# Maximal pronilfactors and a topological Wiener-Wintner theorem

Yonatan Gutman & Zhengxing Lian  To Benjamin Weiss with great respect.

###### Abstract.

For strictly ergodic systems, we introduce the class of CF-Nil($k$) systems:
systems for which the maximal measurable and maximal topological $k$-step
pronilfactors coincide as measure-preserving systems. Weiss’ theorem implies
that such systems are abundant in a precise sense. We show that the CF-
Nil$(k)$ systems are precisely the class of minimal systems for which the
$k$-step nilsequence version of the Wiener-Wintner average converges
everywhere. As part of the proof we establish that pronilsystems are
coalescent both in the measurable and topological categories. In addition, we
characterize a CF-Nil$(k)$ system in terms of its $(k+1)$-th dynamical
cubespace. In particular, for $k=1$, this provides for strictly ergodic
systems a new condition equivalent to the property that every measurable
eigenfunction has a continuous version.

††The authors were partially supported by the National Science Centre (Poland)
grant 2016/22/E/ST1/00448. Y.G. was partially supported by the National
Science Centre (Poland) grant 2020/39/B/ST1/02329. Z.L. was partially
supported by the Xiamen Youth Innovation Foundation No. 3502Z20206037; the
presidential research fund of Xiamen University No. 20720210034 and NNSF of
China No. 1210010472.††Keywords: coalescence; cubespace; nilsequence; maximal
pronilfactor; strictly ergodic; topological model; universality; topological
Wiener-Wintner theorem.††Mathematics Subject Classification (2020): 37A05,
37B05.

###### Contents

  1. 1 Introduction.
  2. 2 Preliminaries.
    1. 2.1 Dynamical systems.
    2. 2.2 Topological models.
    3. 2.3 Conditional expectation.
    4. 2.4 Pronilsystems and nilsequences.
    5. 2.5 Host-Kra structure theory machinery.
    6. 2.6 Maximal measurable pronilfactors.
    7. 2.7 Maximal topological pronilfactors.
    8. 2.8 CF-Nil$(k)$ systems.
    9. 2.9 A CF-Nil$(k)$ topological model.
  3. 3 Coalescence and universality for maximal pronilfactors.
    1. 3.1 Coalescence
    2. 3.2 Universality
  4. 4 Cubespace characterization of CF-Nil($k$).
  5. 5 A topological Wiener-Wintner theorem.

##  1\. Introduction.

In recent years there has been an increase in interest in pronilfactors both
for measure-preserving systems (m.p.s.) and topological dynamical systems
(t.d.s.). Pronilfactors of a given system are either measurable or topological
(depending on the category) factors given by an inverse limit of nilsystems. A
t.d.s. (m.p.s.) is called a topological (measurable) $d$-step pronilsystem if
it is a topological (measurable) inverse limit of nilsystems of degree at most
$d$.111 It is a classical fact that every (measurable) ergodic $d$-step
pronilsystem is isomorphic as m.p.s. to a (topological) minimal $d$-step
pronilsystem. In the theory of measure preserving systems
$(X,\mathcal{X},\mu,T)$ maximal measurable pronilfactors appear in connection
with the $L^{2}$-convergence of the nonconventional ergodic averages

(1) $\frac{1}{N}\sum f_{1}(T^{n}x)\ldots f_{k}(T^{kn}x)$

for $f_{1},\ldots,f_{k}\in L^{\infty}(X,\mu)$ ([HK05, Zie07]). In the theory
of topological dynamical systems maximal topological pronilfactors appear in
connection with the higher order regionally proximal relations ([HKM10, SY12,
GGY18]).

When a system possesses both measurable and topological structure, it seems
worthwhile to investigate pronilfactors both from a measurable and topological
point of view. A natural meeting ground are strictly ergodic systems - minimal
topological dynamical systems $(X,T)$ possessing a unique invariant measure
$\mu$. For $k\in\mathbb{Z}$ let us denote by
$(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$ respectively $(W_{k}(X),T)$ the
maximal $k$-step measurable respectively topological pronilfactor222Both these
objects exist and are unique in a precise sense. See Subsection 3.2. of
$(X,T)$. Clearly $(W_{k}(X),T)$ has a unique invariant measure $\omega_{k}$.
We thus pose the question when is $(W_{k}(X),\mathcal{W}_{k}(X),\omega_{k},T)$
isomorphic to $(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$ as m.p.s.? We call a
t.d.s. which is strictly ergodic and for which
$(W_{k}(X),\mathcal{W}_{k}(X),\omega_{k},T)$ is isomorphic to
$(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$ as m.p.s., a CF-Nil$(k)$
system333This terminology is explained in Subsection 2.8.. Note that
$(W_{k}(X),\mathcal{W}_{k}(X),\omega_{k},T)$ is always a measurable factor of
$(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$. At first glance it may seem that
CF-Nil$(k)$ systems are rare however a theorem by Benjamin Weiss regarding
topological models for measurable extensions implies that every ergodic m.p.s.
is measurably isomorphic to a CF-Nil$(k)$ system444See Subsection 2.9..

We give two characterizations of CF-Nil$(k)$ systems. The first
characterization is related to the Wiener-Wintner theorem while the second
characterization is related to $k$-cube uniquely ergodic systems - a class of
topological dynamical systems introduced in [GL19].

The Wiener-Wintner theorem ([WW41]) states that for an ergodic system
$(X,\mathcal{X},\mu,T)$, for $\mu$-a.e. $x\in X$, any
$\lambda\in\mathbb{S}^{1}$ and any $f\in L^{\infty}(\mu)$, the following limit
exists:

(2) $\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\lambda^{n}f(T^{n}x)$

Denote by $M_{T}\subset\mathbb{S}^{1}$ the set of measurable
eigenvalues555Measurable and topological eigenvalues are defined in Subsection
2.1. of $(X,\mathcal{X},\mu,T)$. Let $P_{\lambda}f$ be the projection of $f$
to the eigenspace corresponding to $\lambda$ (in particular for $\lambda\notin
M_{T}$, $P_{\lambda}f\equiv 0$). For fixed $\lambda\in\mathbb{S}^{1}$, one can
show (2) converges a.s. to $P_{\lambda}f$.

