# Polynomial Furstenberg joinings and its applications

Wen Huang  ,  Song Shao  and  Xiangdong Ye  School of Mathematical Sciences,
University of Science and Technology of China, Hefei, Anhui, 230026, P.R.
China<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, a polynomial version of Furstenberg joining is introduced and
its structure is investigated. Particularly, it is shown that if all
polynomials are non-linear, then almost every ergodic component of the joining
is a direct product of an infinity-step pro-nilsystem and a Bernoulli system.

As applications, some new convergence theorems are obtained. Particularly, it
is proved that if $T$ and $S$ are ergodic measure preserving transformations
on a probability space $(X,\mathcal{X},\mu)$ and $T$ has zero entropy, then
for all $c_{i}\in\mathbb{Z}\setminus\\{0\\}$, all integral polynomials $p_{j}$
with $\deg{p_{j}}\geq 2$, and for all $f_{i},g_{j}\in L^{\infty}(X,\mu)$,
$1\leq i\leq m$ and $1\leq j\leq d$,

$\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}f_{1}(T^{c_{1}n}x)\cdots
f_{m}(T^{c_{m}n}x)\cdot g_{1}(S^{p_{1}(n)}x)\cdots g_{d}(S^{p_{d}(n)}x),$

exists in $L^{2}(X,\mu)$, which extends the recent result by Host and
Frantzikinakis.

Moreover, it is shown that for an ergodic measure-preserving system
$(X,\mathcal{X},\mu,T)$, a non-linear integral polynomial $p$ and $f\in
L^{\infty}(X,\mu)$, the Furstenberg systems of
$\big{(}f(T^{p(n)})x\big{)}_{n\in\mathbb{Z}}$ are ergodic and isomorphic to
direct products of infinite-step pro-nilsystems and Bernoulli systems for
almost every $x\in X$, which answers a problem by Frantzikinakis.

###### 2010 Mathematics Subject Classification:

Primary: 37A05; 37A30

This research is supported by National Natural Science Foundation of China
(12031019, 12090012, 11971455, 11731003).

###### Contents

  1. 1 Introduction
    1. 1.1 Furstenberg Joining
    2. 1.2 Polynomial version of the Furstenberg joining
    3. 1.3 Main results
    4. 1.4 Organization of the paper
  2. 2 Preliminaries
    1. 2.1 Some notion in ergodic theory and topological dynamics
      1. 2.1.1 Measurable systems
      2. 2.1.2 Topological dynamical systems
      3. 2.1.3 ${\mathcal{M}}(X)$ and ${\mathcal{M}}_{T}(X)$
    2. 2.2 Topological models
      1. 2.2.1 Topological models
      2. 2.2.2 Weiss’s Theorem
    3. 2.3 The Rohlin skew-product theorem
    4. 2.4 Conditional expectations
      1. 2.4.1 Conditional expectations
      2. 2.4.2
    5. 2.5 Joinings
      1. 2.5.1 Joinings and conditional product measures
      2. 2.5.2 Relatively independent joining
    6. 2.6 Furstenberg Joinings
      1. 2.6.1 Furstenberg Joinings
      2. 2.6.2 The ergodic decomposition of $\mu_{\vec{a}}^{(d)}$
    7. 2.7 Host-Kra seminorms
      1. 2.7.1 The measure $\mu^{[k]}$
      2. 2.7.2 Host-Kra seminorms
      3. 2.7.3
    8. 2.8 Pro-nilsystems
      1. 2.8.1 Nilpotent groups
      2. 2.8.2 Nilsystems
      3. 2.8.3
      4. 2.8.4 Nilfactors
      5. 2.8.5 $\infty$-step pro-nilsystems
    9. 2.9 Topological pro-nilsystems
      1. 2.9.1
      2. 2.9.2
    10. 2.10 Nilsequencs
    11. 2.11 Characteristic factors
    12. 2.12 Furstenberg’s Theorem and Bourgain’s Theorem
      1. 2.12.1
      2. 2.12.2
    13. 2.13 Multiple polynomial ergodic theorems
  3. 3 Polynomial Furstenberg joinings
    1. 3.1 Standard measures on product spaces
      1. 3.1.1
      2. 3.1.2
      3. 3.1.3
    2. 3.2 A self-joining $\mu_{\mathcal{A}}^{(\infty)}$
      1. 3.2.1 Some notations
      2. 3.2.2 An $\infty$-joining with respect to $\mathcal{A}$
      3. 3.2.3
    3. 3.3 Some special cases
      1. 3.3.1 The case when $\mathcal{A}=\\{p\\}$ with $p(n)=n$
      2. 3.3.2 The case when $\mathcal{A}=\\{p_{1},p_{2},\ldots,p_{d}\\}$ with $p_{i}(n)=a_{i}n$, $1\leq i\leq d$
      3. 3.3.3 Weakly mixing m.p.s.
    4. 3.4 An equivalent way to see $\mu^{(\infty)}_{\mathcal{A}}$
      1. 3.4.1
      2. 3.4.2 Definition of $\widetilde{\mu}^{(\infty)}_{\mathcal{A}}$
      3. 3.4.3 $\mu^{(\infty)}_{\mathcal{A}}$ and $\widetilde{\mu}^{(\infty)}_{\mathcal{A}}$ are isomorphic
      4. 3.4.4 Compare $\mu^{(\infty)}_{\mathcal{A}}$ with $\widetilde{\mu}^{(\infty)}_{\mathcal{A}}$
      5. 3.4.5
    5. 3.5 Topological correspondences
      1. 3.5.1 $N_{\infty}(X,\mathcal{A})$ and $M_{\infty}(X,\mathcal{A})$
      2. 3.5.2
  4. 4 The $\sigma$-algebra of invariant sets of $((X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}},\sigma)$
    1. 4.1 Multiple ergodic averages along polynomials of several variable
    2. 4.2 Condition $(\spadesuit)$
    3. 4.3 Some lemmas
    4. 4.4 The $\sigma$-algebra of invariant sets of $((X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}},\sigma)$
  5. 5 Polynomial orbits in nilsystems
    1. 5.1 The measure $\mu_{\mathcal{A}}^{(\infty)}$ on nilsystems
      1. 5.1.1
      2. 5.1.2
      3. 5.1.3
    2. 5.2 The ergodic decomposition
    3. 5.3 The ergodic decomposition of $\nu_{\mathcal{A}}^{(\infty)}$ under $\sigma$
      1. 5.3.1
      2. 5.3.2
  6. 6 The ergodic decomposition of $\mu^{(\infty)}_{\mathcal{A}}$
    1. 6.1 The ergodic decomposition theorem of $\mu_{\mathcal{A}}^{(\infty)}$ under $\sigma$
      1. 6.1.1
    2. 6.2 The structure of $\big{(}X^{s}\times(X^{d-s})^{\mathbb{Z}},\mathcal{X}^{s}\times(\mathcal{X}^{d-s})^{\mathbb{Z}},\widetilde{\mu}^{(\infty)}_{\mathcal{A},x},\widetilde{\sigma}\big{)}$
  7. 7 Convergence theorems
    1. 7.1 Equivalent conditions for pointwise convergence of multiple ergodic averages
    2. 7.2 Mean convergence
      1. 7.2.1 Pinsker factor
      2. 7.2.2 A convergence theorem
      3. 7.2.3
    3. 7.3 Pointwise convergence
      1. 7.3.1 The main result
      2. 7.3.2 Proof of Theorem 7.11
  8. 8 Relations with Furstenberg systems
    1. 8.1 Furstenberg systems of sequences
      1. 8.1.1
      2. 8.1.2 Furstenberg system associated with $z$
      3. 8.1.3 Problems about Furstenberg systems
    2. 8.2 A corollary of Thouvenot’s and Host-Kra’s theorem
    3. 8.3 A lemma about topological models
    4. 8.4 Relation between $\infty$-joining and Furstenberg systems of sequences
  9. 9 Questions
    1. 9.1 $\mu^{(\infty)}_{\mathcal{A}}$ for general scheme $\mathcal{A}$
    2. 9.2
  10. A From continuous functions to measurable functions
    1. A.1 From continuous functions to measurable ones
    2. A.2 Proof of Proposition 3.4
  11. B A weighted ergodic average and proof of Theorem 7.6
  12. C A lemma about topological models

section

##  1\. Introduction

In the article, the set of integers (resp. natural numbers $\\{1,2,\ldots\\}$)
is denoted by $\mathbb{Z}$ (resp. $\mathbb{N}$). An integral polynomial is a
polynomial taking integer values at the integers. Two polynomials $p(n)$ and
$q(n)$ are essentially distinct if $p(n)-q(n)$ is not a constant function. If
$n<m\in\mathbb{Z}$, then $[n,m]$ denotes the set $\\{n,n+1,\ldots,m\\}$.

Throughout this paper, all probability spaces $(X,\mathcal{X},\mu)$ considered
are Lebesgue, meaning $X$ can be given the structure of a compact metric space
and $\mathcal{X}$ is its Borel $\sigma$-algebra. By a measure-preserving
system (m.p.s. for short), we mean a quadruple $(X,\mathcal{X},\mu,T)$, where
$(X,\mathcal{X},\mu)$ is a Lebesgue probability space and $T:X\rightarrow X$
is an invertible measure preserving transformation.

###  1.1. Furstenberg Joining

The notion of joinings was introduced by Furstenberg in his seminal paper
[15], and now the theory of joinings is an important part of ergodic theory
[19]. Some joinings play a key role in the study of multiple recurrence and
multiple ergodic averages. To give a dynamical proof of the well-known theorem
of Szemerédi [39], Furstenberg proved the following multiple recurrence
theorem [16], which says that for a m.p.s. $(X,\mathcal{X},\mu,T)$,
$d\in\mathbb{N}$ and $A\in\mathcal{X}$ with positive measure, there is some
$n\in\mathbb{N}$ such that

$\mu(A\cap T^{-n}A\cap T^{-2n}A\cap\cdots\cap T^{-dn}A)>0.$

In the same paper, Furstenberg introduced the following measure (now called
Furstenberg joining):

(1.1)
$\mu^{(d)}(\prod_{i=1}^{d}A_{i})=\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\mu(T^{-n}A_{1}\cap
T^{-2n}A_{2}\cap\cdots\cap T^{-dn}A_{d}),$

for all $A_{1},\ldots,A_{d}\in\mathcal{X}$. Note that the existence of the
limit was proved in [20, 43], and in [16] $\mu^{(d)}$ was in fact defined as a
limit of a subsequence. It is easy to verify that $\mu^{(d)}$ is invariant
under the actions of $T^{(d)}=T\times\cdots\times T$ ($d$ times) and
$\tau_{d}=T\times T^{2}\times\cdots\times T^{d}$. Thus we have a m.p.s.
$(X^{d},\mathcal{X}^{d},\mu^{(d)},\tau_{d})$.

In [16], it was shown that $\mu^{(d)}$ is a conditional product measure in
$X^{d}$ relative to $(Y_{d-1},\mu_{d-1})$, where $\\{(Y_{n},\mu_{n})\\}_{n}$
is the distal sequence of $(X,\mathcal{X},\mu,T)$ ([17]). That is,

(1.2)
$\mu^{(d)}=\int_{Y_{d-1}^{d}}\mu_{y_{1}}\times\mu_{y_{2}}\times\cdots\times\mu_{y_{d}}d\mu_{d-1}^{(d)}(y_{1},y_{2},\ldots,y_{d}),$

where $\mu=\int\mu_{y}\ d\mu_{d-1}(y)$ is the disintegration of $\mu$ with
respect to $\mu_{d-1}$, and $\mu_{d-1}^{(d)}$ is defined as (1.1) related to
$Y_{d-1}$. This leads to the notion of “characteristic factors”, which was
first explicitly defined by Furstenberg and Weiss in [18]. Let
$(X,\mathcal{X},\mu,T)$ be a m.p.s. and $(Y,\mathcal{Y},\nu,T)$ be a factor of
$X$. For $d\in\mathbb{N}$, we say that $Y$ is a characteristic factor of $X$
for $\tau_{d}$ if for all $f_{1},\ldots,f_{d}\in L^{\infty}(X,\mu)$,

$\begin{split}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i=1}^{d}f_{i}(T^{in}x)-\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i=1}^{d}\mathbb{E}(f_{i}|\mathcal{Y})(T^{in}x)\to
0,\ N\to\infty\end{split}$

in $L^{2}(X,\mu)$. By (1.2) it is easy to see $Y_{d-1}$ is a characteristic
factor of $X$ for $\tau_{d}$. This enables one to give a reduction of the
problem on the multiple recurrence to its distal factors.

In [20] and in [43] the authors improved the above result significantly by
showing that for an ergodic m.p.s. $(X,\mathcal{X},\mu,T)$, its $(d-1)$-step
pro-nilfactor is a characteristic factor for $\tau_{d}$, and using this result
the authors proved that

$\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}f_{1}(T^{n}x)f_{2}(T^{2n}x)\cdots
f_{d}(T^{dn}x)$

exists in $L^{2}(X,\mu)$.

###  1.2. Polynomial version of the Furstenberg joining

Let $(X,\mathcal{X},\mu,T)$ be a m.p.s. and let
$\mathcal{A}=\\{p_{1},p_{2},\ldots,p_{d}\\}$ be a family of non-constant
integral polynomials. There is a natural way to generalize the Furstenberg
joining to the polynomial setting as follows:

(1.3)
$\mu_{\mathcal{A}}^{(d)}(\prod_{i=1}^{d}A_{i})=\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\mu(T^{-p_{1}(n)}A_{1}\cap
T^{-p_{2}(n)}A_{2}\cap\cdots\cap T^{-p_{d}(n)}A_{d}),$

for all $A_{1},\ldots,A_{d}\in\mathcal{X}$, where the limit exists [21, 31].
Of course, $\mu_{\mathcal{A}}^{(d)}$ is a probability measure on $X^{d}$ and
it is $T^{(d)}$-invariant (in fact, it can be proved to be a self-joining).
Unlike Furstenberg joining, we can not find some transformation $\tau$ related
to $\mathcal{A}$ like $\tau_{d}$ such that
$(X^{d},\mathcal{X}^{d},\mu_{\mathcal{A}}^{(d)},\tau)$ becomes a m.p.s. Thus
to study the dynamical behavior related to polynomials, we need to find
another way to generalize the Furstenberg joining.

To overcome the difficulty, when studying the dynamical behavior concerning
polynomials $\\{p_{1},\ldots,p_{d}\\}$, we will consider infinitely many
polynomials $p_{i}^{[k]},1\leq i\leq d,k\in\mathbb{Z}$ defined by
$p_{i}^{[k]}(n)=p_{i}(n+k)-p_{i}(k),n\in\mathbb{Z}$ at the same time. For
example, for $\mathcal{A}=\\{n^{2},n^{3}\\}$, we define a measure
$\mu^{(\infty)}_{\mathcal{A}}$ on $(X^{2})^{\mathbb{Z}}$ as follows: for all
$l\in\mathbb{N}$, $A_{i}^{(1)},A_{i}^{(2)}\in\mathcal{X}$ and $-l\leq i\leq
l$, put

$\begin{split}&\mu^{(\infty)}_{\mathcal{A}}\Big{(}(\prod_{j=-\infty}^{-l-1}X^{2})\times\prod_{i=-l}^{l}(A^{(1)}_{i}\times
A^{(2)}_{i})\times(\prod_{j=l+1}^{\infty}X^{2})\Big{)}\\\
&\quad\quad\quad\quad\quad\quad=\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\mu\Big{(}\bigcap_{i=-l}^{l}(T^{-(n+i)^{2}}A^{(1)}_{i}\cap
T^{-(n+i)^{3}}A^{(2)}_{i})\Big{)},\end{split}$

where the existence of the limit is guaranteed by [21, 31].

Similarly, for $\mathcal{A}=\\{p_{1},\ldots,p_{d}\\}$, we define a measure
$\mu^{(\infty)}_{\mathcal{A}}$ on $(X^{d})^{\mathbb{Z}}$. This measure
$\mu^{(\infty)}_{\mathcal{A}}$ is invariant under the product map
$T^{\infty}=\cdots\times T^{(d)}\times T^{(d)}\times\cdots$ and the shift map
$\sigma:(X^{d})^{\mathbb{Z}}\rightarrow(X^{d})^{\mathbb{Z}}$. Thus for a given
m.p.s. $(X,\mathcal{X},\mu,T)$ and $\mathcal{A}=\\{p_{1},\ldots,p_{d}\\}$, we
associate two m.p.s.

$\big{(}(X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu_{\mathcal{A}}^{(\infty)},\sigma\big{)}\
\text{and}\
\big{(}(X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu_{\mathcal{A}}^{(\infty)},\langle
T^{\infty},\sigma\rangle\big{)},$

where $\langle T^{\infty},\sigma\rangle$ is the group generated by $\sigma$
and $T^{\infty}$. The first one is a $\mathbb{Z}$-system and the second one is
a $\mathbb{Z}^{2}$-system, as $\sigma\circ T^{\infty}=T^{\infty}\circ\sigma$.
The measure $\mu^{(\infty)}_{\mathcal{A}}$ is a joining on
$(X^{d})^{\mathbb{Z}}$, and we call it an $\infty$-joining with respect to
$\mathcal{A}$, see Section 3 for detailed discussions. In this paper, we will
focus on the dynamical properties of this joining and give its applications.

Note that in a companion paper by the same authors [26], for a given
topological system $(X,T)$ and $\mathcal{A}=\\{p_{1},\ldots,p_{d}\\}$, we
associate two topological dynamical systems $(W_{x}^{\mathcal{A}},\sigma)$,
and $(M_{\infty}(X,\mathcal{A}),\langle T^{\infty},\sigma\rangle)$. It is
proved that if $(X,T)$ is minimal, then $(M_{\infty}(X,\mathcal{A}),\langle
T^{\infty},\sigma\rangle)$ is transitive and has dense minimal points, which
can be used to answer some open question.

###  1.3. Main results

Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s. and let
$\mathcal{A}=\\{p_{1},\ldots,p_{d}\\}$ be a family of non-constant integral
polynomials. First we will study some basic properties of
$((X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu_{\mathcal{A}}^{(\infty)},\sigma)$
and
$((X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu_{\mathcal{A}}^{(\infty)},\langle
T^{\infty},\sigma\rangle)$. It is shown that:

  * $\blacklozenge$

$\big{(}(X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu_{\mathcal{A}}^{(\infty)},\langle
T^{\infty},\sigma\rangle\big{)}$ is ergodic (Theorem 6.1-(1)).

  * $\blacklozenge$

One can characterize the ergodic decomposition of
$\mu_{\mathcal{A}}^{(\infty)}$ under $\sigma$ (Theorem 6.1-(2), (3)).

  * $\blacklozenge$

If the degree of each polynomial in $\mathcal{A}$ is greater than 1, then for
$\mu$ a.e. $x\in X$,
$\big{(}(X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},{\mu}^{(\infty)}_{\mathcal{A},x},\sigma\big{)}$
is a direct product of an infinity-step pro-nilsystem and a Bernoulli system
(Theorem 6.4).

  * $\blacklozenge$

If $(X,T)$ is a topological system, $\mu$ is an invariant Borel probability
measure on $X$ with full support, then $\mu_{\mathcal{A}}^{(\infty)}$ is
supported on $(N_{\infty}(X,\mathcal{A}),\langle T^{\infty},\sigma\rangle)$
(Theorem 3.7).

Note that $(N_{\infty}(X,\mathcal{A}),\langle T^{\infty},\sigma\rangle)$ is an
equivalent form of $(M_{\infty}(X,\mathcal{A}),\langle
T^{\infty},\sigma\rangle)$, see [26, Section 4] for details.

Also we will use
$\big{(}(X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu_{\mathcal{A}}^{(\infty)},\sigma\big{)}$
to study the convergence of multiple averages along polynomials. Particularly
it is proved that

  * $\blacktriangledown$

Let $T$ and $S$ be ergodic measure preserving transformations on a probability
space $(X,\mathcal{X},\mu)$ and with $T$ having zero entropy. Let
$c_{1},\ldots,c_{m}\in\mathbb{Z}\setminus\\{0\\}$ and $p_{i}$ be an integral
polynomial with $\deg{p_{i}}\geq 2,1\leq i\leq d$. Then for all
$f_{1},\ldots,f_{m},g_{1},\ldots,g_{d}\in L^{\infty}(X,\mu)$,

$\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}f_{1}(T^{c_{1}n}x)\cdots
f_{m}(T^{c_{m}n}x)\cdot g_{1}(S^{p_{1}(n)}x)\cdots g_{d}(S^{p_{d}(n)}x)$

exists in $L^{2}(X,\mu)$ (Theorem 7.8).

When $m=d=1$, the result above was given in [14]. The main point here is that
$T$ and $S$ do not satisfy any commutativity assumptions.

Moreover, we will compare the system defined in this paper with Furstenberg
systems of sequences defined in [13, 12] (see Section 8 for definitions), and
show the following result, which answers Problem 1 in [12].

  * $\blacktriangle$

Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s., $p$ be a non-linear integral
polynomial, and $f\in L^{\infty}(X,\mu)$. Then for any strictly increasing
sequence of positive integers $\\{N_{k}\\}_{k\in\mathbb{N}}$ there is a
subsequence $\\{N^{\prime}_{k}\\}_{k\in\mathbb{N}}$ such that for a.e. $x\in
X$ the Furstenberg systems of $\big{(}f(T^{p(n)})x\big{)}_{n\in\mathbb{Z}}$
along $\\{N^{\prime}_{k}\\}_{k\in\mathbb{N}}$ are ergodic and isomorphic to
the direct products of infinite-step pro-nilsystems and Bernoulli systems
(Theorem 8.7).

###  1.4. Organization of the paper

We organize the paper as follows. In Section 2 we state some necessary notions
and some known facts used in the paper. In Section 3, we define
$\infty$-joinings $\mu^{(\infty)}_{\mathcal{A}}$ and
$\widetilde{\mu}^{(\infty)}_{\mathcal{A}}$ with respect to a given m.p.s. and
a family of integral polynomials, and explain them through certain examples.
In Section 4, we study the $\sigma$-algebra of invariant subsets under the
shift map, and show how to relate it to the one in pro-nilfactors. In Section
5 and Section 6, we give the ergodic decomposition theorem of
$\mu_{\mathcal{A}}^{(\infty)}$ under the shift map $\sigma$. In the next two
sections, we derive some applications of the theory we have developed. Namely,
in Section 7, we provide some new convergence theorems along integral
polynomials, and in Section 8, we compare our notion with Furstenberg systems
related to sequences. In the last section, we present some questions.

Acknowledgments: We thank Hanfeng Li for useful discussions concerning Theorem
8.6. Thank Hui Xu for very careful readings and corrections, and Thank Nikos
Frantzikinakis for useful comments.

##  2\. Preliminaries

In this section we give some necessary notions and some known facts used in
the paper. Note that, instead of just considering a single transformation $T$,
sometimes we study commuting transformations $T_{1}$, $\ldots$ , $T_{k}$ of
$X$. We only recall some basic definitions and properties of systems for one
transformation, and the extensions to the general case are straightforward.

###  2.1. Some notion in ergodic theory and topological dynamics

####  2.1.1. Measurable systems

Recall that in this paper we consider invertible m.p.s.
$(X,\mathcal{X},\mu,T)$ defined on a Lebesgue probability space. Thus $X$ can
be given the structure of a compact metric space and $\mathcal{X}$ is its
Borel $\sigma$-algebra.

For a m.p.s. $(X,\mathcal{X},\mu,T)$, the associated unitary operator (Koopman
operator) of $T$ is $U_{T}:L^{2}(X,\mu)\rightarrow L^{2}(X,\mu),f\mapsto
f\circ T$. In the sequel, we usually denote $U_{T}f$ by $Tf$.

