5.13] or [37, Chapter VI]).

######  Theorem 7.1.

Given a factor map $\pi:(X,T)\rightarrow(Y,T)$ of minimal systems, there
exists a commutative diagram of factor maps (called RIC-diagram)

$\begin{CD}X@<{\theta^{\prime}}<{}<X^{\prime}\\\
@V{}V{\pi}V@V{}V{\pi^{\prime}}V\\\ Y@<{\theta}<{}<Y^{\prime}\end{CD}$

such that:

  1. (a)

$\theta^{\prime}$ and $\theta$ are proximal extensions;

  2. (b)

$\pi^{\prime}$ is a RIC extension;

  3. (c)

$X^{\prime}$ is the unique minimal set in $R_{\pi\theta}=\\{(x,y)\in X\times
Y^{\prime}:\pi(x)=\theta(y)\\}$, and $\theta^{\prime}$ and $\pi^{\prime}$ are
the restrictions to $X^{\prime}$ of the projections of $X\times Y^{\prime}$
onto $X$ and $Y^{\prime}$ respectively.

####  7.1.4. Regular extensions

######  Definition 7.2.

Let $\pi:(X,T)\rightarrow(Y,T)$ be an extension of minimal t.d.s. One says
$\pi$ is regular if for any minimal point $(x_{1},x_{2})$ in $R_{\pi}$ (i.e.
$\pi(x_{1})=\pi(x_{2})$) there exists $\chi\in{\rm Aut}_{\pi}(X,T)$ such that
$\chi(x_{1})=x_{2}$. It is equivalent to: for any point $(x_{1},x_{2})\in
R_{\pi}$ there exists $\chi\in{\rm Aut}_{\pi}(X,T)$ such that
$(\chi(x_{1}),x_{2})\in{\bf P}(X,T)$.

The notion of regularity was introduced by Auslander [3]. Examples of regular
extensions are proximal extensions, group extensions. For more information
about regularity, refer to [3, 4, 17].

######  Theorem 7.3.

[37, Chapter V, 6.3-7] Let $(X,T)$ and $(Y,T)$ be minimal t.d.s. and let
$\phi$ be the factor map. Then there is an extension
$\theta:(X^{*},T)\rightarrow(X,T)$ such that
$\phi^{*}=\phi\circ\theta:(X^{*},T)\rightarrow(Y,T)$ is regular. Moreover,
$\theta$ is an isomorphism if and only $\phi$ is regular.

---
$\textstyle{{X}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\scriptstyle{\phi^{*}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{Y}$

The extension $\phi^{*}:X^{*}\rightarrow Y$ is called a regularizer of $\phi$.

######  Remark 7.4.

Let $(Y,T)$ be a minimal t.d.s., and let $\psi:({\bf M},T)\rightarrow(Y,T)$ be
the factor map from the universal minimal t.d.s. $({\bf M},T)$ to $(Y,T)$. By
Theorem 7.3, we have the following diagram.

---
$\textstyle{{{\bf
M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\scriptstyle{\psi^{*}}$$\textstyle{{\bf
M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{Y}$

Since ${\bf M}$ is the universal minimal t.d.s., $\theta$ is an isomorphism.
Thus $\psi$ is regular.

###  7.2. Some lemmas

In the next subsection we show that $(M_{\infty}({X},\mathcal{A}),\langle
T^{\infty},\sigma\rangle)$ is an $M$-system. To do this, we need some
preparation.

The following lemma is well known (for example see [2, Theorem 5.5-(2)]). Note
that in these lemmas, all spaces in t.d.s. are assumed to be compact and
Hausdorff.

######  Lemma 7.5.

Let $\pi:(X,T)\rightarrow(Y,T)$ be a proximal extension of minimal t.d.s.
$(X,T)$ and $(Y,T)$. Then for each $y\in Y$ and any
$x,x_{1},x_{2},\ldots,x_{k}\in\pi^{-1}(y)$, there is a net
$\\{n_{i}\\}\subseteq\mathbb{N}$ such that
$T^{n_{i}}(x_{1},x_{2},\ldots,x_{k})\rightarrow(x,x,\ldots,x),\ i\to\infty$.

The following lemma is also well known (see e.g. [37, Chapter V-(2.9)]).

######  Lemma 7.6.

Let $\pi_{i}:(X_{i},T)\rightarrow(Y_{i},T),i\in I$ be proximal extensions,
where $I$ is an index set. Then

$\prod_{i\in I}\pi_{i}:\prod_{i\in I}X_{i}\rightarrow\prod_{i\in I}Y_{i},\
(x_{i})_{i\in I}\mapsto(\pi_{i}(x_{i}))_{i\in I}$

is proximal.

Now we have

######  Lemma 7.7.

Let $\pi:(X,T)\rightarrow(Y,T)$ be a factor map of minimal t.d.s. and let
$\mathcal{A}=\\{p_{1},p_{2},\ldots,p_{d}\\}$ be a set of integral polynomials.
If $\pi$ is proximal, then so is
$\pi^{\infty}:(M_{\infty}(X,\mathcal{A}),\mathcal{G})\rightarrow(M_{\infty}(Y,\mathcal{A}),\mathcal{G})$,
where $\mathcal{G}=\langle T^{\infty},\sigma\rangle$; and
$(M_{\infty}(X,\mathcal{A}),\mathcal{G})$ is minimal if and only if
$(M_{\infty}(Y,\mathcal{A}),\mathcal{G})$ is minimal.

###### Proof.

If $\pi$ is proximal, then by Lemma 7.6
$\pi^{\infty}:(M_{\infty}(X,\mathcal{A}),T^{\infty})\rightarrow(M_{\infty}(Y,\mathcal{A}),T^{\infty})$
is proximal, and hence
$\pi^{\infty}:(M_{\infty}(X,\mathcal{A}),\mathcal{G})\rightarrow(M_{\infty}(Y,\mathcal{A}),\mathcal{G})$
is also proximal.