In [Les96] Lesigne proved that a.s. convergence in (2) still holds when the
term $\lambda^{n}$ is replaced by a (continuous function) of a real-valued
polynomial $P(n)$, $P\in\mathbb{R}[t]$. In [Fra06] Frantzikinakis established
a uniform version666In the context of the Wiener-Wintner theorem, uniform
versions are a.s. convergence results involving a supremum over weights
belonging to a given class. The first result of this type was obtained by
Bourgain in [Bou90]. of this theorem. In [HK09], Host and Kra showed that a.s.
convergence in (2) still holds when the term $\lambda^{n}$ is replaced by a
nilsequence. In [EZK13] Eisner and Zorin-Kranich established a uniform version
of this theorem.

For topological dynamical systems one may investigate the question of
everywhere convergence in the Wiener-Wintner theorem. In [Rob94], Robinson
proved that for an uniquely ergodic system $(X,\mu,T)$, for any $f\in C(X)$,
if every measurable eigenfunction of $(X,\mathcal{X},\mu,T)$ has a continuous
version then the limit (2) converges everywhere. He noted however that if
$P_{\lambda}f\neq 0$ for some $\lambda\in M_{T}$, then the convergence of (2)
is not uniform in $(x,\lambda)$, since the limit function $P_{\lambda}f(x)$ is
not continuous on $X\times\mathbb{S}^{1}$.777Note $M_{T}$ is countable.
Moreover Robinson constructed a strictly ergodic system $(X,T)$ such that (2)
does not converge for some continuous function $f\in C(X)$, some
$\lambda\in\mathbb{C}$ and some $x\in X$. Other topological versions of the
Wiener-Wintner theorem may be found in [Ass92, Fan18]888One should also note
that topological Wiener-Winter theorems have been investigated in the
generality of operator semigroups by Schreiber and Bartoszek and Śpiewak
([Sch14, BŚ17])..

The first main result of this article is the following theorem:

######  Theorem A.

Let $(X,T)$ be a minimal system. Then for $k\geq 0$ the following are
equivalent:

  * (I).

$(X,T)$ is a CF-Nil$(k)$ system.

  * (II).

For any $k$-step nilsequence $\\{a(n)\\}_{n\in\mathbb{Z}}$, any continuous
function $f\in C(X)$ and any $x\in X$,

(3) $\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}a(n)f(T^{n}x)$

exists.

We remark that the direction (I)$\Rightarrow$(II) of Theorem A follows from
[HK09] whereas the case $k=1$ of Theorem A follows from [Rob94, Theorem 1.1].

As part of the proof of Theorem A we established a fundamental property for
pronilsystems:

######  Theorem B.

Let $(Y,\nu,T)$ be a minimal (uniquely ergodic) $k$-step pronilsystem. Then

  * (I).

$(Y,\nu,T)$ is measurably coalescent, i.e. if
$\pi:(Y,\nu,T)\rightarrow(Y,\nu,T)$ is a measurable factor map, then $\pi$ is
a measurable isomorphism.

and

  * (II).

$(Y,T)$ is topologically coalescent, i.e. if $\Phi:(Y,T)\rightarrow(Y,T)$ is a
topological factor map, then $\Phi$ is a topological isomorphism.

As part of the the theory of higher order regionally proximal relations, Host,
Kra and Maass introduced in [HKM10] the dynamical cubespaces
$\operatorname{C}_{\operatorname{T}}^{n}(X)\subset X^{2^{n}}$,
$n\in\mathbb{N}:=\\{1,2,\ldots\\}$. These compact sets enjoy a natural action
by the Host-Kra cube groups $\mathcal{HK}^{n}(T)$. According to the
terminology introduced in [GL19], a t.d.s. $(X,T)$ is called $k$-cube uniquely
ergodic if $(\operatorname{C}_{\operatorname{T}}^{k}(X),\mathcal{HK}^{k}(T))$
is uniquely ergodic. The third main result of this article is the following
theorem:

######  Theorem C.

Let $(X,T)$ be a minimal t.d.s. Then the following are equivalent for any
$k\geq 0$:

  * (I).

$(X,T)$ is a CF-Nil$(k)$ system.

  * (II).

$(X,T)$ is $(k+1)$-cube uniquely ergodic.

We remark that the direction (I) $\Rightarrow$ (II) follows from [HSY17].

In the context of various classes of strictly ergodic systems, several authors
have investigated the question of whether every measurable eigenfunction has a
continuous version. Famously in [Hos86] (see also [Que10, Page 170]), Host
established this is the case for admissible substitution dynamical systems. In
[BDM10, Theorem 27] an affirmative answer was given for strictly ergodic
Toeplitz type systems of finite rank. In [DFM19], the continuous and
measurable eigenvalues of minimal Cantor systems were studied.

It is easy to see that for strictly ergodic systems $(X,T)$ the condition that
every measurable eigenfunction has a continuous version is equivalent to the
fact that $(X,T)$ is CF-Nil($1$). Thus Theorem C provides for strictly ergodic
systems a new condition equivalent to the property that every measurable
eigenfunction has a continuous version. Namely this holds iff
$(\operatorname{C}_{\operatorname{T}}^{2}(X),\mathcal{HK}^{2}(T))$ is uniquely
ergodic. As the last condition seems quite manageable one wonders if this new
equivalence may turn out to be useful in future applications.