For a m.p.s. $(X,\mathcal{X},\mu,T)$ we write
$\mathcal{I}(X,\mathcal{X},\mu,T)$ for the $\sigma$-algebra
$\\{A\in\mathcal{X}:T^{-1}A=A\\}$ consisting of invariant sets. Sometimes we
will use $\mathcal{I}$ or $\mathcal{I}(T)$ for short. A m.p.s. is ergodic if
all the $T$-invariant sets have measures either $0$ or $1$.
$(X,\mathcal{X},\mu,T)$ is weakly mixing if the product system $(X\times
X,\mathcal{X}\times\mathcal{X},\mu\times\mu,T\times T)$ is erdogic.

Let $X,Y$ be spaces and $\phi:X\to Y$ be a map. For $n\geq 2$ let

$\phi^{(n)}=\underbrace{\phi\times\cdots\times\phi}_{\text{($n$
times)}}:X^{n}\rightarrow Y^{n}.$

For a m.p.s. $(X,\mathcal{X},\mu,T)$, we write
$(X^{n},\mathcal{X}^{n},\mu^{d},T^{(n)})$ for the $n$-fold product system
$(X\times\cdots\times
X,\mathcal{X}\times\cdots\times\mathcal{X},\mu\times\cdots\times\mu,T\times\cdots\times
T)$. The diagonal of $X^{n}$ is

$\Delta_{n}(X)=\\{x^{\otimes n}=(x,\ldots,x)\in X^{n}:x\in X\\}.$

When $n=2$ we write $\Delta(X)=\Delta_{2}(X)$.

A Bernoulli system has the form
$(Y^{\mathbb{Z}},\mathcal{Y}^{\mathbb{Z}},\nu^{\mathbb{Z}},\sigma)$, where
$(Y,\mathcal{Y},\nu)$ is a probability space, $\sigma$ is the shift on
$Y^{\mathbb{Z}}$, $\mathcal{Y}^{\mathbb{Z}}$ is the product $\sigma$-algebra
of $Y^{\mathbb{Z}}$, and $\nu^{\mathbb{Z}}$ is the product measure.

A factor map or homomorphism from $(X,\mathcal{X},\mu,T)$ to
$(Y,\mathcal{Y},\nu,S)$ is a measurable map $\pi:X_{0}\rightarrow Y_{0}$,
where $X_{0}\in\mathcal{X}$ is $T$-invariant and $Y_{0}\in\mathcal{Y}$ is
$S$-invariant, both of full measure, such that
$\pi_{*}\mu\triangleq\mu\circ\pi^{-1}=\nu$ and $S\circ\pi(x)=\pi\circ T(x)$
for $x\in X_{0}$. When we have such a homomorphism we say that
$(Y,\mathcal{Y},\nu,S)$ is a factor of $(X,\mathcal{X},\mu,T)$. If the factor
map $\pi:X_{0}\rightarrow Y_{0}$ can be chosen to be bijective, then we say
that $(X,\mathcal{X},\mu,T)$ and $(Y,\mathcal{Y},\nu,S)$ are (measure
theoretically) isomorphic (bijective maps on Lebesgue spaces have measurable
inverses), and denote it by $(X,\mathcal{X},\mu,T)\cong(Y,\mathcal{Y},\nu,S)$.

Let $(Y,\mathcal{Y},\nu,S)$ be a factor of $(X,\mathcal{X},\mu,T)$. A factor
can be characterized (modulo isomorphism) by $\pi^{-1}(\mathcal{Y})$, which is
a $T$-invariant sub-$\sigma$-algebra of $\mathcal{X}$, and conversely any
$T$-invariant sub-$\sigma$-algebra of $\mathcal{X}$ defines a factor. By a
classical abuse of terminology we denote by the same letter the
$\sigma$-algebra $\mathcal{Y}$ and its inverse image by $\pi$,
$\pi^{-1}(\mathcal{Y})$. In other words, if $(Y,\mathcal{Y},\nu,S)$ is a
factor of $(X,\mathcal{X},\mu,T)$, we think of $\mathcal{Y}$ as a
sub-$\sigma$-algebra of $\mathcal{X}$. Thus we may regard
$L^{2}(Y,\mathcal{Y},\nu)$ as a closed subspace of $L^{2}(X,\mathcal{X},\mu)$.
When there is no danger of confusion, we may abuse the notation and denote the
transformation $S$ on $Y$ by $T$.

We say that $(X,\mathcal{X},\mu,T)$ is an inverse limit of a sequence of
factors $\\{(X_{j},\mathcal{X}_{j},\mu_{j},T)\\}_{j\in\mathbb{N}}$ if
$(\mathcal{X}_{j})_{j\in\mathbb{N}}$ is an increasing sequence of
$T$-invariant sub-$\sigma$-algebras such that
$\bigvee_{j\in\mathbb{N}}\mathcal{X}_{j}=\mathcal{X}$ up to sets of measure
zero.

####  2.1.2. Topological dynamical systems

A topological dynamical system (t.d.s. for short) is a pair $(X,T)$, where $X$
is a compact metric space and $T$ is a homeomorphism from $X$ to itself. A
t.d.s. $(X,T)$ is transitive if there exists some point $x\in X$ whose orbit
$\mathcal{O}(x,T)=\\{T^{n}x:n\in\mathbb{Z}\\}$ is dense in $X$ and we call
such a point a transitive point. A t.d.s. $(X,T)$ is minimal if the orbit of
any point is dense in $X$. This property is equivalent to saying that X is the
only closed non-empty invariant subset in $X$.

A topological factor of a t.d.s. $(X,T)$ is another t.d.s. $(Y,S)$ such that
there exists a continuous and onto map $\phi:X\rightarrow Y$ satisfying
$S\circ\phi=\phi\circ T$. In this case, $(X,T)$ is called an topological
extension  of $(Y,S)$. The map $\phi$ is called a topological factor map.

####  2.1.3. ${\mathcal{M}}(X)$ and ${\mathcal{M}}_{T}(X)$

For a t.d.s. $(X,T)$, denote by ${\mathcal{M}}(X)$ the set of all Borel
probability measures on $X$. Let
${\mathcal{M}}_{T}(X)=\\{\mu\in{\mathcal{M}}(X):T_{*}\mu=\mu\circ
T^{-1}=\mu\\}$. With the weak∗-topology, ${\mathcal{M}}(X)$ and
${\mathcal{M}}_{T}(X)$ are compact convex spaces. By the Krylov-Bogolioubov
theorem ${\mathcal{M}}_{T}(X)\neq\emptyset$. Denote by
${\mathcal{M}}^{erg}_{T}(X)$ the set of ergodic measures of $(X,T)$, then
${\mathcal{M}}^{erg}_{T}(X)$ is the set of extreme points of
${\mathcal{M}}_{T}(X)$ and one can use the Choquet representation theorem to
express each member of ${\mathcal{M}}_{T}(X)$ in terms of the ergodic members
of ${\mathcal{M}}_{T}(X)$. That is, for each $\mu\in{\mathcal{M}}_{T}(X)$
there is a unique measure $\tau$ on the Borel subsets of the compact space
${\mathcal{M}}_{T}(X)$ such that
$\tau\big{(}{\mathcal{M}}^{erg}_{T}(X)\big{)}=1$ and
$\displaystyle\mu=\int_{{\mathcal{M}}^{erg}_{T}(X)}md\tau(m)$, which is called
the ergodic decomposition of $\mu$. In particular,
${\mathcal{M}}_{T}^{erg}(X)\neq\emptyset$.

A t.d.s. $(X,T)$ is called uniquely ergodic if there is a unique $T$-invariant
Borel probability measure on $X$. It is called strictly ergodic if it is
uniquely ergodic and minimal.

###  2.2. Topological models

####  2.2.1. Topological models

Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s. We say that
$(\hat{X},\hat{T})$ is a topological model (or just a model) for
$(X,\mathcal{X},\mu,T)$ if $(\hat{X},\hat{T})$ is a t.d.s. and there exists an
invariant probability measure $\hat{\mu}$ on the Borel $\sigma$-algebra
$\mathcal{B}(\hat{X})$ such that the systems $(X,\mathcal{X},\mu,T)$ and
$(\hat{X},\mathcal{B}(\hat{X}),\hat{\mu},\hat{T})$ are measure theoretically
isomorphic.

The well-known Jewett-Krieger’s theorem [27, 28] states that every ergodic
system has a strictly ergodic model.

####  2.2.2. Weiss’s Theorem

We say that $\hat{\pi}:\hat{X}\rightarrow\hat{Y}$ is a topological model for a
factor map $\pi:(X,\mathcal{X},\mu,T)\rightarrow(Y,\mathcal{Y},\nu,S)$ if
$\hat{\pi}$ is a topological factor map and there exist measure theoretical
isomorphisms $\phi$ and $\psi$ such that the diagram

$\begin{CD}X@>{\phi}>{}>\hat{X}\\\ @V{\pi}V{}V@V{}V{\hat{\pi}}V\\\
Y@>{\psi}>{}>\hat{Y}\end{CD}$

is commutative, i.e. $\hat{\pi}\phi=\psi\pi$. Weiss [42] generalized Jewett-
Krieger Theorem to the relative case. Namely, he proved that if
$\pi:(X,\mathcal{X},\mu,T)\rightarrow(Y,\mathcal{Y},\nu,S)$ is a factor map
with $(X,\mathcal{X},\mu,T)$ ergodic and
$(\hat{Y},\hat{\mathcal{Y}},\hat{\nu},\hat{S})$ is a uniquely ergodic model
for $(Y,\mathcal{Y},\nu,T)$, then there is a uniquely ergodic model
$(\hat{X},\hat{\mathcal{X}},\hat{\mu},\hat{T})$ for $(X,\mathcal{X},\mu,T)$
and a factor map $\hat{\pi}:\hat{X}\rightarrow\hat{Y}$ which is a model for
$\pi:X\rightarrow Y$. We will refer this theorem as Weiss’s Theorem. We note
that in [42] Weiss pointed that the relative case holds for commutative group
actions.

###  2.3. The Rohlin skew-product theorem

Let $(Y,\mathcal{Y},\nu,S)$ be a m.p.s. and $(U,{\mathcal{U}},\rho)$ a
probability space. Let $\alpha:Y\rightarrow{\rm Aut}(U,\rho)$ be a measurable
map ($\alpha$ is called a cocycle), where ${\rm Aut}(U,\rho)$ is the Polish
group of all invertible measure-preserving transformations of
$(U,{\mathcal{U}},\rho)$. We define the skew-product system $(Y\times
U,\mathcal{Y}\times{\mathcal{U}},\nu\times\rho,T_{\alpha})$ as follows:

$T_{\alpha}(y,u)=(Sy,\alpha(y)u),\quad\forall(y,u)\in Y\times U.$

The following is the well known Rohlin skew-product theorem (see [19, Theorem
3.18]).

######  Theorem 2.1 (Rohlin skew-product theorem).

Let $\pi:(X,\mathcal{X},\mu,T)\rightarrow(Y,\mathcal{Y},\nu,S)$ be a factor
map of ergodic m.p.s. Then $(X,\mathcal{X},\mu,T)$ is isomorphic to a skew-
product over $(Y,\mathcal{Y},\nu,S)$. That is, there exist a probability space
$(U,{\mathcal{U}},\rho)$ and a measurable cocycle $\alpha:Y\rightarrow{\rm
Aut}(U,\rho)$ such that

$(X,\mathcal{X},\mu,T)\cong(Y\times
U,\mathcal{Y}\times{\mathcal{U}},\nu\times\rho,T_{\alpha}).$

###  2.4. Conditional expectations

####  2.4.1. Conditional expectations

Let $\pi:(X,\mathcal{X},\mu,T)\rightarrow(Y,\mathcal{Y},\nu,T)$ be a factor
map. As mentioned above, $\mathcal{Y}$ is also regarded as a $T$-invariant
sub-$\sigma$-algebra of $\mathcal{X}$. For each $f\in L^{1}(X,\mu)$, we write
$\mathbb{E}(f|\mathcal{Y})$, or $\mathbb{E}_{\mu}(f|\mathcal{Y})$ if needed,
for the conditional expectation of $f$ with respect to $\mathcal{Y}$. We will
frequently make use of the identities

$\int_{X}\mathbb{E}(f|\mathcal{Y})\ d\mu=\int_{X}f\ d\mu\quad\text{and}\quad
T\mathbb{E}(f|\mathcal{Y})=\mathbb{E}(Tf|\mathcal{Y}).$

We say that a function $f\in L^{2}(X,\mu)$ is orthogonal to $\mathcal{Y}$, and
we write $f\perp\mathcal{Y}$, when it has a zero conditional expectation on
$\mathcal{Y}$.

#### 2.4.2.

The disintegration of $\mu$ over $\nu$, written as
$\displaystyle\mu=\int_{Y}\mu_{y}\ d\nu(y)$, is given by a measurable map
$y\mapsto\mu_{y}$ from $Y$ to the space of Borel probability measures on $X$
such that

(2.1) $\mathbb{E}(f|\mathcal{Y})(y)=\int_{X}fd\mu_{y}$

$\nu$-almost everywhere.

###  2.5. Joinings

####  2.5.1. Joinings and conditional product measures

The notions of joining and conditional product measure were introduced by
Furstenberg in [15] and [16]. Let
$(X_{i},\mathcal{X}_{i},\mu_{i},T_{i}),i\in\\{1,2,\ldots,k\\}$ be m.p.s. A
Borel probability measure $\lambda$ on $X=\prod_{i=1}^{k}X_{i}$ defines a
joining of the measures on $X_{i}$ if it is invariant under
$\prod_{i=1}^{k}T_{i}=T_{1}\times\cdots\times T_{k}$ and maps onto $\mu_{j}$
under the natural map $\prod_{i=1}^{k}X_{i}\rightarrow X_{j},\forall
j\in\\{1,2,\ldots,k\\}$. When
$(X_{1},\mathcal{X}_{1},\mu_{1},T_{1})=(X_{2},\mathcal{X}_{2},\mu_{2},T_{2})=\cdots=(X_{k},\mathcal{X}_{k},\mu_{k},T_{k})$,
we then say that $\lambda$ is a $k$-fold self-joining. One can generalize the
notion to infinitely many m.p.s. easily.

Let $(X_{i},\mathcal{X}_{i},\mu_{i},T_{i})$ be a m.p.s.,
$(Y_{i},\mathcal{Y}_{i},\nu_{i},S_{i})$ be a corresponding factor, and
$\pi_{i}:X_{i}\rightarrow Y_{i}$ be a factor map, $1\leq i\leq k$. Let $\nu$
be a joining of the measures on $Y_{i},i=1,\cdots,k$, and let
$\mu_{i}=\int\mu_{X_{i},y_{i}}\ d\nu_{i}(y_{i})$ represent the disintegration
of $\mu_{i}$ with respect to $\nu_{i}$. Let $\mu$ be a measure on
$X=\prod_{i}X_{i}$ defined by

(2.2)
$\mu=\int_{Y}\mu_{X_{1},y_{1}}\times\mu_{X_{2},y_{2}}\times\cdots\times\mu_{X_{k},y_{k}}\
d\nu(y_{1},y_{2},\cdots,y_{k}).$

Then $\mu$ is called the conditional product measure with respect to $\nu$.

Equivalently, $\mu$ is a conditional product measure relative to $\nu$ if and
only if for all $k$-tuple $f_{i}\in L^{\infty}(X_{i},\mu_{i}),i=1,\ldots,k$

(2.3) $\begin{split}\int_{X}&f_{1}(x_{1})f_{2}(x_{2})\cdots f_{k}(x_{k})\
d\mu(x_{1},x_{2},\cdots,x_{k})\\\
&\quad=\int_{Y}\mathbb{E}(f_{1}|\mathcal{Y}_{1})(y_{1})\mathbb{E}(f_{2}|\mathcal{Y}_{2})(y_{2})\cdots\mathbb{E}(f_{k}|\mathcal{Y}_{k})(y_{k})\
d\nu(y_{1},y_{2},\cdots,y_{k}).\end{split}$

####  2.5.2. Relatively independent joining

Let
$(X_{1},\mathcal{X}_{1},\mu_{1},T_{1}),(X_{2},\mathcal{X}_{2},\mu_{2},T_{1})$
be m.p.s. and let $(Y,\mathcal{Y},\nu,S)$ be a common factor of them with
$\pi_{i}:X_{i}\rightarrow Y$ for $i=1,2$ the factor maps. Let
$\mu_{i}=\int_{Y}\mu_{i,y}\ d\nu(y)$ represent the disintegration of $\mu_{i}$
with respect to $\nu$. Let $\mu_{1}\times_{Y}\mu_{2}$ denote the measure
defined by

$\mu_{1}\times_{Y}\mu_{2}(A)=\int_{Y}\mu_{1,y}\times\mu_{2,y}(A)\ d\nu(y),$

for all $A\in\mathcal{X}_{1}\times\mathcal{X}_{2}$. The system $(X_{1}\times
X_{2},\mathcal{X}_{1}\times\mathcal{X}_{2},\mu_{1}\times_{Y}\mu_{2},T_{1}\times
T_{2})$ is called the relative product of $X_{1}$ and $X_{2}$ with respect to
$Y$ and is denoted $X_{1}\times_{Y}X_{2}$. $\mu_{1}\times_{Y}\mu_{2}$ is also
called a relatively independent joining of $X_{1}$ and $X_{2}$ over $Y$.

###  2.6. Furstenberg Joinings

####  2.6.1. Furstenberg Joinings

Let $T:X\rightarrow X$ be a map and $d\in\mathbb{N}$. Set

$\tau_{\vec{a}}=T^{a_{1}}\times T^{a_{2}}\times\cdots\times T^{a_{d}},$

where $\vec{a}=(a_{1},\ldots,a_{d})\in\mathbb{Z}^{d}$.

######  Definition 2.2.

Let $(X,\mathcal{X},\mu,T)$ be a m.p.s. For $d\geq 1$ let
$\mu^{(d)}_{\vec{a}}$ be the measure on $(X^{d},\mathcal{X}^{d})$ defined by

$\int_{X^{d}}f_{1}(x_{1})\cdots
f_{d}(x_{d})d\mu^{(d)}_{\vec{a}}(x_{1},\ldots,x_{d})=\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}\prod_{j=1}^{d}f_{j}(T^{a_{j}n}x)d\mu(x)$

for $f_{i}\in L^{\infty}(X,\mu)$, $1\leq j\leq d$, where the limits exists by
[20, Theorem 1.1].

We call $\mu^{(d)}_{\vec{a}}$ the Furstenberg self-joining. Clearly, it is
invariant under $\tau_{\vec{a}}$ and $T^{(d)}$.

When in addition we assume that $X$ is a compact metric space and $T$ is a
homeomorphism, it is easy to see that

$\frac{1}{N}\sum_{n=0}^{N-1}(\tau_{\vec{a}})_{*}^{n}\mu_{\Delta}^{d}\longrightarrow\mu^{(d)}_{\vec{a}},\
N\to\infty,\quad\text{weak${}^{*}$ in ${\mathcal{M}}(X^{d})$},$

where $\mu_{\Delta}^{d}$ is the diagonal measure on $X^{d}$ as defined in
[16], i.e. it is defined on $X^{d}$ as follows

$\int_{X^{d}}f_{1}(x_{1})\cdots f_{d}(x_{d})\
d\mu_{\Delta}^{d}(x_{1},\ldots,x_{d})=\int_{X}f_{1}(x)\cdots f_{d}(x)\
d\mu(x),$

where $f_{1},\ldots,f_{d}\in C(X)$.

####  2.6.2. The ergodic decomposition of $\mu_{\vec{a}}^{(d)}$

######  Theorem 2.3.

[25, Theorem D.] Let $(X,\mathcal{X},\mu,T)$ be an ergodic system and
$d\in\mathbb{N}$. Let $a_{1},\ldots,a_{d}$ be distinct non-zero integers. Then
there exists a family $\\{\mu^{(d)}_{\vec{a},x}\\}_{x\in X}$ of Borel
probability measures on $(X^{d},\mathcal{X}^{d})$ such that

  1. (1)

the disintegrations of $\mu_{\vec{a}}^{(d)}$ over $\mu$ is
$\displaystyle\mu_{\vec{a}}^{(d)}=\int_{X}\mu_{\vec{a},x}^{(d)}d\mu(x),$

  2. (2)

for $\mu$ a.e. $x\in X$, $\mu^{(d)}_{\vec{a},x}$ is ergodic under
$\tau_{\vec{a}}=T^{a_{1}}\times T^{a_{2}}\times\cdots\times T^{a_{d}}$,

  3. (3)

for all $f_{1},\ldots,f_{d}\in L^{\infty}(X,\mu)$,

$\begin{split}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i=1}^{d}f_{i}(T^{a_{i}n}x)\overset{L^{2}(\mu)}{\longrightarrow}\int_{X^{d}}\prod_{i=1}^{d}f_{i}(x_{i})\
d\mu^{(d)}_{\vec{a},x}(x_{1},\ldots,x_{d}),\ N\to\infty.\end{split}$

###  2.7. Host-Kra seminorms

When $f_{i}$, $i\in I$, are functions on the set $X$, we define a function
$\bigotimes_{i\in I}f_{i}$ on $X^{I}$ by

(2.4) $\bigotimes_{i\in I}f_{i}({\bf x})=\prod_{i\in I}f_{i}(x_{i}),$

where ${\bf x}=(x_{i})_{i\in I}\in X^{I}$ and $I$ is an index set.

####  2.7.1. The measure $\mu^{[k]}$

Let $(X,\mathcal{X},\mu,T)$ be an ergodic system and $k\in\mathbb{N}$. We
define a probability measure $\mu^{[k]}$ on $X^{2^{k}}$ invariant under
$T^{[k]}=T\times T\times\cdots\times T$ ($2^{k}$ times), by

$\mu^{[1]}=\mu\times_{\mathcal{I}(T)}\mu=\mu\times\mu;$

and for $k\geq 1$,

$\mu^{[k+1]}=\mu^{[k]}\times_{\mathcal{I}(T^{[k]})}\mu^{[k]}.$

####  2.7.2. Host-Kra seminorms

Write ${\bf x}=(x_{0},x_{1},\ldots,x_{2^{k}-1})$ for a point of $X^{2^{k}}$,
we define a seminorm $\interleave\cdot\interleave_{k}$ on $L^{\infty}(\mu)$ by

(2.5) $\interleave
f\interleave_{k}=\Big{(}\int_{X^{2^{k}}}\bigotimes_{i\in\\{0,1,\ldots,2^{k}-1\\}}f({\bf
x})d\mu^{[k]}({\bf
x})\Big{)}^{1/2^{k}}=\Big{(}\int_{X^{2^{k}}}\prod_{i=0}^{2^{k}-1}f(x_{i})d\mu^{[k]}({\bf
x})\Big{)}^{1/2^{k}}.$

That $\interleave\cdot\interleave_{k}$ is a seminorm 111Here for simplicity we
give the formula for real functions, and one can give the formula for complex
functions similarly. can be proved as in [20, Lemma 3.2], and we call it Host-
Kra seminorm (HK seminorm for short). As $X$ is assumed to be ergodic, the
$\sigma$-algebra $\mathcal{I}^{[0]}$ is trivial and $\mu^{[1]}=\mu\times\mu$.
We therefore have

$\interleave
f\interleave_{1}=\Big{(}\int_{X^{2}}f(x_{0}){f(x_{1})}d(\mu\times\mu)(x_{0},x_{1})\Big{)}^{1/2}=\Big{|}\int
fd\mu\Big{|}.$

#### 2.7.3.

It was shown in the proof of [20, Theorem 12.1] that

######  Lemma 2.4.

For every integer $k\geq 0$ and every $f\in L^{\infty}(\mu)$, one has

(2.6) $\interleave
f\interleave_{k+1}=\Big{(}\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}\interleave
f\cdot T^{n}{f}\interleave_{k}^{2^{k}}\Big{)}^{1/2^{k+1}}.$

Note that (2.6) can be considered as an alternate definition of the seminorms.
For each $f\in L^{\infty}(X,\mu)$, one has that [20, Lemma 3.9]

(2.7) $\interleave f\interleave_{k}\leq\interleave f\interleave_{k+1},\
\forall k\in\mathbb{N}.$

######  Lemma 2.5.

[25, Lemma 3.2] Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s. and $d\geq 1$
be an integer. Suppose that $\lambda$ is a $d$-fold self-joining of
$(X,\mathcal{X},\mu,T)$. Assume that $f_{j}\in L^{\infty}(X,\mu)$ with
$\|f_{j}\|_{\infty}\leq 1$ for $j=1,\ldots,d$. Then

(2.8)
$\limsup_{N\to\infty}\Big{\|}\frac{1}{N}\sum_{n=0}^{N-1}f_{1}(T^{n}x_{1})f_{2}(T^{2n}x_{2})\cdots
f_{d}(T^{dn}x_{d})\Big{\|}_{L^{2}(X^{d},\lambda)}\leq\min_{1\leq l\leq
d}\\{l\cdot\interleave f_{l}\interleave_{d}\\}$

In particular, let $\lambda$ be the diagonal joining, and we have

######  Lemma 2.6.

[20, Theorem 12.1] Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s. and $d\geq
1$ be an integer. Assume that $f_{j}\in L^{\infty}(X,\mu)$ with
$\|f_{j}\|_{\infty}\leq 1$ for $j=1,\ldots,d$. Then

(2.9)
$\limsup_{N\to\infty}\Big{\|}\frac{1}{N}\sum_{n=0}^{N-1}f_{1}(T^{n}x)f_{2}(T^{2n}x)\cdots
f_{d}(T^{dn}x)\Big{\|}_{L^{2}(X,\mu)}\leq\min_{1\leq l\leq
d}\\{l\cdot\interleave f_{l}\interleave_{d}\\}$

###  2.8. Pro-nilsystems

####  2.8.1. Nilpotent groups

Let $G$ be a group. For $g,h\in G$, we write $[g,h]=ghg^{-1}h^{-1}$ for the
commutator of $g$ and $h$ and we write $[A,B]$ for the subgroup spanned by
$\\{[a,b]:a\in A,b\in B\\}$. The commutator subgroups $G_{j}$, $j\geq 1$, are
defined inductively by setting $G_{1}=G$ and $G_{j+1}=[G_{j},G]$. Let $k\geq
1$ be an integer. We say that $G$ is $k$-step nilpotent if $G_{k+1}$ is the
trivial subgroup.