It remains to show that $(M_{\infty}(X),\mathcal{G})$ is minimal if
$(M_{\infty}(Y),\mathcal{G})$ is minimal. By Subsection 4.3,
$M_{\infty}(X,\mathcal{A})$ is isomorphic to $N_{\infty}(X,\mathcal{A}).$ So
it suffices to show that $(N_{\infty}(X,\mathcal{A}),\mathcal{G})$ is minimal
if $(N_{\infty}(Y,\mathcal{A}),\mathcal{G})$ is minimal.

Since
$N_{\infty}(X,\mathcal{A})=\overline{\mathcal{O}}(\omega_{x}^{\mathcal{A}},\mathcal{G})$
for each $x\in X$, to show that $(N_{\infty}(X,\mathcal{A}),\mathcal{G})$ is
minimal, we need to show that for any ${\bf x}\in N_{\infty}(X,\mathcal{A})$,
there is some $x\in X$ such that
$\omega_{x}^{\mathcal{A}}\in\overline{\mathcal{O}}({\bf x},\mathcal{G})$.

Let ${\bf x}=({\bf
x}_{n})_{n\in{\mathbb{Z}}}=\Big{(}(x^{(1)}_{n},x^{(2)}_{n},\cdots,x^{(d)}_{n})\Big{)}_{n\in{\mathbb{Z}}}\in
N_{\infty}(X,\mathcal{A}).$ Then ${\bf y}=\pi^{\infty}({\bf x})\in
N_{\infty}(Y,\mathcal{A})$. By assumption,
$(N_{\infty}(Y,\mathcal{A}),\mathcal{G})$ is minimal, and there is some $y\in
Y$ and a net $\\{g_{i}\\}_{i}\subseteq\mathcal{G}$ such that

$g_{i}{\bf y}\to\omega_{y}^{\mathcal{A}}=(\ldots,T^{\vec{p}(-1)}(y^{\otimes
d}),\underset{\bullet}{T^{\vec{p}(0)}y^{\otimes d}},T^{\vec{p}(1)}(y^{\otimes
d}),T^{\vec{p}(2)}(y^{\otimes d}),\cdots),$

where $T^{\vec{p}(n)}=T^{p_{1}(n)}\times T^{p_{2}(n)}\times\cdots\times
T^{p_{d}(n)}.$ Without loss of generality, assume that

$g_{i}{\bf
x}\to{\bf\tilde{x}}=({\bf\tilde{x}}_{n})_{n\in{\mathbb{Z}}}=\Big{(}(\tilde{x}^{(1)}_{n},\tilde{x}^{(2)}_{n},\cdots,\tilde{x_{n}}^{(d)})\Big{)}_{n\in{\mathbb{Z}}}.$

Then ${\bf\tilde{x}}\in\overline{\mathcal{O}}({\bf x},\mathcal{G})$ and

$\pi^{\infty}({\bf\tilde{x}})=\lim_{i}\pi^{\infty}(g_{i}{\bf
x})=\lim_{i}g_{i}\pi^{\infty}({\bf x})=\lim_{i}g_{i}{\bf
y}=\omega_{y}^{\mathcal{A}}.$

Hence

$T^{-p_{j}(n)}\tilde{x_{n}}^{(j)}\in\pi^{-1}(y),\ \text{for all }\ 1\leq j\leq
d,\ n\in{\mathbb{Z}}.$

Fix a point $x\in\pi^{-1}(y)$. By Lemma 7.5, for any $k\in\mathbb{N}$ there is
some net $\\{n_{i}\\}_{i}\subseteq\mathbb{N}$ such that

$\big{(}{T^{((2k+1)d)}}\big{)}^{n_{i}}\Big{(}(T^{\vec{p}(-k)})^{-1}({\bf\tilde{x}}_{-k}),(T^{\vec{p}(-k+1)})^{-1}({\bf\tilde{x}}_{-k+1}),\ldots,(T^{\vec{p}(k)})^{-1}({\bf\tilde{x}}_{k})\Big{)}\to
x^{\otimes(2k+1)d}.$

Since $T^{\vec{p}(-k)}\times\ldots\times T^{\vec{p}(-1)}\times
T^{\vec{p}(0)}\times T^{\vec{p}(1)}\times\ldots\times T^{\vec{p}(k)}$ is
continuous, it follows that

$\big{(}{T^{((2k+1)d)}}\big{)}^{n_{i}}\Big{(}{\bf\tilde{x}}_{-k},{\bf\tilde{x}}_{-k+1},\ldots,{\bf\tilde{x}}_{k}\Big{)}$

converges to

$(T^{\vec{p}(-k)}x^{\otimes d},T^{\vec{p}(-k+1)}x^{\otimes
d},\ldots,T^{\vec{p}(k)}x^{\otimes d}).$

We may assume that $(T^{\infty})^{n_{i}}{\bf\tilde{x}}$ converges to

${\bf\tilde{x}}^{[k]}=\big{(}({\bf\tilde{x}}_{n}^{k})_{n=-\infty}^{-k-1},T^{\vec{p}(-k)}x^{\otimes
d},T^{\vec{p}(-k+1)}x^{\otimes
d},\ldots,\underset{\bullet}{T^{\vec{p}(0)}x^{\otimes
d}},\ldots,T^{\vec{p}(k)}x^{\otimes
d},({\bf\tilde{x}}_{n}^{k})_{n=k+1}^{\infty}\big{)},$

where ${\bf\tilde{x}}_{n}^{k}\in X^{d},|n|\geq k+1.$ Thus for all
$k\in\mathbb{N}$,

${\bf\tilde{x}}^{[k]}\in\overline{\mathcal{O}}({\bf\tilde{x}},T^{\infty})\subset\overline{\mathcal{O}}({\bf
x},\langle T^{\infty},\sigma\rangle).$

It is clear that
$\lim_{k\to\infty}{\bf\tilde{x}}^{[k]}=(T^{\vec{p}(n)}x^{\otimes
d})_{n\in\mathbb{Z}}=\omega_{x}^{\mathcal{A}}$, and we have

$\omega_{x}^{\mathcal{A}}\in\overline{\mathcal{O}}({\bf x},\langle
T^{\infty},\sigma\rangle).$

The proof is complete. ∎

###  7.3. The proof of an equivalence form of theorem E

Now we are ready to prove an equivalence form of Theorem E. In the rest of the
paper we put $\mathcal{G}=\langle T^{\infty},\sigma\rangle$. By Theorem 4.10,
we show that for a minimal t.d.s., for any family
$\mathcal{A}=\\{p_{1},p_{2},\ldots,p_{d}\\}$ of integral polynomials
satisfying $(\spadesuit)$, $(M_{\infty}({X},\mathcal{A}),\mathcal{G})$ is an
$M$-system. Since we will use Theorem 4.9 which requires $X$ being metric, in
this Subsection we assume that $X$ is a metric space. But in the proof of the
theorem, we will use the universal minimal t.d.s. ${\bf M}$ which is not a
metric space.