Structure of the paper. In Subsections 2.1–2.3 we review some definitions and
classical facts; In Subsections 2.4–2.8, we introduce the topological and
measurable maximal pronilfactors and define the CF-Nil$(k)$ systems; In
Subsection 2.9, we use Weiss’s Theorem to show that the CF-Nil$(k)$ systems
are abundant; In Section 3, we prove Theorem B and then establish universality
for maximal pronilfactors; In Section 4, we prove Theorem C; In Section 5, we
prove Theorem A.

Acknowledgements. We are grateful to Bernard Host, Mariusz Lemańczyk and and
anonymous referee for helpful comments.

##  2\. Preliminaries.

###  2.1. Dynamical systems.

Throughout this article we assume every topological space to be metrizable. A
$\mathbb{Z}$-topological dynamical system (t.d.s.) is a pair $(X,T)$, where
$X$ is a compact space and $T$ is a homeomorphism on $X$. Denote by $C(X)$ the
set of real-valued continuous functions on $X$. The orbit $\mathcal{O}(x)$ of
$x\in X$ is the set $\mathcal{O}(x)=\\{T^{n}x:n\in\mathbb{Z}\\}$. Its closure
is denoted by $\operatorname{\overline{\mathcal{O}}}(x)$ A t.d.s. is minimal
if $\operatorname{\overline{\mathcal{O}}}(x)=X$ for all $x\in X$. A t.d.s.
$(X,T)$ is distal if for a compatible metric $d_{X}$ of $X$, for any $x\neq
y\in X$, $\inf_{n\in\mathbb{Z}}d_{X}(T^{n}x,T^{n}y)>0$. We say
$\pi:(Y,S)\rightarrow(X,T)$ is a topological factor map if $\pi$ is a
continuous and surjective map such that for any $x\in X$, $\pi(Sx)=T\pi(x)$.
Given such a map, $(X,T)$ is called a topological factor of $(Y,S)$ and
$(X,T)$ is said to factor continuously on $(Y,S)$. If in addition $\pi$ is
injective then it is called a topological isomorphism and $(Y,S)$ and $(X,T)$
are said to be isomorphic as t.d.s. A factor map $\pi:(Y,S)\rightarrow(X,T)$
is called a topological group extension by a compact group $K$ if there exists
a continuous action $\alpha:K\times Y\rightarrow Y$ such that the actions $S$
and $K$ commute and for all $x,y\in Y$, $\pi(x)=\pi(y)$ iff there exists a
unique $k\in K$ such that $kx=y$. A (topological) eigenvalue of a t.d.s.
$(X,T)$ is a complex number $\lambda\in\mathbb{S}^{1}$ such that an equation
of the form $f(Tx)=\lambda f(x)$ holds for some $f\in C(X,\mathbb{C})$ and all
$x\in X$. The function $f$ is referred to as a continuous or topological
eigenfunction.

Let $\\{(X_{m},T_{m})\\}_{m\in\mathbb{N}}$ be a sequence of t.d.s. and for any
$m\geq n$, $\pi_{m,n}:(X_{n},T_{n})\rightarrow(X_{m},T_{m})$ factor maps such
that $\pi_{i,l}=\pi_{i,j}\circ\pi_{j,l}\text{ for all }1\leq i\leq j\leq l.$
The inverse limit of $\\{(X_{m},T_{m})\\}_{m\in\mathbb{N}}$ is defined to be
the system $(X,T)$, where

$X=\\{(x_{m})_{m\in\mathbb{N}}\in\prod_{m\in\mathbb{N}}X_{m}:\
\pi_{m+1}(x_{m+1})=x_{m}\text{ for }m\geq 1\\}$

equipped with the product topology and
$T(x_{m})_{m\in\mathbb{N}}\triangleq(T_{m}x_{m})_{m\in\mathbb{N}}$. We write
$(X,T)=\underleftarrow{\lim}(X_{m},T_{m})$.

A measure preserving probability system (m.p.s.) is a quadruple
$(X,\mathcal{X},\mu,T)$, where $(X,\mathcal{X},\mu)$ is a standard Borel
probability space (in particular $X$ is a Polish space and $\mathcal{X}$ is
its Borel $\sigma$-algebra) and $T$ is an invertible Borel measure-preserving
map ($\mu(TA)=\mu(A)$ for all $A\in\mathcal{X}$). An m.p.s.
$(X,\mathcal{X},\mu,T)$ is ergodic if for every set $A\in\mathcal{X}$ such
that $T(A)=A$, one has $\mu(A)=0$ or $1$. A measurable factor map is a Borel
map $\pi:(X,\mathcal{X},\mu,T)\rightarrow(Y,\mathcal{Y},\nu,S)$ which is
induced by a $G$-invariant sub-$\sigma$-algebra of $\mathcal{X}$ ([Gla03,
Chapter 2.2]). Given such a map, $(Y,\mathcal{Y},\nu,S)$ is called a
measurable factor of $(X,\mathcal{X},\mu,T)$. If $\pi$ is in addition
invertible on a set of full measure then $\pi$ is called a measurable
isomorphism and $(X,\mathcal{X},\mu,T)$ and $(Y,\mathcal{Y},\nu,S)$ are said
to be isomorphic as m.p.s. Let $(Y,\mathcal{Y},\nu,S)$ be an m.p.s. and $A$ a
compact group with Borel $\sigma$-algebra $\mathcal{A}$ and Haar measure $m$.
A skew-product $(Y\times A,\mathcal{Y}\otimes\mathcal{A},\nu\times m,T)$ is
given by the action $T(y,u)=(Sy,\beta(y)u)$, where $\beta:Y\rightarrow A$ is a
Borel map, the so-called cocycle of the skew-product. The projection $(Y\times
A,\mathcal{Y}\otimes\mathcal{A},\nu\times
m,T)\rightarrow(Y,\mathcal{Y},\nu,S)$ given by $(y,a)\mapsto y$ is called a
measurable group extension (cf. [Gla03, Theorem 3.29]).