####  2.8.2. Nilsystems

Let $G$ be a $k$-step nilpotent Lie group and $\Gamma$ a discrete cocompact
subgroup of $G$. The compact manifold $X=G/\Gamma$ is called a $k$-step
nilmanifold. The group $G$ acts on $X$ by left translations and we write this
action as $(g,x)\mapsto gx$. The Haar measure $\nu$ of $X$ is the unique
probability measure on $X$ invariant under this action. Let $t\in G$ and $T$
be the transformation $x\mapsto tx$ of $X$. Then $(X,\nu,T)$ is called a
$k$-step nilsystem.

######  Definition 2.7.

Let $k\in\mathbb{N}$. A m.p.s. $(X,\mathcal{X},\mu,T)$ is called a $k$-step
pro-nilsystem or a system of order $k$ if it is a (measure theoretic) inverse
limit of $k$-step nilsystems.

#### 2.8.3.

Here are some basic properties of nilsystems, and they also hold for pro-
nilsystems.

######  Theorem 2.8.

[36, 29] Let $(X=G/\Gamma,\nu,T)$ be a $k$-step nilsystem with $T$ the
translation by the element $t\in G$. Then:

  1. (1)

$(X,T)$ is uniquely ergodic if and only if $(X,\nu,T)$ is ergodic if and only
if $(X,T)$ is minimal if and only if $(X,T)$ is transitive.

  2. (2)

Let $Y$ be the closed orbit of some point $x\in X$. Then $Y$ can be given the
structure of a nilmanifold, $Y=H/\Lambda$, where $H$ is a closed subgroup of
$G$ containing $t$, and $\Lambda$ is a closed cocompact subgroup of $H$.

######  Remark 2.9.

One can generalize the above results to the action of several translations.
For example, let $X=G/\Gamma$ be a nilmanifold with Haar measure $\nu$ and let
$t_{1},\ldots,t_{k}$ be commuting elements of $G$. If the group spanned by the
translations $t_{1},\ldots,t_{k}$ acts ergodically on $(X,\nu)$, then $X$ is
uniquely ergodic for this group, see [29] for details.

####  2.8.4. Nilfactors

By the Host-Kra seminorm we can define factors
$(Z_{d-1},\mathcal{Z}_{d-1},\mu_{d-1},T)$.

######  Definition 2.10.

Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s. For $d\in\mathbb{N}$, there
exists a $T$-invariant $\sigma$-algebra $\mathcal{Z}_{d-1}$ of $\mathcal{X}$
such that for $f\in L^{\infty}(\mu)$,

$\interleave f\interleave_{d}=0\ \text{ if and only if}\
\mathbb{E}(f|\mathcal{Z}_{d-1})=0.$

Let $(Z_{d-1},\mathcal{Z}_{d-1},\mu_{d-1},T)$ be the factor of $X$ associated
to the sub $\sigma$-algebra $\mathcal{Z}_{d-1}$.

By (2.7), the factors $Z_{k},k\geq 1,$ form an increasing sequence of factors
of $X$. In [20], it was showed that $(Z_{d-1},\mathcal{Z}_{d-1},\mu_{d-1},T)$
has a very nice structure.

######  Theorem 2.11.

[20, Theorem 10.1] Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s. and
$d\in\mathbb{N}$. Then the system $(Z_{d-1},\mathcal{Z}_{d-1},\mu_{d-1},T)$ is
a $(d-1)$-step pro-nilsystem.

####  2.8.5. $\infty$-step pro-nilsystems

An $\infty$-step pro-nilsystem was introduced and studied in [11].

######  Definition 2.12.

Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s. The factors
$\mathcal{Z}_{k},k\in\mathbb{N}$, form an increasing sequence of $T$-invariant
sub-$\sigma$-algebras of $\mathcal{X}$, and we let
$\mathcal{Z}_{\infty}\triangleq\bigvee_{k\in\mathbb{N}}\mathcal{Z}_{k}$ and
$(Z_{\infty},\mathcal{Z}_{\infty},\mu_{\infty},T)$ be the factor associated
with the $\mathcal{Z}_{\infty}$. Then this system is the inverse limit of the
systems $(Z_{k},\mathcal{Z}_{k},\mu_{k},T),k\in\mathbb{N}$. We say
$(Z_{\infty},\mathcal{Z}_{\infty},\mu_{\infty},T)$ is an $\infty$-step pro-
nilsystem.

Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s. and $f\in L^{\infty}(X,\mu)$.
We define $\interleave
f\interleave_{\infty}\triangleq\sup_{k\in\mathbb{N}}\interleave
f\interleave_{k}$. Note that $\interleave f\interleave_{\infty}=0$ if and only
if $\mathbb{E}(f|\mathcal{Z}_{\infty})=0$.

###  2.9. Topological pro-nilsystems

#### 2.9.1.

One also has the topological version of this notion, i.e. the topological
inverse limit of nilsystems. First recall the definition of an inverse limit
of t.d.s. If $(X_{i},T_{i})_{i\in\mathbb{N}}$ are t.d.s. with $diam(X_{i})\leq
1$ and $\pi_{i}:X_{i+1}\rightarrow X_{i}$ are factor maps, the topological
inverse limit of the systems is defined to be the compact subset of
$\prod_{i\in\mathbb{N}}X_{i}$ given by
$\\{(x_{i})_{i\in\mathbb{N}}:\pi_{i}(x_{i+1})=x_{i},i\in\mathbb{N}\\}$, and we
denote it by $\lim\limits_{\longleftarrow}(X_{i},T_{i})_{i\in\mathbb{N}}$. It
is a compact metric space endowed with the distance
$\rho((x_{i})_{i\in\mathbb{N}},(y_{i})_{i\in\mathbb{N}})=\sum_{i\in\mathbb{N}}1/2^{i}\rho_{i}(x_{i},y_{i})$,
where $\rho_{i}$ is the metric in $X_{i}$. We note that the maps $T_{i}$
induce naturally a transformation $T$ on the inverse limit.

######  Definition 2.13.

[24] Let $d\in\mathbb{N}$. A topological inverse limit of $d$-step minimal
nilsystems is called a topological $d$-step pro-nilsystem or a topological
system of order $d$.

See [24] and [38] for more about topological pro-nilsystems.

#### 2.9.2.

A minimal t.d.s. is a topological $\infty$-step pro-nilsystem if and only if
it is a topological inverse limit of minimal nilsystems [11].

By Theorem 2.8, a topological $d$-step pro-nilsystem is uniquely ergodic for
each $d\in\mathbb{N}\cup\\{\infty\\}$.

######  Theorem 2.14.

[24, Subsection 5.1] Let $d\in\mathbb{N}\cup\\{\infty\\}$. Any $d$-step pro-
nilsystem is isomorphic in the measure theoretic sense to a topological
$d$-step pro-nilsystem.

######  Lemma 2.15.

[11, Lemma A.3] Let $d\in\mathbb{N}\cup\\{\infty\\}$. Let $(X,T)$ be a
topological $d$-step nilsystem, then the maximal measurable and topological
$j$-step pro-nilfactors of $(X,T)$ coincide, where $j\leq d$.

###  2.10. Nilsequencs

######  Definition 2.16.

[4] Let $d\geq 1$ be an integer and let $X=G/\Gamma$ be a $d$-step
nilmanifold. Let $\phi$ be a continuous real (or complex) valued function on
$X$ and let $t\in G$ and $x\in X$. The sequence $\\{\phi(t^{n}\cdot
x)\\}_{n\in\mathbb{Z}}$ is called a basic $d$-step nilsequence. A $d$-step
nilsequence is a uniform limit of basic $d$-step nilsequences.

We will need the following theorem.

######  Theorem 2.17.

[23, Theorem 16.10] Let $(X,\mathcal{X},\mu,T)$ be a m.p.s. (not necessarily
ergodic) and let $k\in\mathbb{N}$. Then for all $p\in[1,\infty)$ and
$\epsilon>0$, every $f\in L^{\infty}(X,\mu)$ admits a decomposition

$f=f_{\rm unif}+f_{\rm nil}+f_{\rm sml},$

where

  1. (1)

the function $f_{\rm unif}$ satisfies $\interleave f_{\rm
unif}\interleave_{k+1}=0$;

  2. (2)

for $\mu$-almost every $x\in X$, the sequence $\big{(}f_{\rm
nil}(T^{n}x)\big{)}_{n\in\mathbb{Z}}$ is a $k$-step nilsequence;

  3. (3)

the function $f_{\rm sml}\in L^{\infty}(X,\mu)$ satisfies $\|f_{\rm
sml}\|_{L^{p}(X,\mu)}\leq\epsilon$.

Furthermore, $\|f_{\rm
nil}\|_{L^{\infty}(X,\mu)}\leq\|f\|_{L^{\infty}(X,\mu)}$ and $\|f_{\rm
nil}+f_{\rm sml}\|_{L^{\infty}(X,\mu)}\leq\|f\|_{L^{\infty}(X,\mu)}$.

######  Remark 2.18.

  1. (1)

Let $(X,\mathcal{X},\mu,T)$ be a m.p.s. and let $k\in\mathbb{N}$. As a
corollary of Theorem 2.17, for all $\epsilon>0$, every $f\in
L^{\infty}(X,\mu)$ admits a decomposition

$f=f_{\rm unif}+f_{\rm nil},$

such that for $\mu$-almost every $x\in X$, the sequence $\big{(}f_{\rm
nil}(T^{n}x)\big{)}_{n\in\mathbb{Z}}$ is a $k$-step nilsequence and the
function $f_{\rm unif}\in L^{\infty}(X,\,u)$ satisfies $\interleave f_{\rm
unif}\interleave_{k+1}<\epsilon$ ([23, Theorem 16.11]).

  2. (2)

Let $(X,\mathcal{X},\mu,T)$ be a m.p.s. and let $k\in\mathbb{N}$. If $f\in
L^{\infty}(\mathcal{Z}_{k},\mu)$, then for every $\epsilon>0$ there is a
function $\widetilde{f}\in L^{\infty}(X,\mu)$ such that

     * •

$\widetilde{f}\in L^{\infty}(\mathcal{Z}_{k},\mu)$,
$\|\widetilde{f}\|_{L^{\infty}(\mu)}\leq\|f\|_{L^{\infty}(X,\mu)}$ and
$\|f-\widetilde{f}\|_{L^{1}(\mu)}\leq\epsilon$,

     * •

for $\mu$-almost every $x\in X$, the sequence
$\big{(}\widetilde{f}(T^{n}x)\big{)}_{n\in\mathbb{Z}}$ is a $k$-step
nilsequence.

(see [8, Proposition 3.1])

###  2.11. Characteristic factors

In the study of multiple ergodic averages, the idea of characteristic factors
plays a very important role. This idea was suggested by Furstenberg in [16],
and the notion of “characteristic factors” was first introduced in a paper by
Furstenberg and Weiss [18].

######  Definition 2.19.

Let $(X,\mathcal{X},\mu,T)$ be a m.p.s. and $(Y,\mathcal{Y},\mu,T)$ be a
factor of $X$. Let $\\{p_{1},\cdots,p_{d}\\}$ be a family of integral
polynomials, $d\in\mathbb{N}$. We say that $Y$ is a $L^{2}$(resp.
a.e.)-characteristic factor of $X$ for the scheme $\\{p_{1},\ldots,p_{d}\\}$
if for all $f_{1},\ldots,f_{d}\in L^{\infty}(X,\mathcal{X},\mu)$,

$\begin{split}\frac{1}{N}\sum_{n=0}^{N-1}&f_{1}(T^{p_{1}(n)}x)f_{2}(T^{p_{2}(n)}x)\cdots
f_{d}(T^{p_{d}(n)}x)\\\
-&\frac{1}{N}\sum_{n=0}^{N-1}\mathbb{E}(f_{1}|\mathcal{Y})(T^{p_{1}(n)}x)\mathbb{E}(f_{2}|\mathcal{Y})(T^{p_{2}(n)}x)\cdots\mathbb{E}(f_{d}|\mathcal{Y})(T^{p_{d}(n)}x)\to
0\end{split}$

in $L^{2}(X,\mathcal{X},\mu)$ (resp. almost everywhere).

In [20] the authors showed that the factor $Z_{d-1}$ is characteristic for the
scheme $\\{n,2n,\cdots,dn\\}$, and hence proved that the averages
$\frac{1}{N}\sum_{n=0}^{N-1}f_{1}(T^{n}x)f_{2}(T^{2n}x)\cdots f_{d}(T^{dn}x)$
converge in $L^{2}(X,\mu)$.

###  2.12. Furstenberg’s Theorem and Bourgain’s Theorem

#### 2.12.1.

Let $(X,\mathcal{X},\mu,T)$ be a m.p.s. Let

${\mathcal{H}}_{\rm rat}=\overline{\\{f\in L^{2}(X,\mu):\exists
a\in\mathbb{N}\ s.t.\ T^{a}f=f\\}},$

and let $\mathcal{X}_{\rm rat}\subseteq\mathcal{X}$ such that
${\mathcal{H}}_{\rm rat}=L^{2}(X,\mathcal{X}_{\rm rat},\mu).$

######  Theorem 2.20 (Furstenberg).

[17, Lemma 3.14] Let $(X,\mathcal{X},\mu,T)$ be a m.p.s. and let $q(n)$ be a
non-constant integral polynomial. Then for all $f\in L^{2}(X,\mu)$,

$\frac{1}{N}\sum_{n=0}^{N-1}T^{q(n)}f-\frac{1}{N}\sum_{n=0}^{N-1}T^{q(n)}\mathbb{E}(f|\mathcal{X}_{\rm
rat})\longrightarrow 0,\ N\to\infty,$

in $L^{2}(X,\mu)$. In particular, if $(X,\mathcal{X},\mu,T)$ is totally
ergodic 222A m.p.s. $(X,\mathcal{X},\mu,T)$ is totally ergodic if
$(X,\mathcal{X},\mu,T^{k})$ is ergodic for all $k\in\mathbb{N}$, then

$\frac{1}{N}\sum_{n=0}^{N-1}T^{q(n)}f\longrightarrow\int_{X}fd\mu,\
N\to\infty,$

in $L^{2}(X,\mu)$.

######  Theorem 2.21 (Bourgain).

[6, Theorem 1] Let $(X,\mathcal{X},\mu,T)$ be a m.p.s. and let $q(n)$ be a
non-constant integral polynomial and $p>1$. Then there exists a constant
$C(p,q)>0$ such that for all $f\in L^{p}(X,\mu)$,

$\Big{\|}\sup_{N\in\mathbb{N}}\big{|}\frac{1}{N}\sum_{n=0}^{N-1}T^{q(n)}f\big{|}\Big{\|}_{L^{p}(\mu)}\leq
C(p,q)\|f\|_{L^{p}(\mu)}.$

Moreover, $\displaystyle\frac{1}{N}\sum_{n=0}^{N-1}T^{q(n)}f$ converges almost
surely, as $N\to\infty$.

######  Remark 2.22.

By Furstenberg’s theorem and Bourgain’s theorem, if $(X,\mathcal{X},\mu,T)$ is
totally ergodic (in particular, when $T$ is weakly mixing), then for all $f\in
L^{2}(X,\mu)$

$\frac{1}{N}\sum_{n=0}^{N-1}T^{q(n)}f\longrightarrow\int_{X}fd\mu,\
N\to\infty,$

converges almost surely.

#### 2.12.2.

By Theorem 2.21 one can show that:

######  Proposition 2.23.

[10, Corollary 2.2] Let $T_{1},T_{2},\cdots,T_{d}$ be invertible measure
preserving transformations acting on a probability space
$(X,\mathcal{X},\mu)$, and let $p_{1},\ldots,p_{d}$ be integral polynomials.
Let $q_{1},\ldots,q_{d}\in\mathbb{R}$ be positive real numbers such that
$\frac{1}{q}=\sum_{i=1}^{d}\frac{1}{q_{i}}<1$. Then all the $d$-tuples
$(f_{1},\ldots,f_{d})$ for which the limit of the averages

(2.10)
$\frac{1}{N}\sum_{n=0}^{N-1}f_{1}(T_{1}^{p_{1}(n)}x)f_{2}(T_{2}^{p_{2}(n)}x)\cdots
f_{d}(T_{2}^{p_{d}(n)}x)$

exits a.e. is closed in $L^{q_{1}}(X,\mu)\times\cdots\times L^{q_{d}}(X,\mu)$.

This proposition says that if
$\\{(f_{1}^{(k)},\ldots,f_{d}^{(k)})\\}_{k\in\mathbb{N}}\subseteq\prod_{i=1}^{d}L^{q_{i}}(X,\mu)$
and the limit of the averages $(\ref{MEA-pol})$ exists a.e. for each
$(f_{1}^{(k)},\ldots,f_{d}^{(k)}),k\in\mathbb{N}$, then for any limit point
$(f_{1},\ldots,f_{d})$ of
$\\{(f_{1}^{(k)},\ldots,f_{d}^{(k)})\\}_{k\in\mathbb{N}}$ in
$\prod_{i=1}^{d}L^{q_{i}}(X,\mu)$,
$\displaystyle\frac{1}{N}\sum_{n=0}^{N-1}f_{1}(T_{1}^{p_{1}(n)}x)\cdots
f_{d}(T_{d}^{p_{d}(n)}x)$ exists a.e. Hence by this result, to study
$(\ref{MEA-pol})$ in $\prod_{i=1}^{d}L^{q_{i}}(X,\mu)$, one may assume that
$f_{1},\ldots,f_{d}\in L^{\infty}(X,\mu)$ or other dense subsets.

###  2.13. Multiple polynomial ergodic theorems

The following theorem was proven under some supplementary assumption about the
polynomials in [21] and was proven in its general form in [31]. See [41] for a
more general result.

######  Theorem 2.24 (Multiple Polynomial Ergodic Theorem).

Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s. and $d\in\mathbb{N}$. Let
$p_{1},p_{2},\ldots,p_{d}$ be integral polynomials. Then for any Følner
sequence $\\{\Phi_{N}\\}_{N=1}^{\infty}$ in $\mathbb{Z}$ and for any
$f_{1},f_{2},\ldots,f_{d}\in L^{\infty}(X)$, the averages

$\frac{1}{|\Phi_{N}|}\sum_{{n}\in\Phi_{N}}T^{p_{1}({n})}f_{1}\cdots
T^{p_{d}({n})}f_{d}$

converge in $L^{2}(X,\mu)$ as $N\to\infty$.

The final result we state here is the following

######  Theorem 2.25.

[5, Theorem A0] Let $(X,\mathcal{X},\mu)$ be a probability space, let
$T_{1},\ldots,T_{d}$ be commuting measure preserving transformations of $X$,
let $p_{1},p_{2},\cdots,p_{d}$ be integral polynomials with
$p_{1}(0)=\cdots=p_{d}(0)=0$, and let $A\in X$ with $\mu(A)>0$. Then

$\liminf_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}\mu(T_{1}^{-p_{1}(n)}A\cap
T_{2}^{-p_{2}(n)}A\cap\cdots\cap T_{d}^{-p_{d}(n)}A)>0.$

##  3\. Polynomial Furstenberg joinings

In this section we introduce polynomial Furstenberg joinings and give some
basic properties.

###  3.1. Standard measures on product spaces

In this subsection, we explain why we need to assume that $X$ is a compact
metric space in a m.p.s. $(X,\mathcal{X},\mu,T)$. For more details see [16]
and [17].

#### 3.1.1.

Recall that two probability spaces $(X,\mathcal{X},\mu)$,
$(X^{\prime},\mathcal{X}^{\prime},\mu^{\prime})$ are isomorphic if there exist
null sets $N\subseteq X,\widetilde{N}\subseteq\widetilde{X}$ and an one to one
measurable, measure preserving map $\phi:X\setminus
N\rightarrow\widetilde{X}\setminus\widetilde{N}$. We say that
$(X,\mathcal{X},\mu)$ is a regular measure space333Here we use definitions
from [16, 17]. This notion consists with the Lebesgue space used in the
previous sections. if it is isomorphic to a measure space
$(\widetilde{X},\widetilde{\mathcal{X}},\widetilde{\mu})$, where
$\widetilde{X}$ is a compact metric space, $\widetilde{\mathcal{X}}$ the
$\sigma$-algebra of Borel sets, and $\widetilde{\mu}$ a regular Borel measure
(in this case, we say
$(\widetilde{X},\widetilde{\mathcal{X}},\widetilde{\mu})$ is a topological
model of $(X,\mathcal{X},\mu)$). The advantage in dealing with compact metric
spaces is that in this situation measures are determined by positive linear
functionals on the algebra of continuous functions $C(X)$. If $X$ is compact
metric, this algebra is separable and the functional is determined by its
values on a dense countable set.

A m.p.s. $(X,\mathcal{X},\mu,T)$ is called regular if $(X,\mathcal{X},\mu)$ is
a regular measure space. A regular m.p.s, always has a compact metric model
$(\widetilde{X},\widetilde{\mathcal{X}},\widetilde{\mu},\widetilde{T})$, where
$\widetilde{T}$ is a homeomorphism of the compact metric space $\widetilde{X}$
[17, Theorem 5.1.5].

#### 3.1.2.

Let $(X_{i},\mathcal{X}_{i},\mu_{i})$ be regular probability spaces, $i\in I$
and $I$ be a finite or countable index set. Let $\Omega=\prod_{i\in I}X_{i}$
with the product $\sigma$-algebra $\mathcal{B}=\prod_{i\in I}\mathcal{X}_{i}$,
i.e. the least $\sigma$-algebra with respect to which the projections
$\Omega\rightarrow X_{i}$ are measurable for all $i\in I$.

