text:
\begin{document}
\title[pointwise convergence of multiple ergodic
averages]{pointwise convergence of multiple ergodic
averages and strictly ergodic models}

\author{Wen Huang}
\author{Song Shao}
\author{Xiangdong Ye}

\address{Department of Mathematics, University of Science and Technology of China,
Hefei, Anhui, 230026, P.R. China.}

\email{wenh@mail.ustc.edu.cn}\email{songshao@ustc.edu.cn}
\email{yexd@ustc.edu.cn}

\subjclass[2000]{Primary: 37A05, 37B05} \keywords{ergodic averages, distal systems}

\thanks{Huang is partially supported by NNSF for Distinguished Young School (11225105),
Shao and Ye is partially supported by NNSF of China (11171320), and Huang and Ye is partially supported by NNSF of China (11371339).}

\date{June 14, 2014}
\date{June 22, 2014}

\begin{abstract}
By building some suitable strictly ergodic models, we prove that for an
ergodic system $(X,\X,\mu, T)$, $d\in\N$, $f_1, \ldots, f_d \in
L^{\infty}(\mu)$, the averages
\begin{equation*}
    \frac{1}{N^2} \sum_{(n,m)\in [0,N-1]^2}
    f_1(T^nx)f_2(T^{n+m}x)\ldots f_d(T^{n+(d-1)m}x)
\end{equation*}
converge $\mu$ a.e.

Deriving some results from the construction, for distal systems we answer positively the question if the multiple ergodic averages converge
a.e. That is, we show that if $(X,\X,\mu, T)$ is an ergodic distal system,
and $f_1, \ldots, f_d \in L^{\infty}(\mu)$, then multiple ergodic averages
\begin{equation*}
    \frac 1 N\sum_{n=0}^{N-1}f_1(T^nx)\ldots f_d(T^{dn}x)
\end{equation*}
converge $\mu$ a.e..

\end{abstract}

\maketitle

\section{Introduction}

\subsection{Main results}

Throughout this paper, by a {\it topological dynamical
system} (t.d.s. for short) we mean a pair $(X, T)$, where $X$ is a
compact metric space  and $T$ is a homeomorphism from $X$ to itself.
A {\em measurable system} (m.p.t. for short) is a quadruple $(X,\X,
\mu, T)$, where $(X,\X,\mu )$ is a Lebesgue probability space and $T
: X \rightarrow X$ is an invertible measure preserving
transformation.

\medskip

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

\medskip

The well-known Jewett-Krieger's theorem  \cite{Jewett, Krieger}
states that every ergodic system has a strictly ergodic model. We
note that one can add some additional properties to the topological
model. For example, in \cite{Lehrer} Lehrer showed that the strictly
ergodic model can be required to be a topological (strongly) mixing
system in addition.

\medskip

Now let $\h{\tau}_d=\h{T}\times \ldots\times \h{T} \ (d \ \text{times})$ and
$\h{\sigma}_d =\h{T}\times \ldots \times \h{T}^{d}$. The group generated by
$\h{\tau}_d$ and $\h{\sigma}_d$ is denoted $\langle \h{\tau}_d, \h{\sigma}_d\rangle$.
For any $x\in \h{X}$, let $N_d(\h{X},x)=\overline{\O((x,\ldots,x),
\langle\h{\tau}_d, \h{\sigma}_d\rangle)}$, the orbit closure of
$(x,\ldots,x)$ ($d$ times) under the action of the group
$\langle\h{\tau}_d, \h{\sigma}_d\rangle$. We remark that if $(\h{X},\h{T})$ is
minimal, then all $N_d(\h{X},x)$ coincide, which will be denoted by
$N_d(\h{X})$. It was shown by Glasner \cite{G94} that if $(\h{X},\h{T})$
is minimal, then $(N_d(\h{X}), \langle\h{\tau}_d, \h{\sigma}_d\rangle)$ is
minimal.

\medskip

In this paper, first we will show the following theorem.

\medskip

\noindent {\bf Theorem A:} {\em Let $(X,\X, \mu, T)$ be an ergodic
m.p.t. and $d\in\N$. Then it has a strictly ergodic model $(\h{X}, \h{T})$ such that
$(N_d(\h{X}), \langle\h{\tau}_d, \h{\sigma}_d\rangle)$ is strictly ergodic.
}

\medskip

As a consequence, we have:

\medskip

\noindent {\bf Theorem B:} {\em Let $(X,\X,\mu, T)$ be an ergodic
m.p.t. and $d\in\N$. Then for $f_1, \ldots, f_d \in L^{\infty}(\mu)$
the averages
\begin{equation}\label{D}
    \frac{1}{N^2} \sum_{n,m\in [0,N-1]}
    f_1(T^nx)f_2(T^{n+m}x)\ldots f_d(T^{n+(d-1)m}x)
\end{equation}
converge $\mu$ a.e.}

\medskip

We remark that similar theorems as Theorems A and B can be established for cubes.
Moreover, the convergence in Theorem B can be stated for any tempered F{\rm ${\o}$}lner sequence
$\{F_N\}_{N\ge 1}$ of $\Z^2$ instead of $[0,N-1]^2$.

\medskip

It is a long open question if the multiple ergodic averages $\frac 1 N\sum_{n=0}^{N-1}\Pi_{j=1}^d f_j(T^{jn}x)$ converge a.e.
Using some results developed when proving Theorem A, we answer the question positively for distal systems.
Namely, we have

\medskip

\noindent {\bf Theorem C:} {\em Let $(X,\X,\mu, T)$ be an ergodic distal system,
and $d\in \N$. Then for all $f_1, \ldots, f_d \in L^{\infty}(\mu)$
\begin{equation*}
    \frac 1 N\sum_{n=0}^{N-1}f_1(T^nx)\ldots f_d(T^{dn}x)
\end{equation*}
converge $\mu$ a.e..}

\medskip

Note that Furstenberg's structure theorem \cite{F} states that each ergodic systems is a weakly mixing extension of
an ergodic distal system. Thus, by Theorem C the open question is reduced to deal with the weakly mixing extensions.
Moreover, we have the following conjecture.

\medskip
\noindent{\bf Conjecture:} {\em Let $(X,\X,\mu, T)$ be an ergodic system, then it has a topological model $(\h{X}, \h{T})$
such that for a.e. $x\in \h{X}$, $(x, \ldots,x)$ is a generic point of some ergodic measures $\mu_x^{(d)}$ invariant under
$\h{T}\times \ldots \times \h{T}^d$.}

\medskip

We also conjecture that the measures $\mu_x^{(d)}$ are the ones defined in Theorem \ref{decomposition}.

Once the conjecture is proven then the multiple ergodic averages converge a.e.
by a similar argument we used to prove Theorem B.
\subsection{Backgrounds}

In this subsection we will give backgrounds of our research.

\subsubsection{Ergodic averages}

In this subsection we recall some
results related to pointwise ergodic averages.

The first pointwise ergodic theorem was proved by Birkhoff in 1931.
Followed from Furstenberg's beautiful work on the dynamical proof of Szemeradi's theorem in 1977, problems concerning the convergence of
multiple ergodic averages (in $L^2$ or pointwisely) attracts a lot of attention.

\medskip

The convergence of the averages
\begin{equation}\label{multiple1}
    \frac 1 N\sum_{n=0}^{N-1}f_1(T^nx)\ldots
f_d(T^{dn}x)
\end{equation}
in $L^2$ norm was established by Host and Kra \cite{HK05} (see also Ziegler \cite{Z}). We note that in their
proofs, the characteristic factors play a great role. The convergence of the multiple ergodic average for commuting
transformations was obtained by Tao \cite{Tao} using the finitary ergodic method, see \cite{Austin,H} for
more traditional ergodic proofs by Austin and Host respectively. Recently, the convergence of multiple ergodic averages for nilpotent group actions
was proved by Walsh \cite{Walsh}.

\medskip

The first breakthrough on pointwise convergence of (\ref{multiple1}) for $d > 1$ is due to Bourgain,
who showed in \cite{B90} that for $d = 2$, for $p,q\in \N$ and for
all $f_1, f_2 \in  L^\infty$,
the limit $\frac{1}{N}\sum_{n=0}^{N-1} f_1(T^px)f_2(T^qx)$ exists a.e.
Before Bourgain's work, Lesigne showed this convergence holds if the
system is distal, with $T^p$, $T^q$ and $T^{p-q}$ ergodic \cite{L87}. Also in \cite{DL,Assani98}, it was shown
that the problem of the almost everywhere convergence of (\ref{multiple1}) can be deduced
to the case when the m.p.t. has zero entropy. One can also find some results for
weakly mixing transformations in \cite{Abd, Assani98, Assani13}.

\medskip

Recently there are some studies on the limiting behavior of the averages along cubes,
and please refer to \cite{Bergelson00,HK05,Assani,CF} for details.

Also in \cite{CF}, they obtained the following result.
For $i=1,2,\ldots,d$, let $T_i: X \rightarrow X$ be m.p.t., $f_i\in L^\infty(\mu)$ be functions,
$p_i\in \mathbb{Z}[t]$ be non-constant polynomials such that $p_i-p_j$ is non-constant for $i\neq j$,
and $b: \mathbb{N} \rightarrow \mathbb{N}$ be a sequence such that $b(N)\to \infty$ and $b(N)/N^{1/d}\to 0$
as $N\to \infty$, where $d$ is the maximum degree of the polynomials $p_i$. Then the averages
$$\frac{1}{Nb(N)}\sum_{1\le m\le N, 1\le n\le b(N)} f_1(T_1^{m+p_1(n)}x)\ldots f_d (T_d^{m+p_l(n)} x) $$
converge pointwise as $N\to \infty$.

\subsubsection{Topological model}

The pioneering work on topological model was done by Jewett in
\cite{Jewett}. He proved the theorem under the additional assumption
that $T$ is weakly mixing,  and the general case was proved by Krieger in \cite{Krieger} soon.

The papers of Hansel and Raoult \cite{HanRao}, Bellow and Furstenberg \cite{BeF}
and Denker \cite{Denker}, gave different proofs of the theorem in
the general ergodic case (see also \cite{DCS}).

One can add some additional properties to the topological model. For example, in \cite{Lehrer}
Lehrer showed that the strictly ergodic model can be required as a
topological (strongly) mixing system in addition. Our Theorems A strengthen Jewett-Krieger' Theorem
in other direction, i.e. we can require the model to be well behavioral under some group actions
on some subsets of the product space.

It is well known that each m.p.t. has a topological model
\cite{F}. There are universal models, models for some group actions and models for some special classes.
Weiss \cite{Weiss89} showed the following nice result:
there exists a minimal t.d.s. $(X, T)$ with the property that for
every aperiodic ergodic m.p.t. $(Y,\Y,\nu, S)$ there exists a
$T$-invariant Borel probability measure $\mu$ on $X$ such that the
systems $(Y,\Y,\nu, S)$ and $(X,\B(X),\mu, T)$ are measure
theoretically isomorphic.

Weiss \cite{Weiss85} (see also \cite{Weiss00, Glasner, GW06}) showed that Jewett-Krieger' Theorem can be
generalized from $\Z$-actions to commutative group actions.

An ergodic system has a doubly
minimal model if and only if it has zero entropy \cite{Weiss95}
(other topological models for zero entropy systems can be found in
\cite{HM, DownL}); and an ergodic system has a strictly ergodic, UPE
(uniform positive entropy) model if and only if it has positive
entropy \cite{GW94}.

Note that not any dynamical properties can be added in the uniquely ergodic models. For example,
Lindernstrauss showed that every ergodic measure distal system $(X,\X,\mu,T )$ has a minimal
topologically distal model \cite{Lindenstrauss}.
This topological model need not, in general, be uniquely ergodic. In other words there
are measure distal systems for which no uniquely ergodic topologically distal model
exists \cite{Lindenstrauss}. We refer to \cite{GW06} for more information on the topic.

\medskip

We say that $\h{\pi}: \h{X}\rightarrow \h{Y}$ is a {\em topological
model} for $\pi: (X,\X, \mu, T)\rightarrow (Y,\Y, \nu, S)$ if
$\h{\pi}$ is a topological factor map and there exist measure
theoretical isomorphisms $\phi$ and $\psi$ such that the diagram
\[
\begin{CD}
X @>{\phi}>> \h{X}\\
@V{\pi}VV      @VV{\h{\pi}}V\\
Y @>{\psi }>> \h{Y}
\end{CD}
\]
is commutative, i.e. $\h{\pi}\phi=\psi\pi$. Weiss \cite{Weiss85}
generalized the theorem of Jewett-Krieger to the relative case.
Namely, he proved that if $\pi: (X,\X, \mu, T)\rightarrow (Y,\Y, \nu,
S)$ is a factor map with $(X,\X, \mu, T)$ ergodic and
$(\h{Y},\h{\Y}, \h{\nu}, \h{S})$ is a uniquely ergodic model for $(Y,\Y,
\nu, T)$, then there is a uniquely ergodic model $(\h{X}, \h{\X},
\h{\mu}, \h{T})$ for $(X,\X, \mu, T)$ and a factor map $\h{\pi}:
\h{X}\rightarrow \h{Y}$ which is a model for $\pi: X\rightarrow Y$.
We will refer this theorem as {\it Weiss's Theorem}.
We note that in \cite{Weiss85} Weiss pointed that the relative case holds
for commutative group actions.

\subsection{Main ideas of the proofs}

Now we describe the main ideas and ingredients in the proofs.

To prove Theorem A
the first fact we face is that for an ergodic m.p.t. $(X,\X, \mu, T)$,
not every strictly ergodic model is the one we need in Theorem A.
This indicates that
to obtain Theorem A, Jewett-Krieger' Theorem is not enough for our purpose. Fortunately, we find that
Weiss's Theorem is a right tool.

Precisely, for $d\ge 3$ let $\pi_{d-2}: X\rightarrow Z_{d-2}$ be the factor map from
$X$ to its $d-2$-step nilfactor $Z_{d-2}$. By the results in \cite{HKM}, $Z_{d-2}$ may be
regarded as a topological system in the natural way. Using Weiss's Theorem
there is a uniquely ergodic model $(\h{X}, \h{\X},
\h{\mu}, \h{T})$ for $(X,\X, \mu, T)$ and a factor map $\h{\pi}_{d-2}:
\h{X}\rightarrow Z_{d-2}$ which is a model for $\pi_{d-2}: X\rightarrow
Z_{d-2}$.
\[
\begin{CD}
X @>{\phi}>> \h{X}\\
@V{\pi_{d-2}}VV      @VV{\h{\pi}_{d-2}}V\\
Z_{d-2} @>{ }>> Z_{d-2}
\end{CD}
\]

We then show (though it is not easy) that $(\h{X},\h{T})$ is what we need. To this aim,
we need understand well the ergodic decomposition of $d$-fold self-joinings of $X$. We first study
the $\sigma$-algebra of $\sigma_d$-invariant sets under $\sigma_d$, and show that we always can deduce
this $\sigma$-algebra to the one on its nilfactors. Then via studying nilsystems,
we get the ergodic decomposition of Furstenberg self-joinings under the action $ \sigma_d$.
This is the main tool we develop to prove Theorem A.

Once Theorem A  is proven, Theorem B will follow by an argument using some
well known theorems related to pointwise convergence for $\Z^d$ actions and for uniquely ergodic systems.

Let $(X,\X,\mu,T)$ be an ergodic distal system. Then $\pi_{d-1}: X\rightarrow Z_{d-1}$ is
a distal extension. By Furstenberg's Structure Theorem, $\pi_{d-1}$ is decomposed into
isometric extensions and inverse limit. We show that the property of almost everywhere
convergence of the multiple ergodic averages (\ref{multiple1}) is preserved by these
isometric extensions, and then we conclude Theorem C. This argument is inspired by Lesigne's work in \cite{L87}.

\subsection{Organization of the paper}

We organize the paper as follows. In Section \ref{section-pre} we introduce some basic notions and
leave most related notions and results in the appendix. In Section \ref{section-furstenberg-joining}
we study the ergodic decomposition of self-joinings under $T\times T^2\times \ldots \times T^d$. Then in Sections \ref{section-thA},\ref{section-thB}
and \ref{section-thC}, we prove Theorems A, B and C respectively.

\section{Preliminaries}\label{section-pre}

\subsection{Ergodic theory and topological dynamics}

In this subsection we introduce some basic notions in ergodic theory
and topological dynamics. For more information, see Appendix.

\medskip

In this paper, instead of just considering a single transformation $T$,
we will consider commuting
transformations $T_1,\ldots , T_k$ of $X$. We only recall some basic
definitions and properties of systems for
one transformation. Extensions to the general case are
straightforward.