A (measurable) eigenvalue of a m.p.s. $(X,\mathcal{X},\mu,T)$ is a complex
number $\lambda\in\mathbb{S}^{1}$ such that an equation of the form
$f(Tx)=\lambda f(x)$ holds for $\mu$-a.e. $x\in X$ for some Borel function
$f:X\rightarrow\mathbb{C}$. The function $f$ is referred to as a measurable
eigenfunction.

Denote by $\operatorname{P_{T}}(X)$ the set of $T$-invariant Borel probability
measures of $X$. A t.d.s. $(X,T)$ is called uniquely ergodic if
$|\operatorname{P_{T}}(X)|=1$. If in addition it is minimal then it is called
strictly ergodic. For a strictly ergodic system $(X,T)$ with a (unique)
invariant measure $\mu$, we will use the notation $(X,\mu,T)$. When considered
as a m.p.s. it is with respect to its Borel $\sigma$-algebra.

Occasionally in this article we will consider more general group actions than
$\mathbb{Z}$-actions. Thus a $G$-topological dynamical system (t.d.s.) is a
pair $(G,X)$ consisting of a (metrizable) topological group $G$ acting on a
(metrizable) compact space $X$. For $g\in G$ and $x\in X$ we denote the action
both by $gx$ and $g.x$. We will need the following proposition:

######  Proposition 2.1.

Let $G$ be an amenable group. Let $(G,X)$ be uniquely ergodic and let
$(G,X)\rightarrow(G,Y)$ be a topological factor map. Then $(G,Y)$ is uniquely
ergodic.

###### Proof.

See proof of Proposition 8.1 of [AKL14]. ∎

###  2.2. Topological models.

######  Definition 2.2.

Let $(X,\mathcal{X},\mu,T)$ be a m.p.s. We say that a t.d.s.
$(\hat{X},\hat{T})$ is a topological model for $(X,\mathcal{X},\mu,T)$ w.r.t.
to a $\hat{T}-$invariant probability measure $\hat{\mu}$ on
$\hat{\mathcal{X}}$, the Borel $\sigma$-algebra of $X$, if the system
$(X,\mathcal{X},\mu,T)$ is isomorphic to
$(\hat{X},\hat{\mathcal{X}},\hat{\mu},\hat{T})$ as m.p.s., that is, there
exist a $T$-invariant Borel subset $C\subset X$ and a $\hat{T}$-invariant
Borel subset $\hat{C}\subset\hat{X}$ of full measure and a (bi)measurable and
equivariant measure preserving bijective Borel map $p:C\rightarrow\hat{C}$.
Notice that oftentimes in this article $(\hat{X},\hat{T})$ will be uniquely
ergodic so that $\hat{\mu}$ will be the unique $\hat{T}-$invariant probability
measure of $X$.

######  Definition 2.3.

Let $(X,\mathcal{X},\mu,T)$, $(Y,\mathcal{Y},\nu,S)$ be m.p.s. Let
$(\hat{X},\hat{T})$, $(\hat{Y},\hat{S})$ be t.d.s. which are topological
models of $(X,\mathcal{X},\mu,T)$ and $(Y,\mathcal{Y},\nu,S)$ w.r.t. measures
$\hat{\mu}$ and $\hat{\nu}$ as witnessed by maps $\phi$ and $\psi$
respectively. We say that
$\hat{\pi}:(\hat{X},\hat{T})\rightarrow(\hat{Y},\hat{S})$ is a topological
model for a factor map
$\pi:(X,\mathcal{X},\mu,T)\rightarrow(Y,\mathcal{Y},\nu,S)$ if $\hat{\pi}$ is
a topological factor and the following diagram

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

is commutative, i.e. $\hat{\pi}\phi=\psi\pi$

###  2.3. Conditional expectation.

Let $(X,\mathcal{X},\mu)$ be a probability space and let $\mathcal{B}$ be a
sub-$\sigma$-algebra of $\mathcal{X}$. For $f\in L^{1}(\mu)$, the conditional
expectation of $f$ w.r.t. $\mathcal{B}$ is the unique function
$\mathbb{E}(f|\mathcal{B})\in L^{1}(X,\mathcal{B},\mu)$ satisfying

(4) $\int_{B}fd\mu=\int_{B}\mathbb{E}(f|\mathcal{B})d\mu$

for every $B\in\mathcal{B}$. For $f\in L^{1}(\mu)$ and $g\in
L^{\infty}(X,\mathcal{B},\mu)$, it holds (see [HK18, Chapter 2, Section 2.4]):

(5) $\int_{X}fgd\mu=\int_{X}\mathbb{E}(f|\mathcal{B})gd\mu.$

Let $(X,\mathcal{X},\mu)$ and $(Y,\mathcal{Y},\nu)$ be probability spaces and
let $\pi:X\rightarrow Y$ be a measurable map such that $\pi_{*}\mu=\nu$.
Denote by $\mathbb{E}(f|Y)\in L^{1}(Y,\nu)$ the function such that
$\mathbb{E}(f|Y)=\mathbb{E}(f|\pi^{-1}(\mathcal{Y}))\circ\pi^{-1}$. Note this
is well-defined. Thus the difference between $\mathbb{E}(f|Y)$ and
$\mathbb{E}(f|\pi^{-1}(\mathcal{Y}))$ is that the first function is considered
as a function on $Y$ and the second as a function on $X$.