####  7.3.1. Idea of the proof

Since the proof is long, we describe the idea of the proof first. Let $(X,T)$
be minimal and $\pi:X\longrightarrow X_{\infty}$ be the factor map. By the
discussion in Section 5, $(M_{\infty}({X_{\infty}},\mathcal{A}),\mathcal{G})$
is minimal. Using the RIC-diagram (modifications by proximal extensions), we
obtain $\pi^{*}:X^{*}\longrightarrow X_{\infty}^{*}$ which is RIC. By Lemma
7.7, $(M_{\infty}({X_{\infty}^{*}},\mathcal{A}),\mathcal{G})$ is also minimal.

The natural idea to show the density of minimal point in
$(M_{\infty}(X^{*},\mathcal{A}),\mathcal{G})$ is that for a minimal point
${\bf x}$ of $\mathcal{G}$ and a non-empty open set $U\subset
M_{\infty}(X^{*},\mathcal{A})$, one finds some $\phi\in{\rm Aut}(X,T)$ such
that $\phi({\bf x})\in U$ is the limit of minimal points. This idea is
realized in the proof of Theorem 5.7 for the distal case. When we apply the
same idea for the RIC extension, we find that we do not have enough elements
in ${\rm Aut}(X,T)$ to realize the idea. Thus, we need to pass our discussion
to the universal minimal system to get suitable elements in ${\rm Aut}(X,T)$
for our purpose.

The proof is done in four steps: firstly we construct the extension (Step 1
below), secondly we transfer the question into some extension of
$(M_{\infty}(X^{*},\mathcal{A}),\mathcal{G})$ (Step 2 below); then we show the
set of minimal points is dense in a certain region (Step 3 below), and finally
we use the RIC property to spread minimal points to the whole space (Step 4
below).

#### 7.3.2.

Now we give the main result of this section.

######  Theorem 7.8.

Let $({X},T)$ be a minimal t.d.s. and let
$\mathcal{A}=\\{p_{1},p_{2},\ldots,p_{d}\\}$ be a family of integral
polynomials satisfying $(\spadesuit)$. Then
$(M_{\infty}({X},\mathcal{A}),\mathcal{G})$ is an $M$-system.

###### Proof.

We split our proof into the following steps.

Step 1: The construction of the extensions.

Let $X^{*},X_{\infty}^{*},\pi^{*}$ etc. be as in Theorem 4.9. Let
$\eta^{\prime}:({\bf M},T)\rightarrow(X^{*},T)$ be the factor map from the
universal minimal t.d.s. $({\bf M},T)$ to $(X^{*},T)$, and let
$\xi^{\prime}=\pi^{*}\circ\eta^{\prime}:{\bf M}\rightarrow X_{\infty}^{*}$.
Now using the RIC-diagram for $\xi^{\prime}$ (Theorem 7.1), we have the
following commutative diagram, where $\theta^{\prime},\theta$ are proximal and
$\psi:{\bf M}\rightarrow Y$ is RIC. Also by Remark 7.4, $\psi$ is regular.

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{{X^{*}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varsigma^{*}}$$\scriptstyle{\pi^{*}}$$\textstyle{{\bf
M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta^{\prime}}$$\scriptstyle{\xi^{\prime}}$$\textstyle{{\bf
M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta^{\prime}}$$\scriptstyle{\psi}$$\textstyle{X_{\infty}}$$\textstyle{{X}^{*}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varsigma}$$\textstyle{{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$

The diagram above can be rewritten as follows:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{{X^{*}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varsigma^{*}}$$\scriptstyle{\pi^{*}}$$\textstyle{{\bf
M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\scriptstyle{\xi}$$\scriptstyle{\psi}$$\textstyle{X_{\infty}}$$\textstyle{{X}^{*}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varsigma}$$\textstyle{{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$

where $\eta=\eta^{\prime}\circ\theta^{\prime}$ and
$\xi=\xi^{\prime}\circ\theta^{\prime}$. Now let

$W=(\psi^{\infty})^{-1}(M_{\infty}(Y,\mathcal{A}))\subseteq{\bf
M}^{s}\times({\bf M}^{d-s})^{\mathbb{Z}},$

where $s,d$ are the numbers appearing in the definition of Condition
$(\spadesuit)$. Then $W$ is a $\mathcal{G}$-invariant closed subset, that is
$(W,\mathcal{G})$ is a t.d.s. It is clear that $M_{\infty}({\bf
M},\mathcal{A})\subseteq W$. We claim that we have the following commutative
diagram:

$\textstyle{M_{\infty}(X^{*},\mathcal{A})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\pi^{*}}^{\infty}}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta^{\infty}}$$\scriptstyle{\psi^{\infty}}$$\textstyle{M_{\infty}(X_{\infty}^{*},\mathcal{A})}$$\textstyle{M_{\infty}(Y,\mathcal{A})\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\theta^{\infty}}$

To see this, note first that
$\theta^{\infty}:(M_{\infty}(Y,\mathcal{A}),\mathcal{G})\rightarrow(M_{\infty}(X_{\infty}^{*},\mathcal{A}),\mathcal{G})$
is a factor map, and it follows that