\subsubsection{Measurable systems}

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

\medskip

A {\em homomorphism} from m.p.t. $(X,\X, \mu, T)$ to $(Y,\Y, \nu,
S)$ is a measurable map $\pi : X_0 \rightarrow  Y_0$, where $X_0$ is
a $T$-invariant subset of $X$ and $Y_0$ is an $S$-invariant subset
of $Y$, both of full measure, such that $\pi_*\mu=\mu\circ
\pi^{-1}=\nu$ and $S\circ \pi(x)=\pi\circ T(x)$ for $x\in X_0$. When
we have such a homomorphism we say that $(Y,\Y, \nu, S)$
is a {\em factor} of  $(X,\X, \mu , T)$. If the factor map
$\pi: X_0\rightarrow  Y_0$ can be chosen to be bijective, then we
say that $(X,\X, \mu, T)$ and $(Y,\Y, \nu, S)$ are {\em
(measure theoretically) isomorphic} (bijective maps on Lebesgue
spaces have measurable inverses). A factor can be characterized
(modulo isomorphism) by $\pi^{-1}(\Y)$, which is a $T$-invariant
sub- $\sigma$-algebra of $\X$, and conversely any $T$-invariant
sub-$\sigma$-algebra of $\X$ defines a factor. By a classical result abuse
of terminology we denote by the same letter the $\sigma$-algebra
$\Y$ and its inverse image by $\pi$. In other words, if $(Y,\Y, \nu,
S)$ is a factor of $(X,\X, \mu, T)$, we think of $\Y$ as a
sub-$\sigma$-algebra of $\X$.

\subsubsection{Topological dynamical systems}

A t.d.s. $(X, T)$ is {\em transitive} if there exists
some point $x\in X$ whose orbit $\O(x,T)=\{T^nx: n\in \Z\}$ is dense
in $X$ and we call such a point a {\em transitive point}. The system
is {\em minimal} if the orbit of any point is dense in $X$. This
property is equivalent to saying that X and the empty set are the
only closed invariant sets in $X$. $(X,T)$ is {\em topologically
weakly mixing} if the product system $(X\times X, T\times T)$ is
transitive.

\medskip

A {\em 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 {\em extension } of $(Y,S)$.
The map $\phi$ is called a {\em factor map}.

\subsubsection{$M(X)$ and $M_T(X)$}

For a t.d.s. $(X,T)$, denote by $M(X)$ the set of all
probability measure on $X$. Let $M_T(X)=\{\mu\in M(X):
T_*\mu=\mu\circ T^{-1}=\mu\}$ be the set of all $T$-invariant
Borel probability measures of $X$ and $M_T^e(X)$ be the set of ergodic elements of $M_T(X)$.
It is well known that $M_T^e(X)\neq \emptyset$.

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

\subsubsection{Topological distal systems}

A t.d.s. $(X,T)$ (with metric $\rho$) is
called {\em topologically distal} if $\inf_{n\in \Z} \rho (T^nx,T^nx')>0$ whenever $x,x'\in X$ are distinct.

\subsection{System of order $d-1$}

A factor $Z$ of $X$ is {\em characteristic} for averages
\begin{equation}\label{F}
    \frac 1 N\sum_{n=0}^{N-1}f_1(T^nx)\ldots
f_d(T^{dn}x)
\end{equation}
if the limiting behavior of (\ref{F}) only depends on the
conditional expectation of $f_i$ with respect to $Z$:
\begin{equation*}
    ||\lim_{N\to \infty}\frac{1}{N}\sum_{n=1}^N(T^nf_1 T^{2n}f_2 \ldots T^{dn}f_d -
    T^n\E(f_1|\mathcal{Z}) T^{2n}\E(f_2|\mathcal{Z}) \ldots T^{dn}\E(f_d|\mathcal{Z}))||_{L^2}=0
\end{equation*}
for any $f_1,\ldots,f_d\in L^{\infty}(X,\X,\mu)$. The minimal characteristic factor of (\ref{F}) always exists \cite{HK05, Z}, and it is denoted by $(Z_{d-1},\mathcal{Z}_{d-1},\mu_{d-1})$.

\begin{thm}\cite{HK05}
Let $(X,\X,\mu , T)$ be an ergodic system and $d\in \N$. Then the
system $(Z_{d-1}, \ZZ_{d-1}, \mu_{d-1}, T)$ is a (measure theoretic) inverse
limit of $d-1$-step nilsystems.
$(Z_{d-1}, \ZZ_{d-1}, \mu_{d-1}, T)$ is called a {\it system of order $d-1$}.
\end{thm}

\subsection{Topological system of order $(d-1)$}

We need to use the inverse limit of nilsystems, so we recall the
definition of an inverse limit of t.d.s. If
$(X_i,T_i)_{i\in \N}$ are t.d.s. with $diam(X_i)\le 1$ and
$\pi_i: X_{i+1}\rightarrow X_i$ are factor maps, the {\em inverse
limit} of the systems is defined to be the compact subset of
$\prod_{i\in \N}X_i$ given by $\{ (x_i)_{i\in \N }: \pi_i(x_{i+1}) =
x_i, i\in \N\}$, and we denote it by
$\lim\limits_{\longleftarrow}(X_i,T_i)_{i\in\N}$. It is a compact
metric space endowed with the distance $\rho((x_{i})_{i\in\N}, (y_{i})_{i\in
\N}) = \sum_{i\in \N} 1/2^i \rho_i(x_i, y_i )$, 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.

\begin{de}\cite{HKM}
An inverse limit of $(d-1)$-step minimal
nilsystems is called a {\em topological system of order $(d-1)$}.
\end{de}

\begin{de}
Let $(X, T)$ be a t.d.s. and let $d\in \N$. The points $x, y \in X$ are
said to be {\em regionally proximal of order $d$} if for any $\d  >
0$, there exist $x', y'\in X$ and a vector ${\bf n} = (n_1,\ldots ,
n_d)\in\Z^d$ such that $\rho (x, x') < \d, \rho (y, y') <\d$, and $$
\rho (T^{{\bf n}\cdot \ep}x', T^{{\bf n}\cdot \ep}y') < \d\
\text{for any $\ep\in \{0,1\}^d\setminus \{(0,0,\ldots,0)\}$}.$$
The set of regionally proximal pairs of
order $d$ is denoted by $\RP^{[d]}$ (or by $\RP^{[d]}(X,T)$ in case of
ambiguity), and is called {\em the regionally proximal relation of
order $d$}.
\end{de}

The above definition was introduced in \cite{HKM} by Host-Kra-Maass and it was proved that for
a minimal distal system, $\RP^{[d]}$ is an equivalence relation and $X / \RP^{[d]}$ is a
topological system of order $d$. Later it was shown
that it is an equivalence relation for any minimal systems by Shao-Ye in \cite{SY}.
We will use the following theorems in the paper.

\begin{thm}\cite[Theorem 1.2]{HKM}\label{thm-SY}
Let $(X, T)$ be a minimal topologically distal system and let $d\in \N$. Then
$(X,T)$ is a topological system of order $d$ if and only if $\RP^{[d]}=\Delta_X$.
\end{thm}

\begin{thm}\cite[Theorem A.1]{HKM}\label{thm-HKM}
Any system of order $d$ is isomorphic in the measure theoretic sense to a topological system of order $d$.
\end{thm}

\begin{lem}\cite[Lemma A.3]{DDMSY}\label{DDMSY}
Let $(X,T)$ be a system of order $d$,
then the maximal measurable and topological factors of order $j$
coincide, where $j\leq d$.
\end{lem}

\section{ergodic decomposition of self-joinings under $T\times T^2\times \ldots \times T^d$}\label{section-furstenberg-joining}

In this section we study ergodic decomposition of self-joinings under $T\times T^2\times \ldots \times T^d$. The theorems in this section are important for our proofs, and also they have their own interest.

\subsection{Furstenberg self-joining}

Let $T: X\rightarrow X$ be a map and $d\in \N$. Set
$$\tau_d=T\times \ldots\times T \ (d \ \text{times}),$$
$$\sigma_d=T\times T^2 \times \ldots \times T^{d}$$ and
$$\sigma_d'=\id \times T\times \ldots \times T^{d-1}=\id\times \sigma_{d-1}.$$
Note that $\langle\tau_d, \sigma_d\rangle=\langle\tau_d,\sigma_d'\rangle$.
For any $x\in {X}$, let $N_d({X},x)=\overline{\O((x,\ldots,x),
\langle\tau_d, \sigma_d\rangle)}$, the orbit closure of
$(x,\ldots,x)$ ($d$ times) under the action of the group
$\langle\tau_d, \sigma_d\rangle$. We remark that if $({X},T)$ is
minimal, then all $N_d({X},x)$ coincide, which will be denoted by
$N_d({X})$. It was shown by Glasner \cite{G94} that if $({X},T)$
is minimal, then $(N_d({X}), \langle\tau_d, \sigma_d\rangle)$ is
minimal. Hence if $(N_d(X), \langle\tau_d, \sigma_d\rangle)$ is uniquely ergodic, then it is strictly
ergodic.

\begin{de}\label{de-Furstenberg-selfjoining}
Let $(X,T)$ be a minimal system with $\mu\in M_T(X)$ and $d\ge 1$.
If $(N_d(X), \langle\tau_d,\sigma_d\rangle)$ is uniquely ergodic, then we denote
the unique measure by $\mu^{(d)}$, and call it the {\em Furstenberg
self-joining}.
\end{de}

For a t.d.s. $(X,T,\mu)$ and $d\in\N$, we define $\mu^{(d)}$ as one of the weak limit points of sequence
$\{\frac {1}{N} \sum_{n=0}^{N-1} \sigma_d^n \mu_\D^d\}$ in $M(X^d)$,
where $\mu_\D^d$ is the diagonal measure on $X^d$ as defined in
\cite{F77}, i.e. it is defined on $X^d$ as follows
\begin{equation*}
    \int_{X^d} f_1(x_1)\ldots f_d(x_d)\ d \mu_\D^d(x_1,\ldots,x_d)=
    \int_X f_1(x)\ldots f_d(x)\ d\mu(x).
\end{equation*}

When $(N_d(X), \langle\tau_d,\sigma_d\rangle)$ is uniquely ergodic, it is easy
to see that
\begin{equation*}
\frac {1}{N} \sum_{n=0}^{N-1} \sigma_d^n \mu_\D^d\longrightarrow
\mu^{(d)},\ N\to \infty, \quad \text{weakly in $M(X^d)$},
\end{equation*}

\subsection{The $\sigma$-algebra of invariant sets under $\sigma_d=T\times T^2\times \ldots \times T^d$}

\

In this subsection we study the $\sigma$-algebra of invariant sets under $\sigma_d=T\times T^2\times \ldots \times T^d$. We will show we always can deduce this $\sigma$-algebra to the one on its nilfactors.

The proof of the following lemma is similar to the proof of Theorem 12.1 in \cite{HK05}.

\begin{lem}\label{AP-lem-vdc}
Let $(X,\X,\mu,T)$ be an ergodic system and $d\ge 1$ be an integer.
Suppose that $\lambda$ is a $d$-fold self-joining of $X$.
Assume that $f_1,\ldots,f_d\in L^\infty(X,\mu)$ with
$\|f_j\|_\infty\le 1$ for $j=1,\ldots,d$. Then
\begin{equation}\label{AP-VDC}
\lim_{N\to\infty}\Big\|
\frac{1}{N}\sum_{n=0}^{N-1}f_1(T^nx_1)f_2(T^{2n}x_2)\ldots
f_d(T^{dn}x_d) \Big\|_{L^2(X^d, \lambda)}\le \min_{1\le l\le d}\{l\cdot
\interleave f_l\interleave_d \}
\end{equation}
\end{lem}

\begin{proof}
We proceed by induction. For $d=1$, by the Ergodic Theorem,
$$\|\frac{1}{N}\sum_{n=0}^{N-1}T^n f_1 \|_{L^2(\mu)}\to |\int f_1d\mu|=\HK f_1 \HK_1.$$
Let $d\ge 1$ and assume that (\ref{AP-VDC}) holds for $d$ and any $d$-fold self-joining of $X$. Let
$f_1,\ldots,f_{d+1}\in L^\infty(\mu)$ with $\|f_j\|_\infty\le 1$ for
$j=1,\ldots,d+1$. Let $\lambda$ be any $d+1$-fold self-joining of $X$. Choose $l\in \{2,3,\ldots,d+1\}$. (The case $l=1$
is similar). Write
$$\xi_n=\bigotimes_{j=1}^{d+1}T^jf_j=f_1(T^nx_1)f_2(T^{2n}x_2)\ldots
f_{d+1}(T^{(d+1)n}x_{d+1}).$$

By the van der Corput lemma (Lemma \ref{vanderCoput}),
\begin{equation*}
    \limsup_{N\to \infty} \big\| \frac{1}{N}\sum _{n=0}^{N-1}
    \xi_n\big\|^2_{L^2(\lambda)}
    \le \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 \overline{\xi}_{n+h}\cdot\xi_n d\lambda \right|.
\end{equation*}
Letting $M$ denote the last $\limsup$, we need to show that $M\le
l^2\HK f_l\HK^2_{d+1}$. For any $h\ge 1$,
\begin{equation*}
\begin{split}
    & \ \ \ \left|\frac{1}{N}\sum_{n=0}^{N-1} \int \overline{\xi}_{n+h}\cdot\xi_n d\lambda
    \right| \\ &=\left | \int(f_1\cdot T^h\overline{f}_1)\otimes \frac{1}{N}\sum_{n=0}^{N-1}
    (\sigma_d)^n \bigotimes_{j=2}^{d+1}f_j\cdot T^{jh}\overline{f}_j d
    \lambda(x_1,\ldots,x_{d+1})\right|\\
    &\le \Big\|f_1\cdot T^h\overline{f}_1 \Big\|_{L^2(\lambda)}\cdot
    \Big\|\frac{1}{N}\sum_{n=0}^{N-1}
    (\sigma_d)^n \bigotimes_{j=2}^{d+1}f_j\cdot T^{jh}\overline{f}_j
    \Big\|_{L^2(\lambda)}\\
    &= \Big\|f_1\cdot T^h\overline{f}_1 \Big\|_{L^2(\mu)}\cdot
    \Big\|\frac{1}{N}\sum_{n=0}^{N-1}
    (\sigma_d)^n \bigotimes_{j=2}^{d+1}f_j\cdot T^{jh}\overline{f}_j
    \Big\|_{L^2(\lambda')}
\end{split}
\end{equation*}
where $\lambda'$ is the image of $\lambda$ to the last $d$ coordinates.
It is clear $\lambda'$ is a $d$-fold self-joining of $X$, and by the inductive assumption,
\begin{equation*}
   \left|\frac{1}{N}\sum_{n=0}^{N-1} \int \overline{\xi}_{n+h}\cdot\xi_n d\lambda
    \right|\le l\HK f_l\cdot T^{lh}\overline{f}_l\HK_d.
\end{equation*}
We get
\begin{equation*}
\begin{split}
    M& \le l\cdot \limsup_{H\to\infty}
    \frac{1}{H}\sum_{h=0}^{H-1}\HK f_l\cdot T^{lh}\overline{f}_l\HK_d
    \le l^2\cdot\limsup_{H\to\infty}\frac{1}{H}\sum_{h=0}^{H-1} \HK f_l\cdot T^h
    \overline{f}_l\HK_d\\
    &\le l^2\cdot \limsup_{H\to\infty} \Big( \frac{1}{H}\sum_{h=0}^{H-1} \HK f_l\cdot T^h
    \overline{f}_l\HK_d^{2^d} \Big)^{1/2^d}\\
    &=l^2\cdot \HK f_l\HK_{d+1}^2.
\end{split}
\end{equation*}
The last equation follows from Lemma \ref{lemmaE1}. The proof is completed.
\end{proof}

\begin{lem}\label{AP-lem-vdc2}
Let $(X,\X,\mu,T)$ be an ergodic system and $d\in \N$.Suppose that $\lambda$ is a $d$-fold self-joining of $X$ and it is $\sigma_d$-invariant. Assume that
$f_1,\ldots,f_d\in L^\infty(X,\mu)$. Then
\begin{equation}\label{}
    \E\Big(\bigotimes_{j=1}^d f_j\Big|\I(X^d, \lambda,\sigma_d)\Big)  =\E\Big( \bigotimes_{j=1}^d
    \E(f_j|\ZZ_{d-1})\Big |\I(X^d, \lambda,\sigma_d) \Big).
\end{equation}