###  2.4. Pronilsystems and nilsequences.

######  Definition 2.4.

A (real) Lie group is a group that is also a finite dimensional real smooth
manifold such that the group operations of multiplication and inversion are
smooth. Let $G$ be a Lie group. Let $G_{1}=G$ and $G_{k}=[G_{k-1},G]$ for
$k\geq 2$, where $[G,H]=\\{[g,h]:g\in G,h\in H\\}$ and $[g,h]=g^{-1}h^{-1}gh$.
If there exists some $d\geq 1$ such that $G_{d+1}=\\{e\\}$, $G$ is called a
$d$-step nilpotent Lie group. We say that a discrete subgroup $\Gamma$ of a
Lie group $G$ is cocompact if $G/\Gamma$, endowed with the quotient topology,
is compact. We say that quotient $X=G/\Gamma$ is a $d$-step nilmanifold if $G$
is a $d$-step nilpotent Lie group and $\Gamma$ is a discrete, cocompact
subgroup. The nilmanifold $X$ admits a natural action by $G$ through
translations $g.a\Gamma=ga\Gamma$, $g,a\in G$. The Haar measure of $X$ is the
unique Borel probability measure on $X$ which is invariant under this action.
A nilsystem of degree at most $d$, $(X,T)$, is given by an $d$-step
nilmanifold $X=G/\Gamma$ and $T\in G$ with action $T.a\Gamma=Ta\Gamma$. When a
nilsystem is considered as a m.p.s. it is always w.r.t. its Haar measure.

######  Definition 2.5.

A t.d.s. (m.p.s) is called a topological (measurable) $d$-step pronilsystem if
it is a topological (measurable) inverse limit of nilsystems of degree at most
$d$. By convention a $0$-step pronilsystem is the one-point trivial system.

######  Remark 2.6.

By [HK18, p. 233] if an ergodic measurable $d$-step pronilsystem is presented
as the inverse limit
$(X,\mathcal{X},\nu,T)=\underleftarrow{\lim}(X_{m},\mathcal{X}_{m},\nu_{m},T_{m})$
given by the measurable factor maps
$\pi_{m}:(X_{m},\mathcal{X}_{m},\nu_{m},T_{m})\rightarrow(X_{m-1},\mathcal{X}_{m-1},\nu_{m-1},T_{m-1})$
between nilsystems of degree at most $d$ then there exist topological factor
maps $\tilde{\pi}_{m}:(X_{m},T_{m})\rightarrow(X_{m-1},T_{m-1})$ such that
$\tilde{\pi}=\pi$ $\nu_{m}$-a.e. and so effectively one can consider
$(X,\mathcal{X},\nu,T)$ as a (minimal) topological pronilsystem. Moreover any
two $d$-step pronilsystem topological models of $(X,\mathcal{X},\nu,T)$ are
isomorphic as t.d.s. (Theorem 3.3).

######  Definition 2.7.

([HKM10, Definition 2.2]) A bounded sequence $\\{a(n)\\}_{n\in\mathbb{Z}}$ is
called a $d$-step nilsequence if there exists a $d$-step pronilsystem $(X,T)$,
$x_{0}\in X$ and a continuous function $f\in C(X)$ such that
$a(n)=f(T^{n}x_{0})$ for $n\in\mathbb{Z}$.

######  Theorem 2.8.

([HK09, Theorem 3.1]) Let $(X,T)$ be a nilsystem. Then $(X,T)$ is uniquely
ergodic if and only if $(X,T)$ is ergodic w.r.t. the Haar measure if and only
if $(X,T)$ is minimal.

The following proposition is an immediate corollary of the previous theorem.

######  Proposition 2.9.

Let $(X,T)$ be a pronilsystem. Then $(X,T)$ is uniquely ergodic if and only if
$(X,T)$ is minimal.

######  Definition 2.10.

Let $(X,\mu,T)$ be a strictly ergodic t.d.s. We say that a t.d.s. $(Y,T)$ is a
topological $k$-step pronilfactor of $(X,T)$ if it is a topological factor of
$(X,T)$ and if it is isomorphic to a $k$-step pronilsystem as a t.d.s. We say
that a m.p.s. $(Y,\mathcal{Y},\nu,T)$ is a measurable $k$-step pronilfactor of
$(X,T)$ if it is a measurable factor of $(X,\mathcal{X},\mu,T)$ and if it is
isomorphic to a $k$-step pronilsystem as a m.p.s.

###  2.5. Host-Kra structure theory machinery.

By a face of the discrete cube $\\{0,1\\}^{k}$ we mean a subcube obtained by
fixing some subset of the coordinates. For $k\in\mathbb{N}$, let
$[k]=\\{0,1\\}^{k}$. Thus $X^{[k]}=X\times\cdots\times X$, $2^{k}$ times and
similarly $T^{[k]}=T\times\cdots\times T$, $2^{k}$ times. For $x\in X$,
$x^{[k]}=(x,\ldots,x)\in X^{[k]}$. Let
$[k]_{*}=\\{0,1\\}^{k}\setminus\\{\vec{0}\\}$ and define
$X_{*}^{[k]}=X^{[k]_{*}}$.

######  Definition 2.11.

([HK05]) Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s. For $1\leq j\leq k$,
let $\overline{\alpha}_{j}=\\{v\in\\{0,1\\}^{k}:v_{j}=1\\}$ be the $j$-th
upper face of $\\{0,1\\}^{k}$. For any face $F\subset\\{0,1\\}^{k}$, define

$(T^{F})_{v}=\begin{cases}T&v\in F\\\ \operatorname{Id}&v\notin F.\end{cases}$

Define the face group $\mathcal{F}^{k}(T)\subset\operatorname{Homeo}(X^{[k]})$
to be the group generated by the elements $\\{T^{\overline{\alpha}_{j}}:1\leq
j\leq k\\}$. Define the the $k$-th Host-Kra cube group $\mathcal{HK}^{k}(T)$
to be the subgroup of $\operatorname{Homeo}(X^{[k]})$ generated by
$\mathcal{F}^{k}(T)$ and $T^{[k]}$.