$\begin{split}W&=(\psi^{\infty})^{-1}(M_{\infty}(Y,\mathcal{A}))\subseteq(\psi^{\infty})^{-1}(\theta^{\infty})^{-1}(M_{\infty}(X_{\infty}^{*},\mathcal{A}))\\\
&=(\theta^{\infty}\circ\psi^{\infty})^{-1}(M_{\infty}(X_{\infty}^{*},\mathcal{A}))=({\pi^{*}}^{\infty}\circ\eta^{\infty})^{-1}(M_{\infty}(X_{\infty}^{*},\mathcal{A}))\\\
&=(\eta^{\infty})^{-1}\Big{(}({\pi^{*}}^{\infty})^{-1}(M_{\infty}(X_{\infty}^{*},\mathcal{A}))\Big{)}.\end{split}$

Thus

$M_{\infty}(X^{*},\mathcal{A})=\eta^{\infty}(M_{\infty}({\bf
M},\mathcal{A}))\subseteq\eta^{\infty}(W)\subseteq({\pi^{*}}^{\infty})^{-1}M_{\infty}(X_{\infty}^{*},\mathcal{A}).$

By Theorem 4.9,
$M_{\infty}(X^{*},\mathcal{A})=({\pi^{*}}^{\infty})^{-1}(M_{\infty}(X_{\infty}^{*},\mathcal{A}))$.
Hence

$\eta^{\infty}:(W,\mathcal{G})\rightarrow(M_{\infty}(X^{*},\mathcal{A}),\mathcal{G})$

is a factor map.

Step 2: Transitions of the question.

By Theorem 4.9, $(M_{\infty}(X^{*},\mathcal{A}),\mathcal{G})$ is a transitive
t.d.s. To show $(M_{\infty}(X^{*},\mathcal{A}),\mathcal{G})$ is an $M$-system,
it suffices to show that $\mathcal{G}$-minimal points are dense in
$(W,\mathcal{G})$, as $(M_{\infty}(X^{*},\mathcal{A}),\mathcal{G})$ is a
factor of $(W,\mathcal{G})$.

For convenience, sometimes we denote the point ${\bf x}$ in ${\bf
M}^{s}\times({\bf M}^{d-s})^{\mathbb{Z}}$ by ${\bf x}=(x_{i})_{i\in I}$
instead of $\small{\bf
x}=\Big{(}(x_{1},x_{2},\ldots,x_{s}),\big{(}(x^{(1)}_{n},x^{(2)}_{n},\ldots,x^{(d-s)}_{n})\big{)}_{n\in{\mathbb{Z}}}\Big{)}$,
and similarly for the points in $Y^{s}\times(Y^{d-s})^{\mathbb{Z}}$.

By Corollary 5.6, $(M_{\infty}(X_{\infty},\mathcal{A}),\mathcal{G})$ is
minimal. Since $\varsigma$ and $\theta$ are proximal, by Lemma 7.7,
$(M_{\infty}(Y,\mathcal{A}),\mathcal{G})$ is minimal, and hence
$T^{\infty}$-minimal points are dense in $M_{\infty}(Y,\mathcal{A})$ (Lemma
2.2). Since $\psi$ is open, thus to show that $\mathcal{G}$-minimal points are
dense in $(W,\mathcal{G})$, it suffices to show for each $T^{\infty}$-minimal
point ${\bf y}\in M_{\infty}(Y,\mathcal{A})$, and each ${\bf
x}\in(\psi^{\infty})^{-1}({\bf y})$, there is a net $\\{{\bf
x}_{\alpha}\\}_{\alpha}\subseteq W$ such that each ${\bf x}_{\alpha}$ is
$\mathcal{G}$-minimal and ${\bf x}_{\alpha}\rightarrow{\bf x}$.

Now fix a $T^{\infty}$-minimal point ${\bf y}=(y_{i})_{i\in I}\in
M_{\infty}(Y,\mathcal{A})$. Since $(M_{\infty}(Y,\mathcal{A}),\mathcal{G})$ is
minimal, ${\bf y}$ is also $\mathcal{G}$-minimal. Thus there exists a
$\mathcal{G}$-minimal point ${\bf x}^{\prime}=(x^{\prime}_{i})_{i\in
I}\in(\psi^{\infty})^{-1}({\bf y})$ by Lemma 2.2. For all $i\in I$, since
$x_{i}^{\prime}\in{\bf M}$, there is some minimal idempotent $u_{i}\in{\bf M}$
such that $u_{i}x_{i}^{\prime}=x_{i}^{\prime}$. Thus

${\bf x^{\prime}}\in\prod_{i\in I}u_{i}\psi^{-1}(y_{i}),$

since $x_{i}^{\prime}\in\psi^{-1}(y_{i})$ for each $i\in I$. We remark that if
$\psi$ is distal, then $u_{i}\psi^{-1}(y_{i})=\psi^{-1}(y_{i})$ for each $i\in
I$.

Step 3: For each ${\bf\widetilde{x}}\in\prod_{i\in I}u_{i}\psi^{-1}(y_{i})$,
there is a sequence $\\{{\bf z}_{k}\\}_{k\in\mathbb{N}}\subseteq\prod_{i\in
I}u_{i}\psi^{-1}(y_{i})$ such that ${\bf z}_{k}$ is $\mathcal{G}$-minimal and
${\bf z}_{k}\rightarrow{\bf\widetilde{x}},k\to\infty$. In particular,
$\mathcal{G}$-minimal points are dense in $\prod_{i\in
I}u_{i}\psi^{-1}(y_{i})$.

First we show the case for $\mathcal{A}=\\{p\\}$ with $\deg p\geq 2$. In this
case, $I={\mathbb{Z}}$, and the proof is clearer than the general case. Then
we give the proof of the general case.