A measure $\lambda$ on $(\Omega,\mathcal{B})$ is called standard if its image
in each $X_{i}$ is $\mu_{i}$.444Let $(X_{i},\mathcal{X}_{i},\mu_{i},T_{i})$ be
m.p.s., where $i\in I$ and $I$ is an index set. Then a Borel probability
measure $\lambda$ on $\prod_{i\in I}X_{i}$ is a joining of
$\\{(X_{i},\mathcal{X}_{i},\mu_{i},T_{i})\\}_{i\in I}$ if it is a $\prod_{i\in
I}T_{i}$-invariant standard measure on $\prod_{i\in I}X_{i}$. For
$i_{1},\ldots,i_{n}\in I$ and $f_{i_{j}}:X_{i_{j}}\rightarrow\mathbb{C},1\leq
j\leq n$, let $\displaystyle f_{i_{1}}\otimes f_{i_{2}}\otimes\cdots\otimes
f_{i_{n}}:\Omega=\prod_{i\in I}X_{i}\rightarrow\mathbb{C}$ be defined as
follows:

(3.1) $f_{i_{1}}\otimes f_{i_{2}}\otimes\cdots\otimes
f_{i_{n}}\big{(}(x_{i})_{i\in
I}\big{)}=f_{i_{1}}(x_{i_{1}})f_{i_{2}}(x_{i_{2}})\cdots
f_{i_{n}}(x_{i_{n}}).$

It is easy to see that if $f_{i_{j}}\in
L^{\infty}(X_{i_{j}},\mu_{i_{j}}),1\leq j\leq n$, then $\displaystyle
f_{i_{1}}\otimes f_{i_{2}}\otimes\cdots\otimes f_{i_{n}}\in
L^{\infty}(\Omega,\lambda)$ since $\lambda$ is standard.

Note that if each $(X_{i},\mathcal{X}_{i},\mu_{i})$ is regular, and $\lambda$
on $\Omega$ is standard, then $(\Omega,\mathcal{B},\lambda)$ is also regular.
Let
$\\{(\widetilde{X}_{i},\widetilde{\mathcal{X}}_{i},\widetilde{\mu}_{i})\\}_{i\in
I}$ be compact metric models of $\\{(X_{i},\mathcal{X}_{i},\mu_{i})\\}_{i\in
I}$. Let $\\{O_{i}\\}_{i\in I}$ and $\\{\widetilde{O}_{i}\\}_{i\in I}$ be null
sets in each $X_{i}$ and $\widetilde{X}_{i}$ respectively such that
$\phi_{i}:X_{i}\setminus
O_{i}\rightarrow\widetilde{X}_{i}\setminus\widetilde{O}_{i}$ is isomorphic.
Since $\mu_{i}(O_{i})=0$, we have

$\lambda\big{(}\prod_{i\in I}(X_{i}\setminus O_{i})\big{)}=1.$

Thus $\lambda$ may be carried over to a unique regular Borel measure in
$\prod_{i\in I}\widetilde{X}_{i}$.

#### 3.1.3.

Now assume that each $X_{i}$ is a compact metric space and $\Omega=\prod_{i\in
I}X_{i}$. Let

$\mathcal{F}=\Big{\\{}\sum_{t=1}^{k}f^{(t)}_{i^{(t)}_{1}}\otimes
f^{(t)}_{i^{(t)}_{2}}\otimes\cdots\otimes
f^{(t)}_{i^{(t)}_{n_{t}}}:k\in\mathbb{N},i^{(t)}_{j}\in
I,f^{(t)}_{i^{(t)}_{j}}\in C(X_{i^{(t)}_{j}}),1\leq j\leq n_{t},1\leq t\leq
k\Big{\\}}.$

$\mathcal{F}$ is a subalgebra of $C(\Omega)$ and separates points. By the
Stone-Weierstrass theorem, $\mathcal{F}$ is dense in $C(\Omega)$. Thus each
positive linear functional on $C(\Omega)$ is determined uniquely by its value
on $\mathcal{F}$. In particular, if we want to define a measure $\lambda$ on
$\Omega$, we only need to check $\lambda(f_{i_{1}}\otimes
f_{i_{2}}\otimes\cdots\otimes f_{i_{n}})$ for all $n\in\mathbb{N}$, and for
all $f_{i_{1}}\in C(X_{i_{1}}),\ldots,f_{i_{n}}\in C(X_{i_{n}})$.

In the sequel we always assume that $(X,\mathcal{X},\mu,T)$ is regular, i.e.
$X$ is a compact metric space.

###  3.2. A self-joining $\mu_{\mathcal{A}}^{(\infty)}$

####  3.2.1. Some notations

Let $(X,\mathcal{X},\mu,T)$ be a m.p.s. and $d\in\mathbb{N}$. Let
$\mathcal{A}=\\{p_{1},p_{2},\ldots,p_{d}\\}$ be a family of non-constant
integral polynomials. We will define a joining $\mu^{(\infty)}_{\mathcal{A}}$
on $(X^{d})^{\mathbb{Z}}$.

The point of $(X^{d})^{\mathbb{Z}}$ is denoted by

${\bf x}=({\bf x}_{n})_{n\in\mathbb{Z}}=(\ldots,{\bf
x}_{-1},\underset{\bullet}{\bf x_{0}},{\bf
x}_{1},\ldots)=\Big{(}(x^{(1)}_{n},x^{(2)}_{n},\ldots,x^{(d)}_{n})\Big{)}_{n\in\mathbb{Z}},$

where $``\ \underset{\bullet}{\ }"$ means the $0$-th coordinate. Let
$\vec{p}=(p_{1},p_{2},\ldots,p_{d})$ and let $T^{\vec{p}(n)}:X^{d}\rightarrow
X^{d}$ be defined by

(3.2)
$T^{\vec{p}(n)}\Big{(}(x_{1},x_{2},\ldots,x_{d})\Big{)}=(T^{p_{1}(n)}x_{1},T^{p_{2}(n)}x_{2},\ldots,T^{p_{d}(n)}x_{d}).$

Define $T^{\infty}:(X^{d})^{\mathbb{Z}}\rightarrow(X^{d})^{\mathbb{Z}}$ such
that $T^{\infty}=\cdots\times T^{(d)}\times T^{(d)}\times\cdots\ (\infty\
\text{times}),$ i.e.

$T^{\infty}({\bf x}_{n})_{n\in\mathbb{Z}}=(T^{(d)}{\bf
x}_{n})_{n\in\mathbb{Z}}.$

Let $\sigma:(X^{d})^{\mathbb{Z}}\rightarrow(X^{d})^{\mathbb{Z}}$ be the shift
map, i.e., for all ${\bf x}=({\bf
x}_{n})_{n\in\mathbb{Z}}\in(X^{d})^{\mathbb{Z}}$,

$(\sigma{\bf x})_{n}={\bf x}_{n+1},\ \forall n\in\mathbb{Z}.$

Let $x^{\otimes d}=(x,x,\ldots,x)\in X^{d}$ and

$\Delta_{\infty}(X)=\\{x^{(\infty)}\triangleq(\ldots,x^{\otimes d},x^{\otimes
d},\ldots)\in(X^{d})^{\mathbb{Z}}:x\in X\\}.$

For each $x\in X$, put

(3.3)
$\begin{split}\omega_{x}^{\mathcal{A}}&\triangleq(T^{\vec{p}(n)}x^{\otimes
d})_{n\in\mathbb{Z}}\\\ &=(\cdots,T^{\vec{p}(-1)}(x^{\otimes
d}),\underset{\bullet}{T^{\vec{p}(0)}x^{\otimes d}},T^{\vec{p}(1)}(x^{\otimes
d}),T^{\vec{p}(2)}(x^{\otimes d}),\cdots)\in(X^{d})^{\mathbb{Z}}.\end{split}$

####  3.2.2. An $\infty$-joining with respect to $\mathcal{A}$

Now we will define an $\infty$-joining with respect to a given m.p.s.
$(X,\mathcal{X},\mu,T)$ and $\mathcal{A}=\\{p_{1},\ldots,p_{d}\\}$.

######  Definition 3.1.

Let $(X,\mathcal{X},\mu,T)$ be a m.p.s., $d\in\mathbb{N}$ and
$\mathcal{A}=\\{p_{1},p_{2},\ldots,p_{d}\\}$ be a set of non-constant integral
polynomials. Let $\mu_{\mathcal{A}}^{(\infty)}$ be the measure on
$(X^{d})^{\mathbb{Z}}$ defined by

(3.4) $\int_{(X^{d})^{\mathbb{Z}}}F({\bf x})d\mu_{\mathcal{A}}^{(\infty)}({\bf
x})=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}F(\sigma^{n}(\omega_{x}^{\mathcal{A}}))d\mu(x)$

for all $F\in C((X^{d})^{\mathbb{Z}})$. We call $\mu^{(\infty)}_{\mathcal{A}}$
the $\infty$-joining with respect to $\mathcal{A}$.

Note that the limits exist by the remark below and the equality

$\begin{split}\sigma^{n}(\omega_{x}^{\mathcal{A}})=(T^{\vec{p}(n+j)}x^{\otimes
d})_{j\in\mathbb{Z}}=(\ldots,T^{\vec{p}(n-1)}x^{\otimes
d},\underset{\bullet}{T^{\vec{p}(n)}x^{\otimes d}},T^{\vec{p}(n+1)}x^{\otimes
d},\ldots).\end{split}$

Clearly, the measure $\mu_{\mathcal{A}}^{(\infty)}$ is invariant under
$T^{\infty}$ and $\sigma$.

######  Remark 3.2.

Now we explain the existence of (3.4). Let $f^{\otimes}$ be elements of
$C(X^{d})$ with the following forms:

$f^{\otimes}\triangleq f^{(1)}\otimes f^{(2)}\otimes\cdots\otimes f^{(d)},$

where $f^{(1)},f^{(2)},\cdots,f^{(d)}\in C(X)$. By the Stone-Weierstrass
theorem, the set

$\mathcal{F}=\left\\{\bigotimes_{j=-l}^{l}f_{j}^{\otimes}:l\in\mathbb{N},f^{\otimes}_{1},f^{\otimes}_{2},\ldots,f^{\otimes}_{l}\in
C(X^{d})\right\\}$

is dense in $C((X^{d})^{\mathbb{Z}})$. Note that by (3.1),
$\bigotimes_{j=-l}^{l}f_{j}^{\otimes}$ can be regarded as an element of
$C((X^{d})^{\mathbb{Z}})$.

Thus (3.4) holds if and only if for all $l\in\mathbb{N}$ and all
$f_{j}^{\otimes}=f_{j}^{(1)}\otimes f_{j}^{(2)}\otimes\cdots\otimes
f_{j}^{(d)}\in C(X^{d}),-l\leq j\leq l$,

(3.5)
$\begin{split}&\int_{(X^{d})^{\mathbb{Z}}}\Big{(}\bigotimes_{j=-l}^{l}f_{j}^{\otimes}\Big{)}({\bf
x})d\mu_{\mathcal{A}}^{(\infty)}({\bf x})\\\
&=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}\prod_{j=-l}^{l}f_{j}^{\otimes}(T^{\vec{p}(n+j)}x^{\otimes
d})d\mu(x)\\\
&=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}\prod_{j=-l}^{l}f_{j}^{(1)}(T^{p_{1}(n+j)}x)f_{j}^{(2)}(T^{p_{2}(n+j)}x)\cdots
f_{j}^{(d)}(T^{p_{d}(n+j)}x)d\mu(x),\end{split}$

where the limits exist by Theorem 2.24. Thus $\mu^{(\infty)}_{\mathcal{A}}$ is
well defined.

Another way to see $\mu_{\mathcal{A}}^{(\infty)}$ is as follows:

(3.6) $\frac{1}{N}\sum_{n=0}^{N-1}(\cdots\times
T^{\vec{p}(n-1)}\times\underset{\bullet}{T^{\vec{p}(n)}}\times
T^{\vec{p}(n+1)}\times\cdots)_{*}\mu_{\Delta}^{\infty}\longrightarrow\mu_{\mathcal{A}}^{(\infty)},$

as $N\to\infty$ in the weak∗ topology of
${\mathcal{M}}((X^{d})^{\mathbb{Z}})$, where $\mu_{\Delta}^{\infty}$ is the
diagonal measure on $(X^{d})^{\mathbb{Z}}$ as defined as follows

$\int_{(X^{d})^{\mathbb{Z}}}\Big{(}\bigotimes_{j=-l}^{l}f_{j}^{\otimes}\Big{)}({\bf
x})\ d\mu_{\Delta}^{\infty}({\bf
x})=\int_{X}\prod_{j=-l}^{l}f_{j}^{\otimes}(x^{\otimes d})\ d\mu(x),$

for all $l\in\mathbb{N}$ and all $f_{j}^{\otimes}=f_{j}^{(1)}\otimes
f_{j}^{(2)}\otimes\cdots\otimes f_{j}^{(d)}\in C(X^{d}),-l\leq j\leq l$.

######  Remark 3.3.

Note that $\mu_{\mathcal{A}}^{(\infty)}$ is a standard measure. To see this
fact, let $\Pi:(X^{d})^{\mathbb{Z}}\rightarrow X$ be the projection to some
coordinate. Let $\widetilde{\mu}=(\Pi)_{*}\mu_{\mathcal{A}}^{(\infty)}$. Then
for all $f\in C(X)$,

$\widetilde{\mu}(f)=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}f(T^{q(n)}x)d\mu(x)=\int_{X}fd\mu=\mu(f),$

where $q(n)=p(n+m)$ for some $p\in\mathcal{A},m\in\mathbb{Z}$. Thus
$\widetilde{\mu}=\mu$, and $\mu^{(\infty)}_{\mathcal{A}}$ is a standard
measure.

#### 3.2.3.

Now we show (3.5) still holds if we replace the continuous functions by
bounded measurable ones, and the definition of $\mu^{(\infty)}_{\mathcal{A}}$
is independent of the choice of topological models of $(X,\mathcal{X},\mu,T)$.
That is, we will show

######  Proposition 3.4.

Let $(X,\mathcal{X},\mu,T)$ be a m.p.s., and let
$\mathcal{A}=\\{p_{1},\ldots,p_{d}\\}$ be a family of non-constant integral
polynomials. Then for all $l\in\mathbb{N}$ and all
$f_{j}^{\otimes}=f_{j}^{(1)}\otimes f_{j}^{(2)}\otimes\cdots\otimes
f_{j}^{(d)}$ (where $f^{(i)}_{j}\in L^{\infty}(X,\mu),1\leq i\leq d,-l\leq
j\leq l$)

(3.7)
$\begin{split}\int_{(X^{d})^{\mathbb{Z}}}\Big{(}\bigotimes_{j=-l}^{l}f_{j}^{\otimes}\Big{)}({\bf
x})d\mu_{\mathcal{A}}^{(\infty)}({\bf
x})=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}\prod_{j=-l}^{l}f_{j}^{\otimes}(T^{\vec{p}(n+j)}x^{\otimes
d})d\mu(x),\end{split}$

where $f_{j}^{\otimes}(T^{\vec{p}(n+j)}x^{\otimes
d})=f_{j}^{(1)}(T^{p_{1}(n+j)}x)f_{j}^{(2)}(T^{p_{2}(n+j)}x)\cdots
f_{j}^{(d)}(T^{p_{d}(n+j)}x).$

See Appendix A for a proof.

######  Proposition 3.5.

Let $(X_{i},\mathcal{X}_{i},\mu_{i},T_{i}),i=1,2$, be two isomorphic m.p.s.
such that

$\phi:(X_{1},\mathcal{X}_{1},\mu_{1},T)\rightarrow(X_{2},\mathcal{X}_{2},\mu_{2},T)$

is an isomorphism. Let $\mathcal{A}=\\{p_{1},\ldots,p_{d}\\}$ be a set of non-
constant integral polynomials. Then

$\phi^{\infty}:((X_{1}^{d})^{\mathbb{Z}},(\mathcal{X}_{1}^{d})^{\mathbb{Z}},(\mu_{1})_{\mathcal{A}}^{(\infty)},\langle
T_{1}^{\infty},\sigma\rangle)\rightarrow((X_{2}^{d})^{\mathbb{Z}},(\mathcal{X}_{2}^{d})^{\mathbb{Z}},(\mu_{2})_{\mathcal{A}}^{(\infty)},\langle
T_{2}^{\infty},\sigma\rangle)$

is also an isomorphism.

###### Proof.

Note that $\phi^{\infty}$ is defined by $\phi^{\infty}({\bf
x}_{n})_{n\in\mathbb{Z}}=(\phi^{(d)}{\bf x}_{n})_{n\in\mathbb{Z}},\forall({\bf
x}_{n})_{n\in\mathbb{Z}}\in(X_{1}^{d})^{\mathbb{Z}}.$ It is easy to verify
that $\phi^{\infty}\circ T_{1}^{\infty}=T_{2}^{\infty}\circ\phi^{\infty}$ and
$\sigma\circ\phi^{\infty}=\phi^{\infty}\circ\sigma$.

It suffices to show that
$\phi^{\infty}_{*}(\mu_{1})_{\mathcal{A}}^{(\infty)}=(\mu_{2})_{\mathcal{A}}^{(\infty)}$.
By Proposition 3.4, for all $l\in\mathbb{N}$ and all
$f_{j}^{\otimes}=f_{j}^{(1)}\otimes f_{j}^{(2)}\otimes\cdots\otimes
f_{j}^{(d)}$ (where $f^{(i)}_{j}\in L^{\infty}(X_{2},\mu_{2}),1\leq i\leq
d,-l\leq j\leq l$)

$\begin{split}&\int_{(X_{2}^{d})^{\mathbb{Z}}}\Big{(}\bigotimes_{j=-l}^{l}f_{j}^{\otimes}\Big{)}({\bf
x})d(\mu_{2})_{\mathcal{A}}^{(\infty)}({\bf
x})=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X_{2}}\prod_{j=-l}^{l}f_{j}^{\otimes}(T_{2}^{\vec{p}(n+j)}x^{\otimes
d})d\mu_{2}(x)\\\
&\quad=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X_{2}}\prod_{j=-l}^{l}f_{j}^{(1)}(T_{2}^{p_{1}(n+j)}x)\cdots
f_{j}^{(d)}(T_{2}^{p_{d}(n+j)}x)d\mu_{2}(x)\\\
&\quad=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X_{2}}\prod_{j=-l}^{l}f_{j}^{(1)}(T_{2}^{p_{1}(n+j)}x)\cdots
f_{j}^{(d)}(T_{2}^{p_{d}(n+j)}x)d(\phi_{*}\mu_{1})(x)\\\
&\quad=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X_{1}}\prod_{j=-l}^{l}f_{j}^{(1)}\circ\phi(T_{1}^{p_{1}(n+j)}z)\cdots
f_{j}^{(d)}\circ\phi(T_{1}^{p_{d}(n+j)}z)d\mu_{1}(z)\\\
&\quad=\int_{(X_{1}^{d})^{\mathbb{Z}}}\Big{(}\bigotimes_{j=-l}^{l}f_{j}^{\otimes}\Big{)}\circ\phi^{\infty}({\bf
z})d(\mu_{1})_{\mathcal{A}}^{(\infty)}({\bf
z})=\int_{(X_{2}^{d})^{\mathbb{Z}}}\Big{(}\bigotimes_{j=-l}^{l}f_{j}^{\otimes}\Big{)}({\bf
x})d\big{(}\phi^{\infty}_{*}(\mu_{1})_{\mathcal{A}}^{(\infty)}\big{)}({\bf
x}).\end{split}$

The proof is complete. ∎

###  3.3. Some special cases

To get a better understanding of $\mu^{(\infty)}_{\mathcal{A}}$, we explain it
by some examples.

####  3.3.1. The case when $\mathcal{A}=\\{p\\}$ with $p(n)=n$

In this case we show that

(3.8)
$(X^{\mathbb{Z}},\mathcal{X}^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}},\sigma)\cong(X,\mathcal{X},\mu,T),$

where $\cong$ means that two m.p.s. are isomorphic.

Now $p(n)=n$, and we have for $x\in X$

$\omega_{x}^{\mathcal{A}}=(\ldots,T^{-1}x,\underset{\bullet}{x},Tx,T^{2}x,\ldots)=(T^{n}x)_{n\in\mathbb{Z}}.$

Note that

$\sigma\omega_{x}^{\mathcal{A}}=(\ldots,T^{-1}x,x,\underset{\bullet}{Tx},T^{2}x,\ldots)=T^{\infty}\omega_{x}^{\mathcal{A}}.$

For all $l\in\mathbb{N}$, and all $f_{-l},f_{-l+1},\ldots,f_{l}\in
L^{\infty}(X,\mu)$,

$\begin{split}&\int_{X^{\mathbb{Z}}}f_{-l}(x_{-l})f_{-l+1}(x_{-l+1})\cdots
f_{l}(x_{l})d\mu_{\mathcal{A}}^{(\infty)}(\ldots,x_{-1},x_{0},x_{1},\ldots)\\\
&\quad=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}f_{-l}(T^{n-l}x)f_{-l+1}(T^{n-l+1}x)\cdots
f_{l}(T^{n+l}x)d\mu(x)\\\
&\quad=\int_{X}f_{-l}(T^{-l}x)f_{-l+1}(T^{-l+1}x)\cdots
f_{l}(T^{l}x)d\mu(x).\end{split}$

Equivalently, for all $l\in\mathbb{N}$ and
$A_{-l},A_{-l+1},\ldots,A_{l}\in\mathcal{X}$,

$\mu_{\mathcal{A}}^{(\infty)}(A_{-l}\times A_{-l+1}\times\cdots\times
A_{l})=\mu(T^{l}A_{-l}\cap T^{l-1}A_{-l+1}\cap\cdots\cap T^{-l}A_{l}).$

That means

(3.9) $\mu_{\mathcal{A}}^{(\infty)}=(\cdots\times T^{-1}\times{\rm id}\times
T\times T^{2}\times\cdots)_{*}\mu^{\infty}_{\Delta}.$

Thus $(X^{\mathbb{Z}},\mathcal{X}^{\mathbb{Z}},\mu_{\mathcal{A}}^{(\infty)})$
is isomorphic to
$(X^{\mathbb{Z}},\mathcal{X}^{\mathbb{Z}},\mu_{\Delta}^{\infty})$, and hence
it is isomorphic to $(X,\mathcal{X},\mu)$. That is, if we define

$\phi:(X,\mathcal{X},\mu)\rightarrow(X^{\mathbb{Z}},\mathcal{X}^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}}),\
x\mapsto\omega_{x}^{\mathcal{A}},\ \forall x\in X$

then it is an isomorphism between them. So
$(X^{\mathbb{Z}},\mathcal{X}^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}},\sigma)\cong(X,\mathcal{X},\mu,T).$

####  3.3.2. The case when $\mathcal{A}=\\{p_{1},p_{2},\ldots,p_{d}\\}$ with
$p_{i}(n)=a_{i}n$, $1\leq i\leq d$

Let $\mathcal{A}=\\{p_{1},p_{2},\ldots,p_{d}\\}$ with $p_{i}(n)=a_{i}n,1\leq
i\leq d$, where $a_{1},\ldots,a_{d}$ are distinct non-zero integers. In this
case we show that

(3.10)
$((X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}},\sigma)\cong(X^{d},\mathcal{X}^{d},\mu^{(d)}_{\vec{a}},\tau_{\vec{a}}),$

where $\tau_{\vec{a}}=T^{a_{1}}\times\cdots\times T^{a_{d}}$ and
$\mu^{(d)}_{\vec{a}}$ is the Furstenberg joining.