######  Definition 2.12.

([HK05]) Let $(X,\mathcal{B},\mu,T)$ be an ergodic m.p.s. Let
$\mu^{[1]}=\mu\times\mu$. For $k\in\mathbb{N}$, let $\mathcal{I}_{T^{[k]}}$ be
the $T^{[k]}$-invariant $\sigma$-algebra of
$(X^{[k]},\mathcal{X}^{[k]},\mu^{[k]})$. Define $\mu^{[k+1]}$ to be the
relative independent joining of two copies of $\mu^{[k]}$ over
$\mathcal{I}_{T^{[k]}}$. That is, for $f_{v}\in L^{\infty}(\mu)$,
$v\in\\{0,1\\}^{k+1}$:

$\int_{X^{[k+1]}}\prod_{v\in\\{0,1\\}^{k+1}}f_{v}(x_{v})d\mu^{[k+1]}(x)=\\\
\int_{X^{[k]}}\mathbb{{E}}(\prod_{v\in\\{0,1\\}^{k}}f_{v0}|\mathcal{I}_{T^{[k]}})(x)\mathbb{{E}}(\prod_{v\in\\{0,1\\}^{k}}f_{v1}|\mathcal{I}_{T^{[k]}})(x)d\mu^{[k]}(x).$

In particular, from Equation (5), it follows that for all measurable functions
$H_{1},H_{2}\in L^{\infty}(X^{[k]},\mu^{[k]})$,

(6)
$\int_{X^{[k]}}\mathbb{{E}}(H_{1}|\mathcal{I}_{T^{[k]}})(c)\mathbb{{E}}(H_{2}|\mathcal{I}_{T^{[k]}})(c)d\mu^{[k]}(c)=\\\
\int_{X^{[k]}}\mathbb{{E}}(H_{1}|\mathcal{I}_{T^{[k]}})(c)H_{2}(c)d\mu^{[k]}(c).$

Note $\mu^{[k]}$ is $\mathcal{HK}^{k}(T)$-invariant ([HK18, Chapter 9,
Proposition 2]).

######  Definition 2.13.

[HK18, Chapter 9, Section 1] For $k\in\mathbb{N}$, let $\mathcal{J}_{*}^{k}$
be the $\sigma$-algebras of sets invariant under $\mathcal{F}^{k}(T)$ on
$X_{*}^{[k]}$.

######  Definition 2.14.

[HK18, Subsection 9.1] Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s. For
$k\in\mathbb{N}$, define $\mathcal{Z}_{k}(X)$ to be the $\sigma$-algebra
consisting of measurable sets $B$ such that there exists a
$\mathcal{J}_{*}^{k+1}$-measurable set $A\subset X_{*}^{[k+1]}$ so that up to
$\mu^{[k+1]}$\- measure zero it holds:

$X\times A=B\times X_{*}^{[k+1]}$

Define the $k$-th Host-Kra factor $Z_{k}(X)$ as the measurable factor of $X$
induced by $\mathcal{Z}_{k}(X)$ and denote by $\pi_{k}:X\rightarrow Z_{k}(X)$
the (measurable) canonical $k$-th projection. Let $\mu_{k}$ be the projection
of $\mu$ w.r.t. $\pi_{k}$.

######  Definition 2.15.

Let $(X,\mathcal{X},\mu,T)$ be an m.p.s. and $k\in\mathbb{N}$. The Host-Kra-
Gowers seminorms on $L^{\infty}(\mu)$ are defined as follows:

$|||f|||_{k}=(\int\prod_{v\in\\{0,1\\}^{k}}\mathcal{C}^{|v|}fd\mu^{[k]})^{1/2^{k}},$

where $|(v_{1},\ldots,v_{k+1})|=\Sigma_{i=1}^{k+1}v_{i}$ and
$\mathcal{C}^{n}z=z$ if $n$ is even and $\mathcal{C}^{n}z=\overline{z}$ if $n$
is odd. By [HK18, Subsection 8.3], $|||\cdot|||_{k}$ is a seminorm.

######  Lemma 2.16.

[HK18, Chapter 9, Theorem 7] Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s.
and $k\in\mathbb{N}$. Then for $f\in L^{\infty}(\mu)$, $|||f|||_{k+1}=0$ if
and only if $\mathbb{E}(f|\mathcal{Z}_{k}(X))=0$.

###  2.6. Maximal measurable pronilfactors.

######  Definition 2.17.

Let $k\in\mathbb{N}$. A m.p.s. $(X,\mathcal{X},\mu,T)$ is called a
(measurable) system of order $k$ if it is isomorphic to
$(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$.

######  Theorem 2.18.

([HK05, Theorem 10.1], [HK18, Chapter 16, Theorem 1], for an alternative proof
see [GL19, Theorem 5.3]) An ergodic m.p.s. is a system of order $k$ iff it is
isomorphic to a minimal $k$-step pronilsystem as m.p.s.

######  Remark 2.19.

Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s. In the literature
$(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$ is referred to as the maximal
measurable $k$-step pronilfactor or as the maximal factor which is a system of
order $k$ (see [HK18, Chapter 9, Theorem 18]). By this it is meant that any
measurable factor map
$\phi:(X,\mathcal{X},\mu,T)\rightarrow(Y,\mathcal{Y},\nu,S)$ where
$(Y,\mathcal{Y},\nu,S)$ is a minimal $k$-step pronilsystem, factors through
the canonical $k$-th projection
$\pi_{k}:(X,\mathcal{X},\mu,T)\rightarrow(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$,
i.e., there exists a unique (up to measure zero)
$\psi:(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)\rightarrow(Y,\mathcal{Y},\nu,S)$
such that $\phi=\psi\circ\pi_{k}$ a.s. In section 3 we establish the
complementary property of universality for
$(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$.

######  Remark 2.20.

In [HKM14, Corollary 2.2] a criterion for an ergodic m.p.s.
$(X,\mathcal{X},\mu,T)$ to have $Z_{k}(X)=Z_{1}(X)$ for all $k\geq 1$ is
given. Indeed this is the case for ergodic systems whose spectrum does not
admit a Lebesgue component with infinite multiplicity. In particular this
holds true for weakly mixing systems, systems with singular maximal spectral
type and systems with finite spectral multiplicity.