Let
${\bf\widetilde{x}}=(\widetilde{x}_{i})_{i=-\infty}^{\infty}\in\prod_{i=-\infty}^{\infty}u_{i}\psi^{-1}(y_{i})$.
For each $i\in{\mathbb{Z}}$, $x_{i}^{\prime},\ \widetilde{x}_{i}\in u_{i}{\bf
M}$. Since $\psi$ is regular, there is some $\phi_{i}\in{\rm Aut}_{\psi}({\bf
M})$ such that $\widetilde{x}_{i}=\phi_{i}(x_{i}^{\prime})$. For each
$k\in\mathbb{N}$, let

$\begin{split}\Phi_{k}&=(\phi_{-k}\times\phi_{-k+1}\times\cdots\times\phi_{k})^{\infty}\\\
&=\cdots\times(\phi_{-k}\times\phi_{-k+1}\times\ldots\times\phi_{k})\times(\phi_{-k}\times\phi_{-k+1}\times\ldots\times\phi_{k})\times\cdots.\end{split}$

Then

$\begin{split}\Phi_{k}({\bf
x^{\prime}})&=(\ldots,\phi_{k}(x_{-k-1}^{\prime}),\widetilde{{x}}_{-k},\ldots,\underset{\bullet}{\widetilde{x_{0}}},\ldots,\widetilde{{x}_{k}},\phi_{-k}(x_{k+1}^{\prime}),\ldots,\phi_{k}(x_{3k+1}^{\prime}),\phi_{-k}(x_{3k+2}^{\prime}),\ldots)\\\
&\rightarrow\widetilde{{\bf x}},\ k\to\infty.\end{split}$

Since $\phi_{j}\in{\rm Aut}_{\psi}({\bf M})$ for each $j\in\mathbb{Z}$, we
have that $\phi_{j}\psi^{-1}(y_{i})=\psi^{-1}(y_{i})$ and

$\phi_{j}(u_{i}\psi^{-1}(y_{i}))=u_{i}\phi_{j}(\psi^{-1}(y_{i}))=u_{i}\psi^{-1}(y_{i})$

for each $i\in\mathbb{Z}$ and hence $\Phi_{k}({\bf
x^{\prime}})\in\prod_{i=-\infty}^{\infty}u_{i}\psi^{-1}(y_{i})\subseteq W.$

It is left to show that $\Phi_{k}({\bf x^{\prime}})$ is $\mathcal{G}$-minimal.
For all $k\in\mathbb{N}$, by Lemma 2.1, ${\bf x^{\prime}}$ is a $\langle
T^{\infty},\sigma^{2k+1}\rangle$-minimal point as ${\bf x^{\prime}}$ is
$\mathcal{G}$-minimal. Note that

$\sigma^{2k+1}\Phi_{k}=\Phi_{k}\sigma^{2k+1}.$

It follows that $\Phi_{k}({\bf x^{\prime}})$ is a $\langle
T^{\infty},\sigma^{2k+1}\rangle$-minimal point. Again by Lemma 2.1, it is
$\mathcal{G}$-minimal.

Now we show the general case. Now

$\small{\bf y}=(y_{i})_{i\in
I}=\Big{(}(y_{1},\ldots,y_{s}),\big{(}(y^{(1)}_{n},\cdots,y^{(d-s)}_{n})\big{)}_{n\in{\mathbb{Z}}}\Big{)},$
$\small{\bf\widetilde{x}}=(\widetilde{x}_{i})_{i\in
I}=\Big{(}(\widetilde{x}_{1},\ldots,\widetilde{x}_{s}),\big{(}(\widetilde{x}^{(1)}_{n},\cdots,\widetilde{x}^{(d-s)}_{n})\big{)}_{n\in{\mathbb{Z}}}\Big{)},$
$\small{\bf
x^{\prime}}=\Big{(}(x^{\prime}_{1},\ldots,x^{\prime}_{s}),\big{(}(x^{\prime(1)}_{n},\cdots,x^{\prime(d-s)}_{n})\big{)}_{n\in{\mathbb{Z}}}\Big{)}$

and

$\small\prod_{i\in
I}u_{i}\psi^{-1}(y_{i})=\prod_{i=1}^{s}u_{i}\psi^{-1}(y_{i})\times\prod_{n=-\infty}^{\infty}\big{(}u_{n}^{(1)}\psi^{-1}(y_{n}^{(1)})\times\cdots\times
u_{n}^{(d-s)}\psi^{-1}(y_{n}^{(d-s)})\big{)}.$

For each $i\in I$, $x_{i}^{\prime},\ \widetilde{x}_{i}\in u_{i}{\bf M}$. Since
$\psi$ is regular, there is some $\phi_{i}\in{\rm Aut}_{\psi}({\bf M})$ such
that $\widetilde{x}_{i}=\phi_{i}(x_{i}^{\prime})$. For each $k\in\mathbb{N}$,
let

$\Phi_{k}=(\phi_{1}\times\cdots\times\phi_{s})\times\Big{(}\prod_{j=-k}^{k}\big{(}\phi_{j}^{(1)}\times\cdots\times\phi_{j}^{(d-s)}\big{)}\Big{)}^{\infty},$

where

$\begin{split}\small&\Big{(}\prod_{j=-k}^{k}\big{(}\phi_{j}^{(1)}\times\cdots\times\phi_{j}^{(d-s)}\big{)}\Big{)}^{\infty}\\\
&=\cdots\times\prod_{j=-k}^{k}\big{(}\phi_{j}^{(1)}\times\cdots\times\phi_{j}^{(d-s)}\big{)}\times\prod_{j=-k}^{k}\big{(}\phi_{j}^{(1)}\times\cdots\times\phi_{j}^{(d-s)}\big{)}\times\cdots.\end{split}$

Note since $\phi_{i}\in{\rm Aut}_{\psi}({\bf M})$,
$\phi_{i}(u_{j}\psi^{-1}(y_{j}))=u_{j}\psi^{-1}(y_{j})$ for all $i,j\in I$.
Thus