As $p_{i}(n)=a_{i}n,1\leq i\leq d$, we have that

$\omega_{x}^{\mathcal{A}}=\Big{(}(T^{a_{1}n}x,T^{a_{2}n}x,\ldots,T^{a_{d}n}x)\Big{)}_{n\in\mathbb{Z}}\in(X^{d})^{\mathbb{Z}}.$

Note that

$\begin{split}\sigma\omega_{x}^{\mathcal{A}}&=\Big{(}(T^{a_{1}(n+1)}x,T^{a_{2}(n+1)}x,\ldots,T^{a_{d}(n+1)}x)\Big{)}_{n\in\mathbb{Z}}\\\
&=\Big{(}\tau_{\vec{a}}(T^{a_{1}n}x,T^{a_{2}n}x,\ldots,T^{a_{d}n}x)\Big{)}_{n\in\mathbb{Z}}=\tau_{\vec{a}}^{\infty}\omega_{x}^{\mathcal{A}},\end{split}$

where
$\tau_{\vec{a}}^{\infty}=\cdots\times\tau_{\vec{a}}\times\tau_{\vec{a}}\times\cdots.$

Now we calculate the $\infty$-joining $\mu_{\mathcal{A}}^{(\infty)}$. For all
$l\in\mathbb{N}$ and all $f_{j}^{\otimes}=f_{j}^{(1)}\otimes\cdots\otimes
f_{j}^{(d)}\in C(X^{d}),-l\leq j\leq l$,

$\begin{split}&\int_{(X^{d})^{\mathbb{Z}}}\Big{(}\bigotimes_{j=-l}^{l}f_{j}^{\otimes}\Big{)}({\bf
x})d\mu_{\mathcal{A}}^{(\infty)}({\bf x})\\\
&\quad=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}\prod_{j=-l}^{l}f_{j}^{\otimes}(T^{a_{1}(n+j)}x,\ldots,T^{a_{d}(n+j)}x)d\mu(x)\\\
&\quad=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}(T^{a_{1}}\times
T^{a_{2}}\times\cdots\times
T^{a_{d}})^{n}\Big{(}\prod_{j=-l}^{l}f_{j}^{\otimes}(T^{a_{1}j}x,\ldots,T^{a_{d}j}x)\Big{)}d\mu(x)\\\
&\quad=\int_{X^{d}}\prod_{j=-l}^{l}f_{j}^{\otimes}(T^{a_{1}j}x_{1},\ldots,T^{a_{d}j}x_{d})d\mu^{(d)}_{\vec{a}}(x_{1},x_{2},\ldots,x_{d})\\\
&\quad=\int_{X^{d}}\prod_{j=-l}^{l}f_{j}^{(1)}(T^{a_{1}j}x_{1})f_{j}^{(2)}(T^{a_{2}j}x_{2})\cdots
f_{j}^{(d)}(T^{a_{d}j}x_{d})d\mu^{(d)}_{\vec{a}}(x_{1},x_{2},\ldots,x_{d}).\end{split}$

Equivalently, for all $l\in\mathbb{N}$ and
$A_{j}^{\otimes}=A_{j}^{(1)}\times\cdots\times
A_{j}^{(d)}\in\mathcal{X}^{d},-l\leq j\leq l$,

(3.11) $\mu^{(\infty)}_{\mathcal{A}}(A_{-l}^{\otimes}\times
A_{-l+1}^{\otimes}\times\cdots\times
A_{l}^{\otimes})=\mu^{(d)}_{\vec{a}}\big{(}\tau_{\vec{a}}^{l}A^{\otimes}_{-l}\cap\tau_{\vec{a}}^{l-1}A_{-l+1}^{\otimes}\cap\cdots\cap\tau_{\vec{a}}^{-l}A^{\otimes}_{l}\big{)}.$

Similar to the definition of $\mu_{\Delta}^{\infty}$, we define
$(\mu^{(d)}_{\vec{a}})^{\infty}_{\Delta}$ on $(X^{d})^{\mathbb{Z}}$ as
follows:

$\int_{(X^{d})^{\mathbb{Z}}}{\bf f}_{-l}({\bf x}_{-l})\cdots{\bf f}_{l}({\bf
x}_{l})d(\mu^{(d)}_{\vec{a}})^{\infty}_{\Delta}(\ldots,{\bf x}_{-1},{\bf
x}_{0},{\bf x}_{1},\ldots)=\int_{X^{d}}{\bf f}_{-l}({\bf x})\cdots{\bf
f}_{l}({\bf x})d\mu^{(d)}_{\vec{a}}({\bf x}),$

for all $l\in\mathbb{N}$ and all ${\bf f}_{-l},\ldots,{\bf f}_{l}\in
C(X^{d})$. It is clear that
$((X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},(\mu^{(d)}_{\vec{a}})^{\infty}_{\Delta})$
is isomorphic to $(X^{d},\mathcal{X}^{d},\mu^{(d)}_{\vec{a}})$.

Thus (3.11) means that

$\mu^{(\infty)}_{\mathcal{A}}=(\cdots\times\tau_{\vec{a}}^{-1}\times{\rm
id}_{X^{d}}\times\tau_{\vec{a}}\times\tau_{\vec{a}}^{2}\times\cdots)_{*}(\mu^{(d)}_{\vec{a}})_{\Delta}^{\infty}.$

It follows that
$\big{(}(X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu_{\mathcal{A}}^{(\infty)}\big{)}$
is isomorphic to
$((X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},(\mu^{(d)}_{\vec{a}})^{\infty}_{\Delta})$,
and hence also isomorphic to $(X^{d},\mathcal{X}^{d},\mu^{(d)}_{\vec{a}})$.
That is, if we define

$\begin{split}\phi:(X^{d},\mathcal{X}^{d},\mu^{(d)}_{\vec{a}},\tau_{\vec{a}})&\rightarrow\big{(}(X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu_{\mathcal{A}}^{(\infty)},\sigma\big{)},\\\
{\bf y}&\mapsto(\ldots,\tau^{-1}_{\vec{a}}{\bf y},{\bf y},\tau_{\vec{a}}{\bf
y},\tau_{\vec{a}}^{2}{\bf y},\ldots),\ \forall{\bf y}\in X^{d}\end{split}$

then it is an isomorphism.

####  3.3.3. Weakly mixing m.p.s.

##### (I) Case $\mathcal{A}=\\{n^{2}\\}$

Let $(X,\mathcal{X},\mu,T)$ be a weakly mixing m.p.s. and
$\mathcal{A}=\\{p\\}$ with $p(n)=n^{2}$. In this case we show that

(3.12)
$(X^{\mathbb{Z}},\mathcal{X}^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}},\sigma)\cong(X^{\mathbb{Z}},\mathcal{X}^{\mathbb{Z}},\mu^{\mathbb{Z}},\sigma),$

that is, it is a Bernoulli system.

Recall that for $x\in X$,
$\omega_{x}^{\mathcal{A}}=(\ldots,T^{p(-1)}x,\underset{\bullet}{x},T^{p(1)}x,T^{p(2)}x,\ldots)=(T^{p(n)}x)_{n\in\mathbb{Z}}.$
For all $l\in\mathbb{N}$, and all $f_{-l},\ldots,f_{l}\in C(X)$,

$\begin{split}&\int_{X^{\mathbb{Z}}}f_{-l}(x_{-l})f_{-l+1}(x_{-l+1})\cdots
f_{l}(x_{l})d\mu_{\mathcal{A}}^{(\infty)}(\ldots,x_{-1},x_{0},x_{1},\ldots)\\\
&\quad=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}\prod_{j=-l}^{l}f_{j}(T^{p(n+j)}x)d\mu(x)\\\
&\quad=\big{(}\int_{X}f_{-l}d\mu\big{)}\big{(}\int_{X}f_{-l+1}d\mu\big{)}\cdots\big{(}\int_{X}f_{l}d\mu\big{)},\end{split}$

where the last equation holds by [2, Theorem 1.1]. Thus
$\mu^{(\infty)}_{\mathcal{A}}=\cdots\times\mu\times\mu\times\cdots=\mu^{\mathbb{Z}}.$

##### (II) Case $\mathcal{A}=\\{n,n^{2}\\}$

Now we study the case when $\mathcal{A}=\\{p_{1},p_{2}\\}$ with
$p_{1}(n)=n,p_{2}(n)=n^{2}$. In this case we show that

(3.13)
$((X^{2})^{\mathbb{Z}},(\mathcal{X}^{2})^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}},\sigma)\cong(X\times
X^{\mathbb{Z}},\mathcal{X}\times\mathcal{X}^{\mathbb{Z}},\mu\times\mu^{\mathbb{Z}},T\times\sigma).$

For $x\in X$,
$\omega_{x}^{\mathcal{A}}=\Big{(}(T^{n}x,T^{n^{2}}x)\Big{)}_{n\in\mathbb{Z}}.$
Thus, for all $l\in\mathbb{N}$, and all
$f_{-l}^{(1)},f_{-l}^{(2)},\ldots,f_{l}^{(1)},f_{l}^{(2)}\in C(X)$, ${\bf
x}=\big{(}(x_{n}^{(1)},x_{n}^{(2)})\big{)}_{n\in\mathbb{Z}}$,

$\begin{split}&\int_{(X^{2})^{\mathbb{Z}}}\prod_{j=-l}^{l}f_{j}^{(1)}(x_{j}^{(1)})f_{j}^{(2)}(x_{j}^{(2)})d\mu_{\mathcal{A}}^{(\infty)}({\bf
x})\\\
&\quad=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}\prod_{j=-l}^{l}f_{j}^{(1)}(T^{n+j}x)f_{j}^{(2)}(T^{(n+j)^{2}}x)d\mu(x))\\\
&\quad=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}\prod_{j=-l}^{l}\big{(}f_{j}^{(1)}\circ
T^{j}\big{)}(T^{n}x)\prod_{j=-l}^{l}f_{j}^{(2)}(T^{(n+j)^{2}}x)d\mu(x)\\\
&\quad=\Big{(}\int_{X}\prod_{j=-l}^{l}\big{(}f_{j}^{(1)}\circ
T^{j}\big{)}d\mu\Big{)}\prod_{j=-l}^{l}\int_{X}f_{j}^{(2)}d\mu,\end{split}$

where the last equation holds by [2, Theorem 1.1]. Hence

$\mu^{(\infty)}_{\mathcal{A}}=\big{(}(\cdots\times
T^{-1}\times\underset{\bullet}{{\rm id}}\times
T\times\cdots)_{*}\mu^{\infty}_{\Delta}\big{)}\times\mu^{\mathbb{Z}}.$

It follows that $\mu^{(\infty)}_{\mathcal{A}}$ is isomorphic to
$\mu\times\mu^{\mathbb{Z}}$. If we define

$\begin{split}\phi:(X\times
X^{\mathbb{Z}},&\mathcal{X}\times\mathcal{X}^{d},\mu\times\mu^{\mathbb{Z}})\rightarrow((X^{2})^{\mathbb{Z}},(\mathcal{X}^{2})^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}}),\\\
&\big{(}x,(x_{n})_{n\in\mathbb{Z}}\big{)}\mapsto\big{(}T^{n}x,x_{n}\big{)}_{n\in\mathbb{Z}},\
\forall\big{(}x,(x_{n})_{n\in\mathbb{Z}}\big{)}\in X\times
X^{\mathbb{Z}},\end{split}$

then it is an isomorphism between them.

###  3.4. An equivalent way to see $\mu^{(\infty)}_{\mathcal{A}}$

In previous subsections we see that when
$\mathcal{A}=\\{p_{1},\ldots,p_{d}\\}$ with $p_{i}(n)=a_{i}n$, (where
$a_{1},\ldots,a_{d}$ are distinct non-zero integers),
$((X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}},\sigma)\cong(X^{d},\mathcal{X}^{d},\mu^{(d)}_{\vec{a}},\tau_{\vec{a}})$.
In this subsection, we give an equivalent way to see
$\mu^{(\infty)}_{\mathcal{A}}$, and it will be more convenient to deal with
certain problems when $\mathcal{A}$ contains linear elements.

#### 3.4.1.

We assume that $\mathcal{A}=\\{p_{1},\ldots,p_{s},p_{s+1},\ldots,p_{d}\\}$,
where $s\geq 0$, $a_{1},\ldots,a_{s}$ are distinct non-zero integers,
$p_{i}(n)=a_{i}n$ ($1\leq i\leq s$) and $\deg p_{i}\geq 2,p_{i}(0)=0,s+1\leq
i\leq d$. Recall that

$\omega_{x}^{\mathcal{A}}=\Big{(}(T^{a_{1}n}x,\ldots,T^{a_{s}n}x,T^{p_{s+1}(n)}x,\ldots,T^{p_{d}(n)}x)\Big{)}_{n\in\mathbb{Z}}\in(X^{d})^{\mathbb{Z}}.$

For all $l\in\mathbb{N}$ and all $f_{j}^{\otimes}=f_{j}^{(1)}\otimes
f_{j}^{(2)}\otimes\cdots\otimes f_{j}^{(d)}\in C(X^{d}),-l\leq j\leq l$,
denote

$f_{j}^{\otimes_{I}}=f_{j}^{(1)}\otimes f_{j}^{(2)}\otimes\cdots\otimes
f_{j}^{(s)},\ f_{j}^{\otimes_{II}}=f_{j}^{(s+1)}\otimes
f_{j}^{(s+2)}\otimes\cdots\otimes f_{j}^{(d)}.$

Thus $f_{j}^{\otimes}=f_{j}^{\otimes_{I}}\otimes f_{j}^{\otimes_{II}}$. Now

(3.14)
$\begin{split}&\quad\int_{(X^{d})^{\mathbb{Z}}}\Big{(}\bigotimes_{j=-l}^{l}f_{j}^{\otimes}\Big{)}({\bf
x})d\mu_{\mathcal{A}}^{(\infty)}({\bf x})\\\
&\quad=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}\prod_{j=-l}^{l}f_{j}^{\otimes}(T^{a_{1}(n+j)}x,T^{a_{2}(n+j)}x,\ldots,T^{p_{d}(n+j)}x)d\mu(x)\\\
&\quad=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}\prod_{j=-l}^{l}f_{j}^{\otimes_{I}}(T^{a_{1}(n+j)}x,\ldots,T^{a_{s}(n+j)}x)\cdot\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
f_{l}^{\otimes_{II}}(T^{p_{s+1}(n+j)}x,\ldots,T^{p_{d}(n+j)}x)d\mu(x)\\\
&\quad=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}(T^{a_{1}}\times\cdots\times
T^{a_{s}})^{n}\Big{(}\prod_{j=-l}^{l}f_{j}^{\otimes_{I}}(T^{a_{1}j}x,\ldots,T^{a_{s}j}x)\Big{)}\cdot\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\prod_{j=-l}^{l}f_{j}^{\otimes_{II}}(T^{p_{s+1}(n+j)}x,\ldots,T^{p_{d}(n+j)}x)d\mu(x).\end{split}$

####  3.4.2. Definition of $\widetilde{\mu}^{(\infty)}_{\mathcal{A}}$

Now we define the new system.

Let $\Omega=X^{s}\times(X^{d-s})^{\mathbb{Z}}$. Define
$\widetilde{\sigma}:\Omega\rightarrow\Omega$ as follow: for
$(x_{1},x_{2},\ldots,x_{s})\times{\bf
x}\in\Omega=X^{s}\times(X^{d-s})^{\mathbb{Z}},$

(3.15) $\widetilde{\sigma}\Big{(}(x_{1},x_{2},\ldots,x_{s})\times{\bf
x}\Big{)}=\Big{(}(T^{a_{1}}x_{1},T^{a_{2}}x_{2},\ldots,T^{a_{s}}x_{s})\times\sigma_{II}{\bf
x},$

where $\sigma_{II}$ is the shift on $(X^{d-s})^{\mathbb{Z}}$. That is,
$\widetilde{\sigma}=\tau_{\vec{a}}\times\sigma_{II}$, where
$\tau_{\vec{a}}=T^{a_{1}}\times\cdots\times T^{a_{s}}$. Let

$\xi_{x}^{\mathcal{A}}=\Big{(}(x,\ldots,x),(T^{p_{s+1}(j)}x,\ldots,T^{p_{d}(j)}x)_{j\in\mathbb{Z}}\Big{)}\in
X^{s}\times(X^{d-s})^{\mathbb{Z}}.$

Then

$\widetilde{\sigma}^{n}\xi^{\mathcal{A}}_{x}=\Big{(}(T^{a_{1}n}x,\ldots,T^{a_{s}n}x),(T^{p_{s+1}(n+j)}x,\ldots,T^{p_{d}(n+j)}x)_{j\in\mathbb{Z}}\Big{)}\in
X^{s}\times(X^{d-s})^{\mathbb{Z}}.$

Note that when $s=0$, $\xi_{x}^{\mathcal{A}}=\omega_{x}^{\mathcal{A}}$.

Next we define a measure $\widetilde{\mu}^{(\infty)}_{\mathcal{A}}$ on
$\Omega$. For all $f_{1},f_{2},\ldots,f_{s}\in C(X)$, all $l\in\mathbb{N}$ and
all $f_{j}^{\otimes_{II}}=f_{j}^{(s+1)}\otimes
f_{j}^{(s+2)}\otimes\cdots\otimes f_{j}^{(d)}\in C(X^{d-s}),-l\leq j\leq l$,

(3.16)
$\begin{split}&\int_{X^{s}\times(X^{d-s})^{\mathbb{Z}}}\big{(}f_{1}\otimes\cdots\otimes
f_{s}\big{)}\otimes\Big{(}\bigotimes_{j=-l}^{l}f_{j}^{\otimes_{II}}\Big{)}((x_{1},\ldots,x_{s}),{\bf
x})d\widetilde{\mu}_{\mathcal{A}}^{(\infty)}\\\
&\quad=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}\prod_{i=1}^{s}f_{i}(T^{a_{i}n}x)\prod_{j=-l}^{l}f_{j}^{\otimes_{II}}(T^{{p_{s+1}}(n+j)}x,\ldots,T^{p_{d}(n+j)}x)d\mu(x)\\\
&\quad=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}(T^{a_{1}}\times\cdots\times
T^{a_{s}})^{n}(f_{1}\otimes\cdots\otimes f_{s})(x,\ldots,x)\\\
&\quad\quad\quad\quad\quad\prod_{j=-l}^{l}f_{j}^{\otimes_{II}}(T^{{p_{s+1}}(n+j)}x,\ldots,T^{p_{d}(n+j)}x)d\mu(x),\end{split}$

where $((x_{1},x_{2},\ldots,x_{s}),{\bf x})=((x_{1},x_{2},\ldots,x_{s}),({\bf
x}_{n})_{n\in\mathbb{Z}})\in X^{s}\times(X^{d-s})^{\mathbb{Z}}$.

####  3.4.3. $\mu^{(\infty)}_{\mathcal{A}}$ and
$\widetilde{\mu}^{(\infty)}_{\mathcal{A}}$ are isomorphic

Compared (3.14) with (3.16), we see that

$((X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}},\sigma)\cong(X^{s}\times(X^{d-s})^{\mathbb{Z}},\mathcal{X}^{s}\times(\mathcal{X}^{d-s})^{\mathbb{Z}},\widetilde{\mu}^{(\infty)}_{\mathcal{A}},\widetilde{\sigma}).$

That is, if we define

(3.17)
$\begin{split}\phi:&(X^{s}\times(X^{d-s})^{\mathbb{Z}},\mathcal{X}^{s}\times(\mathcal{X}^{d-s})^{\mathbb{Z}},\widetilde{\mu}^{(\infty)}_{\mathcal{A}})\rightarrow((X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}}),\\\
&({\bf y},{\bf x})\mapsto\big{(}\tau_{\vec{a}}^{j}{\bf y},{\bf
x}_{j}\big{)}_{j\in\mathbb{Z}},\ \forall({\bf y},{\bf x})=\Big{(}{\bf
y},\big{(}{\bf x}_{j}\big{)}_{j\in\mathbb{Z}}\Big{)}\in
X^{s}\times(X^{d-s})^{\mathbb{Z}},\end{split}$

then it is an isomorphism.

Note that if $s=0$, then
$\widetilde{\mu}^{(\infty)}_{\mathcal{A}}=\mu^{(\infty)}_{\mathcal{A}}$; and
if $s=d$, then $\widetilde{\mu}^{(\infty)}_{\mathcal{A}}=\mu^{(d)}_{\vec{a}}$.

####  3.4.4. Compare $\mu^{(\infty)}_{\mathcal{A}}$ with
$\widetilde{\mu}^{(\infty)}_{\mathcal{A}}$

We have shown that
$((X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}},\sigma)$
is isomorphic to
$(X^{s}\times(X^{d-s})^{\mathbb{Z}},\mathcal{X}^{s}\times(\mathcal{X}^{d-s})^{\mathbb{Z}},\widetilde{\mu}^{(\infty)}_{\mathcal{A}},\widetilde{\sigma}).$
It is clear that the definition of $\mu^{(\infty)}_{\mathcal{A}}$ is cleaner
than $\widetilde{\mu}^{(\infty)}_{\mathcal{A}}$, as we do not care whether
there are linear terms in $\mathcal{A}$ or not. But when we study certain
dynamical properties, $\widetilde{\mu}^{(\infty)}_{\mathcal{A}}$ is easier to
handle than $\mu^{(\infty)}_{\mathcal{A}}$. For example, in the next section,
we use $\widetilde{\mu}^{(\infty)}_{\mathcal{A}}$ in Lemma 4.9, and it much
more complicated if we use $\mu^{(\infty)}_{\mathcal{A}}$ instead.

#### 3.4.5.

Similar to Remark 3.3, we have that $\mu_{\mathcal{A}}^{(\infty)}$ is a
standard measure. And similar to the proof of Proposition 3.4, in (3.16) we
can replace continuous functions by measurable ones. That is, we have the
following proposition.

######  Proposition 3.6.

Let $(X,\mathcal{X},\mu,T)$ be a m.p.s. and $d\in\mathbb{N}$. Let
$\mathcal{A}=\\{p_{1},\ldots,p_{s},p_{s+1},\ldots,p_{d}\\}$ be a family of
non-constant integral polynomials, where $s\geq 0$, $p_{i}(n)=a_{i}n$ ($1\leq
i\leq s$), $a_{1},a_{2},\ldots,a_{s}$ are distinct non-zero integers, and
$\deg p_{i}\geq 2,p_{i}(0)=0,s+1\leq i\leq d$. For all
$f_{1},f_{2},\ldots,f_{s}\in L^{\infty}(X,\mu)$, all $l\in\mathbb{N}$ and all
$f_{j}^{\otimes_{II}}=f_{j}^{(s+1)}\otimes f_{j}^{(s+2)}\otimes\cdots\otimes
f_{j}^{(d)}$ (where $f^{(t)}_{j}\in L^{\infty}(X,\mu)$, $s+1\leq t\leq
d,-l\leq j\leq l$),

(3.18)
$\begin{split}&\quad\int_{X^{s}\times(X^{d-s})^{\mathbb{Z}}}\big{(}\bigotimes_{i=1}^{s}f_{i}\big{)}\otimes\Big{(}\bigotimes_{j=-l}^{l}f_{j}^{\otimes_{II}}\Big{)}((x_{1},x_{2},\ldots,x_{s}),{\bf
x})d\widetilde{\mu}_{\mathcal{A}}^{(\infty)}\\\
&\quad=\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}\prod_{i=1}^{s}f_{i}(T^{a_{i}n}x)\prod_{j=-l}^{l}f_{j}^{\otimes_{II}}(T^{{p_{s+1}}(n+j)}x,\ldots,T^{p_{d}(n+j)}x)d\mu(x),\end{split}$

where $((x_{1},x_{2},\ldots,x_{s}),{\bf x})=((x_{1},x_{2},\ldots,x_{s}),({\bf
x}_{n})_{n\in\mathbb{Z}})\in X^{s}\times(X^{d-s})^{\mathbb{Z}}$.