\end{lem}

\begin{proof}
By Lemma \ref{lem-product}, it suffices to show that
\begin{equation}\label{}
\E\Big(\bigotimes_{j=1}^d f_j\Big|\I(X^d, \lambda,\sigma_d)\Big)
=0
\end{equation}
whenever $\E(f_k|\ZZ_{d-1})=0$ for some $k\in \{1,2,\ldots,d\}$.
This condition implies that $\HK f_k\HK_d=0$. By the Ergodic Theorem
and Lemma \ref{AP-lem-vdc}, we have
\begin{equation*}
\begin{split}
&\ \ \ \ \Big\|\E\Big(\bigotimes_{j=1}^d f_j\Big|\I(X^d,
\mu^{(d)},\sigma_d)\Big)\Big\|_{L^2(\lambda)} \\
&=\lim_{N\to\infty}\Big\|
\frac{1}{N}\sum_{n=0}^{N-1}f_1(T^nx_1)f_2(T^{2n}x_2)\ldots
f_d(T^{dn}x_d) \Big\|_{L^2(\lambda)}\le k \cdot\interleave
f_k\interleave_d =0.
\end{split}
\end{equation*}
So the lemma follows.
\end{proof}

\begin{prop}
Let $(X,\X,\mu,T)$ be ergodic and $d\in \N$. Suppose that
$\lambda$ is a $d$-fold self-joining of $X$ and
it is $\sigma_d$-invariant. Then the
$\sigma$-algebra $\I(X^d, \lambda,\sigma_d)$ is measurable with
respect to $\ZZ_{d-1}^{d}$.
\end{prop}

\begin{proof}
Every bounded function on $X^{d}$ which is measurable with respect
to \linebreak $\I(X^d, \lambda,\sigma_d)$ can be approximated in
$L^2(\lambda)$ by finite sums of functions of the form
$\E(\bigotimes_{j=1}^d f_j | \I( X^d, \lambda, \sigma_d))$ where
$f_1,\ldots,f_d$ are bounded functions on $X$. By Lemma
\ref{AP-lem-vdc2}, one can assume that these functions are
measurable with respect to $Z_{d-1}$. In this case
$\bigotimes_{j=1}^df_j$ is measurable with respect to
$\ZZ_{d-1}^{d}$. Since this $\sigma$-algebra $\ZZ_{d-1}^{d}$ is
invariant under $\sigma_d$, $\E(\bigotimes_{j=1}^df_j|\I(X^d,
\lambda,\sigma_d))=\lim \limits_{N\rightarrow +\infty}\frac{1}{N}\sum \limits_{n=0}^{N-1}
\big(\bigotimes_{j=1}^df_j\big)\circ \sigma_d^n$ is also measurable with respect to
$\ZZ_{d-1}^{d}$. Therefore $\I(X^d, \lambda,\sigma_d)$ is
measurable with respect to $\ZZ_{d-1}^{d}$.
\end{proof}

\begin{cor}\label{AP-cor-ergodic}
Let $(X, \X, \mu,T)$ be an ergodic system and $d\in \N$. Suppose that $\lambda$
is a $d$-fold self-joining of $X$ and it is $\sigma_d$-invariant. Then the
factor map $\pi_{d-1}^d: (X^{d}, \lambda, \sigma_d)\rightarrow
(Z_{d-1}^d, \widetilde{\lambda},\sigma_d)$ is ergodic, where $\widetilde{\lambda}$ is the image of $\lambda$.

In particular, one has that $\I(X^{d}, \lambda, \sigma_d)$ is isomorphic to $\I(Z_{d-1}^d, \widetilde{\lambda},\sigma_d)$.
\end{cor}

\subsection{Ergodic decomposition of Furstenberg self-joining of nilsystems under action $ \sigma_d$}
\

In the previous subsection we show that to study the $\sigma$-algebra of invariant sets
under $\sigma_d=T\times T^2\times \ldots \times T^d$, we only need to study the one on its nilfactors.
Hence in this subsection we study the ergodic decomposition of Furstenberg self-joinings of nilsystems under
the action $ \sigma_d$.

In this subsection $d\ge 2$ is an integer, and $(X =
Z_{d-1},\mu_{d-1}, T )$ is a topological system of order $d-1$. Recall
$$N_d=N_d(X)=\overline{\O(\Delta_d({X}),
\sigma_d)}=\overline{\O((x,\ldots,x), \langle\tau_d, \sigma_d\rangle)}\subset
X^d$$ and $$N_d[x]=\overline{\O((x,\ldots,x), \sigma'_d)},$$ where
$x\in X$.

\medskip

\subsubsection{Basic properties}

First we recall some basic properties.

\begin{thm}\cite{BHK05, Z05}\label{ziegler}
With the notations above, we have
\begin{enumerate}
  \item The $(N_d, \langle\tau_d,\sigma_d\rangle)$ is ergodic (and thus uniquely
  ergodic) with some measure $\mu^{(d)}_{d-1}$.
  \item For each $x\in X$, the system $(N_d[x],\sigma_d')$ is
uniquely ergodic with some measure $\delta_x\times \mu^{(d)}_{d-1,x}$.
  \item $\displaystyle \mu^{(d)}_{d-1} = \int_X \d_x\times \mu^{(d)}_{d-1,x}\
  d\mu(x)$.
  \item (Ziegler) Let $f_1, f_2, \ldots , f_{d-1}$ be continuous functions on
$X$ and let $\{M_i\}$ and $\{N_i\}$ be two sequences of integers
such that $N_i\to \infty$. For $\mu_{d-1}$-almost every $x\in X$,
\begin{equation}
\begin{split}
\frac{1}{N_i} \sum_{n=M_i}^{N_i+M_i-1}
   &f_1(T^nx)f_2(T^{2n}x)\ldots f_{d-1}(T^{(d-1)n}x)\\
   \rightarrow & \int f_1(x_1)f_2(x_2)\ldots f_{d-1}(x_{d-1})\
   d\mu^{(d)}_{d-1,x}(x_1,x_2,\ldots,x_{d-1})
\end{split}
\end{equation}
as $i\to \infty$.
\end{enumerate}
\end{thm}

\begin{rem}
In fact, in \cite{BHK05, Z05} Theorem \ref{ziegler} is for nilsystems. Via an inverse limit argument,
it is easy to see that Theorem \ref{ziegler} holds for topological systems of order $d$.
\end{rem}

\subsubsection{The ergodic decomposition of $\mu^{(d)}_{d-1}$ under $\sigma_d$}

Now we study the ergodic decomposition of $\mu^{(d)}_{d-1}$ under
$\sigma_d$. For each $x\in X$, let $\nu^{(d)}_{d-1,x}$ be the unique
$\sigma_d$-invariant measure on $\overline{\O(x^d,\sigma_d)}$, where
$x^d=(x,x,\ldots,x)\in X^d$. Then
$$\varphi: X \longrightarrow
M(N_d);\ \ \ x\mapsto \nu^{(d)}_{d-1,x}$$ is a Borel map and $\varphi(X)\subseteq M_{\sigma_d}^{e}(N_d)$. This
fact follows from that $x \mapsto
\frac{1}{N}\sum_{n<N}\d_{\sigma_d^n x^d}$ is continuous and
$\frac{1}{N}\sum_{n<N}\d_{\sigma_d^nx^d}$ converges to
$\nu^{(d)}_{d-1,x}$ weakly.

It is easy to check that $\displaystyle \int_{X}
\nu^{(d)}_{d-1,x} \ d \mu(x)$ is $\langle\tau_d,\sigma_d\rangle$-invariant and
hence it is equal to $\mu^{(d)}_{d-1}$ by the uniqueness. Hence we have
\begin{equation}\label{h9}
    \mu^{(d)}_{d-1}=\int_{X} \nu^{(d)}_{d-1,x} \ d \mu(x).
\end{equation}

\medskip

Now we will prove the following result:

\begin{thm}\label{de-Zd}
$\mu^{(d)}_{d-1}=\int_{X} \nu^{(d)}_{d-1,x} \ d \mu(x)$ is the ergodic decomposition of
$\mu^{(d)}_{d-1}$ under $\sigma_d$.
\end{thm}

First we show that
$\nu^{(d)}_{d-1,x}\neq \nu^{(d)}_{d-1,y}$ whenever $x\neq y$. This
result will follows from the following fact:
$\overline{\O(x^d,\sigma_d)}\cap
\overline{\O(y^d,\sigma_d)}=\emptyset$ for all $x\neq y$.
In fact, if $\overline{\O(x^d,\sigma_d)}\cap
\overline{\O(y^d,\sigma_d)}\not =\emptyset$, then $y^d\in \overline{\O(x^d,\sigma_d)}$ since
both $\overline{\O(x^d,\sigma_d)}$ and $\overline{\O(y^d,\sigma_d)}$ are minimal.
It is easy to check that $(x,y)\in \RP^{[d-1]}(X)$. Hence $x=y $ by Theorem \ref{thm-SY}.

\medskip

By the above discussion, one has that $\varphi$ is a one-to-one Borel map. Hence by Souslin Theorem (See e.g.  \cite[Theorem 2.8 (2)]{Glasner}),
$\varphi(X)$ is a Borel subset of $M(N_d)$ and $\varphi$ is a Borel isomorphism from $X$ to $\varphi(X)$.
Let $\kappa=\varphi_*(\mu)$. Then $\kappa$ is a Borel probability measure on the $G_\delta$ subset
$M_{\sigma_d}^{e}(N_d)$ of $M(N_d)$, $\kappa(\varphi(X))=1$ and
$$\mu_{d-1}^{(d)}\stackrel{\eqref{h9}}{=}\int_{X} \varphi(x) \ d \mu(x)=\int_{M_{\sigma_d}^{e}(N_d)} \theta d \kappa(\theta)$$
is the ergodic decomposition of $\mu_{d-1}^{(d)}$ under $\sigma_d$.

Let $\mathcal{N}_d$ be  the Borel $\sigma$-algebra  of $N_d$.
By the ergodic decomposition Theorem (see e.g. \cite[Theorem 4.2]{V}), there exists a Borel map $\xi: N_d\rightarrow M_{\sigma_d}^{e}(N_d)$
such that

(i) $\xi(\sigma_d x)=\xi(x)$ for any $x\in N_d$,

(ii) for any $\theta\in M_{\sigma_d}^{e}(N_d)$, $\theta(\xi^{-1}(\theta))=1$,

(iii) for any $\eta\in M_{\sigma_d}(N_d)$,
\begin{equation}\label{eeeee11}
\eta(A)=\int_{N_d} \xi(x)(A)d\eta(x)
\end{equation}
for any $A\in \mathcal{N}_d$.

Let $\mathcal{M}_{\sigma_d}^{e}(N_d)$ be the Borel $\sigma$-algebra  of $M_{\sigma_d}^{e}(N_d)$ and
$\nu=\xi_*(\mu_{d-1}^{(d)})$. Then $\xi: (N_d,\mathcal{N}_d, \mu_{d-1}^{(d)})\rightarrow (M_{\sigma_d}^{e}(N_d),
\mathcal{M}_{\sigma_d}^{e}(N_d),\nu)$  is a measure-preserving map. By \cite[Lemma 4.2]{V},
$\xi^{-1}(\mathcal{M}_{\sigma_d}^{e}(N_d))=\I(N_d,\mathcal{N}_d , \mu_{d-1}^{(d)},\sigma_d)$ (mod  $\mu_{d-1}^{(d)})$
and  $$\mu_{d-1}^{(d)}=\int_{N_d} \xi(x)\ d\mu_{d-1}^{(d)}(x)=\int_{M_{\sigma_d}^{e}(N_d)} \theta d \nu(\theta)$$ is
the disintegration of $\mu_{d-1}^{(d)}$ over $\nu$ by $\xi$.

Hence the uniqueness of the representation in Choquet's theorem implies $\nu=\kappa$ so that $\nu(\varphi(X))=1$.
Now $\varphi^{-1}:(M_{\sigma_d}^{e}(N_d),\mathcal{M}_{\sigma_d}^{e}(N_d),\nu)\rightarrow (X,\mathcal{X},\mu_{d-1})$ is an isomorphism.

\medskip

To sum up, we have

\begin{prop}\label{Zd-erg-decom}
Let $\psi=\varphi^{-1}\circ \xi$. Then $\psi: (N_d,\mathcal{N}_d, \mu_{d-1}^{(d)})
\rightarrow (X,\mathcal{X},\mu_{d-1})$ is a measure preserving map such that
\begin{enumerate}
\item $\psi^{-1}(\mathcal{X})=\I(N_d,\mathcal{N}_d , \mu_{d-1}^{(d)},\sigma_d)$.

\item  the disintegration $\mu^{(d)}_{d-1}= \int_X (\mu_{d-1}^{(d)})_x d\mu_{d-1}(x)$ of $\mu_{d-1}^{(d)}$ over $\mu_{d-1}$ by $\psi$ is the
ergodic decomposition of $\mu^{(d)}_{d-1}$ under $\sigma_d$.

\item  $(\mu_{d-1}^{(d)})_x=\nu^{(d)}_{d-1,x}$ for $\mu_{d-1}$-a.e $x\in X$.
\end{enumerate}
\end{prop}

It is easy to see that Theorem \ref{de-Zd} follows from Proposition \ref{Zd-erg-decom}.

\subsection{Ergodic decomposition of Furstenberg self-joining under the action $ \sigma_d$}
\

Let $(X,T,\mu)$ be a minimal t.d.s, and let $\pi_{d-1}: (X,T,\mu)\rightarrow (Z_{d-1}, T,\mu_{d-1})$
be the topological factor map, where $Z_{d-1}$ is both a topological system of order $d-1$
and a system of order $d-1$. Notice that in the next section we will show that for each ergodic system,
one always can find such a minimal topological model.

\medskip

Recall that for a t.d.s. $(X,T,\mu)$ and $d\in\N$, we define $\mu^{(d)}$ as one of the weak limit points
of sequence $\{\frac {1}{N} \sum_{n=0}^{N-1} \sigma_d^n \mu_\D^d\}$ in $M(X^d)$.
Then $\mu^{(d)}$ is a $d$-fold self-joining of $X$ and it is $\langle\tau_d,\sigma_d\rangle$-invariant.
By definition, the image of $\mu^{(d)}$ under $\pi_{d-1}^d$ is $\mu_{d-1}^{(d)}$.

\medskip

By Corollary \ref{AP-cor-ergodic}, the factor map $\pi_{d-1}^d:
(X^{d}, \mu^{(d)},\sigma_d)\rightarrow (Z_{d-1}^d,
\mu_{d-1}^{(d)},\sigma_d)$ is ergodic. Hence $\I(X^d,
\mu^{(d)},\sigma_d)=\I(Z_{d-1}^d, \mu_{d-1}^{(d)},\sigma_d)$. By
(\ref{h9}),
$$\displaystyle \mu^{(d)}_{d-1} = \int_{Z_{d-1}} \nu^{(d)}_{d-1,x}\ d \mu_{d-1}(x) $$
is the ergodic decomposition of $\mu^{(d)}_{d-1}$ under $\sigma_d$.

Let $\phi=\pi_{d-1}^d|_{N_d(X)}$ and $\psi: (N_d(Z_{d-1}),\mathcal{N}_d(Z_{d-1}),
\mu_{d-1}^{(d)})\rightarrow (Z_{d-1},\mathcal{Z}_{d-1},\mu_{d-1})$ be the
measure-preserving map defined in Proposition \ref{Zd-erg-decom}. Then by  Corollary \ref{AP-cor-ergodic} and the fact $\mu^{(d)}\big(N_d(X)\big)=1$ and $\mu_{d-1}^{(d)}\big(N_d(Z_{d-1})\big)=1$, one has $$\phi: (N_d(X), \mathcal{N}_d(X),\mu^{(d)},\sigma_d)\rightarrow (N_d(Z_{d-1}),\mathcal{N}_d(Z_{d-1}),\mu_{d-1}^{(d)},\sigma_d) $$ is
a factor map with
\begin{equation}\label{ddbb-d}
\I(N_d(X), \mu^{(d)},\sigma_d)= \phi^{-1}\I(N_d(Z_{d-1}), \mu_{d-1}^{(d)},\sigma_d).
\end{equation}
 Combining this with Proposition \ref{Zd-erg-decom} (1), we have
\begin{align}\label{h4}
    (N_d(X), \mathcal{N}_d(X),\mu^{(d)})&\stackrel{\phi}{\lra} (N_d(Z_{d-1}),\mathcal{N}_d(Z_{d-1}), \mu_{d-1}^{(d)})\stackrel{\psi}{\lra}
    (Z_{d-1},\ZZ_{d-1},\mu_{d-1})\nonumber\\
    {\bf x}&\lra \phi({\bf x}) \lra s=\psi(\phi({\bf x}))
\end{align}
and $\phi^{-1}(\psi^{-1}(\ZZ_{d-1}))=\I(N_d(X),\mu^{(d)},\sigma_d)$.
From this, let
\begin{equation}\label{1111}
    \mu^{(d)}=\int_{Z_{d-1}}\nu^{(d)}_s\ d \mu_{d-1}(s)
\end{equation}
be the disintegration of $\mu^{(d)}$
over $\mu_{d-1}$ by  $\psi\circ \phi$. Since $$\phi^{-1}(\psi^{-1}(\ZZ_{d-1}))=\I(N_d(X),\mu^{(d)},\sigma_d),$$
\eqref{1111} is the ergodic decompositions of $\mu^{(d)}$ under $\sigma_d$ (see e.g. \cite[Theorem 8.7]{Glasner}).  Moreover, for $\mu_{d-1}$-a.e. $s\in Z_{d-1}$,
$$\phi_*(\nu^{(d)}_s)=(\mu_{d-1}^{(d)})_s=\nu^{(d)}_{d-1,s}$$  by \cite[Corollary 5.24]{EW}, \eqref{ddbb-d}   and Proposition \ref{Zd-erg-decom}, where  $\mu^{(d)}_{d-1}= \int_{Z_{d-1}} (\mu_{d-1}^{(d)})_s d\mu_{d-1}(s)$ is the disintegration of $\mu_{d-1}^{(d)}$ over $\mu_{d-1}$ by $\psi$.