###  2.7. Maximal topological pronilfactors.

Recall the Definition of $\mathcal{HK}^{k}(T)$ and $\mathcal{F}^{k}(T)$
(Definition 2.11).

######  Definition 2.21.

Let $(X,T)$ be a minimal t.d.s. Define the induced $(k+1)$-th dynamical
cubespace by:

$\operatorname{C}_{\operatorname{T}}^{k+1}(X)=\overline{\\{gx^{[k+1]}|\,g\in\mathcal{HK}^{k+1}(T)\\}}.$

######  Definition 2.22.

([HKM10, Definition 3.2]) Let $(X,T)$ be a topological dynamical system and
$k\geq 1$. The points $x,y\in X$ are said to be regionally proximal of order
$k$, denoted $(x,y)\in\operatorname{RP}^{[k]}(X)$, if there are sequences of
elements $f_{i}\in\mathcal{F}^{k}(T)$, $x_{i},y_{i}\in X$, $a_{*}\in
X_{*}^{[k]}$ such that

$\lim_{i\rightarrow\infty}(f_{i}x_{i}^{[k]},f_{i}y_{i}^{[k]})=(x,a_{*},y,a_{*}).$

######  Theorem 2.23.

([SY12, Theorem 3.5]999This theorem was generalized to arbitrary minimal group
actions in [GGY18, Theorem 3.8].) Let $(X,T)$ be a minimal t.d.s. and $k\geq
1$. Then $\operatorname{RP}^{[k]}(X)$ is a closed $T$-invariant equivalence
relation.

######  Definition 2.24.

A t.d.s. $(X,T)$ is called a (topological) system of order $k$ if
$\operatorname{RP}^{[k]}(X)=\\{(x,x)\,|\,x\in X\\}$.

######  Theorem 2.25.

([HKM10, Theorem 1.2], for an alternative proof see [GMV20, Theorem 1.30]) A
minimal t.d.s. is a topological system of order $k$ iff it is isomorphic to a
minimal $k$-step pronilsystem as t.d.s.

Theorem 2.23 allows us to give the following definition.

######  Definition 2.26.

Let $(X,T)$ be a minimal t.d.s. Define the maximal $k$-step nilfactor by
$W_{k}(X)=X/\operatorname{RP}^{[k]}(X)$. Denote the associated map
$\operatorname{\pi_{k}^{top}}:X\rightarrow W_{k}(X)$ as the (topological)
canonical $k$-th projection.

######  Remark 2.27.

The terminology of Definition 2.26 is justified by the following property: Any
topological factor map $\phi:(X,T)\rightarrow(Y,T)$ where $(Y,T)$ is a system
of order $k$, factors through the canonical $k$-th projection
$\operatorname{\pi_{k}^{top}}:(X,T)\rightarrow(W_{k}(X),T)$, i.e., there
exists a unique $\psi:(W_{k}(X),T)\rightarrow(Y,T)$ such that
$\phi=\psi\circ\operatorname{\pi_{k}^{top}}$ ([HKM10, Proposition 4.5]). In
section 3 we establish the complementary property of universality for
$(W_{k}(X),T)$.

######  Definition 2.28.

([GL19, Definition 3.1]) A t.d.s. $(X,T)$ is called $k$-cube uniquely ergodic
if $(\operatorname{C}_{\operatorname{T}}^{k}(X),\mathcal{HK}^{k}(T))$ is
uniquely ergodic.

###  2.8. CF-Nil$(k)$ systems.

######  Definition 2.29.

For $k\geq 0$, we say $(X,T)$ is a CF-Nil($k$) system if $(X,T)$ is strictly
ergodic and $(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$ is isomorphic to
$(W_{k}(X),\omega_{k},T)$ as m.p.s.where $\mu_{k}$ and $\omega_{k}$ are the
images of the unique invariant measure of $(X,T)$ under the measurable,
respectably topological canonical $k$-th projections.

######  Remark 2.30.

By convention $Z_{0}(X)=W_{0}(X)=\\{\bullet\\}$. Thus every strictly ergodic
$(X,T)$ is CF-Nil($0$).

The term "$(X,\mu,T)$ is CF-Nil($k$)" is an abbreviation of

"$(X,\mu,T)$ Continuously Factors on a $\mathbf{k}$-step proNilsystem which is
isomorphic to $(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$ as m.p.s."

Indeed if $(W_{k}(X),\omega_{k},T)$ is isomorphic to
$(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$ as m.p.s. then obviously this
condition holds. The reverse implication is given by the following proposition
which has been (implicitly) used several times in the literature ([HK09,
HKM14, HSY19]). Its proof is given at the end of Subsection 3.2.

######  Proposition 2.31.

Let $(X,T)$ be a strictly ergodic t.d.s. which topologically factors on a
(minimal) $k$-step pronilsystem $(\hat{Z}_{k},T)$ with the unique ergodic
measure $\gamma_{k}$. If $(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$ is
isomorphic to $(\hat{Z}_{k},\gamma_{k},T)$ as m.p.s., then $(\hat{Z}_{k},T)$
and $(W_{k}(X),T)$ are isomorphic as t.d.s. In particular $(X,\mu,T)$ is CF-
Nil($k$).

Theorem C allows us to give a remarkable simple proof of the following
Theorem.

######  Theorem 2.32.

Let $(X,T)$ be a CF-Nil$(k)$ system. The following holds:

  1. (1)

If $\pi:(X,T)\rightarrow(Y,T)$ is a topological factor map, then $(Y,T)$ is a
CF-Nil$(k)$ system.

  2. (2)

$(X,T)$ is a CF-Nil($i$) system for $0\leq i\leq k$.