Also similar to Proposition 3.5, if
$(X_{i},\mathcal{X}_{i},\mu_{i},T_{i}),i=1,2$, are two isomorphic m.p.s., then
$(X_{1}\times(X_{1}^{d-s})^{\mathbb{Z}},\mathcal{X}_{1}\times(\mathcal{X}_{1}^{d-s})^{\mathbb{Z}},(\widetilde{\mu_{1}})_{\mathcal{A}}^{(\infty)},\langle
T_{1}^{\infty},\widetilde{\sigma}\rangle)$ and

$(X_{2}\times(X_{2}^{d-s})^{\mathbb{Z}},\mathcal{X}_{2}\times(\mathcal{X}_{2}^{d-s})^{\mathbb{Z}},(\widetilde{\mu_{2}})_{\mathcal{A}}^{(\infty)},\langle
T_{2}^{\infty},\widetilde{\sigma}\rangle)$

are also isomorphic.

###  3.5. Topological correspondences

####  3.5.1. $N_{\infty}(X,\mathcal{A})$ and $M_{\infty}(X,\mathcal{A})$

Let $(X,T)$ be a t.d.s. We set

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

and

$M_{\infty}(X,\mathcal{A})=\overline{\bigcup\\{\mathcal{O}(\xi_{x}^{\mathcal{A}},\widetilde{\sigma}):x\in
X\\}}=\overline{\bigcup\\{\widetilde{\sigma}^{n}\xi_{x}^{\mathcal{A}}:x\in
X,n\in\mathbb{Z}\\}}\subseteq X^{s}\times(X^{d-s})^{\mathbb{Z}}.$

It is clear that $N_{\infty}(X,\mathcal{A})$ is invariant under the action of
$T^{\infty}$ and $\sigma$, and $M_{\infty}(X,\mathcal{A})$ is invariant under
the action of $T^{\infty}$ and $\widetilde{\sigma}$. Thus
$(N_{\infty}(X,\mathcal{A}),\langle T^{\infty},{\sigma}\rangle)$ and
$(M_{\infty}(X,\mathcal{A}),\langle T^{\infty},\widetilde{\sigma}\rangle)$ are
$\mathbb{Z}^{2}$-t.d.s., which were introduced and investigated in [26].

#### 3.5.2.

The following theorem indicates the connection between them and the systems we
introduced in the current paper.

######  Theorem 3.7.

Let $({X},T)$ be a t.d.s. and let $\mathcal{A}=\\{p_{1},\cdots,p_{d}\\}$ be a
family of non-constant integral polynomials with $p_{i}(0)=0,1\leq i\leq d$.
Then for all $\mu\in M_{T}(X)$, we have

${\rm supp}\ \mu_{\mathcal{A}}^{(\infty)}\subseteq N_{\infty}(X,\mathcal{A})\
\text{and}\ {\rm supp}\ \widetilde{\mu}_{\mathcal{A}}^{(\infty)}\subseteq
M_{\infty}(X,\mathcal{A}).$

If in addition ${\rm supp}\ \mu=X$, then ${\rm supp}\
\mu_{\mathcal{A}}^{(\infty)}=N_{\infty}(X,\mathcal{A})$ and ${\rm supp}\
\widetilde{\mu}_{\mathcal{A}}^{(\infty)}=M_{\infty}(X,\mathcal{A}).$

###### Proof.

We only show the first inclusion, and the second one can be proved similarly.

The measure $\mu_{\mathcal{A}}^{(\infty)}$ is a weak limit of averages of
Dirac masses at points of the form $\sigma^{n}\omega_{x}^{\mathcal{A}}$ for
$n\in\mathbb{Z}$ and $x\in X$. Since all of these points belong to
$N_{\infty}(X,\mathcal{A})$, the measure $\mu^{(\infty)}_{\mathcal{A}}$ is
concentrated on this set, i.e. ${\rm supp}\
\mu_{\mathcal{A}}^{(\infty)}\subseteq N_{\infty}(X,\mathcal{A})$.

It remains to show that $N_{\infty}(X,\mathcal{A})={\rm supp}\
\mu_{\mathcal{A}}^{(\infty)}$ if ${\rm supp}\ \mu=X$. Since ${\rm supp}\
\mu_{\mathcal{A}}^{(\infty)}$ is invariant under the actions of $T^{\infty}$
and $\sigma$, it suffices to show that for any $x\in X$, we have
$\omega_{x}^{\mathcal{A}}\in{\rm supp}\ \mu_{\mathcal{A}}^{(\infty)}$. Thus we
need to show that if $U$ is an open neighbourhood of
$\omega_{x}^{\mathcal{A}}$, then $\mu_{\mathcal{A}}^{(\infty)}(U)>0$. Without
loss of generality, we assume that
$U=\big{(}\prod_{j=-\infty}^{-l-1}X^{d}\big{)}\times\big{(}\prod_{j=-l}^{l}U^{\otimes}_{j}\big{)}\times\big{(}\prod_{j=l+1}^{\infty}X^{d}\big{)},$
where $l\in\mathbb{N}$, $U_{j}^{\otimes}=U_{j}^{(1)}\times
U_{j}^{(2)}\times\cdots\times U_{j}^{(d)}$, and $U_{j}^{(t)}$ is open in $X$,
$1\leq t\leq d,-l\leq j\leq l$. Since
$\omega_{x}^{\mathcal{A}}=(T^{\vec{p}(n)}x^{\otimes d})_{n\in\mathbb{Z}}\in
U$, we have $T^{p_{t}{(j)}}x\in U^{(t)}_{j}$, $1\leq t\leq d,-l\leq j\leq l$.
Choose a non-empty open subset $V$ of $X$ such that

$V\subseteq T^{-p_{t}(j)}U_{j}^{(t)},\ \forall 1\leq t\leq d,-l\leq j\leq l.$

Thus we have

$\begin{split}\mu_{\mathcal{A}}^{(\infty)}(U)&=\mu_{\mathcal{A}}^{(\infty)}\Big{(}\big{(}\prod_{j=-\infty}^{-l-1}X^{d}\big{)}\times\big{(}\prod_{j=-l}^{l}U^{\otimes}_{j}\big{)}\times\big{(}\prod_{j=l+1}^{\infty}X^{d}\big{)}\Big{)}\\\
&=\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}\mu\Big{(}\bigcap_{j=-1}^{l}\big{(}T^{-p_{1}(n+j)}U_{j}^{(1)}\cap\cdots\cap
T^{-p_{d}(n+j)}U_{j}^{(d)}\big{)}\Big{)}\\\
&=\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}\mu\Big{(}\bigcap_{j=-1}^{l}\bigcap_{k=1}^{d}T^{-(p_{k}(n+j)-p_{k}(j))-p_{k}(j)}U_{j}^{(k)}\big{)}\\\
&\geq\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}\mu\Big{(}\bigcap_{j=-1}^{l}\big{(}T^{-(p_{1}(n+j)-p_{1}(j))}V\cap\cdots\cap
T^{-(p_{d}(n+j)-p_{d}(j))}V\big{)}\Big{)}.\end{split}$

Since ${\rm supp}\ \mu=X$, we have $\mu(V)>0$. By Theorem 2.25,

$\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}\mu\Big{(}\bigcap_{j=-1}^{l}\big{(}T^{-(p_{1}(n+j)-p_{1}(j))}V\cap\cdots\cap
T^{-(p_{d}(n+j)-p_{d}(j))}V\big{)}\Big{)}>0,$

which implies that $\mu_{\mathcal{A}}^{(\infty)}(U)>0$. The proof is complete.
∎

######  Remark 3.8.

We have

  1. (1)

By the proof of Theorem 3.7, we actually proved the following equality:

${\rm supp}\
\mu_{\mathcal{A}}^{(\infty)}=\overline{\\{\sigma^{n}\omega_{x}^{\mathcal{A}}:x\in{\rm
supp}\mu,\ n\in\mathbb{Z}\\}}.$

When $(X,T)$ is minimal, we have ${\rm supp}\ \mu=X$ for each
$\mu\in{\mathcal{M}}_{T}(X)$ and ${\rm supp}\
\mu_{\mathcal{A}}^{(\infty)}=N_{\infty}(X,\mathcal{A})$. Similar statement
holds for $\widetilde{\mu}_{\mathcal{A}}^{(\infty)}$.

  2. (2)

By Theorem 3.7, if $(X,T)$ is minimal, then
$(N_{\infty}({X},\mathcal{A}),\langle T^{\infty},\sigma\rangle)$ is an
$E$-system, i.e. it is transitive and have an invariant measure with a full
support. In fact we can say more, i.e. it was proved [26] that
$(N_{\infty}({X},\mathcal{A}),\langle T^{\infty},\sigma\rangle)$ is an
$M$-system, i.e., it is transitive and the set of $\langle
T^{\infty},\sigma\rangle$-minimal points is dense.

##  4\. The $\sigma$-algebra of invariant sets of
$((X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}},\sigma)$

In this section the $\sigma$-algebra of invariant measurable sets of
$((X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}},\sigma)$
will be studied. The main result of this section is that its $\sigma$-algebra
of invariant measurable sets is isomorphic to the $\sigma$-algebra of
invariant measurable sets of its $\infty$-step pro-nilfactors.

###  4.1. Multiple ergodic averages along polynomials of several variable

We need the following multiple ergodic averages along polynomials of several
variable.

######  Theorem 4.1.

[31, Theorem 3] Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s. and
$m,d\in\mathbb{N}$. Let
$p_{1},p_{2},\ldots,p_{d}:\mathbb{Z}^{m}\rightarrow\mathbb{Z}$ be non-constant
essentially distinct polynomials (i.e. $p_{i}-p_{j}$ is not constant for
$i\neq j$). Then there is $k\in\mathbb{N}$ such that for any
$f_{1},f_{2},\ldots,f_{d}\in L^{\infty}(X)$ with $\interleave
f_{j}\interleave_{k}=0$ for some $j\in\\{1,2,\ldots,d\\}$, one has

$\lim_{N\to\infty}\frac{1}{|\Phi_{N}|}\sum_{{\bf n}\in\Phi_{N}}T^{p_{1}({\bf
n})}f_{1}\cdots T^{p_{d}({\bf n})}f_{d}=0$

in $L^{2}(X,\mu)$ for any Følner sequence $\\{\Phi_{N}\\}_{N=1}^{\infty}$ in
$\mathbb{Z}^{m}$.

######  Remark 4.2.

The integer $k$ in Theorem 4.1 depends only on the weight of polynomials
$\\{p_{1},\ldots,p_{d}\\}$ (see [31] for the definition of the weight and the
proof of this fact).

###  4.2. Condition $(\spadesuit)$

For a given integral polynomial $p$ with $p(0)=0$, for each $j\in\mathbb{Z}$
let $p^{[j]}$ be defined by

$p^{[j]}(n)=p(n+j)-p(j),\ \forall n\in\mathbb{Z}.$

######  Definition 4.3.

Let $\mathcal{A}=\\{p_{1},p_{2},\ldots,p_{d}\\}$ be a family of integral
polynomials. We say $\mathcal{A}$ satisfies condition $(\spadesuit)$ if
$p_{1}(0)=\ldots=p_{d}(0)=0$ and

  1. (1)

$p_{i}(n)=a_{i}n,1\leq i\leq s$, where $s\geq 0$, and
$a_{1},a_{2},\ldots,a_{s}$ are distinct non-zero integers;

  2. (2)

$\deg p_{j}\geq 2,s+1\leq j\leq d$;

  3. (3)

for each $i\neq j\in\\{s+1,s+2,\ldots,d\\}$, $p_{j}^{[k]}\neq p_{i}^{[t]}$ for
any $k,t\in\mathbb{Z}$.

######  Remark 4.4.

The key point in the definition of condition $(\spadesuit)$ is the term (3).
As we explained in the introduction, when we study $\mathcal{A}=\\{p\\}$ with
$p(n)=n^{2}$, we actually consider infinitely many polynomials $p^{[k]}$
($k\in\mathbb{Z}$) defined by
$p^{[k]}(n)=p(n+k)-p(k)=n^{2}+2kn,n\in\mathbb{Z}$. But when we study
$\mathcal{A}=\\{n^{2},n^{2}+2n,n^{2}+8n\\}$, we still consider the same
polynomials $p^{[k]},k\in\mathbb{Z}$. To exclude this kind of situation, we
add the term (3) when we study $\mathcal{A}$.

We have the following simple observation concerning the condition
$(\spadesuit)$.

######  Lemma 4.5.

Let $k\in\mathbb{N}$ and $Q=\\{q_{1},q_{2},\ldots,q_{k}\\}$ be integral
polynomials with $q_{i}(0)=0,1\leq i\leq k$. Then there is $1\leq d\leq k$ and
$\mathcal{A}=\\{p_{1},\ldots,p_{d}\\}\subset\\{q_{1},\ldots,q_{k}\\}$ such
that $\mathcal{A}$ satisfies condition $(\spadesuit)$, and
$\\{q_{1},\ldots,q_{k}\\}\subseteq\\{p_{i}^{[m]}:1\leq i\leq
d,m\in\mathbb{Z}\\}$.

###### Proof.

If there are $1\leq i\not=j\leq k$ and $k_{1},k_{2}\in\mathbb{Z}$ such that
$q_{i}^{[k_{1}]}=q_{j}^{[k_{2}]}$ we just remove one of them (say $q_{j}$) to
get a proper subset $Q^{\prime}=Q\setminus\\{q_{j}\\}$ of $Q$. Continuing this
process for finitely many steps, the argument will stop, and the resulting
$\mathcal{A}$ is what we want. ∎

######  Example 4.6.

Let $\mathcal{A}^{\prime}=\\{n^{2},n^{2}+6n,n^{2}+10n\\}$ and
$\mathcal{A}=\\{n^{2}\\}$. Then by the definition, $\mathcal{A}^{\prime}$ does
not satisfy the condition $(\spadesuit)$, $\mathcal{A}$ satisfies the
condition $(\spadesuit)$ and
$\mathcal{A}^{\prime}\subseteq\\{p^{[m]}:p(n)=n^{2}\in\mathcal{A},m\in\mathbb{Z}\\}$.
Let $(X,\mathcal{X},\mu,T)$ be a m.p.s. Now we show that
$(X^{\mathbb{Z}},\mathcal{X}^{\mathbb{Z}},\mu_{\mathcal{A}}^{(\infty)},\sigma)$
is isomorphic to
$((X^{3})^{\mathbb{Z}},(\mathcal{X}^{3})^{\mathbb{Z}},\mu_{\mathcal{A}^{\prime}}^{(\infty)},\sigma)$.

Recall that

$\omega_{x}^{\mathcal{A}}=(T^{n^{2}}x)_{n\in\mathbb{Z}}\in X^{\mathbb{Z}},\
\text{and}\
\omega_{x}^{\mathcal{A}^{\prime}}=\Big{(}(T^{n^{2}}x,T^{n^{2}+6n}x,T^{n^{2}+10n}x)\Big{)}_{n\in\mathbb{Z}}\in(X^{3})^{\mathbb{Z}}.$

Note that

$\sigma^{k}\omega_{x}^{\mathcal{A}}=(T^{(n+k)^{2}}x)_{n\in\mathbb{Z}}=(T^{n^{2}+2kn+k^{2}}x)_{n\in
Z}=(T^{\infty})^{k^{2}}(T^{n^{2}+2kn}x)_{n\in\mathbb{Z}},\forall
k\in\mathbb{Z}.$

We have
$\omega_{x}^{\mathcal{A}^{\prime}}=\Big{(}\omega_{x}^{\mathcal{A}},(T^{\infty})^{-9}\sigma^{3}\omega_{x}^{\mathcal{A}},(T^{\infty})^{-25}\sigma^{5}\omega_{x}^{\mathcal{A}}\Big{)}$
for all $x\in X$.

Define $\Psi={\rm
id}\times(T^{\infty})^{-9}\sigma^{3}\times(T^{\infty})^{-25}\sigma^{5}$:

$\Psi:X^{\mathbb{Z}}\rightarrow(X^{3})^{\mathbb{Z}},\ {\bf x}\mapsto({\bf
x},(T^{\infty})^{-9}\sigma^{3}{\bf x},(T^{\infty})^{-25}\sigma^{5}{\bf x}),\
\forall{\bf x}\in X^{\mathbb{Z}}.$

It is easy to verify that

$\Psi_{*}(\mu_{\mathcal{A}}^{(\infty)})=\mu_{\mathcal{A}^{\prime}}^{(\infty)},$

and $\Psi$ is an isomorphism from
$(X^{\mathbb{Z}},\mathcal{X}^{\mathbb{Z}},\mu_{\mathcal{A}}^{(\infty)},\sigma)$
to
$((X^{3})^{\mathbb{Z}},(\mathcal{X}^{3})^{\mathbb{Z}},\mu_{\mathcal{A}^{\prime}}^{(\infty)},\sigma)$.

Similar to Example 4.6, we have the following general result:

######  Theorem 4.7.

Let $(X,\mathcal{X},\mu,T)$ be a m.p.s. and
$\mathcal{A}^{\prime}=\\{q_{1},\ldots,q_{k}\\}$ be a family of integral
polynomials with $q_{i}(0)=0,1\leq i\leq k$. Then there is a family of
integral polynomials $\mathcal{A}=\\{p_{1},\ldots,p_{d}\\}$ satisfying the
condition $(\spadesuit)$ such that

$((X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}},\sigma)\cong((X^{k})^{\mathbb{Z}},(\mathcal{X}^{k})^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}^{\prime}},\sigma),$

and

$(X^{s}\times(X^{d-s})^{\mathbb{Z}},\mathcal{X}^{s}\times(\mathcal{X}^{d-s})^{\mathbb{Z}},\widetilde{\mu}^{(\infty)}_{\mathcal{A}},\widetilde{\sigma})\cong(X^{t}\times(X^{k-t})^{\mathbb{Z}},\mathcal{X}^{t}\times(\mathcal{X}^{k-t})^{\mathbb{Z}},\widetilde{\mu}^{(\infty)}_{\mathcal{A}^{\prime}},\widetilde{\sigma}),$

where $t$ is the number of linear polynomials in $\mathcal{A}^{\prime}$.

###  4.3. Some lemmas

######  Lemma 4.8 (van der Corput lemma).

[2] Let $\\{\zeta_{n}\\}$ be a bounded sequence in a Hilbert space
$\mathcal{H}$ with norm $\parallel\cdot\parallel$ and inner product
$\langle\cdot,\cdot\rangle$. Then

$\limsup_{N\to\infty}\Big{\|}\frac{1}{N}\sum_{n=1}^{N}\zeta_{n}\Big{\|}^{2}\leq\limsup_{H\to\infty}\frac{1}{H}\sum_{h=1}^{H}\limsup_{N\to\infty}\left|\frac{1}{N}\sum_{n=1}^{N}\langle\zeta_{n},\zeta_{n+h}\rangle\right|.$

Recall that $\widetilde{\sigma}:X^{s}\times(X^{d-s})^{\mathbb{Z}}\rightarrow
X^{s}\times(X^{d-s})^{\mathbb{Z}}$ is defined as follows: for
$\big{(}(x_{1},\ldots,x_{s}),{\bf x}\big{)}\in
X^{s}\times(X^{d-s})^{\mathbb{Z}},$
$\widetilde{\sigma}\Big{(}(x_{1},\ldots,x_{s}),{\bf
x}\Big{)}=\Big{(}\tau_{\vec{a}}(x_{1},\ldots,x_{s}),\sigma_{II}{\bf
x}\Big{)},$ where $\tau_{\vec{a}}=T^{a_{1}}\times\cdots\times T^{a_{s}}$ and
$\sigma_{II}$ is the shift on $(X^{d-s})^{\mathbb{Z}}$.

We now show

######  Lemma 4.9.

Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s. and $d\in\mathbb{N}$. Let
$\mathcal{A}=\\{p_{1},\ldots,p_{d}\\}$ be a family of non-constant integral
polynomials satisfying the condition $(\spadesuit)$. Then for all
$f_{1},f_{2},\ldots,f_{s}\in L^{\infty}(X,\mu)$, all $l\in\mathbb{N}$ and all
$f^{(t)}_{j}\in L^{\infty}(X,\mu)$ with $\interleave
f_{i}\interleave_{\infty}=0$ for some $1\leq i\leq s$ or $\interleave
f_{j}^{(t)}\interleave_{\infty}=0$ for some $s+1\leq t\leq d,-l\leq j\leq l$,
one has

(4.1)
$\limsup_{N\to\infty}\left\|\frac{1}{N}\sum_{n=0}^{N-1}\Big{(}\big{(}\bigotimes_{j=1}^{s}f_{j}\big{)}\otimes\bigotimes_{j=-l}^{l}f_{j}^{\otimes_{II}}\Big{)}\Big{(}\tau_{\vec{a}}^{n}(x_{1},\ldots,x_{s}),\sigma_{II}^{n}{\bf
x}\Big{)}\right\|_{L^{2}(\widetilde{\mu}_{\mathcal{A}}^{(\infty)})}=0.$

Note that here $f_{j}^{\otimes_{II}}=f_{j}^{(s+1)}\otimes
f_{j}^{(s+2)}\otimes\cdots\otimes f_{j}^{(d)}$, and
$\big{(}\bigotimes_{j=1}^{s}f_{j}\big{)}\otimes\bigotimes_{j=-l}^{l}f_{j}^{\otimes_{II}}$
is regarded as an element of
$L^{\infty}(X^{s}\times(X^{d-s})^{\mathbb{Z}},\widetilde{\mu}_{\mathcal{A}}^{(\infty)})$.

###### Proof.

Let $\mathcal{A}=\\{p_{1},\ldots,p_{s},p_{s+1},\ldots,p_{d}\\}$ be a family of
non-constant integral polynomials, where $s\geq 0$, $p_{i}(n)=a_{i}n$ ($1\leq
i\leq s$) satisfying $(\spadesuit)$. Let $f_{1},\ldots,f_{s}\in
L^{\infty}(X,\mu)$, $l\in\mathbb{N}$ and $f^{(t)}_{j}\in L^{\infty}(X,\mu)$
with $\interleave f_{i}\interleave_{\infty}=0$ for some $1\leq i\leq s$ or
$\interleave f_{j}^{(t)}\interleave_{\infty}=0$ for some $s+1\leq t\leq
d,-l\leq j\leq l$. Without loss of generality, we assume that
$\|f_{i}\|_{\infty}\leq 1$ for all $i$ and $\|f_{j}^{(t)}\|_{\infty}\leq 1$
for all $j,t$. Write

$\begin{split}\zeta_{n}&=\Big{(}\big{(}f_{1}\otimes\cdots\otimes
f_{s}\big{)}\otimes\bigotimes_{j=-l}^{l}f_{j}^{\otimes_{II}}\Big{)}\Big{(}\tau_{\vec{a}}^{n}(x_{1},\ldots,x_{s}),\sigma_{II}^{n}{\bf
x}\Big{)}\\\ &=f_{1}(T^{a_{1}n}x_{1})f_{2}(T^{a_{2}n}x_{2})\cdots
f_{s}(T^{a_{s}n}x_{s})f_{-l}^{\otimes_{II}}({\bf
x}_{n-l})f_{-l+1}^{\otimes_{II}}({\bf x}_{n-l+1})\cdots
f_{l}^{\otimes_{II}}({\bf x}_{n+l})\\\
&=f_{1}(T^{a_{1}n}x_{1})f_{2}(T^{a_{2}n}x_{2})\cdots
f_{s}(T^{a_{s}n}x_{s})\prod_{j=-l}^{l}f_{j}^{(s+1)}(x^{(s+1)}_{n+j})f_{j}^{(s+2)}(x_{n+j}^{(s+2)})\cdots
f_{j}^{(d)}(x_{n+j}^{(d)}),\end{split}$

where ${\bf x}=({\bf
x}_{n})_{n\in\mathbb{Z}}=\Big{(}(x^{(s+1)}_{n},x^{(s+2)}_{n},\ldots,x^{(d)}_{n})\Big{)}_{n\in\mathbb{Z}}\in(X^{d-s})^{\mathbb{Z}}$.