\medskip

To sum up, we have the following result:

\begin{thm}\label{ergodic-deco-Fur}
$\mu^{(d)}=\int_{Z_{d-1}}\nu^{(d)}_s\ d \mu_{d-1}(s)$ is the ergodic decomposition of
$\mu^{(d)}$ under $\sigma_d$.
\end{thm}

\section{Proof of Theorem A}\label{section-thA}

In this section we show Theorem A. First we give the proofs for weakly mixing systems and for the case when $d=2$
to illustrate the basic ideas. Then we will give the whole proof by using the tools developed in Section \ref{section-furstenberg-joining}.

\subsection{Weakly mixing systems and the case $d=2$}

In this subsection we show Theorem A holds for weakly
mixing systems and for the case when $d=2$.

\subsubsection{Weakly mixing systems}
We start from the following proposition.

\begin{prop}\label{prop-wm-AP} Let $(X,T, \mu)$ be a uniquely ergodic t.d.s. with $\mu$ weakly mixing and $d\in
\N$.

Then $(X^{d}, \X^{d},
\mu^{d}, \langle\tau_d, \sigma_d\rangle)$ is uniquely ergodic, where $\mu^{d}=
\mu\times \ldots \times \mu$ ($d$ times).
\end{prop}

\begin{proof}
We prove the result inductively. It is trivial when $d=1$, since
$\tau_1=\sigma_1=T$.

Now assume the statements hold for $d-1$ ($d\ge 2$), and we show the
case for $d$. Let $\lambda$ be a $\langle\tau_d, \sigma_d\rangle$-invariant measure
of $(X^{d}, \X^{d})$. Let $$p_1: X^d=X\times X^{d-1}\rightarrow X; \
{\bf x} =(x_1, {\bf x'})\mapsto x_1$$
$$p_2: X^d=X\times X^{d-1} \rightarrow X^{d-1}; \ {\bf x} =(x_1, {\bf x'})\mapsto {\bf x'}$$  be
the projections. Note that $(p_2)_*(\lambda)$ is a
$\langle\tau_{d-1},\sigma_{d-1}\rangle$-invariant measure of $X^{d-1}$. By the
inductive assumption, $(p_2)_*(\lambda)=\mu^{d-1}$. Let
\begin{equation}\label{b3}
\lambda =\int \lambda_{{\bf x}}\times \d_{\bf x}\ d \mu^{d-1}({\bf x})
\end{equation}
be the disintegration of $\lambda$ over $\mu^{d-1}$. Since $\lambda$ is
$\sigma_d'=\id \times \sigma_{d-1}$-invariant, we have
\begin{eqnarray*}
    \lambda &= &\sigma'_d \lambda=\id \times \sigma_{d-1} \lambda=
    \int \lambda_{\bf x}\times \sigma_{d-1}\d_{\bf x} \ d \mu^{d-1}({\bf x})\
    \\ & = & \int \lambda_{\bf x}\times \d_{\sigma_{d-1}\bf x} \ d \mu^{d-1}({\bf x})\\
    \\&=& \int \lambda_{(\sigma_{d-1})^{-1}\bf x}\times \d_{\bf x} \ d \mu^{d-1}({\bf x}).
\end{eqnarray*}
The uniqueness of disintegration implies that
\begin{equation}\label{b4}
    \lambda_{(\sigma_{d-1})^{-1}\bf x}=\lambda_{\bf x},\quad \mu^{d-1} \ a.e.
\end{equation}

Define $$F: (X^{d-1},\X^{d-1},\sigma_{d-1}) \lra M(X): \ {\bf
x}\mapsto \lambda_{\bf x}.$$ By (\ref{b4}), $F$ is a
$\sigma_{d-1}$-invariant $M(X)$-value function. Since $(X,\X,\mu,
T)$ is weakly mixing,
$(X^{d-1},\X^{d-1},\sigma_{d-1})=(X^{d-1},\X^{d-1}, T\times
T^2\times\ldots \times T^{d-1})$ is ergodic and hence $\lambda_{\bf
x}=\nu, \ \mu^{d-1}\ $ a.e. for some $\nu\in M(X)$. Thus by
(\ref{b3}) one has that
$$\lambda=\int  \lambda_{\bf x}\times \d_{\bf x}\ d\mu^{d-1}({\bf x})=
\int \nu \times\d_{\bf x}\ d\mu^{d-1}({\bf x})= \nu\times\mu^{d-1}
.$$ Then we have that $\nu=(p_1)_*(\lambda)$ is a $T$-invariant measure
of $X$. By assumption, $\nu=(p_1)_*(\lambda)=\mu$. Thus $\lambda=\mu\times
\mu^{d-1}=\mu^{d}$. Hence $(X^{d}, \X^{d}, \mu^{d}, \langle\tau_d,
\sigma_d\rangle)$ is uniquely ergodic. The proof is completed.
\end{proof}

\begin{de}
Let $(X,\X, \mu, T)$ be an ergodic m.p.t. and $(\h{X}, \h{T})$
be its model.
For $d\in \N$, $(\h{X}, \h{T})$ is called a

{\em $\langle\tau_d, \sigma_d\rangle-$strictly ergodic model} for $(X,\X, \mu,
T)$ if $(\h{X}, \h{T})$ is a strictly ergodic model and $(N_d(\h{X}),
\langle\h{\tau}_d, \h{\sigma}_d\rangle)$ is strictly ergodic.
\end{de}

\begin{thm}\label{Thm-wm-AP}
Let $(X,\X, \mu, T)$ be a m.p.t. with $\mu$ weakly mixing, then it has
a $\langle\tau_d, \sigma_d\rangle-$strictly ergodic model for all $d\in\N$.
\end{thm}

\begin{proof}
By Jewett-Krieger's Theorem, $(X,\X,
\mu, T)$ has a strictly ergodic model. Without loss of generality,
we may assume that $(X,T)$ itself is a topological minimal system
and $\mu$ is its unique $T$-invariant measure. By Proposition
\ref{prop-wm-AP}, $(X^d, \langle\tau_d,\sigma_d\rangle)$ is uniquely ergodic for
all $d\in \N$.
\end{proof}

\subsubsection{Case when $d=2$}\label{AP-d=2}

Note that $X^2=X\times X$, $\tau_2=T\times T$, $\sigma_2=T\times
T^2$ and $\sigma_2'=\id \times T$. It is easy to see that $N_2(X)=X\times X$. Now we show
each $\langle\tau_2,\sigma_2\rangle$ invariant measure is $\mu\times \mu$.

Let $\lambda$ be a
$\langle\tau_2,\sigma_2\rangle$-invariant measure of $(X\times X,
\X\times \X)$. Since $\lambda$ is $T\times T$-invariant, it is a
self-joining of $(X,\X,\mu,T)$ and has $\mu$ as its marginal. Let
\begin{equation}\label{a2}
\lambda =\int_X \d_x\times \lambda_x\ d \mu(x)
\end{equation}
be the disintegration of $\lambda$ over $\mu$. Since $\lambda$ is $\id
\times T$-invariant, we have
\begin{equation*}
    \lambda=\id \times T \lambda=\int_X \d_x\times T\lambda_x \ d \mu(x).
\end{equation*}
The uniqueness of disintegration implies that
\begin{equation*}
    T\lambda_x=\lambda_x, \mu \ a.e.
\end{equation*}
Since $(X,\X,\mu, T)$ is uniquely ergodic, $\lambda_x=\mu, \ \mu\ $ a.e. Thus
by (\ref{a2}) one has that
$$\lambda=\int_X \d_x\times \lambda_x\ d\mu(x)=\int_X \d_x\times \mu \ d\mu(x)=\mu\times \mu.$$
Hence we finish the proof of the case $d=2$.

\subsection{Another form of Theorem A}

From the statement of Theorem A one does not know  how the model looks like. The following statement
avoids this weakness and is suitable for the induction.
Note that we let $Z_0=\{pt\}$ be the trivial system.
\begin{thm}\label{model}
Let $(X,\X, \mu, T)$ be an ergodic  m.p.t, $d\ge 2$  and $\pi_{d-2}: X\lra Z_{d-2}$ be the
factor map to the $Z_{d-2}$ factor. Assume that $Z_{d-2}$ is isomorphic to a topological system
of order $d-2$ (see Theorem \ref{thm-HKM}, still denote it by $Z_{d-2}$). Then any strictly ergodic system $\h{X}$
obtained from Weiss's theorem is a $\langle\tau_d, \sigma_d\rangle-$strictly ergodic model.

\[
\begin{CD}
X @>{}>> \h{X}\\
@V{\pi_{d-2}}VV      @VV{\h{\pi}_{d-2}}V\\
Z_{d-2} @>{ }>> Z_{d-2}
\end{CD}
\]

\end{thm}

It is clear that Theorem A follows from the above theorem immediately.

\subsection{Proof of Theorem \ref{model}}

Let $(X,T)$ be a strictly ergodic
system and let $\mu$ be its unique $T$-invariant measure. Assume that Theorem \ref{model} holds
for $d\ge 2$. We show it also holds for $d+1$.

\medskip

Let $\pi_{d-1}: X\rightarrow Z_{d-1}$ be the factor map from $X$ to
$Z_{d-1}$, the system of order $d-1$. We build $\h{X}$ in the following way
by Weiss's theorem and Theorem \ref{thm-HKM}.
\[
\begin{CD}
X @>{}>> \h{X}\\
@V{\pi_{d-1}}VV      @VV{\h{\pi}_{d-1}}V\\
Z_{d-1} @>{ }>> Z_{d-1}\\

\end{CD}
\]

Without loss of generality we assume that $X=\h{X}$.
Now we show that
$ (N_{d+1}(X), \langle \tau_{d+1}, \sigma_{d+1} \rangle)$ is uniquely ergodic.

Let $\zeta: Z_{d-1}\lra Z_{d-2}$ be the factor map to the maximal topological factor of order $d-2$.

By Theorem \ref{DDMSY}, $\zeta$ is also the factor map to the maximal factor of order $d-2$.
By the inductive assumption, $(N_{d}(X),\langle\tau_{d},\sigma_{d}\rangle)$ is uniquely ergodic,
and we denote its unique measure by $\mu^{(d)}$.

\medskip

By Theorem \ref{ergodic-deco-Fur},
\begin{equation}
    \mu^{(d)}=\int_{Z_{d-1}}\nu^{(d)}_s\ d \mu_{d-1}(s)
\end{equation}
is the ergodic decompositions of $\mu^{(d)}$ under $\sigma_d$.

\medskip

Let $\lambda$ be a
$\langle\tau_{d+1},\sigma_{d+1}\rangle$-invariant measure of $N_{d+1}(X)$ and
$\mu=\int _{Z_{d-1}}\theta_s\ d\mu_{d-1}(s)$ be the disintegration
of $\mu$ over $\mu_{d-1}$. We will show that
\begin{equation}\label{limit-D}
\lambda=\int_{Z_{d-1}}\theta_s\times \nu^{(d)}_s \ d\mu_{d-1}(s)
\end{equation}
which implies that $\lambda$ is unique.

\medskip

To do this let
$$p_1: (N_{d+1}(X), \langle\tau_{d+1},\sigma_{d+1}\rangle)\rightarrow (X, T); \
(x_1,{\bf x})\mapsto x_1$$
$$p_2: (N_{d+1}(X), \langle\tau_{d+1},\sigma_{d+1}\rangle)\rightarrow
(N_d(X), \langle\tau_d, \sigma_d\rangle); \ (x_1,{\bf x})\mapsto
{\bf x}$$  be the projections (here we use the fact that $\langle
\tau_d, T^2\times \ldots \times T^{d+1}\rangle=\langle\tau_d,
\sigma_d\rangle$). Then $(p_2)_*(\lambda)$ is a
$\langle\tau_d,\sigma_d\rangle$-invariant measure of $N_d(X)$. By
the assumption on $d$, $(p_2)_*(\lambda)=\mu^{(d)}$. Hence we may
assume that
\begin{equation}\label{h5}
\lambda =\int_{X^d} \lambda_{{\bf x}}\times \d_{\bf x}\ d \mu^{(d)} ({\bf
x})
\end{equation}
is the disintegration of $\lambda$ over $\mu^{(d)}$. Since $\lambda$ is
$\sigma'_{d+1}=\id\times \sigma_d$-invariant, we have
\begin{eqnarray*}
    \lambda &= &\id \times \sigma_d \lambda=
    \int_{X^d} \lambda_{\bf x}\times \sigma_d\d_{\bf x} \ d \mu^{(d)}({\bf x})\
    \\ & = & \int_{X^d} \lambda_{\bf x}\times \d_{\sigma_d({\bf x})} \ d \mu^{(d)}({\bf x})\\
    \\&=& \int_{X^d} \lambda_{(\sigma_d)^{-1}({\bf x})}\times \d_{\bf x} \ d \mu^{(d)}({\bf x}).
\end{eqnarray*}
The uniqueness of disintegration implies that
\begin{equation}\label{h6}
    \lambda_{(\sigma_d)^{-1}({\bf x})}=\lambda_{\bf x},\quad \mu^{(d)} \ a.e.
\end{equation}

Define $$F: (X^d,\mu^{(d)}, \sigma_d) \lra M(X): \ {{\bf x}}\mapsto
\lambda_{{\bf x}}.$$ By (\ref{h6}), $F$ is a $\sigma_d$-invariant
$M(X)$-valued function. Hence $F$ is $\I(X^d,
\mu^{(d)},\sigma_d)$-measurable, and this implies $\lambda_{\bf
x}=\lambda_{\psi(\phi({\bf x}))}=\lambda_s, \ \mu^{(d)}\ $ a.e.,
where $s, \psi$ and $\phi$ are defined in (\ref{h4}).

Thus by (\ref{h5}) one has that
\begin{equation}\label{add-final}
\begin{split}
    \lambda& =\int_{X^d}  \lambda_{{\bf x}}\times \d_{\bf x}\ d\mu^{(d)} ({\bf x})=
\int_{X^d} \lambda_{\psi(\phi({\bf x}))} \times\d_{\bf x}\ d\mu^{(d)} ({\bf x})\\
       &=\int_{Z_{d-1}}\int_{X^d}\lambda_s\times \d_{\bf x}\ d\nu^{(d)}_s({\bf x})
       d\mu_{d-1}(s)\\
       &= \int_{Z_{d-1}}\lambda_s\times \Big(\int_{X^d} \d_{\bf x}\
       d\nu^{(d)}_s({\bf x})\Big ) d \mu_{d-1}(s)\\
       &=\int_{Z_{d-1}}\lambda_s\times \nu^{(d)}_s \ d\mu_{d-1}(s).
\end{split}
\end{equation}

In the sequel we will show that $\lambda_s=\theta_s$ for $\mu_{d-1}$-a.e. $s\in Z_{d-1}$
and it is clear that (\ref{limit-D}) follows from this fact and (\ref{add-final}) immediately.