$\begin{split}\Phi_{k}({\bf
x^{\prime}})&=(\phi_{1}\times\cdots\times\phi_{s})\times\Big{(}\prod_{j=1}^{k}\big{(}\phi_{j}^{(1)}\times\cdots\times\phi_{j}^{(d-s)}\big{)}\Big{)}^{\infty}(\bf
x^{\prime})\\\
&=\Big{(}(\widetilde{x}_{1},\ldots,\widetilde{x}_{s}),\big{(}(\phi_{j}^{(1)}({x_{j}^{\prime}}^{(1)}),\ldots,\phi_{j}^{(d-s)}({x_{j}^{\prime}}^{(d-s)}))\big{)}_{j=-\infty}^{-k-1}\\\
&\quad\quad\quad\quad\big{(}(\widetilde{x}^{(1)}_{j},\ldots,\widetilde{x}^{(d-s)}_{j})\big{)}_{j=1}^{k},\big{(}(\phi_{j}^{(1)}({x_{j}^{\prime}}^{(1)}),\ldots,\phi_{j}^{(d-s)}({x_{j}^{\prime}}^{(d-s)}))\big{)}_{j=k+1}^{\infty}\Big{)}\\\
&\in u_{1}\psi^{-1}(y_{1})\times\cdots\times
u_{s}\psi^{-1}(y_{s})\times\prod_{n=-\infty}^{\infty}\big{(}u_{n}^{(1)}\psi^{-1}(y_{n}^{(1)})\times\cdots\times
u_{n}^{(d-s)}\psi^{-1}(y_{n}^{(d-s)})\big{)}\end{split}$

and hence

$\Phi_{k}({\bf x^{\prime}})\rightarrow{\bf\widetilde{x}},\ k\to\infty.$

Now we show that $\Phi_{k}({\bf x^{\prime}})$ is $\langle
T^{\infty},\sigma\rangle$-minimal. This follows by the same arguments as the
simple case we just proved. This ends the Step 3.

Step 4: The final arguments of the proof using the RIC property.

Finally, we show that for each ${\bf x}=(x_{i})_{i\in
I}\in(\psi^{\infty})^{-1}({\bf y})$, there is a net $\\{{\bf
x}_{\alpha}\\}_{\alpha}\subseteq W$ such that each ${\bf x}_{\alpha}$ is
$\mathcal{G}$-minimal and ${\bf x}_{\alpha}\rightarrow{\bf x}$.

Since ${\bf y}=(y_{i})_{i\in I}\in M_{\infty}(Y)$ is $T^{\infty}$-minimal,
there is some minimal idempotent $v\in{\bf M}$ such that $v{\bf y}={\bf y}$.
Thus $vy_{i}=y_{i}$ for all $i\in I$. As $\psi$ is RIC, we have (see (7.1))

$v\circ\big{(}u_{j}\psi^{-1}(y_{i})\big{)}=\psi^{-1}(vy_{i})=\psi^{-1}(y_{i}),\
\forall i\in I.$

It follows that

(7.2) $v\circ\Big{(}\prod_{i\in I}u_{i}\psi^{-1}(y_{i})\Big{)}=\prod_{i\in
I}\psi^{-1}(y_{i})=(\psi^{\infty})^{-1}({\bf y}).$

Since ${\bf x}=(x_{i})_{i\in I}\in(\psi^{\infty})^{-1}({\bf y})$, by (7.2),
for any net $\\{n_{\alpha}\\}_{\alpha}\subseteq\mathbb{Z}$ converging to $v$,
there is some ${\bf z}_{\alpha}\in\prod_{i\in I}u_{i}\psi^{-1}(y_{i})$ such
that

$(T^{\infty})^{n_{\alpha}}{\bf z}_{\alpha}\rightarrow{\bf x}.$

By the definition it is easy to see $p\circ A=p\circ B,p\in{\bf M}$, whenever
$A$ is dense in $B$. By Step 3, we may assume that each ${\bf z}_{\alpha}$ is
$\mathcal{G}$-minimal. Let ${\bf x}_{\alpha}=(T^{\infty})^{n_{\alpha}}{\bf
z}_{\alpha}$. Then ${\bf x}_{\alpha}$ is $\mathcal{G}$-minimal and ${\bf
x}_{\alpha}\rightarrow{\bf x}$. The proof is complete. ∎

###  7.4. The proof of Theorem E

Theorem E follows from Lemma 4.6, Theorem 4.10 and Theorem 7.8.

###  7.5. Proof of Corollary F

In this subsection we show Corollary F.

###### Proof of Corollary F.

We will show the corollary for $N_{\infty}(X,\mathcal{A})$, and then we have
the same result for $M_{\infty}(X,\mathcal{A})$ by Lemma 4.2. Without loss of
generality, we assume that $\mathcal{A}$ satisfies $(\spadesuit)$. Recall that
for each $x\in X$

$\omega_{x}^{\mathcal{A}}=(T^{\vec{p}(n)}x^{\otimes
d})_{n\in\mathbb{Z}}=\big{(}(T^{p_{1}(n)}x,T^{p_{2}(n)}x,\ldots,T^{p_{d}(n)}x)\big{)}_{n\in\mathbb{Z}}\in(X^{d})^{{\mathbb{Z}}},$

and

$N_{\infty}(X,\mathcal{A})=\overline{\bigcup\\{\mathcal{O}(\omega_{x}^{\mathcal{A}},\sigma):x\in
X\\}}\subseteq(X^{d})^{{\mathbb{Z}}}.$

(1) First we assume that $(X,T)$ has dense $T$-minimal points and $x\in X$.
Then there are $T$-minimal points $x_{i}\in X$ such that $x_{i}\rightarrow
x,i\to\infty$. It is clear that
$\omega_{x_{i}}^{\mathcal{A}}\rightarrow\omega_{x}^{\mathcal{A}},i\to\infty$
and for each $i\in\mathbb{N}$, $(\overline{\mathcal{O}}(x_{i},T),T)$ is a
minimal t.d.s. By Lemma 4.6, Theorem 4.10 and Theorem 7.8,
$\big{(}N_{\infty}(\overline{\mathcal{O}}(x_{i},T),\mathcal{A}),\mathcal{G}\big{)}$
is an $M$-system for each $i\in\mathbb{Z}$, and thus
$\omega_{x_{i}}^{\mathcal{A}}$ is the limit of $\mathcal{G}$-minimal points
which implies that $\omega_{x}^{\mathcal{A}}$ is the limit of
$\mathcal{G}$-minimal points. Then by the definition, the set of
$\mathcal{G}$-minimal points is dense in $N_{\infty}(X,\mathcal{A}).$

Conversely, assume that $N_{\infty}(X,\mathcal{A})$ has dense
$\mathcal{G}$-minimal points. Applying Lemma 2.2 to the orbit closures of the
minimal points, we get that the $T^{\infty}$-minimal points are dense in
$N_{\infty}(X,\mathcal{A})$. It follows by projection to $0$-th coordinate
that the set of $T$-minimal points is dense in $X$.