By van der Corput lemma (Lemma 4.8),

$\limsup_{N\to\infty}\big{\|}\frac{1}{N}\sum_{n=0}^{N-1}\zeta_{n}\big{\|}^{2}_{L^{2}(\widetilde{\mu}_{\mathcal{A}}^{(\infty)})}\leq\limsup_{H\to\infty}\frac{1}{H}\sum_{h=0}^{H-1}\limsup_{N\to\infty}\left|\frac{1}{N}\sum_{n=0}^{N-1}\int_{X^{s}\times(X^{d-s})^{\mathbb{Z}}}\overline{\zeta}_{n+h}\cdot\zeta_{n}d\widetilde{\mu}_{\mathcal{A}}^{(\infty)}\right|.$

For simplicity, we only consider real-valued functions. Firstly we show two
special cases, and then give the proof of the general case.

Case I: $s=d$.

In this case all polynomials involved are linear and
$\widetilde{\mu}_{\mathcal{A}}^{(\infty)}={\mu}^{(d)}_{\vec{a}}$, where
$\vec{a}=(a_{1},\ldots,a_{d})$. The results follows from Lemma 2.5.

Case II: $s=0$.

In this case there are no linear items in $\mathcal{A}$ and
$\widetilde{\mu}_{\mathcal{A}}^{(\infty)}={\mu}_{\mathcal{A}}^{(\infty)}$. For
$h>2l$, we have

$\begin{split}&\left|\frac{1}{N}\sum_{n=0}^{N-1}\int_{(X^{d})^{\mathbb{Z}}}{\zeta}_{n+h}\cdot\zeta_{n}d\mu_{\mathcal{A}}^{(\infty)}\right|\\\
&\quad=\left|\frac{1}{N}\sum_{n=0}^{N-1}\int_{(X^{d})^{\mathbb{Z}}}\prod_{j=-l}^{l}{f_{j}^{\otimes}}({\bf
x}_{n+h+j})\prod_{j=-l}^{l}f_{j}^{\otimes}({\bf
x}_{n+j})d\mu_{\mathcal{A}}^{(\infty)}({\bf x})\right|\\\
&\quad=\left|\frac{1}{N}\sum_{n=0}^{N-1}\int_{(X^{d})^{\mathbb{Z}}}\prod_{j=-l}^{l}f_{j}^{\otimes}({\bf
x}_{n+j}){f_{j}^{\otimes}}({\bf x}_{n+h+j})d\mu_{\mathcal{A}}^{(\infty)}({\bf
x})\right|.\end{split}$

By the definition of $\mu_{\mathcal{A}}^{(\infty)}$,

(4.2)
$\begin{split}&\int_{(X^{d})^{\mathbb{Z}}}\prod_{j=-l}^{l}f_{j}^{\otimes}({\bf
x}_{n+j}){f_{j}^{\otimes}}({\bf x}_{n+h+j})d\mu_{\mathcal{A}}^{(\infty)}({\bf
x})\\\
&\quad=\lim_{M\to\infty}\frac{1}{M}\sum_{m=0}^{M-1}\int_{X}\prod_{j=-l}^{l}f_{j}^{\otimes}(T^{\vec{p}(m+n+j)}x^{\otimes
d}){f_{j}^{\otimes}}(T^{\vec{p}(m+n+h+j)}x^{\otimes d})d\mu(x)\\\
&\quad=\lim_{M\to\infty}\frac{1}{M}\sum_{m=0}^{M-1}\int_{X}\prod_{j=-l}^{l}f_{j}^{(1)}(T^{p_{1}(m+n+j)}x)f_{j}^{(2)}(T^{p_{2}(m+n+j)}x)\cdots
f_{j}^{(d)}(T^{p_{d}(m+n+j)}x)\cdot\\\
&\quad\quad\quad\quad{f_{j}^{(1)}}(T^{p_{1}(m+n+h+j)}x){f_{j}^{(2)}}(T^{p_{2}(m+n+h+j)}x)\cdots{f_{j}^{(d)}}(T^{p_{d}(m+n+h+j)}x)d\mu(x).\end{split}$

By (3) of the condition $(\spadesuit)$ on $\mathcal{A}$, all $m$-polynomials

$\\{p_{i}(m+n+j),p_{i}(m+n+h+j),1\leq i\leq d,-l\leq j\leq l\\}$

are essentially distinct, and by Theorem 4.1, the equation (4.2) is $0$. Thus
for $h>2l$,

$\left|\frac{1}{N}\sum_{n=0}^{N-1}\int_{(X^{d})^{\mathbb{Z}}}{\zeta}_{n+h}\cdot\zeta_{n}d\mu_{\mathcal{A}}^{(\infty)}\right|=0.$

So by Lemma 4.8,

$\limsup_{N\to\infty}\Big{\|}\frac{1}{N}\sum_{n=0}^{N-1}\zeta_{n}\Big{\|}^{2}_{L^{2}(\mu_{\mathcal{A}}^{(\infty)})}\leq\limsup_{H\to\infty}\frac{1}{H}\sum_{h=0}^{H-1}\limsup_{N\to\infty}\left|\frac{1}{N}\sum_{n=0}^{N-1}\int_{X^{s}\times(X^{d-s})^{\mathbb{Z}}}{\zeta}_{n+h}\cdot\zeta_{n}d{\mu}_{\mathcal{A}}^{(\infty)}\right|=0.$

Case III: $1\leq s\leq d-1$.

It is easy to see

$\Big{\|}\frac{1}{N}\sum_{n=0}^{N-1}\zeta_{n}\Big{\|}^{2}_{L^{2}(\widetilde{\mu}_{\mathcal{A}}^{(\infty)})}=\Big{\langle}\frac{1}{N}\sum_{n=0}^{N-1}\zeta_{n},\frac{1}{N}\sum_{n=0}^{N-1}\zeta_{n}\Big{\rangle}=\frac{1}{N^{2}}\sum_{n_{1},n_{2}=0}^{N-1}\langle\zeta_{n_{1}},\zeta_{n_{2}}\rangle.$

For each $(n_{1},n_{2})\in[0,N-1]^{2}$ with $|n_{1}-n_{2}|>2l$, we have that

(4.3)
$\begin{split}&\langle\zeta_{n_{1}},\zeta_{n_{2}}\rangle=\int_{X^{s}\times(X^{d-s})^{\mathbb{Z}}}{\zeta}_{n_{1}}\cdot\zeta_{n_{2}}d\widetilde{\mu}_{\mathcal{A}}^{(\infty)}\\\
&\quad=\int_{X^{s}\times(X^{d-s})^{\mathbb{Z}}}f_{1}(T^{a_{1}n_{1}}x_{1})\cdots
f_{s}(T^{a_{s}n_{1}}x_{s})f_{-l}^{\otimes_{II}}({\bf x}_{n_{1}-l})\ldots
f_{l}^{\otimes_{II}}({\bf x}_{n_{1}+l})\cdot\\\
&\quad\quad\quad\quad\quad\quad\quad f_{1}(T^{a_{1}n_{2}}x_{1})\cdots
f_{s}(T^{a_{s}n_{2}}x_{s})f_{-l}^{\otimes_{II}}({\bf x}_{n_{2}-l})\cdots
f_{l}^{\otimes_{II}}({\bf
x}_{n_{2}+l})d\widetilde{\mu}_{\mathcal{A}}^{(\infty)}\\\
&\quad=\int_{X^{s}\times(X^{d-s})^{\mathbb{Z}}}\Big{(}\prod_{i=1}^{s}f_{i}(T^{a_{i}n_{1}}x_{i})f_{i}(T^{a_{i}n_{2}}x_{i})\Big{)}\Big{(}\prod_{j=-l}^{l}f_{j}^{\otimes_{II}}({\bf
x}_{n_{1}+j})f_{j}^{\otimes_{II}}({\bf
x}_{n_{2}+j})\Big{)}d\widetilde{\mu}_{\mathcal{A}}^{(\infty)}\\\
&\quad=\lim_{M\to\infty}\frac{1}{M}\sum_{m=0}^{M-1}\int_{X}B(n_{1},n_{2},m)d\mu(x),\end{split}$

where

(4.4) $\begin{split}&B(n_{1},n_{2},m)\\\
&=\Big{(}\prod_{i=1}^{s}f_{i}(T^{a_{i}(m+n_{1})}x)f_{i}(T^{a_{i}(m+n_{2})}x)\Big{)}\cdot\prod_{j=-l}^{l}f_{j}^{\otimes_{II}}(T^{p_{s+1}(m+n_{1}+j)}x,\ldots,T^{p_{d}(m+n_{1}+j)}x)\cdot\\\
&\quad\quad\quad\quad\quad\quad\quad\quad
f_{j}^{\otimes_{II}}(T^{p_{s+1}(m+n_{2}+j)}x,\ldots,T^{p_{d}(m+n_{2}+j)}x).\end{split}$

For each $N\in\mathbb{N}$, by (4.3) we choose an $M(N)\in\mathbb{N}$ such that
$\lim_{N\to\infty}M(N)=+\infty$ and for each $(n_{1},n_{2})\in[0,N-1]^{2}$
with $|n_{1}-n_{2}|>2l$, we have that

(4.5)
$\begin{split}\Big{|}\langle\zeta_{n_{1}},\zeta_{n_{2}}\rangle-\frac{1}{M(N)}\sum_{m=0}^{M(N)-1}\int_{X}B(n_{1},n_{2},m)d\mu(x)\Big{|}<\frac{1}{N^{3}}.\end{split}$

Let $\Phi_{N}=[0,N-1]^{2}\times[0,M(N)]$. Then $\\{\Phi_{N}\\}_{N=1}^{\infty}$
is a Følner sequence of $\mathbb{Z}^{3}$. Let
$C_{l}^{N}=|\\{(n_{1},n_{2})\in[0,N-1]^{2}:|n_{1}-n_{2}|\leq 2l\\}|$. By (4.5)

(4.6) $\begin{split}&\ \ \
\big{\|}\frac{1}{N}\sum_{n=0}^{N-1}\zeta_{n}\big{\|}^{2}_{L^{2}(\widetilde{\mu}_{\mathcal{A}}^{(\infty)})}=\frac{1}{N^{2}}\sum_{n_{1},n_{2}=0}^{N-1}\langle\zeta_{n_{1}},\zeta_{n_{2}}\rangle\leq\frac{1}{N^{2}}\sum_{n_{1},n_{2}\in[0,N-1]^{2}\setminus
C_{l}^{N}}\langle\zeta_{n_{1}},\zeta_{n_{2}}\rangle+\frac{C_{l}^{N}}{N^{2}}\\\
&\quad\leq\frac{1}{N^{2}}\sum_{n_{1},n_{2}\in[0,N-1]^{2}\setminus
C_{l}^{N}}\Big{(}\frac{1}{M(N)}\sum_{m=0}^{M(N)-1}\int_{X}B(n_{1},n_{2},m)d\mu(x)+\frac{1}{N^{3}}\Big{)}+\frac{C_{l}^{N}}{N^{2}}\\\
&\quad\leq\frac{1}{N^{2}}\sum_{n_{1},n_{2}\in[0,N-1]^{2}}\frac{1}{M(N)}\sum_{m=0}^{M(N)-1}\int_{X}B(n_{1},n_{2},m)d\mu(x)+\frac{1}{N^{3}}+\frac{2C_{l}^{N}}{N^{2}}\\\
&\quad=\frac{1}{|\Phi_{N}|}\sum_{(n_{1},n_{2},m)\in\Phi_{N}}\int_{X}B(n_{1},n_{2},m)d\mu(x)+\frac{1}{N^{3}}+\frac{2C_{l}^{N}}{N^{2}}.\end{split}$

Since $\lim_{N\to\infty}\frac{2C_{l}^{N}}{N^{2}}=0$, we have

(4.7)
$\begin{split}&\limsup_{N\to\infty}\big{\|}\frac{1}{N}\sum_{n=0}^{N-1}\zeta_{n}\big{\|}^{2}_{L^{2}(\widetilde{\mu}_{\mathcal{A}}^{(\infty)})}=\limsup_{N\to\infty}\frac{1}{N^{2}}\sum_{n_{1},n_{2}=0}^{N-1}\langle\zeta_{n_{1}},\zeta_{n_{2}}\rangle\\\
&\leq\limsup_{N\to\infty}\frac{1}{|\Phi_{N}|}\sum_{(n_{1},n_{2},m)\in\Phi_{N}}\int_{X}B(n_{1},n_{2},m)d\mu(x)\\\
&\leq\limsup_{N\to\infty}\left\|\frac{1}{|\Phi_{N}|}\sum_{(n_{1},n_{2},m)\in\Phi_{N}}B(n_{1},n_{2},m)\right\|_{L^{1}(\mu)}\\\
&\leq\limsup_{N\to\infty}\left\|\frac{1}{|\Phi_{N}|}\sum_{(n_{1},n_{2},m)\in\Phi_{N}}B(n_{1},n_{2},m)\right\|_{L^{2}(\mu)}\\\
&=\limsup_{N\to\infty}\Big{\|}\frac{1}{|\Phi_{N}|}\sum_{(n_{1},n_{2},m)\in\Phi_{N}}\prod_{i=1}^{s}f_{i}(T^{a_{i}(m+n_{1})}x)f_{i}(T^{a_{i}(m+n_{2})}x)\cdot\\\
&\quad\quad\quad\quad\quad\quad\prod_{j=-l}^{l}f_{j}^{\otimes_{II}}(T^{p_{s+1}(m+n_{1}+j)}x,\ldots,T^{p_{d}(m+n_{1}+j)}x)\cdot\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
f_{j}^{\otimes_{II}}(T^{p_{s+1}(m+n_{2}+j)}x,\ldots,T^{p_{d}(m+n_{2}+j)}x)\Big{\|}_{L^{2}(\mu)}.\end{split}$

Recall that we assume $\interleave f_{i}\interleave_{\infty}=0$ for some
$1\leq i\leq s$ or $\interleave f_{j}^{(t)}\interleave_{\infty}=0$ for some
$s+1\leq t\leq d,1\leq j\leq l$. By the condition $(\spadesuit)$ on
$\mathcal{A}$, all $(n_{1},n_{2},m)$-polynomials

$\\{p_{t}(m+n_{1}+j),p_{t}(m+n_{2}+j),s+1\leq t\leq d,-l\leq j\leq l\\}$

and $\\{a_{i}(m+n_{1}),a_{i}(m+n_{2}):1\leq i\leq s\\}$ as polynomials of
several variable $\mathbb{Z}^{3}\rightarrow\mathbb{Z}$ are essentially
distinct. Thus by Theorem 4.1, the last line in (4.7) is equal to $0$. So by
(4.7),

$\limsup_{N\to\infty}\Big{\|}\frac{1}{N}\sum_{n=0}^{N-1}\zeta_{n}\Big{\|}^{2}_{L^{2}(\widetilde{\mu}_{\mathcal{A}}^{(\infty)})}=0.$

The whole proof is complete. ∎

The final lemma we need is the following

######  Lemma 4.10.

Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s. and $d\in\mathbb{N}$. Let
$\mathcal{A}=\\{p_{1},\cdots,p_{d}\\}$ be a family of non-constant integral
polynomials satisfying $(\spadesuit)$. Let
$\pi_{\infty}:(X,\mathcal{X},\mu,T)\rightarrow(Z_{\infty},\mathcal{Z}_{\infty},\mu_{\infty},T)$
be the factor map from $X$ to its $\infty$-step pro-nilfactor. Then for all
$f_{1},\ldots,f_{s}\in L^{\infty}(X,\mu)$, all $l\in\mathbb{N}$ and all
$f_{j}^{\otimes_{II}}=f_{j}^{(s+1)}\otimes f_{j}^{(s+2)}\otimes\cdots\otimes
f_{j}^{(d)}$ (where $f^{(t)}_{j}\in L^{\infty}(X,\mu)$, $s+1\leq t\leq
d,-l\leq j\leq l$), one has

(4.8)
$\begin{split}&\mathbb{E}_{\widetilde{\mu}_{\mathcal{A}}^{(\infty)}}\Big{(}\big{(}\bigotimes_{k=1}^{s}f_{k}\big{)}\otimes\bigotimes_{j=-l}^{l}f_{j}^{\otimes_{II}}\Big{|}\mathcal{I}(X^{s}\times(X^{d-s})^{\mathbb{Z}},\widetilde{\mu}_{\mathcal{A}}^{(\infty)},\widetilde{\sigma})\Big{)}\\\
&\quad=\mathbb{E}_{\widetilde{\mu}_{\mathcal{A}}^{(\infty)}}\Big{(}\bigotimes_{k=1}^{s}\mathbb{E}_{\mu}(f_{k}|\mathcal{Z}_{\infty})\otimes\bigotimes_{j=-l}^{l}\mathbb{E}_{\mu}(f_{j}^{\otimes_{II}}|\mathcal{Z}_{\infty})\Big{|}\mathcal{I}(X^{s}\times(X^{d-s})^{\mathbb{Z}},\widetilde{\mu}_{\mathcal{A}}^{(\infty)},\widetilde{\sigma})\Big{)},\end{split}$

where

$\mathbb{E}_{\mu}(f^{\otimes_{II}}_{j}|\mathcal{Z}_{\infty})\triangleq\mathbb{E}_{\mu}(f^{(s+1)}_{j}|\mathcal{Z}_{\infty})\otimes\mathbb{E}_{\mu}(f^{(2)}_{j}|\mathcal{Z}_{\infty})\otimes\cdots\otimes\mathbb{E}_{\mu}(f^{(d)}_{j}|\mathcal{Z}_{\infty}),$

$-l\leq j\leq l$, and $\widetilde{\sigma}=\tau_{\vec{a}}\times\sigma_{II}$.

###### Proof.

Let $f_{1},\ldots,f_{s}\in L^{\infty}(X,\mu)$, $l\in\mathbb{N}$ and
$f^{(s+1)}_{j},\cdots,f_{j}^{(d)}\in L^{\infty}(X,\mu)$ with $-l\leq j\leq l$.
Since $\widetilde{\mu}_{\mathcal{A}}^{(\infty)}$ is a standard measure,
$\big{(}\bigotimes_{k=1}^{s}f_{k}\big{)}\otimes\bigotimes_{j=-l}^{l}f_{j}^{\otimes_{II}}$
is an element of
$L^{\infty}(X^{s}\times(X^{d-s})^{\mathbb{Z}},\widetilde{\mu}_{\mathcal{A}}^{(\infty)})$.

By the telescoping, it suffices to show that sum

(4.9)
$\mathbb{E}_{\widetilde{\mu}_{\mathcal{A}}^{(\infty)}}\Big{(}\big{(}\bigotimes_{k=1}^{s}f_{k}\big{)}\otimes\bigotimes_{j=-l}^{l}\bigotimes_{t=s+1}^{d}f_{j}^{(t)}\Big{|}\mathcal{I}(X^{s}\times(X^{d-s})^{\mathbb{Z}},\widetilde{\mu}_{\mathcal{A}}^{(\infty)},\widetilde{\sigma})\Big{)}=0$

whenever $\mathbb{E}_{\mu}(f_{i}|\mathcal{Z}_{\infty})=0$ for some $1\leq
i\leq s$ or $\mathbb{E}_{\mu}(f_{j}^{(t)}|\mathcal{Z}_{\infty})=0$ for some
$-l\leq j\leq l,s+1\leq t\leq d$, i.e. $\interleave
f_{i}\interleave_{\infty}=0$ for some $1\leq i\leq s$ or $\interleave
f_{j}^{(t)}\interleave_{\infty}=0$ for some $-l\leq j\leq l,s+1\leq t\leq d$.

By the Ergodic theorem we have

$\begin{split}&\Big{\|}\mathbb{E}_{\widetilde{\mu}_{\mathcal{A}}^{(\infty)}}\Big{(}\big{(}\bigotimes_{k=1}^{s}f_{k}\big{)}\otimes\bigotimes_{j=-l}^{l}\bigotimes_{t=s+1}^{d}f_{j}^{(t)}\Big{|}\mathcal{I}(X^{s}\times(X^{d-s})^{\mathbb{Z}},\widetilde{\mu}_{\mathcal{A}}^{(\infty)},\widetilde{\sigma})\Big{)}\Big{\|}_{L^{2}(\widetilde{\mu}_{\mathcal{A}}^{(\infty)})}\\\
&\quad=\Big{\|}\mathbb{E}_{\widetilde{\mu}_{\mathcal{A}}^{(\infty)}}\Big{(}\big{(}\bigotimes_{k=1}^{s}f_{k}\big{)}\otimes\bigotimes_{j=-l}^{l}f_{j}^{\otimes_{II}}\Big{|}\mathcal{I}(X^{s}\times(X^{d-s})^{\mathbb{Z}},\widetilde{\mu}_{\mathcal{A}}^{(\infty)},\widetilde{\sigma})\Big{)}\Big{\|}_{L^{2}(\widetilde{\mu}_{\mathcal{A}}^{(\infty)})}\\\
&\quad=\lim_{N\to\infty}\left\|\frac{1}{N}\sum_{n=0}^{N-1}\Big{(}\big{(}\bigotimes_{k=1}^{s}f_{k}\big{)}\otimes\bigotimes_{j=-l}^{l}f_{j}^{\otimes_{II}}\Big{)}\Big{(}\tau_{\vec{a}}^{n}(x_{1},\ldots,x_{s}),\sigma_{II}^{n}{\bf
x}\Big{)}\right\|_{L^{2}(\widetilde{\mu}_{\mathcal{A}}^{(\infty)})}\\\
&\quad=0,\end{split}$

where the last equality holds following from Lemma 4.9 since $\interleave
f_{i}\interleave_{\infty}=0$ for some $1\leq i\leq s$ or $\interleave
f_{j}^{(t)}\interleave_{\infty}=0$ for some $-l\leq j\leq l,s+1\leq t\leq d$.
This ends the proof. ∎

Now we are able to show

######  Proposition 4.11.

Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s. and $d\in\mathbb{N}$. Let
$\mathcal{A}=\\{p_{1},\cdots,p_{d}\\}$ be a family of non-constant integral
polynomials satisfying $(\spadesuit)$. Let
$\pi_{\infty}:(X,\mathcal{X},\mu,T)\rightarrow(Z_{\infty},\mathcal{Z}_{\infty},\mu_{\infty},T)$
be the factor map from $X$ to its $\infty$-step pro-nilfactor. Then the
$\sigma$-algebra
$\mathcal{I}(X^{s}\times(X^{d-s})^{\mathbb{Z}},\widetilde{\mu}_{\mathcal{A}}^{(\infty)},\widetilde{\sigma})$
is measurable with respect to
$\mathcal{Z}_{\infty}^{s}\times(\mathcal{Z}_{\infty}^{d-s})^{\mathbb{Z}}$.