\medskip

Let $\pi_{d-1}^{d+1}: (N_{d+1}(X), \langle\tau_{d+1},\sigma_{d+1}\rangle)\lra
(N_{d+1}(Z_{d-1}), \langle\tau_{d+1},\sigma_{d+1}\rangle)$ be the natural factor
map. By Theorem \ref{ziegler}, $(N_{d+1}(Z_{d-1}),
\langle\tau_{d+1},\sigma_{d+1}\rangle,\mu^{(d+1)}_{d-1})$ is uniquely ergodic.
Hence
\begin{equation*}
\begin{split}
    \int_{Z_{d-1}}(\pi^{d+1}_{d-1})_*(\lambda_s\times \nu^{(d)}_s) \ d\mu_{d-1}(s)=(\pi^{d+1}_{d-1})_*(\lambda) =\mu_{d-1}^{(d+1)}=\int_{Z_{d-1}}\d_{s}\times \mu^{(d+1)}_{d-1,s} \
    d\mu_{d-1}(s).
\end{split}
\end{equation*}
The last equality follows from  Theorem \ref{ziegler}(3), since
for $\mu_{d-1}$-a.e. $s\in Z_{d-1}$, the system $(\overline{\O((s,\ldots,s), \sigma'_{d+1})},\sigma_{d+1}')$ is
uniquely ergodic with some measure $\delta_s\times \mu^{(d+1)}_{d-1,s}$.
Hence $\mu^{(d+1)}_{d-1,s}$
is the unique ergodic measure of $(\overline{\O((s,\ldots,s), \sigma_{d})},\sigma_{d})$,
i.e.  $\mu^{(d+1)}_{d-1,s}=\nu^{(d)}_{d-1,s}$.

Note that $$(\pi^d_{d-1})_*(\nu^{(d)}_s)=\phi_*(\nu^{(d)}_s)=(\mu_{d-1}^{(d)})_s=\nu^{(d)}_{d-1,s}=\mu^{(d+1)}_{d-1,s}$$ and
$(\mu_{d-1}^{(d)})_s(\psi^{-1}(s))=1$
for $\mu_{d-1}$-a.e. $s\in Z_{d-1}$.
We claim that
\begin{equation}\label{meas-two-eq}
(\pi^{d+1}_{d-1})_*(\lambda_s\times \nu^{(d)}_s)=\d_{s}\times \mu^{(d+1)}_{d-1,s}
\end{equation}
for $\mu_{d-1}$-a.e. $s\in Z_{d-1}$. We postpone the verification of (\ref{meas-two-eq})
to the next subsection.

It is clear that (\ref{meas-two-eq}) implies
\begin{equation}\label{proj=s}
(\pi_{d-1})_*(\lambda_s)=\d_s
\end{equation}
for $\mu_{d-1}$-a.e. $s\in Z_{d-1}$. Since $(p_1)_*(\lambda)=\mu$,
it follows from (\ref{add-final}) that
\begin{equation}\label{===}
\mu=\int_{Z_{d-1}}\lambda_s\ d\mu_{d-1}(s).
\end{equation}

(\ref{proj=s}) and (\ref{===}) imply that
$\mu=\int_{Z_{d-1}}\lambda_s\ d\mu_{d-1}(s)$ is also the
disintegration of $\mu$ over $\mu_{d-1}$. So we conclude
$$\lambda_s=\theta_s\  \text{for}\ \mu_{d-1}-a.e.\ s\in Z_{d-1}$$
by the uniqueness of the disintegration. The proof is complete.

\subsubsection{Proof of \eqref{meas-two-eq}}
Assume the contrary that \eqref{meas-two-eq} does not hold.
Then $$\mu_{d-1}(\{s\in Z_{d-1}: (\pi^{d+1}_{d-1})_*(\lambda_s\times \nu^{(d)}_s)\not=\d_{s}\times \mu^{(d+1)}_{d-1,s}\})>0.$$ So
there is some function $f\in C(N_{d+1}(Z_{d-1}))$ such that $\mu_{d-1}(C)>0$, where
\begin{equation*}
   C=\{s\in Z_{d-1}: (\pi^{d+1}_{d-1})_*(\lambda_s\times \nu^{(d)}_s)(f)>\d_{s}\times \mu^{(d+1)}_{d-1,s}(f)\}.
\end{equation*}
Let $B=\psi^{-1}(C)$ and  $A=p_{d-1,2}^{-1}(B)$, where $$p_{d-1,2}:
(N_{d+1}(Z_{d-1}), \langle\tau_{d+1},\sigma_{d+1}\rangle)\rightarrow
(N_d(Z_{d-1}), \langle\tau_d, \sigma_d\rangle); \ (s_1,{\bf
s})\mapsto {\bf s}$$  be the projection. Since
$\big((\pi^d_{d-1})_*(\nu^{(d)}_s)=\mu^{(d+1)}_{d-1,s}=(\mu_{d-1}^{(d)})_s$
and $(\mu_{d-1}^{(d)})_s(\psi^{-1}(s))=1$ for $\mu_{d-1}$-a.e. $s\in
Z_{d-1}$, one has that for $\mu_{d-1}$-a.e. $s\in Z_{d-1}$,
$$\mu^{(d+1)}_{d-1,s}(B)=\begin{cases} 1 &\text { if }s\in C\\ 0 &\text{ if } s\not \in C\end{cases} \text{ and }
(\pi^d_{d-1})_*(\nu^{(d)}_s)(B)=\begin{cases} 1 &\text { if }s\in C\\ 0 &\text{ if } s\not \in C.
\end{cases}$$

Moreover, for $\mu_{d-1}$-a.e. $s\in Z_{d-1}$, one has that for
$\mu_{d-1}$-a.e. $s\in Z_{d-1}$,
$$(\pi^{d+1}_{d-1})_*(\lambda_s\times \nu^{(d)}_s)(A)=(\pi^d_{d-1})_*(\nu^{(d)}_s)(B)=\begin{cases} 1 &\text { if }s\in C\\ 0 &\text{ if } s\not \in C\end{cases}$$
and
$$\d_{s}\times \mu^{(d+1)}_{d-1,s}(A)=\mu^{(d+1)}_{d-1,s}(B)=\begin{cases} 1 &\text { if }s\in C\\ 0 &\text{ if } s\not \in C\end{cases}.$$

Thus
\begin{equation*}
\begin{split}
    \mu^{(d+1)}_{d-1}(f\cdot 1_A)&=\int_{N_{d+1}(Z_{d-1})}f\cdot 1_A \ d\mu^{(d+1)}_{d-1}\\
    &=\int_{Z_{d-1}}\big(\int_{N_{d+1}(Z_{d-1})} f\cdot 1_A \ d \d_{s}\times \mu^{(d+1)}_{d-1,s}\big) \
    d\mu_{d-1}(s)\\
    &=\int_{C}\d_{s}\times \mu^{(d+1)}_{d-1,s}(f)d\mu_{d-1}(s)\\
    &<\int_{C}(\pi^{d+1}_{d-1})_*(\lambda_s\times \nu^{(d)}_s)(f)\ d\mu_{d-1}(s)\\
    &=\int_{Z_{d-1}} \big( \int_{N_{d+1}(Z_{d-1})} f\cdot 1_A \ d (\pi^{d+1}_{d-1})_*(\lambda_s \times \nu^{(d)}_s) \big) \ d\mu_{d-1}(s)\\
    &= \mu^{(d+1)}_{d-1}(f\cdot 1_A),
\end{split}
\end{equation*}
a contradiction! Hence \eqref{meas-two-eq} holds.

\section{Proof of Theorem B}\label{section-thB}

In this section we show how we obtain Theorem B from Theorem A. We need the following formula which is easy to be verified.

\begin{lem}\label{lem-product}
Let $\{a_i\}, \{b_i\}\subseteq \C$. Then
\begin{equation}\label{}
\prod_{i=1}^k a_i-\prod_{i=1}^k b_i=(a_1-b_1)b_2\ldots b_k+
a_1(a_2-b_2)b_3\ldots b_k +a_1\ldots a_{k-1}(a_k-b_k).
\end{equation}
\end{lem}

\medskip

\noindent{\bf The proof of Theorem B:}
Since $(X,\X, \mu, T)$ has a $\langle \tau_d,\sigma_d\rangle-$strictly ergodic model, we
may assume that $(X,T)$ itself is a minimal t.d.s. and
$\mu$ is its unique measure such that $(N_d(X), \langle\tau_d, \sigma_d
\rangle)$ is uniquely ergodic with the unique measure $\mu^{(d)}$.

Fix $f_1,\ldots, f_d\in L^\infty$ and let $\delta>0$. Without loss of generality, we assume that for all $1\le j\le d$,
$\|f_j\|_\infty\le 1$. Choose continuous functions $g_j$ such that
$\|g_j\|_\infty\le 1$ and $\|f_j-g_j\|_1<\d/d$ for all $1\le j\le
d$. We have
\begin{equation}\label{yexd1}
\begin{split}
&  \left | \frac{1}{N^2} \sum_{n\in [0,N-1] \atop{m\in [0,N-1]}} \prod_{j=1}^{d} f_j(T^{n+(j-1)m}x) -
\int_{N_d(X)}\bigotimes_{j=1}^df_jd\mu^{(d)} \right |\\
& \le  \left | \frac{1}{N^2} \sum_{n\in [0,N-1] \atop{m\in [0,N-1]}} \prod_{j=1}^{d} f_j(T^{n+(j-1)m}x) -
\frac{1}{N^2} \sum_{n\in [0,N-1] \atop{m\in [0,N-1]}}\prod_{j=1}^d g_j(T^{n+(j-1)m}x)\right |\\
&+\left | \frac{1}{N^2} \sum_{n\in [0,N-1] \atop{m\in [0,N-1]}}\prod_{j=1}^d g_j(T^{n+(j-1)m}x)-
\int_{N_d(X)}\bigotimes_{j=1}^dg_jd\mu^{(d)} \right |\\
&+\left | \int_{N_d(X)}\bigotimes_{j=1}^dg_jd\mu^{(d)}-
\int_{N_d(X)}\bigotimes_{j=1}^df_jd\mu^{(d)} \right |.

\end{split}
\end{equation}

Now by Pointwise Ergodic Theorem for $\Z^2$ i.e. Theorem
\ref{Lindenstrauss} (applying to $(n,m)\mapsto T^{n+(j-1)m}$) we have that for all $1\le j\le d$
\begin{equation}\label{need}
    \frac {1}{N^2} \sum_{n\in [0,N-1] \atop{m\in
[0,N-1]}}\Big | f_j(T^{n+(j-1)m}x)-g_j(T^{n+(j-1)m}x)\Big | \lra
\|f_j-g_j\|_1, \quad N\to \infty.
\end{equation}
for $\mu$ a.e.
This implies that there is a measurable $X_0^\delta\subset X$ with $\mu(X_0^\delta)=1$ such that for any $x\in X_0^\delta$
(\ref{need}) holds for all $1\le j\le d$.
Fix $x\in X_0^\delta$. Hence by Lemma \ref{lem-product}, when $N$ is large
\begin{equation}\label{yexd2}
\begin{split}
&   \left | \frac{1}{N^2} \sum_{n\in [0,N-1] \atop{m\in [0,N-1]}}
   \prod_{j=1}^{d} f_j(T^{n+(j-1)m}x) -
\frac{1}{N^2} \sum_{n\in [0,N-1] \atop{m\in [0,N-1]}}
   \prod_{j=1}^d g_j(T^{n+(j-1)m}x)\right |\\
& \le \sum_{j=1}^d \Big[\frac {1}{N^2} \sum_{n\in [0,N-1]
\atop{m\in
[0,N-1]}}\Big | f_j(T^{n+(j-1)m}x)-g_j(T^{n+(j-1)m}x)\Big |\Big ]\\
& <2\sum_{j=1}^d \|f_j-g_j\|_1\le 2\d,\ a.e.
\end{split}
\end{equation}

Note that
\begin{equation*}
\begin{split}
&\frac{1}{N^2} \sum_{n\in [0,N-1] \atop{m\in [0,N-1]}}
   \prod_{j=1}^d g_j(T^{n+(j-1)m}x)\\
   =& \frac{1}{N^2} \sum_{n\in [0,N-1] \atop{m\in [0,N-1]}}
    g_1(T^nx)g_2(T^{n+m}x)\ldots g_d(T^{n+(d-1)m}x) \\ =&
    \frac{1}{N^2} \sum_{n\in [0,N-1] \atop{m\in [0,N-1]}}
    g_1\otimes \ldots \otimes g_d \Big( \tau_d^n\sigma_d^m(x,x,\ldots,x)\Big).
\end{split}
\end{equation*}

Since $g_1\otimes \ldots \otimes g_d: X^{d}\rightarrow \R$ is
continuous and $(N_d(X), \langle \tau_d, \sigma_d \rangle)$ is uniquely ergodic,
by Theorem \ref{unique-ergodic}, $\displaystyle \frac{1}{N^2}
\sum_{n\in [0,N-1] \atop{m\in [0,N-1]}}
   \prod_{j=1}^d g_j(T^{n+(j-1)m}x)$ converges pointwisely to $\int_{N_d(X)}\bigotimes_{j=1}^dg_jd\mu^{(d)} $.
So when $N$ is large
\begin{equation}\label{xdye3}
\left | \frac{1}{N^2} \sum_{n\in [0,N-1] \atop{m\in [0,N-1]}}\prod_{j=1}^d g_j(T^{n+(j-1)m}x)-
\int_{N_d(X)}\bigotimes_{j=1}^dg_jd\mu^{(d)} \right |\le \d.
\end{equation}

By Lemma \ref{lem-product},
\begin{equation}\label{yexd4}
\left | \int_{N_d(X)}\bigotimes_{j=1}^dg_jd\mu^{(d)}-
\int_{N_d(X)}\bigotimes_{j=1}^df_jd\mu^{(d)} \right |
\le \sum_{j=1}^d\int_{N_d(X)}|g_j-f_j|d\mu^{(d)} \le \d.
\end{equation}

So combining (\ref{yexd1})-(\ref{yexd4}), when $x\in X_0^\delta$ and $N$ is large, we have
$$ \left | \frac{1}{N^2} \sum_{n\in [0,N-1] \atop{m\in [0,N-1]}} \prod_{j=1}^{d} f_j(T^{n+(j-1)m}x) -
\int_{N_d(X)}\bigotimes_{j=1}^df_jd\mu^{(d)} \right |\le 4\d, a.e. $$

This clearly implies that for all $x\in \bigcap_{k=1}^\infty X_0^{\frac{1}{k}}$,

\begin{equation*}
    \frac{1}{N^2} \sum_{n\in [0,N-1] \atop{m\in [0,N-1]}}
    f_1(T^nx)f_2(T^{n+m}x)\ldots f_d(T^{n+(d-1)m}x)
\end{equation*}
converges. Since $\mu(\bigcap_{k=1}^\infty X_0^{\frac{1}{k}})=1$, the proof is completed.

\section{Proof of Theorem C}\label{section-thC}
In this section we will prove Theorem C. To do this, first we derive some properties from
the result proved in the previous sections. Then using the properties and a lemma we show that
the pointwise convergence can be lifted from a distal system to its isometric extension under some
conditions. Finally we conclude Theorem C by the structure theorem for distal systems.

\subsection{Convergence in $L^2$}
In this subsection we will derive some theorem from the results in
the previous sections. It will be used in the proof of Theorem C and
will be useful in other settings.

\begin{thm}\label{decomposition}
Let $(X,\X,\mu,T)$ be an ergodic system and $d\in \N$. Then there exists a family $\{\mu^{(d)}_x\}_{x\in X}$ of probability measures on $X^d$ such that
\begin{enumerate}
  \item for $\mu$ a.e. $x\in X$, $\mu^{(d)}_x$ is ergodic under $T\times T^2\times \ldots \times T^d$,

  \item for all $f_1,\ldots, f_d\in L^\infty(\mu)$.
  \begin{equation}\label{ss}
\begin{split}
\frac{1}{N} \sum_{n=0}^{N-1}
   &f_1(T^nx)f_2(T^{2n}x)\ldots f_d(T^{dn}x)\\
   \longrightarrow & \int_{X^d} f_1(x_1)f_2(x_2)\ldots f_{d}(x_{d})\
   d \mu^{(d)}_x(x_1,x_2,\ldots,x_{d})
\end{split}
\end{equation}
as $N\to \infty$, where convergence is in $L^2(\mu)$.

 \item  for $\mu$ a.e. $x\in X$, $(p_j)_*(\mu_x^{(d)})\ll \mu$ for $1\le j\le d$, where $p_j: X^d\rightarrow X$ is the projection to the $j$-th coordinate.
\end{enumerate}
\end{thm}

\begin{proof}

First we assume that $(X,\mu, T)$ is the strictly
ergodic system obtained in Theorem \ref{model} by setting $X=\h{X}$

with the unique measure $\mu^{(d+1)}$.