(2) Now we assume that $(X,T)$ is an $M$-system. Then for each transitive
point $x$ of $(X,T)$, $\omega_{x}^{\mathcal{A}}$ is a transitive point of
$(N_{\infty}(X,\mathcal{A}),\mathcal{G})$. By (1) the set of
$\mathcal{G}$-minimal points is dense in $N_{\infty}(X,\mathcal{A})$, and it
follows that $(N_{\infty}(X,\mathcal{A}),\mathcal{G})$ is an $M$-system.

Conversely, suppose that $(N_{\infty}(X,\mathcal{A}),\mathcal{G})$ is an
$M$-system. By (1) the set of $T$-minimal points is dense in $X$. To prove the
transitivity of $(X,T)$ we assume that ${\bf x}$ is a transitive point of
$N_{\infty}(X,\mathcal{A})$. There are a sequence
$\\{x_{i}\\}_{i\in\mathbb{N}}$ of $X$ and a sequence
$\\{n_{i}\\}_{i=1}^{\infty}$ of $\mathbb{Z}$ such that
$\sigma^{n_{i}}\omega_{x_{i}}^{\mathcal{A}}\rightarrow{\bf x},i\to\infty.$
Then it is easy to verify that for each $\epsilon>0$, there is $i=i(\epsilon)$
such that
${\mathcal{O}}(\sigma^{n_{i}}\omega_{x_{i}}^{\mathcal{A}},\mathcal{G})$ is
$\epsilon$-dense in $N_{\infty}(X,\mathcal{A})$.444Recall the definition of
$\epsilon$-dense subsets: if $B$ is a subset of a metric space $S$, then $B$
is $\epsilon$-dense in $S$ for given $\epsilon>0$ if for any $s\in S$, there
is $b\in B$ such that the distance between $s$ and $b$ is less than
$\epsilon$. Hence, ${\mathcal{O}}(x_{i},T)$ is $\epsilon$-dense in $X$ since
the $0$-th coordinate of
$\sigma^{n}(T^{\infty})^{m}\big{(}\sigma^{n_{i}}\omega_{x_{i}}^{\mathcal{A}}\big{)}$
is $(T^{m+p_{1}(n+n_{i})}x_{i},\ldots,T^{m+p_{1}(n+n_{i})}x_{i})$ for each
$n,m\in\mathbb{Z}$.

To sum up, for each $\epsilon>0$, there is some $x_{i}\in X$ such that
${\mathcal{O}}(x_{i},T)$ is $\epsilon$-dense in $X$, which implies that
$(X,T)$ is transitive. ∎

##  8\. Combinatoric consequences

In this section, we give some combinatoric consequences. Theorem A is a
corollary of the following theorem.

######  Theorem 8.1.

Let $S\subseteq\mathbb{Z}$ and $d\in\mathbb{N}$, and let $p_{i}$ be an
integral polynomial with $p_{i}(0)=0$, $1\leq i\leq d$. If
$S\in\mathcal{F}_{ps}(\mathbb{Z})$, then there is
$A\in\mathcal{F}_{ps}(\mathbb{Z}^{2})$ such that for any $N\in\mathbb{N}$,
there is some ${\bf m}_{N}\in\mathbb{Z}^{2}$ satisfying

${\bf
m}_{N}+(A\cap[-N,N]^{2})\subseteq\\{(m,n)\in\mathbb{Z}^{2}:m+p_{1}(n),m+p_{2}(n),\ldots,m+p_{d}(n)\in
S\\}.$

In particular,
$\\{(m,n)\in\mathbb{Z}^{2}:m+p_{1}(n),m+p_{2}(n),\ldots,m+p_{d}(n)\in
S\\}\in\mathcal{F}_{ps}(\mathbb{Z}^{2}).$

###### Proof.

Let $\Sigma_{2}=\\{0,1\\}^{\mathbb{Z}}$ and
$\sigma:\Sigma_{2}\rightarrow\Sigma_{2}$ be the shift, that is, for ${\bf
x}=(x_{n})_{n\in\mathbb{Z}}\in\Sigma_{2}$, $(\sigma{\bf
x})_{n}=x_{n+1},\forall n\in\mathbb{Z}$. Let the metric $\rho$ on $\Sigma_{2}$
be defined as follows: for ${\bf x}=(x_{n})_{n\in\mathbb{Z}},{\bf
y}=(y_{n})_{n\in\mathbb{Z}}$,

$\rho({\bf x},{\bf y})=0\ \text{if}\ {\bf x}={\bf y};\ \ \rho({\bf x},{\bf
y})=\frac{1}{k+1},\ \text{if}\ {\bf x}\not={\bf y}\ \text{and}\
k=\min\\{|i|:x_{i}\neq y_{i}\\}.$

Let ${\bf x}=1_{S}\in\Sigma_{2}$ be the indicator function, i.e., $x_{n}=1$ if
and only if $n\in S$. Let $Z=\overline{\mathcal{O}}({\bf x},\sigma)$. Since
$S$ is piecewise syndetic, $Z$ contains a point $1_{S^{\prime}}$ with
$S^{\prime}$ actually syndetic. Since $S^{\prime}$ is syndetic, no limit point
of translates of $1_{S^{\prime}}$ is equal to ${\bf 0}=(\ldots 000\ldots)$.
Let $X$ be a minimal subset in orbit closure of $1_{S^{\prime}}$. Then for any
${\bf y}\in X$, ${\bf y}\neq{\bf 0}$.