###### Proof.

Every bounded function on $X^{s}\times(X^{d-s})^{\mathbb{Z}}$ which is
measurable with respect to
$\mathcal{I}(X^{s}\times(X^{d-s})^{\mathbb{Z}},\widetilde{\mu}_{\mathcal{A}}^{(\infty)},\widetilde{\sigma})$
can be approximated in
$L^{2}(X^{s}\times(X^{d-s})^{\mathbb{Z}},\widetilde{\mu}_{\mathcal{A}}^{(\infty)})$
by finite sums of functions of the form

$\mathbb{E}_{\widetilde{\mu}_{\mathcal{A}}^{(\infty)}}\Big{(}\big{(}f_{1}\otimes\ldots\otimes
f_{s}\big{)}\otimes\bigotimes_{j=-l}^{l}f_{j}^{\otimes_{II}}\Big{|}\mathcal{I}(X^{s}\times(X^{d-s})^{\mathbb{Z}},\widetilde{\mu}_{\mathcal{A}}^{(\infty)},\widetilde{\sigma})\Big{)},$

where $f_{1},\ldots,f_{s}\in L^{\infty}(X,\mu)$, $l\in\mathbb{N}$ and
$f^{(s+1)}_{j},\ldots,f_{j}^{(d)}\in L^{\infty}(X,\mu),-l\leq j\leq l$. By
Lemma 4.10, one can assume that these functions are measurable with respect to
$\mathcal{Z}_{\infty}$. In this case $\big{(}f_{1}\otimes\ldots\otimes
f_{s}\big{)}\otimes\bigotimes_{j=-l}^{l}f_{j}^{\otimes_{II}}$ is measurable
with respect to
$\mathcal{Z}_{\infty}^{s}\times(\mathcal{Z}_{\infty}^{d-s})^{\mathbb{Z}}$.
Since this $\sigma$-algebra
$\mathcal{Z}^{s}_{\infty}\times(\mathcal{Z}^{d-s}_{\infty})^{\mathbb{Z}}$ is
invariant under $\widetilde{\sigma}=\tau_{\vec{a}}\times\sigma_{II}$, we have

$\begin{split}&\mathbb{E}_{\widetilde{\mu}_{\mathcal{A}}^{(\infty)}}\Big{(}\big{(}f_{1}\otimes\cdots\otimes
f_{s}\big{)}\otimes\bigotimes_{j=-l}^{l}f_{j}^{\otimes_{II}}\Big{|}\mathcal{I}(X^{s}\times(X^{d-s})^{\mathbb{Z}},\widetilde{\mu}_{\mathcal{A}}^{(\infty)},\widetilde{\sigma})\Big{)}\\\
&\quad=\lim\limits_{N\to\infty}\frac{1}{N}\sum\limits_{n=0}^{N-1}\Big{(}\big{(}f_{1}\otimes\cdots\otimes
f_{s}\big{)}\otimes\bigotimes_{j=-l}^{l}f_{j}^{\otimes_{II}}\Big{)}\circ\widetilde{\sigma}^{n}\end{split}$

is also measurable with respect to
$\mathcal{Z}^{s}_{\infty}\times(\mathcal{Z}_{\infty}^{d-s})^{\mathbb{Z}}$.
Therefore
$\mathcal{I}(X^{s}\times(X^{d-s})^{\mathbb{Z}},\widetilde{\mu}_{\mathcal{A}}^{(\infty)},\widetilde{\sigma})$
is measurable with respect to
$\mathcal{Z}_{\infty}^{s}\times(\mathcal{Z}_{\infty}^{d-s})^{\mathbb{Z}}$. ∎

###  4.4. The $\sigma$-algebra of invariant sets of
$((X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}},\sigma)$

Before going on, we recall that when $(Y,\mathcal{Y},\nu,S)$ is a factor of
$(X,\mathcal{X},\mu,T)$, we denote by the same letter the $\sigma$-algebra
$\mathcal{Y}$ and its inverse image by $\pi$, $\pi^{-1}(\mathcal{Y})$. In
other words, if $(Y,\mathcal{Y},\nu,S)$ is a factor of
$(X,\mathcal{X},\mu,T)$, we think of $\mathcal{Y}$ as a sub-$\sigma$-algebra
of $\mathcal{X}$.

######  Definition 4.12.

Let $\pi:(X,\mathcal{X},\mu,T)\rightarrow(Y,\mathcal{Y},\nu,S)$ be a
homomorphism. $\pi$ is ergodic or $(X,\mathcal{X},\mu,T)$ is an ergodic
extension of $(Y,\mathcal{Y},\nu,S)$ if $T$-invariant sets of $\mathcal{X}$ is
contained in $\mathcal{Y}$, i.e.
$\mathcal{I}(X,\mathcal{X},\mu,T)\subseteq\mathcal{Y}$.

By this definition, we can rephrase Proposition 4.11 as follows:

######  Theorem 4.13.

Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s. and $d\in\mathbb{N}$. Let
$\mathcal{A}=\\{p_{1},\ldots,p_{d}\\}$ be a family of non-constant integral
polynomials satisfying $(\spadesuit)$. Let
$\pi_{\infty}:(X,\mathcal{X},\mu,T)\rightarrow(Z_{\infty},\mathcal{Z}_{\infty},\mu_{\infty},T)$
be the factor map from $X$ to its $\infty$-step pro-nilfactor. Then the factor
map

$\begin{split}\pi_{\infty}^{(s)}\times\pi_{\infty}^{\infty}:\
&(X^{s}\times(X^{d-s})^{\mathbb{Z}},\mathcal{X}^{s}\times(\mathcal{X}^{d-s})^{\mathbb{Z}},\widetilde{\mu}^{(\infty)}_{\mathcal{A}},\widetilde{\sigma})\\\
&\quad\quad\quad\quad\longrightarrow(Z_{\infty}^{s}\times({Z_{\infty}^{d-s}})^{\mathbb{Z}},\mathcal{Z}_{\infty}^{s}\times(\mathcal{Z}_{\infty}^{d-s})^{\mathbb{Z}},\widetilde{(\mu_{\infty})}^{(\infty)}_{\mathcal{A}},\widetilde{\sigma})\end{split}$

is ergodic.

In particular, one has that the $\sigma$-algebra
$\mathcal{I}(X^{s}\times(X^{d-s})^{\mathbb{Z}},\widetilde{\mu}_{\mathcal{A}}^{(\infty)},\widetilde{\sigma})$
is equal to (modulo isomorphisms) the $\sigma$-algebra
$\mathcal{I}(Z_{\infty}^{s}\times({Z_{\infty}^{d-s}})^{\mathbb{Z}},(\widetilde{\mu_{\infty}})_{\mathcal{A}}^{(\infty)},\widetilde{\sigma})$.

###### Proof.

By Proposition 4.11, the $\sigma$-algebra
$\mathcal{I}(X^{s}\times(X^{d-s})^{\mathbb{Z}},\widetilde{\mu}_{\mathcal{A}}^{(\infty)},\widetilde{\sigma})$
is measurable with respect to
$\mathcal{Z}_{\infty}^{s}\times(\mathcal{Z}_{\infty}^{d-s})^{\mathbb{Z}}$.
Thus, it means that $\pi_{\infty}^{(s)}\times\pi_{\infty}^{\infty}$ is ergodic
by the definition.

Since
$(Z_{\infty}^{s}\times({Z_{\infty}^{d-s}})^{\mathbb{Z}},\mathcal{Z}_{\infty}^{s}\times(\mathcal{Z}_{\infty}^{d-s})^{\mathbb{Z}},\widetilde{(\mu_{\infty})}^{(\infty)}_{\mathcal{A}},\widetilde{\sigma})$
is the factor of

$(X^{s}\times(X^{d-s})^{\mathbb{Z}},\mathcal{X}^{s}\times(\mathcal{X}^{d-s})^{\mathbb{Z}},\widetilde{\mu}^{(\infty)}_{\mathcal{A}},\widetilde{\sigma}),$

$\mathcal{I}(Z_{\infty}^{s}\times({Z_{\infty}^{d-s}})^{\mathbb{Z}},(\widetilde{\mu_{\infty}})_{\mathcal{A}}^{(\infty)},\widetilde{\sigma})$
is contained in
$\mathcal{I}(X^{s}\times(X^{d-s})^{\mathbb{Z}},\widetilde{\mu}_{\mathcal{A}}^{(\infty)},\widetilde{\sigma})$.
On the other hand, as

$\mathcal{I}(X^{s}\times(X^{d-s})^{\mathbb{Z}},\widetilde{\mu}_{\mathcal{A}}^{(\infty)},\widetilde{\sigma})\subseteq\mathcal{Z}_{\infty}^{s}\times(\mathcal{Z}_{\infty}^{d-s})^{\mathbb{Z}},$

we have that
$\mathcal{I}(X^{s}\times(X^{d-s})^{\mathbb{Z}},\widetilde{\mu}_{\mathcal{A}}^{(\infty)},\widetilde{\sigma})$
is contained in
$\mathcal{I}(Z_{\infty}^{s}\times({Z_{\infty}^{d-s}})^{\mathbb{Z}},(\widetilde{\mu_{\infty}})_{\mathcal{A}}^{(\infty)},\widetilde{\sigma})$.
∎

######  Theorem 4.14.

Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s. and $d\in\mathbb{N}$. Let
$\mathcal{A}=\\{p_{1},p_{2},\cdots,p_{d}\\}$ be a family of non-constant
integral polynomials satisfying $(\spadesuit)$. Let
$\pi_{\infty}:(X,\mathcal{X},\mu,T)\rightarrow(Z_{\infty},\mathcal{Z}_{\infty},\mu_{\infty},T)$
be the factor map from $X$ to its $\infty$-step pro-nilfactor, and let
$\mu=\int_{Z_{\infty}}\nu_{z}\ d\mu_{\infty}(z)$ be the disintegration of
$\mu$ over $\mu_{\infty}$. Then $\widetilde{\mu}_{\mathcal{A}}^{(\infty)}$ is
the conditional product measure with respect to
$(\widetilde{\mu_{\infty}})_{\mathcal{A}}^{(\infty)}$. That is,

$\widetilde{\mu}_{\mathcal{A}}^{(\infty)}=\int_{Z^{s}_{\infty}\times(Z_{\infty}^{d-s})^{\mathbb{Z}}}\big{(}\nu_{z_{1}}\times\cdots\times\nu_{z_{s}}\big{)}\times\prod_{n\in\mathbb{Z}}\Big{(}\nu_{z_{n}^{(s+1)}}\times\cdots\times\nu_{z_{n}^{(d)}}\Big{)}d(\widetilde{\mu_{\infty}})_{\mathcal{A}}^{(\infty)}((z_{1},\ldots,z_{s}),{\bf
z}),$

where $\big{(}(z_{1},\ldots,z_{s}),{\bf
z}\big{)}=\big{(}(z_{1},\ldots,z_{s}),(z_{n}^{(s+1)},\ldots,z_{n}^{(d)})_{n\in\mathbb{Z}}\big{)}\in
Z^{s}_{\infty}\times(Z_{\infty}^{d-s})^{\mathbb{Z}}$.

###### Proof.

We need to show that for all $f_{1},\ldots,f_{s}\in L^{\infty}(X,\mu)$, all
$l\in\mathbb{N}$ and all $f_{j}^{\otimes_{II}}=f_{j}^{(s+1)}\otimes
f_{j}^{(s+2)}\otimes\cdots\otimes f_{j}^{(d)}$ (where $f^{(t)}_{j}\in
L^{\infty}(X,\mu)$, $s+1\leq t\leq d,-l\leq j\leq l$), one has

$\begin{split}&\int_{X^{s}\times(X^{d-s})^{\mathbb{Z}}}\big{(}f_{1}\otimes\cdots\otimes
f_{s}\big{)}\otimes\bigotimes_{j=-l}^{l}f_{j}^{\otimes_{II}}d\widetilde{\mu}_{\mathcal{A}}^{(\infty)}\\\
&=\int_{Z^{s}_{\infty}\times(Z_{\infty}^{d-s})^{\mathbb{Z}}}\mathbb{E}_{\mu}(f_{1}|\mathcal{Z}_{\infty})\otimes\cdots\otimes\mathbb{E}_{\mu}(f_{s}|\mathcal{Z}_{\infty})\otimes\bigotimes_{j=-l}^{l}\mathbb{E}_{\mu}(f_{j}^{\otimes_{II}}|\mathcal{Z}_{\infty})d(\widetilde{\mu_{\infty}})_{\mathcal{A}}^{(\infty)},\end{split}$

where

$\mathbb{E}_{\mu}(f^{\otimes_{II}}_{j}|\mathcal{Z}_{\infty})=\mathbb{E}_{\mu}(f^{(s+1)}_{j}|\mathcal{Z}_{\infty})\otimes\mathbb{E}_{\mu}(f^{(2)}_{j}|\mathcal{Z}_{\infty})\otimes\cdots\otimes\mathbb{E}_{\mu}(f^{(d)}_{j}|\mathcal{Z}_{\infty}),$

$-l\leq j\leq l$. But this follows from Lemma 4.10 by taking integrals both
sides of (4.8). ∎

Since by (3.17) we have the isomorphism

$\phi:((X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}},\sigma)\cong(X^{s}\times(X^{d-s})^{\mathbb{Z}},\mathcal{X}^{s}\times(\mathcal{X}^{d-s})^{\mathbb{Z}},\widetilde{\mu}^{(\infty)}_{\mathcal{A}},\widetilde{\sigma}),$

we have the following commuting diagram

$\textstyle{((X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}},\sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\infty}^{\infty}}$$\textstyle{(X^{s}\times(X^{d-s})^{\mathbb{Z}},\mathcal{X}^{s}\times(\mathcal{X}^{d-s})^{\mathbb{Z}},\widetilde{\mu}^{(\infty)}_{\mathcal{A}},\widetilde{\sigma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\infty}^{(s)}\times\pi_{\infty}^{\infty}}$$\textstyle{((Z_{\infty}^{d})^{\mathbb{Z}},(\mathcal{Z}_{\infty}^{d})^{\mathbb{Z}},(\mu_{\infty})_{\mathcal{A}}^{(\infty)},\sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(Z_{\infty}^{s}\times({Z_{\infty}^{d-s}})^{\mathbb{Z}},\mathcal{Z}_{\infty}^{s}\times(\mathcal{Z}_{\infty}^{d-s})^{\mathbb{Z}},\widetilde{(\mu_{\infty})}^{(\infty)}_{\mathcal{A}},\widetilde{\sigma}).}$

By the diagram, we can replace $\widetilde{\mu}^{(\infty)}_{\mathcal{A}}$ in
results above by $\mu^{(\infty)}_{\mathcal{A}}$. Thus, by Theorem 4.13 we have

######  Theorem 4.15.

Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s. and $d\in\mathbb{N}$. Let
$\mathcal{A}=\\{p_{1},p_{2},\cdots,p_{d}\\}$ be a family of non-constant
integral polynomials satisfying $(\spadesuit)$. Let
$\pi_{\infty}:(X,\mathcal{X},\mu,T)\rightarrow(Z_{\infty},\mathcal{Z}_{\infty},\mu_{\infty},T)$
be the factor map from $X$ to its $\infty$-step pro-nilfactor. Then the factor
map

$\pi_{\infty}^{\infty}:((X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu_{\mathcal{A}}^{(\infty)},\sigma)\rightarrow((Z_{\infty}^{d})^{\mathbb{Z}},(\mathcal{Z}_{\infty}^{d})^{\mathbb{Z}},(\mu_{\infty})_{\mathcal{A}}^{(\infty)},\sigma)$

is ergodic.

In particular, one has that the $\sigma$-algebra
$\mathcal{I}((X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu_{\mathcal{A}}^{(\infty)},\sigma)$
is equal to (modulo isomorphism) the $\sigma$-algebra
$\mathcal{I}((Z_{\infty}^{d})^{\mathbb{Z}},(\mathcal{Z}_{\infty}^{d})^{\mathbb{Z}},(\mu_{\infty})_{\mathcal{A}}^{(\infty)},\sigma)$.

##  5\. Polynomial orbits in nilsystems

Now we work on pro-nilsystems. All results remain true for $\infty$-step pro-
nilsystems. Since
$((X^{d})^{\mathbb{Z}},(\mathcal{X}^{d})^{\mathbb{Z}},\mu^{(\infty)}_{\mathcal{A}},\sigma)\cong(X^{s}\times(X^{d-s})^{\mathbb{Z}},\mathcal{X}^{s}\times(\mathcal{X}^{d-s})^{\mathbb{Z}},\widetilde{\mu}^{(\infty)}_{\mathcal{A}},\widetilde{\sigma}),$
we only give statements concering $\mu^{(\infty)}_{\mathcal{A}}$, and the
corresponding results for $\widetilde{\mu}^{(\infty)}_{\mathcal{A}}$ can be
proved similarly.

###  5.1. The measure $\mu_{\mathcal{A}}^{(\infty)}$ on nilsystems

#### 5.1.1.

Let $(X=G/\Gamma,\nu,T)$ be a nilsystem with $T$ the translation by an element
$\tau\in G$. Then $(X,T)$ is a t.d.s. By Theorem 2.8, $(X,T)$ is uniquely
ergodic if and only if $(X,\nu,T)$ is ergodic if and only if $(X,T)$ is
minimal. This result holds for pro-nilsystems.

#### 5.1.2.

Now let $(X,T)$ be a minimal topological pro-nilsystem (a minimal pro-
nilsystem for short). Let $x\in X$ and $d\in\mathbb{N}$. Let
$\mathcal{A}=\\{p_{1},\cdots,p_{d}\\}$ be a family of non-constant essentially
distinct integral polynomials with $p_{1}(0)=\ldots=p_{d}(0)=0$. Recall that

(5.1)
$\omega_{x}^{\mathcal{A}}=\Big{(}(\tau^{p_{1}(n)}x,\tau^{p_{2}(n)}x,\ldots,\tau^{p_{d}(n)}x)\Big{)}_{n\in\mathbb{Z}}\in(X^{d})^{\mathbb{Z}}.$

Note that
$(\sigma\omega^{\mathcal{A}}_{x})(n)=(\tau^{p_{1}(n+1)}x,\tau^{p_{2}(n+1)}x,\ldots,\tau^{p_{d}(n+1)}x)$
for $n\in\mathbb{Z}$. Recall that

(5.2)
$N_{\infty}(X,\mathcal{A})=\overline{\bigcup\\{\sigma^{n}\omega^{\mathcal{A}}_{x}:n\in\mathbb{Z},x\in
X\\}}\subseteq(X^{d})^{\mathbb{Z}}.$

In [26] we have shown the following result.

######  Theorem 5.1.

[26, Theorem 5.5] Let $(X,T)$ be a minimal pro-nilsystem. Let
$\mathcal{A}=\\{p_{1},\ldots,p_{d}\\}$ be a family of non-constant essentially
distinct integral polynomials with $p_{1}(0)=\cdots=p_{d}(0)=0$. Then we have

  1. (1)

The system $(N_{\infty}(X,\mathcal{A}),\langle T^{\infty},\sigma\rangle)$ is a
minimal pro-nilsystem.

  2. (2)

For each $x\in X$, the system
$(\overline{\mathcal{O}}(\omega_{x}^{\mathcal{A}},\sigma),\sigma)$ is a
minimal pro-nilsystem.

Now we prove the following

######  Theorem 5.2.

Let $(X,T)$ be a minimal pro-nilsystem. Let
$\mathcal{A}=\\{p_{1},\cdots,p_{d}\\}$ be a family of non-constant essentially
distinct integral polynomials. With the notations above, we have

  1. (1)

$(N_{\infty}(X,\mathcal{A}),\langle T^{\infty},\sigma\rangle)$ is uniquely
ergodic with the $\infty$-joining $\nu^{(\infty)}_{\mathcal{A}}$.

  2. (2)

For each $x\in X$,
$(\overline{\mathcal{O}}(\omega^{\mathcal{A}}_{x},\sigma),\sigma)$ is uniquely
ergodic with some measure $\nu^{(\infty)}_{\mathcal{A},x}$.

  3. (3)

$\displaystyle\nu^{(\infty)}_{\mathcal{A}}=\int_{X}\nu^{(\infty)}_{\mathcal{A},x}\
d\nu(x)$.

  4. (4)

For all $l\in\mathbb{N}$ and all $f_{j}^{\otimes}=f_{j}^{(1)}\otimes
f_{j}^{(2)}\otimes\cdots\otimes f_{j}^{(d)}$ (where $f^{(i)}_{j}\in
L^{\infty}(X,\mu),1\leq i\leq d,-l\leq j\leq l$)

(5.3)
$\begin{split}&\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{j=-l}^{l}f_{j}^{\otimes}(T^{\vec{p}(n+j)}x^{\otimes
d})=\int_{(X^{d})^{\mathbb{Z}}}\Big{(}\bigotimes_{j=-l}^{l}f_{j}^{\otimes}\Big{)}({\bf
x})\ d\nu_{\mathcal{A},x}^{(\infty)}({\bf x}),\end{split}$

for $\nu$ a.e. $x\in X$.

###### Proof.

By Theorem 5.1, $(N_{\infty}(X,\mathcal{A}),\langle T^{\infty},\sigma\rangle)$
is a minimal pro-nilsystem, and for each $x\in X$, so is the system
$(\overline{\mathcal{O}}(\omega_{x}^{\mathcal{A}},\sigma),\sigma)$. It follows
by Theorem 2.8 that they are uniquely ergodic. We denote their unique measures
by $\nu_{\mathcal{A}}^{(\infty)}$ and $\nu_{\mathcal{A},x}^{(\infty)}$
respectively.

Now we show that (3) holds, that is,
$\displaystyle\nu^{(\infty)}_{\mathcal{A}}=\int_{X}\nu^{(\infty)}_{\mathcal{A},x}\
d\nu(x)$. Let

$\varphi:X\rightarrow{\mathcal{M}}((X^{d})^{\mathbb{Z}}),\
x\mapsto\nu^{(\infty)}_{\mathcal{A},x}.$

Since the map

$X\rightarrow{\mathcal{M}}((X^{d})^{\mathbb{Z}}),\
x\mapsto\frac{1}{N}\sum_{n=0}^{N-1}(\cdots\times
T^{\vec{p}(n-1)}\times\underset{\bullet}{T^{\vec{p}(n)}}\times
T^{\vec{p}(n+1)}\times\cdots)_{*}\delta_{x^{(\infty)}}$

is continuous (here $x^{(\infty)}=(\ldots,x^{\otimes d},x^{\otimes
d},\ldots)$) and

$\displaystyle\frac{1}{N}\sum_{n=0}^{N-1}(\cdots\times
T^{\vec{p}(n-1)}\times\underset{\bullet}{T^{\vec{p}(n)}}\times
T^{\vec{p}(n+1)}\times\cdots)_{*}\delta_{x^{(\infty)}}$

converges to $\nu^{(\infty)}_{\mathcal{A},x}$ weakly $\big{(}$ as
$(\overline{\mathcal{O}}(\omega_{x}^{\mathcal{A}},\sigma),\nu_{\mathcal{A},x}^{(\infty)},\sigma)$
is uniquely ergodic $\big{)}$, we have that $\varphi$ is a Borel map. Thus we
can define a measure
$\displaystyle\kappa=\int_{X}\nu^{(\infty)}_{\mathcal{A},x}\ d\nu(x)$. It is
easy to check that $\kappa$ is $\langle T^{\infty},\sigma\rangle$-invariant.
By (1), the system $(N_{\infty}(X,\mathcal{A}),\langle
T^{\infty},\sigma\rangle)$ is uniquely ergodic with the $\infty$-joining
$\nu^{(\infty)}_{\mathcal{A}}$, and hence by the uniqueness we have

$\nu^{(\infty)}_{\mathcal{A}}=\kappa=\int_{X}\nu^{(\infty)}_{\mathcal{A},x}\
d\nu(x).$

Since for each $x\in X$,
$(\overline{\mathcal{O}}(\omega^{\mathcal{A}}_{x},\sigma),\sigma)$ is uniquely
ergodic with the measure $\nu^{(\infty)}_{\mathcal{A},x}$, we have that for
all $F\in C((X^{d})^{\mathbb{Z}})$,

$\lim_{n\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}F(\sigma^{n}(\omega_{x}^{\mathcal{A}}))=\int_{(X^{d})^{\mathbb{Z}}}F({\bf
x})\ d\nu^{(\infty)}_{\mathcal{A},x}({\bf x}),\ \ \text{where}\ {\bf x}=({\bf
x}_{n})_{n\in\mathbb{Z}}\in(X^{d})^{\mathbb{Z}}.$

That is, for all $l\in\mathbb{N}$ and all $f_{j}^{\otimes}=f_{j}^{(1)}\otimes
f_{j}^{(2)}\otimes\cdots\otimes f_{j}^{(d)}$ (where $f^{(i)}_{j}\in C(X),1\leq
i\leq d,-l\leq j\leq l$)