Let $\pi_{d-1}: X\lra Z_{d-1}$ be the factor map and $\mu=\int _{Z_{d-1}}\theta_s\ d\mu_{d-1}(s)$ be the disintegration
of $\mu$ over $\mu_{d-1}$. Then by (\ref{limit-D}),
we have
\begin{equation}\label{s1}
\mu^{(d+1)}=\int_{Z_{d-1}}\theta_s\times \nu^{(d)}_s \ d\mu_{d-1}(s).
\end{equation}
For $x\in X$, let
\begin{equation}\label{s2}
    \mu^{(d)}_x=\nu^{(d)}_{\pi_{d-1}(x)}.
\end{equation}
By definition, for $\mu$ a.e. $x\in X$, $\mu^{(d)}_x$ is ergodic under $T\times T^2\times \ldots \times T^d$.

\medskip

Now we verify that $\{\mu^{(d)}_x\}_{x\in X}$ satisfies (\ref{ss}). First together with $\mu=\int _{Z_{d-1}}\theta_s\ d\mu_{d-1}(s)$,
we can rewrite (\ref{s1}) as
\begin{equation}\label{s3}
\mu^{(d+1)}=\int_{X}\d_x\times \mu^{(d)}_x \ d\mu(x).
\end{equation}

In fact,
\begin{equation*}
\begin{split}
\int_{X}\d_x\times \mu^{(d)}_x \ d\mu(x)=&\int_{Z_{d-1}} \int_X \d_x\times \nu^{(d)}_{\pi_{d-1}(x)} \ d \theta_s(x) \ d \mu_{d-1}(s) \\
   = & \int_{Z_{d-1}} \Big ( \int_X \d_x \ d \theta_s(x) \Big) \times \nu^{(d)}_{s} \ d \mu_{d-1}(s)
   \\ = &\int_{Z_{d-1}}\theta_s\times \nu^{(d)}_s \ d\mu_{d-1}(s)= \mu^{(d+1)}.
\end{split}
\end{equation*}

Now we show (\ref{ss}).
Assume that the left side of (\ref{ss}) converge (in $L^2$) to some function $g$. Now we show $g$ is equal to the right side of
(\ref{ss}). Let $f\in L^\infty(X)$, we have
\begin{equation*}
\begin{split}
& \int f(x)g(x)\ d \mu(x)\\ =&\lim_{N\to\infty}\int_X \frac{1}{N}
\sum_{n=0}^{N-1} f(x)
   f_1(T^nx)f_2(T^{2n}x)\ldots f_d(T^{d n}x)d\mu(x)\\
   = & \lim_{N\to \infty} \int_X\frac{1}{N^2}
\sum_{n,m=0}^{N-1} f(T^mx)
   f_1(T^{m+n}x)f_2(T^{m+2n}x)\ldots f_d(T^{m+dn}x)d\mu(x)
   \\ = & \lim_{N\to \infty} \int_X\frac{1}{N^2}
\sum_{n,m=0}^{N-1} (\tau_{d+1})^m(\sigma_{d+1})^n \Big( f\otimes
f_1\otimes \ldots \otimes f_{d} (x,x,\ldots,x)\Big )d\mu(x)
   \\ =& \int_{X^{d+1}} f(x_0)f_1(x_1)\ldots f_{d}(x_{d})d \mu^{(d+1)}(x_0,x_1,\ldots,x_{d-1}) \ (\text{by Theorem B})
   \\ =& \int_X f(x)\Big(\int_{X^{d}} f_1(x_1)f_2(x_2)\ldots f_{d}(x_{d})\
   d \mu^{(d)}_x(x_1,x_2,\ldots,x_{d})\Big) d\mu(x).
\end{split}
\end{equation*}
Thus $g(x)=\displaystyle \int_{X^d} f_1(x_1)f_2(x_2)\ldots
f_{d}(x_{d})\    d \mu^{(d)}_x(x_1,x_2,\ldots,x_{d}), \ \mu \ a.e. \ x\in X$. Note that (\ref{s3}) is
used in the last equality.

\medskip

For $j\in \{1,2,\ldots,d\}$, $(p_j)_*(\mu_x^{(d)})$ is a $T^j$-invariant measure of $X$. Since
$(X,\mu,T)$ is uniquely ergodic, it is easy to see that $(p_j)_*(\mu_x^{(d)})\ll \mu$.
Hence the theorem holds for the system $(X,\mu, T)$.

\medskip

Now we prove the result for any ergodic system $(X,\X,\mu,T)$. By
the proof of Theorem A, $(X,\X,\mu,T)$ has a strictly ergodic model
$(\h{X},\h{\mu},\h{T})$.

Let $\phi: X\rightarrow \h{X}$ be
the isomorphism. It is clear that $\phi^d: X^d\rightarrow \h{X}^d$
is also an isomorphism.

By the proof above, we have show that for $(\h{X},\h{\mu},\h{T})$, there exists a family
$\{\h{\mu}^{(d)}_x\}_{x\in \h{X}}$ of probability measures on ${\h{X}}^d$ such that it
satisfies condition (1)-(3) listed in the theorem. Define $\mu^{(d)}_x=\h{\mu}^{(d)}_{\phi(x)}\circ \phi^d$.
Then by (3) it is easy to check that $\{\mu^{(d)}_x\}_{x\in X}$ is what we need. The proof is completed.
\end{proof}

\subsection{Isometric extensions}

An isometric extension is one of the two basic extensions in the Furstenberg structure theorem
for a m.p.t. For the definition see Appendix A.
It can be shown that if $X$ is an isometric extension of an m.p.t. $(Y, \nu, S)$, then $X$ is isomorphic to a {\em skew product} $X'=Y\times M$,
where $M = G / H$ is a homogeneous compact metric space, $\mu'=\nu\times m_M$
with $m_M$ is the unique probability measure invariant under the transitive group
of isometries $G$. Moreover, the action of $T'$ on $X'$ is given by
$$T'(y,gH)=(Sy, \rho(y)gH),$$ where $\rho: Y\rightarrow G$ is a {\em cocycle}.
We denote $X'$ by $Y\times _\rho G/H$, and $T'$ by $T_\rho$. When $H$ is trivial,
we say $Y\times _\rho G$ is a {\em group extension} of $Y$. We refer to \cite{Glasner}
for the details.

\begin{lem}\label{extension}
Let $\pi:(X=Y\times _\rho G/H,\X,\mu,T)\rightarrow (Y,\Y,\nu,S)$ be an isometric extension
between two ergodic systems with $Z_{d-1}(X)=Z_{d-1}(Y)$, and $d\in \N$. Assume that $\{\mu^{(d)}_x\}_{x\in X}$
and $\{\nu^{(d)}_y\}_{y\in Y}$  are the families of measures defined in Theorem \ref{decomposition} respectively.
Then for given $f_1,\ldots, f_d\in L^\infty(\mu)$, one has that for $\mu$ a.e.  $x=(y,gH)\in X$
\begin{equation}\label{xx}
\begin{split}
\int_{X^d} & f_1(x_1) \ldots f_d(x_d)\ d\mu^{(d)}_x(x_1,\ldots,x_d) \\
   = & \int_{Y^d\times G^d} f_1(y_1, g_1H)\ldots f_{d}(y_{d},g_dH)\
   d \nu^{(d)}_y(y_1,\ldots, y_d) dg_1 \ldots dg_d .
\end{split}
\end{equation}
\end{lem}

\begin{proof}
By Theorem \ref{decomposition},
\begin{equation}\label{x1}
\begin{split}
\frac{1}{N} \sum_{n=0}^{N-1}
   &f_1(T^nx)f_2(T^{2n}x)\ldots f_d(T^{dn}x)\\
   \longrightarrow & \int_{X^d} f_1(x_1)f_2(x_2)\ldots f_{d}(x_{d})\
   d \mu^{(d)}_x(x_1,x_2,\ldots,x_{d})
\end{split}
\end{equation}
as $N\to \infty$, where convergence is in $L^2(\mu)$.

Since $Z_{d-1}(X)=Z_{d-1}(Y)$, by Theorem 12.1 in \cite{HK05}, $Y$ is also a characteristic factor of $X$. That is
\begin{equation}\label{x2}
\begin{split}
\|\frac{1}{N} \sum_{n=0}^{N-1}
   &f_1(T^nx)f_2(T^{2n}x)\ldots f_d(T^{dn}x)\\
   - & \frac{1}{N} \sum_{n=0}^{N-1}
   \E (f_1|\Y)(T^nx)\E(f_2|\Y)(T^{2n}x)\ldots \E(f_d|\Y)(T^{dn}x) \|_{L^2}\to 0
\end{split}
\end{equation}
as $N\to \infty$.

Applying Theorem \ref{decomposition} to $(Y, \Y,\nu, S)$, we have
\begin{equation}\label{x3}
\begin{split}
\frac{1}{N} \sum_{n=0}^{N-1}
   &\E (f_1|\Y)(S^ny)\E(f_2|\Y)(S^{2n}y)\ldots \E(f_d|\Y)(S^{dn}y)\\
   \longrightarrow & \int_{X^d} \E (f_1|\Y)(y_1)\E(f_2|\Y)(y_2)\ldots \E(f_d|\Y)(y_d)\
   d \nu^{(d)}_y(y_1,y_2,\ldots,y_{d})
\end{split}
\end{equation}
as $N\to \infty$, where convergence is in $L^2(\mu)$ and $x=(y, gH)$.

By (\ref{x1})-(\ref{x3}), we have for $\mu$ a.e.  $x=(y,gH)\in X$
\begin{equation}\label{x4}
\begin{split}
\int_{X^d} & f_1(x_1) \ldots f_d(x_d)\ d\mu^{(d)}_x(x_1,\ldots,x_d) \\
   = & \int_{X^d} \E (f_1|\Y)(y_1)\E(f_2|\Y)(y_2)\ldots \E(f_d|\Y)(y_d)\
   d \nu^{(d)}_y(y_1,y_2,\ldots,y_{d}) .
\end{split}
\end{equation}
Now (\ref{xx}) follows from (\ref{x4}) and $\E(f_j|\Y)(y_j)=\int_G f_j(y_j, gH)dg, 1\le j\le d$.

\end{proof}

The following proposition is crucial for our proof.
\begin{prop}\label{IsoExten}
Let $\pi:(X=Y\times _\rho G/H,\X,\mu,T)\rightarrow (Y,\Y,\nu,S)$ be an isometric extension
between two ergodic systems with $Z_{d-1}(X)=Z_{d-1}(Y)$, and $d\in \N$. If
\begin{equation*}
    \frac 1 N\sum_{n=0}^{N-1}f'_1(T^ny)f'_2(T^{2n}y)\ldots f'_d(T^{dn}y)
\end{equation*}
converge $\nu$ a.e. for any given $f'_1,\ldots, f'_d\in
L^\infty(\nu)$, then
\begin{equation*}
    \frac 1 N\sum_{n=0}^{N-1}f_1(T^nx)f_2(T^{2n}x)\ldots f_d(T^{dn}x)
\end{equation*}
converge $\mu$ a.e. for any given $f_1,\ldots, f_d\in
L^\infty(\mu)$.
\end{prop}

\begin{proof}

We may assume that $Y$ is a compact metric space.
By the assumption of the theorem, there is some measurable set $Y_0\in \Y$ with $\nu(Y_0)=1$ such that for $y\in Y_0$
and for all $f'_1,\ldots, f'_d\in C(Y)$
\begin{equation*}
    \frac 1 N\sum_{n=0}^{N-1}f'_1(T^ny)f'_2(T^{2n}y)\ldots f'_d(T^{dn}y)
\end{equation*}
converge since $C(Y)$ is separable. By Theorem \ref{decomposition}, we may assume that for all $y\in Y_0$ and for all $f'_1,\ldots, f'_d\in C(Y)$,
\begin{equation*}
    \frac 1 N\sum_{n=0}^{N-1}f'_1(T^ny)f'_2(T^{2n}y)\ldots f'_d(T^{dn}y)\longrightarrow \int_{Y^d} f'_1(y_1)f'_2(y_2)\ldots f'_{d}(y_{d})\
   d \nu^{(d)}_y(y_1,y_2,\ldots,y_{d})
\end{equation*}
as $N \to \infty$ since the almost everywhere limit coincides with the limit in $L^2$.

Let $X_0=\{(y,gH): y\in Y_0, g\in G\}$. Then $\mu(X_0)=\mu(Y_0\times G/H)=1$.
Since $C(X)$ is a separable space, by Lemma \ref{extension}, there is measurable set $X_1$ such that
$\mu(X_1)=1$ and (\ref{xx}) holds for all continuous functions.

\medskip

Now fix $x=(y,gH)\in X_0\cap X_1$. Let $\lambda$ be a weak limit point of the sequence
$$\big\{\frac {1}{N} \sum_{n=0}^{N-1} (T\times T^2\times \ldots \times T^d)^n_* \d_{(x,x,\ldots,x)}\big\}_N$$ in $M(X^d)$.
It is obvious that $\lambda$ is $T\times \ldots \times T^d$-invariant. We are going to show that $\lambda=\mu^{(d)}_x$.

\medskip

Let $\pi^{d}: X^d\rightarrow Y^{d}, (x_1,\ldots, x_d)\mapsto (\pi(x_1),\ldots,\pi(x_d))$.
Then $\pi^{d}_*\lambda$ is a weak limit point of the sequence
$$\big\{\frac {1}{N} \sum_{n=0}^{N-1} (S\times S^2\times \ldots \times S^d)^n_* \d_{(y,y,\ldots,y)}\big\}_N$$ in $M(Y^d)$.
By the assumption, we know
$$\lim_{N\to \infty}\frac {1}{N} \sum_{n=0}^{N-1} (S\times S^2\times \ldots \times S^d)^n_* \d_{(y,y,\ldots,y)}=\nu^{(d)}_y.$$
Thus $\pi^{d}_*\lambda=\nu^{(d)}_y$.

Let $\psi_1,\ldots, \psi_d\in C(G)$ such that $\psi_i\ge 0$ and $\int_G \psi_i d m=1$
for all $i\in \{1,\ldots, d\}$.
Now define a new measure $\lambda_{\psi_1,\ldots, \psi_d}$ as follows:
\begin{equation}\label{}
\begin{split}
& \lambda_{\psi_1,\ldots, \psi_d}(f_1\otimes \ldots \otimes f_d)  \\
= & \int_{G^d\times X^d} f_1(y_1, h_1g_1H)\ldots f_d(y_d, h_d g_dH)\psi_1(h_1)\ldots
\psi_d(h_d)\ d h_1\ldots d h_d d \lambda(x_1,\ldots,x_d),
\end{split}
\end{equation}
where $x_i=(y_i,g_iH)$ and $f_i\in C(X)$ for all $i\in \{1,2, \ldots, d\}$.

Then by $\pi^{(d)}_*\lambda=\nu^{(d)}_y$ and (\ref{xx}) we have
\begin{equation*}
\begin{split}
& |\lambda_{\psi_1,\ldots, \psi_d}(f_1\otimes \ldots \otimes f_d) | \\
 \le & \prod_{i=1}^d \sup_{g\in G} |\psi_i(g)| \int_{G^d\times X^d} |f_1(y_1, h_1g_1H)
 \ldots f_d(y_d, h_d g_dH)| d h_1\ldots d h_d d \lambda(x_1,\ldots,x_d)\\
  = & \prod_{i=1}^d \sup_{g\in G} |\psi_i(g)| \int_{G^d\times Y^d} |f_1(y_1, h_1H)
  \ldots f_d(y_d, h_d H)| d h_1\ldots d h_d d \nu^{(d)}_y (y_1,\ldots,y_d) \\
  =& \prod_{i=1}^d \sup_{g\in G} |\psi_i(g)| \int_{X^d}
  |f_1(x_1)\ldots f_d(x_d)| d \mu^{(d)}_x (x_1,\ldots,x_d).
\end{split}
\end{equation*}
Thus we have
\begin{equation}\label{}
    \lambda_{\psi_1,\ldots, \psi_d} \ll \mu^{(d)}_x.
\end{equation}

Note $\rho: Y\rightarrow G$ is a cocycle. For each $n\in \N$, let
$$\rho^{(n)}(y)=\rho(S^{n-1}y)\rho(S^{n-2}y)\ldots \rho(y).$$
Then we have $$T^n(y,gH)=(S^n, \rho^{(n)}(y)gH).$$

Now in addition we assume that $\{\psi_i\}$ satisfy $\psi_i(h^{-1}gh)=\psi_i(g)$ for all $1\le i\le d$.
Then for $f_1,\ldots,f_d\in C(X)$, we have
\begin{equation*}
\begin{split}
& \lambda_{\psi_1,\ldots, \psi_d}((f_1\otimes \ldots \otimes f_d)\circ T\times \ldots \times T^d) \\
  = & \int_{G^d\times X^d} f_1(Sy_1, \rho(y_1)h_1g_1H)\ldots f_d(S^dy_d,
  \rho^{(d)}(y_d)h_d g_dH) \psi_1(h_1)\ldots \psi_d(h_d) d h_1\ldots d h_d d \lambda(x_1,\ldots,x_d)\\
  = & \int_{G^d\times X^d} f_1(Sy_1, h_1\rho(y_1)g_1H)\ldots f_d(S^dy_d,
  h_d\rho^{(d)}(y_d) g_dH) \psi_1(h_1)\ldots \psi_d(h_d) d h_1\ldots d h_d d
  \lambda(x_1,\ldots,x_d)\\
  = & \int_{G^d\times X^d} f_1(y_1, h_1g_1H)\ldots f_d(y_d, h_d g_dH)
  \psi_1(h_1)\ldots \psi_d(h_d)\ d h_1\ldots d h_d d \lambda(x_1,\ldots,x_d)\\
  = & \lambda_{\psi_1,\ldots, \psi_d}(f_1\otimes \ldots \otimes f_d) .
\end{split}
\end{equation*}
That is, $ \lambda_{\psi_1,\ldots, \psi_d}$ is $T\times \ldots \times T^d$-invariant. Since $\mu^{(d)}_x$
is ergodic, we have that
$$ \lambda_{\psi_1,\ldots, \psi_d}=\mu^{(d)}_x.$$

Now we will define a sequence $\{\phi_n\}_n$ such that all $\phi_n$ satisfies
the properties which $\psi_i$ hold above, and $$\lambda_{\phi_n,\ldots, \phi_n}\to \lambda,\ n\to \infty .$$
Then we get that $\lambda=\mu^{(d)}_x$.