Let $U=\\{{\bf z}=(z_{n})_{n\in\mathbb{Z}}\in X:z_{0}=1\\}$. By the choice of
$X$, $U$ is a non-empty open subset of $X$. By Theorem E, there is some ${\bf
y}\in U$ such that

$A=N^{\mathbb{Z}^{2}}_{\\{p_{1},\ldots,p_{d}\\}}({\bf
y},U)=\\{(m,n)\in\mathbb{Z}^{2}:\sigma^{m+p_{1}(n)}{\bf y}\in
U,\ldots,\sigma^{m+p_{d}(n)}{\bf y}\in U\\}$

is piecewise syndetic in $\mathbb{Z}^{2}$. That is,

$A=\\{(m,n)\in\mathbb{Z}^{2}:y_{0}=y_{m+p_{1}(n)}=y_{m+p_{2}(n)}=\ldots=y_{m+p_{d}(n)}=1\\}\in\mathcal{F}_{ps}(\mathbb{Z}^{2}).$

Let

$B=\\{(m,n)\in\mathbb{Z}^{2}:m+p_{1}(n),m+p_{2}(n),\ldots,m+p_{d}(n)\in S\\}.$

We show that for any $N\in\mathbb{N}$, there is some ${\bf
m}_{N}\in\mathbb{Z}^{2}$ such that

${\bf m}_{N}+(A\cap[-N,N]^{2})\subseteq B,$

then by Lemma 6.2, $B$ is also a piecewise syndetic subset in
$\mathbb{Z}^{2}$.

Let $N\in\mathbb{N}$ and set

$A\cap[-N,N]^{2}=\\{(m_{i},n_{i})\\}_{i=1}^{l_{N}}.$

Then

$y_{m_{i}+p_{j}(n_{i})}=y_{0}=1,\ \forall 1\leq j\leq d,\ 1\leq i\leq l_{N}.$

Note that ${\bf y}\in X\subseteq Z=\overline{\mathcal{O}}({\bf x},\sigma)$.
Choose $m\in\mathbb{Z}$ such that $\sigma^{m}{\bf x}$ is close to ${\bf y}$
such that $\rho(\sigma^{m}{\bf x},{\bf
y})<\frac{1}{\max\\{|m_{i}|+|p_{j}(n_{i})|:1\leq j\leq d,1\leq i\leq
l_{N}\\}}$. It follows that

$x_{m+k}=y_{k},\ \forall|k|\leq\max\\{|m_{i}|+|p_{j}(n_{i})|:1\leq j\leq
d,1\leq i\leq l_{N}\\}.$

In particular,

$x_{m+m_{i}+p_{j}(n_{i})}=y_{m_{i}+p_{j}(n_{i})}=y_{0}=1,\ \forall 1\leq j\leq
d,\ 1\leq i\leq l_{N}.$

So we have

$m+m_{i}+p_{j}(n_{i})\in S,\ \forall 1\leq j\leq d,\ 1\leq i\leq l_{N},$

which implies that

${\bf m}_{N}+(A\cap[-N,N]^{2})\subseteq B,$

where ${\bf m}_{N}=(m,0)$. Thus $B$ is piecewise syndetic in $\mathbb{Z}^{2}$.
The proof is complete. ∎

As a corollary, we have Furstenberg-Glasner’s theorem which was mentioned in
the introduction. With the almost same proof of Theorem 8.1 using Theorem D
instead of Theorem E, we have the following result.

######  Theorem 8.2.

Let $S\subseteq\mathbb{N}$ and $d\in\mathbb{N}$, and let $p_{i}$ be an
integral polynomial with $p_{i}(0)=0$ for each $1\leq i\leq d$. If
$S\in\mathcal{F}_{ps}(\mathbb{Z})$, then there is
$A\in\mathcal{F}_{ps}(\mathbb{Z})$ such that for any $N\in\mathbb{N}$, there
is some ${m}_{N}\in\mathbb{Z}$ satisfying

$A\cap[-N,N]\subseteq\\{n\in\mathbb{Z}:m_{N}+p_{1}(n),m_{N}+p_{2}(n),\ldots,m_{N}+p_{d}(n)\in
S\\}.$

##  9\. Some open questions

We have some natural questions for the further investigation.

######  Question 3.

Assume that $k,d\in\mathbb{N}$, $S\subset\mathbb{Z}^{k}$ is piecewise
syndetic, and $p_{i,j}$ is an integral polynomial with $p_{i,j}(0)=0$ for each
$1\leq i\leq d$ and $1\leq j\leq k$. Consider the set

(9.1) $\displaystyle\begin{aligned}
&\\{(m_{1},\ldots,m_{k},n)\in\mathbb{Z}^{k+1}:(m_{1}+p_{1,1}(n),\ldots,m_{k}+p_{1,k}(n))\in
S,\\\ &\quad\quad\quad\ldots,(m_{1}+p_{d,1}(n),\ldots,m_{k}+p_{d,k}(n))\in
S\\}.\end{aligned}$

Is it true that the (9.1) is piecewise syndetic in $\mathbb{Z}^{k+1}$?

The corresponding dynamical version is the following

######  Question 4.

Let $k,d\in\mathbb{N}$ and $(X,\langle T_{1},\ldots,T_{k}\rangle)$ be a
minimal t.d.s., where $T_{i}:X\rightarrow X$ is a homeomorphism and
$T_{i}\circ T_{j}=T_{j}\circ T_{i}$ for $1\leq i,j\leq k$. Let $p_{i,j}$ be an
integral polynomial with $p_{i,j}(0)=0$ for each $1\leq i\leq d$ and $1\leq
j\leq k$.

Is it true that for each $x\in X$ and each neighbourhood $U$ of $x$

(9.2) $\displaystyle\begin{aligned}
&\\{(m_{1},\ldots,m_{k},n)\in\mathbb{Z}^{d+1}:T_{1}^{m_{1}+p_{1,1}(n)}\cdots
T_{k}^{m_{k}+p_{1,k}(n)}x\in U,\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\ldots,T_{1}^{m_{1}+p_{d,1}(n)}\cdots
T_{k}^{m_{k}+p_{d,k}(n)}x\in U\\}\end{aligned}$

is piecewise syndetic in $\mathbb{Z}^{k+1}$?

The above two questions can also be asked for a nilpotent group.

We remark that for $k=1$ the two questions have been answered in the current
paper. Moreover, it seems that the tools in the current paper can not be
applied directly to the above questions, namely one needs to work out some
kind “saturation theorem” for commuting homeomorphisms.

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