Since $G$ is a compact metric group, there is an invariant metric $\varrho$. For all $n\in \N$, let
$$\varphi_n(g)= 1/n -\inf \{ 1/n, \varrho(e, g)\}.$$
Then let
$$\phi_n=\frac{\varphi_n}{\int_G\varphi_n d m}.$$
Note that $\phi_n$ is supported on $A_n=\{g\in G: \varrho(e,g)<\frac{1}{n}\}$. It follows that
for given $y_1,\ldots, y_d$, $f_1(y_1, h_1g_1H)\ldots f_d(y_d, h_d g_dH)$ is close to $f_1(y_1, g_1H)\ldots f_d(y_d, g_dH)$
uniformly on $A_n$. Then using the fact that $\int_G \phi_n d m=1$  we deduce that $\phi_n$ is what we need.

\medskip

To sum up, we have proved that for all $x\in X_0\cap X_1$, $\mu^{(d)}_x$
is the unique weak limit point of sequence
$$\big\{\frac {1}{N} \sum_{n=0}^{N-1} (T\times T^2\times \ldots \times T^d)^n_* \d_{(x,x,\ldots,x)}\big\}_N$$ in $M(X^d)$.
Hence for all $x\in X_0\cap X_1$, we have that for all $f_1,\ldots, f_d\in
C(X)$
\begin{equation}\label{s4}
\begin{split}
\frac{1}{N} \sum_{n=0}^{N-1}
   &f_1(T^nx)f_2(T^{2n}x)\ldots f_d(T^{dn}x)\\
   \longrightarrow & \int_{X^d} f_1(x_1)f_2(x_2)\ldots f_{d}(x_{d})\
   d \mu^{(d)}_x(x_1,x_2,\ldots,x_{d})
\end{split}
\end{equation}
as $N\to \infty$. Note that $\mu(X_0\cap X_1)=1$.

\medskip

Now we show that for all $f_1,\ldots, f_d\in L^\infty(\mu)$,
\begin{equation*}
    \frac 1 N\sum_{n=0}^{N-1}f_1(T^nx)f_2(T^{2n}x)\ldots f_d(T^{dn}x)
\end{equation*}
converges $\mu$ a.e.. The proof is similar to the proof of Theorem B.

Without loss of generality, we assume that for all $1\le j\le d$,
$\|f_j\|_\infty\le 1$. For any $\d>0$, choose continuous functions $g_j$ such that
$\|g_j\|_\infty\le 1$ and $$\E(|f_j-g_j||\I(X,T^{j}))<\d / d $$ for all $1\le j\le
d$. Recall that $\I(X,T^j)=\{A\in \X: T^jA=A\}$. Since $T$ is ergodic, $\I(X,T^j)$ is finite which guarantees the existence of such $g_j$.

By Lemma \ref{lem-product}, we have
\begin{equation*}
\begin{split}
&  \left | \frac{1}{N} \sum_{n=0}^{N-1}
   \prod_{j=1}^{d} f_j(T^{jn}x) -
\frac{1}{N} \sum_{n=0}^{N-1}
   \prod_{j=1}^d g_j(T^{jn}x)\right |\\
=&  \left | \frac{1}{N} \sum_{n=0}^{N-1}\Big[ \prod_{j=1}^{d} f_j(T^{jn}x) - \prod_{j=1}^{d}
g_j(T^{jn}x)\Big] \right |\\
\le & \sum_{j=1}^d \Big[\frac {1}{N} \sum_{n=0}^{N-1}\Big | f_j(T^{jn}x)-g_j(T^{jn}x)\Big |\Big ].
\end{split}
\end{equation*}

Now by Birkhoff Pointwise Ergodic Theorem we have that for all $1\le j\le d$
\begin{equation*}
    \frac {1}{N} \sum_{n=0}^{N-1}\Big | f_j(T^{jn}x)-g_j(T^{jn}x)\Big | \lra
\E(|f_j-g_j||\I(T^{j})), \quad N\to \infty.
\end{equation*}
for $\mu$ a.e..
Hence
\begin{equation}\label{s5}
\begin{split}
&   \left | \frac{1}{N} \sum_{n=0}^{N-1}
   \prod_{j=1}^{d} f_j(T^{jn}x) -
\frac{1}{N} \sum_{n=0}^{N-1}
   \prod_{j=1}^d g_j(T^{jn}x)\right |\\
\le &  \sum_{j=1}^d \Big[\frac {1}{N} \sum_{n=0}^{N-1}\Big |
f_j(T^{jn}x)-g_j(T^{jn}x)\Big |\Big ]\\
\lra &  \sum_{j=1}^d \E(|f_j-g_j||\I(T^{j}))\le \d,\  \quad N\to \infty.
\end{split}
\end{equation}
for $\mu$ a.e..

By (\ref{s4}),
\begin{equation}\label{s6}
   \frac{1}{N} \sum_{n=0}^{N-1}
   \prod_{j=1}^d g_j(T^{jn}x)\to \int_{X^d}\bigotimes_{j=1}^d g_j\
   d \mu^{(d)}_x, N\to \infty.
\end{equation}
for $\mu$ a.e..

By Lemma \ref{lem-product},
\begin{equation}\label{s7}
\begin{split}
\left | \int_{X^d}\bigotimes_{j=1}^dg_j d\mu^{(d)}_x-
\int_{X^d}\bigotimes_{j=1}^df_jd\mu^{(d)}_x \right |
\le \sum_{j=1}^d\int_{X}|g_j-f_j|d\mu \le \d.
\end{split}
\end{equation}

So combining (\ref{s5})-(\ref{s7}), we have for $\mu$ a.e. $x\in X$, when $N$ is large enough
\begin{equation*}
   \left |\frac{1}{N} \sum_{n=0}^{N-1}\prod_{j=1}^{d} f_j(T^{jn}x)-\int_{X^d}\bigotimes_{j=1}^df_jd\mu^{(d)}_x\right |<3\d.
\end{equation*}
Since $\d$ is arbitrary, we have for $\mu$ a.e. $x\in X$
\begin{equation*}
\begin{split}
\frac{1}{N} \sum_{n=0}^{N-1}
   &f_1(T^nx)f_2(T^{2n}x)\ldots f_d(T^{dn}x)\\
   \longrightarrow & \int_{X^d} f_1(x_1)f_2(x_2)\ldots f_{d}(x_{d})\
   d \mu^{(d)}_x(x_1,x_2,\ldots,x_{d})
\end{split}
\end{equation*}
as $N\to \infty$. The proof is completed.
\end{proof}

\subsection{Proof of Theorem C}
In this final subsection we will prove Theorem C. We start with the definition of a distal system.

\begin{de}
Let $\pi: (X,X,\mu,T)\rightarrow (Y,\Y,\nu,T)$ be a factor between two ergodic systems.
We call the extension $\pi$ a {\em distal extension} if there exists a countable ordinal $\eta$
and a directed family of factors $(X_\theta, \X_\theta, \mu_\theta, T), \theta \le \eta$ such that
\begin{enumerate}
  \item $X_0=Y$ and $X_\eta=X$.
  \item For $\theta<\eta$ the extension $\pi_\theta : X_{\theta+1}\rightarrow X_\theta$ is isometric and non-trivial (i.e. not an isomorphism).
  \item For a limit ordinal $\lambda \le \eta$, $X_\lambda={\lim\limits_{\longleftarrow}}_{\theta<\lambda} X_\theta$ (i.e. $\X_\lambda =\bigvee X_\theta$).
\end{enumerate}

If $X$ is a distal extension of the trivial system, then $(X,\X,\mu,T)$ is called a {\em distal system}.
\end{de}

\medskip

\noindent {\bf The proof of Theorem C:} We say a system $(X,\X,\mu, T)$ satisfies ($\divideontimes$), if for all $f_1, \ldots, f_d \in L^{\infty}(\mu)$
\begin{equation*}
    \frac 1 N\sum_{n=0}^{N-1}f_1(T^nx)\ldots f_d(T^{dn}x)
\end{equation*}
converge $\mu$ a.e.. The aim is to prove each distal system satisfies ($\divideontimes$). We will use the structure of distal systems and Proposition \ref{IsoExten} to complete the proof.

\medskip

Let $\pi_{d-1}: X\rightarrow Z_{d-1}$ be the factor map. Then $\pi_{d-1}$
is distal since $X$ is distal. By the definition of a distal extension,
there exists a countable ordinal $\eta$ and a directed family of factors
$(X_\theta, \X_\theta, \mu_\theta, T), \theta \le \eta$ such that
\begin{enumerate}
  \item $X_0=Z_{d-1}$ and $X_\eta=X$.
  \item For $\theta<\eta$ the extension $\pi_\theta : X_{\theta+1}\rightarrow X_\theta$ is non-trivial isometric.
  \item For a limit ordinal $\lambda \le \eta$, $X_\lambda={\lim\limits_{\longleftarrow}}_{\theta<\lambda} X_\theta$.
\end{enumerate}
\medskip

Then we have:

($i$) By Theorem \ref{ziegler}, $X_0=Z_{d-1}$ satisfies ($\divideontimes$).

\medskip

($ii$) For $\theta<\eta$ the extension $\pi_\theta : X_{\theta+1}\rightarrow X_\theta$ is
non-trivial isometric. If $X_\theta$ satisfies ($\divideontimes$),
then by Proposition \ref{IsoExten}, $X_{\theta+1}$ satisfies ($\divideontimes$).

\medskip

($iii$) For a limit ordinal $\lambda \le \eta$, if for all $\theta<\lambda$,
$X_\theta$ satisfies ($\divideontimes$), then it is easy to verify that the inverse
limit $X_\lambda={\lim\limits_{\longleftarrow}}_{\theta<\lambda} X_\theta$ also satisfies ($\divideontimes$).

\medskip
($iv$) By ($i$-$iii$), $X_\eta=X$ satisfies ($\divideontimes$).

The proof is completed.

\bigskip

\appendix

\section{Background on Ergodic Theory}

In this Appendix we will state notions and results in ergodic
theory which are used in the article. Let $(X,\X,\mu,T)$ be a m.p.t.

\subsubsection{Ergodicity and weak mixing}

First we list some equivalent conditions for ergodicity and weak
mixing.

\begin{thm}
Let $(X,\X,\mu,T)$ be a m.p.t. Then the following
conditions are equivalent:
\begin{enumerate}
  \item $T$ is ergodic.
  \item Every measurable function $f$ from $X$ to some Polish space
  $ P$ satisfying $f\circ T=f\ a.e.$ is of form $f\equiv p$ $a.e.$
  for some point $p\in P$.
  \item $\displaystyle \lim_{N\to\infty} \sum_{n=0}^{N-1}
  \int f\circ T^n\cdot g \ d\mu=\int f\ d\mu
  \int g\ d\mu  $, for all $f,g\in L^2(\mu)$ (or $L^{1}(\mu)$).
\end{enumerate}
\end{thm}

\begin{thm}
Let $(X,\X,\mu,T)$ be a m.p.t. Then the following
conditions are equivalent:
\begin{enumerate}
  \item $T$ is weakly mixing.
  \item 1 is the only eigenvalue of $T$ and the geometric multiplicity
  of eigenvalue 1 is 1.
  \item The product system with any ergodic system is still ergodic.
\end{enumerate}
\end{thm}

\subsubsection{Conditional expectation}

If $\Y$ is a $T$-invariant sub-$\sigma$-algebra of $\X$ and $f\in
L^1(\mu)$, we write $\E(f|\Y)$, or $\E_\mu(f|\Y)$ if needed, for the
{\em conditional expectation} of $f$ with respect to $\Y$. The
conditional expectation $\E(f|\Y)$ is characterized as the unique
$\Y$-measurable function in $L^2(Y,\Y, \nu)$ such that
\begin{equation}
    \int_Y g \E(f|\Y)d\nu = \int_X  g\circ \pi f d\mu
\end{equation}
for all $g\in L^2(Y,\Y, \nu)$. We will frequently make use of the
identities $$\int \E(f|\Y) \ d\mu = \int f \ d\mu \quad
\text{and}\quad T \E(f|\Y) = \E(Tf|\Y).$$ We say that a function $f$
is {\em orthogonal} to $\Y$, and we write $f \perp \Y$, when it has a zero
conditional expectation on $\Y$. If a function $f\in L^1(\mu)$ is
measurable with respect to the factor $\Y$, we write $f \in L^1(Y,
\Y, \nu)$.

\medskip

The disintegration of $\mu$ over $\nu$, written as $\mu=\int\mu_y\ d\ \nu(y)$,
is given by a measurable map
$y \mapsto \mu_y$ from $Y$ to the space of probability measures on
$X$  such  that
\begin{equation}
    \E(f|\Y)(y)=\int_X f d\mu_y
\end{equation}
$\nu$-almost everywhere.

\subsubsection{Ergodic decomposition}

The ergodic decomposition is a basic result in ergodic. We quote this theorem from \cite{Glasner}.
\begin{thm}\label{ergodic-decomposition}
Let $(X,\X,\mu, T)$ be a dynamical system $\I$ the $\sigma$-algebra of $T$-invariant sets. Let
$\pi: (X,\X,\mu, T)\rightarrow (Y,\Y,\nu, S)$ be the factor defined by $\I$, so that $\I=\pi^{-1}(\Y)$. Let $\mu=\int_Y \mu_y \ d \nu(y)$ be the disintegration of $\mu$ over $\nu$ and for every $y\in Y$ denote $X_y=\pi^{-1}(y)$ and $\X_y=\X\cap X_y$, then:
\begin{enumerate}
  \item $S$ acts as the identity on $Y$.
  \item For $\nu$ a.e. $y\in Y$, the system $(X_y,\X_y,\mu_y, T)$ is ergodic.
  \item This decomposition is unique in the following sense. If $(Z,\ZZ, \eta)$ is a standard probability space and $\psi: z\mapsto \tilde{\mu}_z$ a measurable map from $Z$ into $M^e_T(X)$ such that $\mu=\int_Z \tilde{\mu}_y \ d \eta(z)$, then there exists a measurable map $\phi:(Z,\eta)\rightarrow (Y,\nu)$ such that $\eta$ a.e. $\mu_{\phi(z)}=\tilde{\mu}_z$.
\end{enumerate}

\end{thm}

\subsubsection{Inverse limit}

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

\subsubsection{Group rotation}

All locally compact groups are implicitly assumed to be metrizable
and endowed with their Borel $\sigma$-algebras. Every compact group
$G$ is endowed with its Haar measure, denoted by $m_G$.

For a compact abelian group $Z$ and $t\in Z$, we write $(Z, t)$ for
the probability space $(Z,m_Z)$, endowed with the transformation
given by $z \mapsto tz$. A system of this kind is called a {\em
rotation}.

\subsubsection{Joining and conditional product measure}

The notions of joining and conditional product measure are introduced by Furstenberg in \cite{F77}.
Let $(X_i, \mu_i,T_i), i = 1,\ldots, k$, be m.p.t, and let $(Y_i,\nu_i,S_i)$ be corresponding factors, and
$\pi_i:X_i\rightarrow Y_i$ the factor maps. A measure $\nu$ on
$Y=\prod_i Y_i$ defines a {\em joining} of the measures on $Y_i$ if
it is invariant under $S_1\times \ldots \times S_k$ and maps onto
$\nu_j$ under the natural map $\prod_i Y_i\rightarrow Y_j$. When $S_1=\ldots=S_k$, we then say that
$\nu$ is a $k$-fold self-joining.

Let $\nu$ be a joining of the measures on $Y_i, i=1,\ldots, k$, 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
\begin{equation}\label{}
    \mu=\int_Y \mu_{X_1,y_1}\times \mu_{X_2,y_2}\times \ldots \times
    \mu_{X_k,y_k}\ d\nu(y_1,y_2,\ldots,y_k).
\end{equation}
Then $\mu$ is called the {\em conditional product measure with
respect to $\nu$}.

Equivalently, $\mu$ is conditional product measure relative to $\nu$
if and only if for all $k$-tuple $f_i\in L^\infty(X_i,\mu_i),
i=1,\ldots, k$
\begin{equation}
\begin{split}
\int_X
   &f_1(x_1)f_2(x_2)\ldots f_k(x_k)\ d\mu(x_1,x_2,\ldots,x_k)\\
   = & \int_Y \E(f_1|\Y_1)(y_1)\E(f_2|\Y_2)(y_2)\ldots \E(f_{k}|\Y_k)(y_{k})\
   d\nu (y_1,y_2,\ldots,y_{k}).
\end{split}
\end{equation}

\subsubsection{Relatively independent joining}

Let $(X_1,\X_1,\mu_1,T), (X_2,\X_2,\mu_2,T)$ be two systems and let
$(Y,\Y,\nu, S)$ be a common factor with $\pi_i: X_i\rightarrow Y$ for
$i = 1, 2$ the factor maps. Let $\mu_i=\int \mu_{i,y}\ d\nu(y)$
represent the disintegration of $\mu_i$ with respect to $Y$. Let
$\mu_1\times_Y \mu_2$ denote the measure defined by
$$\mu_1\times_Y \mu_2(A)=\int_Y \mu_{1,y}\times \mu_{2,y}\ d\nu(y),$$
for all $A\in \X_1\times \X_2$. The system $(X_1\times X_2,
\X_1\times \X_2,\mu_1\times_Y \mu_2, T\times T)$ is called the
{\em 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 {\em relatively independent joining} of $X_1$ and $X_2$ over
$Y$.

\subsubsection{Isometric extensions and Ergodic extensions}

Let $\pi: (X,\X,\mu,T)\rightarrow (Y,\Y,\nu,S)$ be a factor map. The
$L^2(X,\X,\mu)$ norm is denoted by $||\cdot||$ and the
$L^2(X,\X,\mu_y)$ norm by $||\cdot||_y$ for $\nu$-almost every $y
\in Y$. Recall $\{\mu_y\}_{y \in Y}$ is the disintegration of $\mu$
relative to $\nu$.

A function $f\in L^2(X,\X,\mu)$ is {\em almost periodic over $\Y$}
if for every $\ep>0$ there exist $g_1,\ldots, g_l \in
L^{2}(X,\X,\mu)$ such that for all $n\in \Z$
$$\min_{1\le j\le l} ||T^nf-g_j||_y<\ep$$
for $\nu$ almost every $y\in Y$. One writes  $f\in AP(\Y)$.
Let $K(X|Y, T)$ be the closed subspace of $L^2(X)$ spanned by the
almost periodic functions over $\Y$. When $\Y$ is trivial,
$K(X,T)=K(X|Y,T)$ is the closed subspace spanned by eigenfunctions
of $T$.

$X$ is an {\em isometric extensions} of $Y$ if $K(X|Y,T)=L^2(X)$ and
it is a {\em (relatively) weak mixing extension} of $Y$ if
$K(X|Y,T)=L^2(Y)$.

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

\section{The pointwise ergodic theorem for amenable groups}

\subsubsection{}

Amenability has many equivalent formulations; for us, the most
convenient definition is that a locally compact group $G$ is {\em
amenable} if for any compact $K \subset G$ and $\d>0$ there is a
compact set $F\subset G$ such that $$|F\Delta KF|<\d |F|,$$ where we
use both $|\cdot |$ and $m$ to denote the left Haar measure on $G$
(for discrete $G$, we take this to be the counting measure on $G$).
Such a set $F$ will be called {\em $(K, \d)$-invariant}. A sequence
$F_1$, $F_2$, $\ldots$ of compact subsets of $G$ will be called a
{\em F{\rm ${\o}$}lner sequence} if for every compact $K$ and $\d >
0$, for all large enough $n$ we have that $F_n$ is $(K,
\d)$-invariant. Here all groups are assumed to be locally compact
second countable.

\subsubsection{}

Suppose now that $G$ acts bi-measurably from the left by measure
preserving transformations on a Lebesgue space $(X,\mathcal{B},
\mu)$ with $\mu(X) = 1$. We will use for any $f : X \rightarrow \R$
the symbol $\A (F, f )(x)=\A_F(f)$ to denote the average
$$\A(F,f)(x)=\frac{1}{|F|}\int_F f(gx)\ d m(g).$$

\begin{de}
A F{\rm ${\o}$}lner sequence $F_n$ will be said to be {\em tempered}
if for some $C > 0$ and all $n$
\begin{equation}\label{}
    \left|\bigcup_{k< n}F^{-1}_kF_n\right |< C\left|F_n\right|.
\end{equation}
\end{de}

\begin{thm}[Lindenstrauss]\cite{Lin}\label{Lindenstrauss} Let $G$ be an amenable
group acting on a measure space $(X,\mathcal{B},\mu)$ by measure
preserving transformation, and let $F_n$ be a tempered F{\rm
{\o}}lner sequence. Then for any $f\in L^1(\mu)$, there is a
$G$-invariant $f^*\in L^1(\mu)$ such that
$$\lim_{n\to \infty} \A(F_n , f)(x) = f^* (x)\quad a.e.$$
In particular, if the $G$ action is ergodic,
$$\lim_{ n\to \infty} \A(F_n , f
)(x) = \int f(x)\ d\mu (x)\quad  a.e.$$

\end{thm}

\section{Uniquely ergodic systems}

In this section we give some conditions for unique ergodicity under
$\Z^d$ actions ($d\in\N$).

\begin{thm}\label{unique-ergodic}
Let $(X,\Gamma)$ be a topological system, where $\Gamma=\Z^d$. The
following conditions are equivalent.
\begin{enumerate}

  \item $(X,\Gamma)$ is uniquely ergodic.

  \item For every continuous function $f\in C(X)$ the sequence of
  functions
  \begin{equation}\label{}
    \A_Nf(x)=\frac {1}{N^d} \sum_{\gamma \in [0,N-1]^d} f(\gamma x)
  \end{equation}
  converges uniformly to a constant function.

  \item For every continuous function $f\in C(X)$ the sequence of
  functions $\A_Nf(x)$ converges pointwise to a constant function.

 \item There exists a $\mu\in M_\Gamma(X)$ such that
 for all continuous function $f\in C(X)$ and all $x\in X$ the sequence of
  functions \begin{equation}\label{}
    \A_Nf(x)\lra \int f\ d\mu, \ N\to \infty.
  \end{equation}
\end{enumerate}
\end{thm}

\section{Nilmanifolds and nilsystems}

\subsubsection{}

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 $G_j$, $j\ge 1$, are defined inductively by
setting $G_1 = G$ and $G_{j+1} = [G_j ,G]$. Let $k \ge 1$ be an
integer. We say that $G$ is {\em $k$-step nilpotent} if $G_{k+1}$ is
the trivial subgroup.

\subsubsection{}

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 {\em $k$-step nilmanifold}. The group $G$ acts on $X$ by
left translations and we write this action as $(g, x)\mapsto gx$.
The Haar measure $\mu$ of $X$ is the unique probability measure on
$X$ invariant under this action. Let $\tau\in G$ and $T$ be the
transformation $x\mapsto \tau x$ of $X$. Then $(X, T, \mu)$ is
called a {\em $k$-step nilsystem}.

\subsubsection{}

For every integer $j\ge 1$, the subgroup $G_j$ and $\Gamma G_j$ are
closed in $G$. It follows that the group $\Gamma_j=\Gamma \cap G_j$
is cocompact in $G_j$.

\subsubsection{}

Here are some basic properties of nilsystems. For the references, please refer to Appendix B in \cite{HK05} or Section 4 in \cite{BHK05} etc.

\begin{thm}\label{nilsystem}
Let $(X = G/\Gamma,\mu , T )$ be a $k$-step nilsystem with $T$ the
translation by the element $t\in G$. Then:

\begin{enumerate}
\item $(X, T )$ is uniquely ergodic if and only if $(X,\mu , T )$ is
ergodic if and only if $(X, T )$ is minimal if and only if $(X, T )$
is transitive.

\item 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 $H$.

\medskip

\noindent Assume furthermore that $G$ is spanned by the connected
component of the identity and the element $t$. Then:

\medskip

\item The groups $G_j$, $j\ge 2$, are connected.

\item The nilsystem $(X,\mu, T )$ is ergodic if and only if the
rotation induced by $t$ on the compact abelian group $G/ G_2\Gamma$
is ergodic.

\item If the nilsystem $(X,\mu , T )$ is ergodic then its Kronecker
factor is $Z = G/G_2\Gamma$ with the rotation induced by $t$ and
with the natural factor map $X = G/\Gamma\rightarrow G/G_2\Gamma =
Z$.
\end{enumerate}
\end{thm}

\begin{thm}
Let X =$ G/\Gamma$ be a nilmanifold with Haar measure $\mu$ 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
$(X,\mu)$, then X is uniquely ergodic for this group.
\end{thm}

\section{HK-seminorms}

Let $(X,\mu,T)$ be an ergodic system and $k\in \N$. We write $C:
\C\rightarrow \C$ for the conjugate map $z\mapsto \overline{z}$. Let
$|\ep|=\ep_1+\ldots+\ep_k$ for $\ep\in V_k= \{0,1\}^k$. It is easy
to verify that for all $f\in L^\infty(\mu)$ the integral
$\int_{X^{[k]}}\bigotimes_{\ep\in V_k}
C^{|\ep|}f(x_\ep)d\mu^{[k]}(\bf x)$ is real and nonnegative. Hence
we can define {\em Host-Kra seminorm} (HK seminorm for short) as follows:
\begin{equation}\label{}
    \HK f\HK_k=\Big( \int_{X^{[k]}} \bigotimes_{\ep\in V_k}
C^{|\ep|}f(x_\ep)d\mu^{[k]}({\bf x})\Big)^{1/2^k}.
\end{equation}

As $X$ is assumed to be ergodic, the $\sigma$-algebra $\I^{[0]}$ is
trivial and $\mu^{[1]}=\mu \times \mu$. We therefore have
$$\HK f\HK_1=\Big(\int_{X^2}f(x_0)\overline{f(x_1)}d\mu \times\mu(x_0,x_1)\Big)^{1/2}
=\Big|\int fd\mu\Big|.$$

It is showed in \cite{HK05} that $\HK\cdot\HK_k$ is a seminorm on
$L^\infty(\mu)$, and for all $f_\ep\in L^\infty(\mu), \ep\in V_k$,
$$\Big|\int \bigotimes_{\ep\in V_k}f_\ep d\mu^{[k]}\Big|\le \prod_{\ep\in V_k}\HK f_\ep\HK_k.$$
The following lemma follows immediately from the definition of the
measures and the Ergodic Theorem.

\begin{lem}\label{lemmaE1}
For every integer $k\ge 0$ and every $f\in L^\infty(\mu)$, one has
\begin{equation}\label{}
    \HK f\HK_{k+1}=\Big(\lim_{N\to \infty} \frac{1}{N}\sum_{n=0}^{N-1}
    \HK f\cdot T^n \overline{f}\HK_k^{2^k}\Big)^{1/2^{k+1}}.
\end{equation}
\end{lem}

An important property is

\begin{prop}\cite[Lemma 4.3]{HK05}
For a $f\in L^\infty(\mu)$, $\HK f\HK_k=0$ if and only if
$\E(f|\ZZ_{k-1})=0$.
\end{prop}

\section{The van der Corput lemma}

\begin{lem}\cite{Bergelson87}\label{vanderCoput}
Let $\{x_n\}$ be a bounded sequence in a Hilbert space $\mathcal{H}$
with norm $\parallel \cdot \parallel$ and inner product $<\cdot,
\cdot>$. Then
\begin{equation*}
    \limsup_{N\to \infty} \big\| \frac{1}{N}\sum _{n=1}^N
    x_n\big\|^2
    \le \limsup_{H\to \infty}\frac{1}{H} \sum_{h=1}^H\limsup_{N\to\infty}
    \left|\frac{1}{N}\sum_{n=1}^N <x_n, x_{n+h}>  \right|.
\end{equation*}

\end{lem}


TITLE: Why are pointwise measurable functions not measurable?
QUESTION [3 upvotes]: Let $(X,\mathcal{X})$, $(Y,\mathcal{Y})$ and $(Z,\mathcal{Z})$ be measurable spaces. Let $(X \times Y, \mathcal{X} \otimes \mathcal{Y})$ be the product of the two spaces, where $\mathcal{X} \otimes \mathcal{Y} := \sigma(\mathcal{X} \times \mathcal{Y})$.
Let $f:X \times Y \to Z$ be a pointwise measurable function (my own term) in the sense that

$\forall x \in X: f(x, \cdot):Y \to Z$ is measurable
$\forall y \in Y: f(\cdot, y):X \to Z$ is measurable

Sufficient conditions for separately measurable functions being jointly measurable. claims that $f:X \times Y \to Z$ is not necessarily measurable. Why? Is there a simple counter-example?

REPLY [5 votes]: Yes, there is. Indeed consider the Borel measure on $[0,1]$ and let $A$ be your favorite non measurable subset of the unit interval. Then define a function $f$ on $[0,1]^2$ as follows:
$$f(x,y) =
\begin{cases}
0 & \text{if } x \notin A \text{ or if } x \neq y , \\
1 & \text{otherwise}.
\end{cases}
$$
One can easily check that $f$ is separately measurable, but if it were measurable on the product space, its composition with the function $g \colon [0,1] \to [0,1]^2$ defined via $g(x) = (x,x)$ would also be measurable (composition of Borel measurable functions is measurable). Clearly this is not the case since $$f \circ g = \chi_{A},$$ which by assumption is not measurable.


TITLE: Partial Derivative notation in a context I don't recognise
QUESTION [0 upvotes]: I'm trying to follow a document describing a 'Perfectly Matched Layer' (an FDTD boundary condition), however it uses partial derivative notation in a context I'm not familiar with:
$$ \psi^n_i = a_i\psi^{n-1}_i + b_i\bigg(\frac{\partial}{\partial i}\bigg)^n $$
I don't know what the numerator-less partial derivative in the parentheses is, it appears to be partial derivative of nothing in respect to $i$.  The authors use partial-derivative-with-no-numerator notation in a context I'm more familiar with later on in the document:
$$ \frac{\partial^2 P}{\partial i^2} = \frac{\partial}{\partial i}\bigg(\frac{\partial P}{\partial i}\bigg)$$
I should point out that $i$ is a basis vector in 3D Cartesian space i.e. $i \epsilon \{x, y, z\}$.
As you may have realised, mathematics is not a strong subject of mine, but I'll try to answer any queries you have.

REPLY [1 votes]: That is an equality of operators. You have to apply it to some function $f$, like in
$$\psi^n_i f = a_i\psi^{n-1}_i f + b_i\bigg(\frac{\partial}{\partial i}\bigg)^n f = a_i\psi^{n-1}_i f + b_i \frac{\partial^n f}{\partial i^n}.$$


TITLE: If $u_n$ are uniformly bounded in $L^\infty$, $u_n \to u$ in $L^2(\Omega)$, is $u$ also bounded?
QUESTION [0 upvotes]: Let $\Omega$ be a bounded domain.
Let $u_n \in L^\infty(\Omega)$ with $|u_n|_{L^\infty} \leq M$. If $u_n \to u$ in $L^2$, is also $u \in L^\infty$?
We know that for a.a $x$, $u_n(x)$ is bounded uniformly in $n$. Does it not follow that $u(x)$ is also bounded?

REPLY [1 votes]: It follows indeed, because there is a subsequence $(u_{n_k})_k$ such that $u_{n_k}(x) \rightarrow u(x)$ almost everywhere, i.e. for all $x \in X\setminus N$, where $N$ is of measure zero (that is a standard result on Lebesgue spaces).
Let $N_n \subset X$ be a null-set with $|u_n(x)| \leq \Vert u_n \Vert_\infty \leq M$ for all $x \in X\setminus N_n$.
Conclude $|u(x)| \leq M$ for all $x \in X\setminus (N \cup \bigcup_{n \in \Bbb{N} N_n})$, i.e. $|u(x)| \leq M$ a.e.