###### Proof.

To prove (1) we note $(Y,T)$ is minimal being a factor of a minimal system and
$(\operatorname{C}_{\operatorname{T}}^{k+1}(Y),\mathcal{HK}^{k+1}(T))$ is
uniquely ergodic being a factor of
$(\operatorname{C}_{\operatorname{T}}^{k+1}(X),\mathcal{HK}^{k+1}(T))$ under
the natural topological factor map induced from $\pi:(X,T)\rightarrow(Y,T)$
(see Proposition 2.1). By Theorem C this implies $(Y,T)$ is a CF-Nil$(k)$
system.

Similarly, to prove (2), we consider
$(\operatorname{C}_{\operatorname{T}}^{k+1}(X),\mathcal{HK}^{k+1}(T))\rightarrow(\operatorname{C}_{\operatorname{T}}^{i+1}(X),\mathcal{HK}^{i+1}(T))$
given by

$(c_{v_{1},\ldots,v_{k+1}})_{(v_{1},\ldots,v_{k+1})\in\\{0,1\\}^{k+1}}\mapsto(c_{v_{1},\ldots,v_{i+1},0,\ldots,0})_{(v_{1},\ldots,v_{i+1})\in\\{0,1\\}^{i+1}}$

∎

###  2.9. A CF-Nil$(k)$ topological model.

Recall the definitions of Subsection 2.2. In [Wei85, Theorem 2] Benjamin Weiss
proved the following theorem:

######  Theorem 2.33.

(Weiss) Let $(Z,\nu,S)$ be a strictly ergodic t.d.s. and
$(X,\mathcal{X},\mu,S)$ an ergodic m.p.s. such that there exists a measurable
factor $\pi:(X,\mathcal{X},\mu,T)\rightarrow(Z,\mathcal{Z},\nu,S)$. Then $\pi$
has a topological model $\hat{\pi}:(\hat{X},\hat{T})\rightarrow(Z,S)$ where
$(\hat{X},\hat{T})$ is strictly ergodic.

The following theorem is already implicit in [HSY19].

######  Theorem 2.34.

Let $k\in\mathbb{Z}$. Every ergodic system $(X,\mathcal{X},\mu,T)$ has a
topological model $(\hat{X},\hat{T})$ such that $(\hat{X},\hat{T})$ is CF-
Nil($k$).

###### Proof.

By Theorem 2.18, $(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$ is measurably
isomorphic to a strictly ergodic inverse limit of $k$-step nilsystems
$(\hat{Z}_{k},\hat{T})$. By Theorem 2.33, $(X,\mathcal{X},\mu,T)$ admits a
strictly ergodic topological model $(\hat{X},\hat{T})$ such that there exists
a topological factor map $(\hat{X},\hat{T})\rightarrow(\hat{Z}_{k},\hat{T})$
which is a topological model of
$(X,\mathcal{X},\mu,T)\rightarrow(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$. By
Proposition 2.31, $(\hat{X},\hat{T})$ is CF-Nil($k$).∎

######  Remark 2.35.

One can easily construct a strictly ergodic system which is not CF-Nil($k$).
Let $(X,\mathcal{X},\mu,T)$ be an irrational rotation on the circle. By
[Leh87], there exists a topologically mixing and strictly ergodic model
$(\hat{X},\hat{\mu},T)$ of $(X,\mu,T)$. As $X$ is an irrational rotation,
$Z_{1}(\hat{X})=\hat{X}$ and therefore for all $k\geq 1$,
$Z_{k}(\hat{X})=\hat{X}$. As $\hat{X}$ is topologically mixing, it is
topologically weakly mixing and therefore for all $k\geq 1$,
$W_{k}(\hat{X})=\\{\bullet\\}$ ([SY12, Theorem 3.13(1)]). It follows for all
$k\geq 1$ one has that $(W_{k}(\hat{X}),T)$ is not isomorphic to
$(Z_{k}(\hat{X}),\hat{\mu}_{1},T)$ as m.p.s.

##  3\. Coalescence and universality for maximal pronilfactors.

###  3.1. Coalescence

In this section we establish Theorem B, i.e., both topological coalescence
(introduced in [Aus63]) and measurable coalescence (introduced in [HP68]) for
minimal pronilsystems101010The definitions of these concepts appear as part of
the statements of Theorems 3.1 and 3.3 respectively.. There is a vast
literature dedicated to coalescence (see [LLT92] and references within).
Coalescence plays an important role in the next subsection.

######  Theorem 3.1.

(Topological coalescence for minimal pronilsystems) Let $(Y,T)$ be a minimal
$k$-step pronilsystem. Then $(Y,T)$ is topologically coalescent, i.e. if
$\Phi:(Y,T)\rightarrow(Y,T)$ is a topological factor map, then $\Phi$ is a
topological isomorphism.

###### Proof.

Recall that the Ellis semigroup is defined as
$E=E(Y,T)=\overline{\\{T^{n}:n\in\mathbb{Z}\\}}$, where the closure is w.r.t.
the product topology on $Y^{Y}$ (see [Ell58] for more details). By a theorem
of Donoso [Don14, Theorem 1.1], $E(Y,T)$ is a $k$-step nilpotent group, i.e.
for $E_{1}=E$, $E_{i+1}=[E_{i},E],i\geq 1$, one has that
$E_{k+1}=\\{\operatorname{Id}\\}$. As $\Phi$ is continuous, one has that $E$
and $\Phi$ commute, i.e. for any $g\in E$, $\Phi\circ g=g\circ\Phi$. For any
$z\in Y$, we define the group $\mathcal{G}(Y,z)=\\{\alpha\in E(Y,T),\alpha
z=z\\}$. Let $x,y\in Y$ such that $\Phi(x)=y$. If $u\in\mathcal{G}(Y,x)$, one
always has that $uy=u(\Phi(x))=\Phi(ux)=\Phi(x)=y$, i.e.
$u\in\mathcal{G}(Y,y)$. Thus $\mathcal{G}(Y,x)\subset\mathcal{G}(Y,y)$.

Assume that $\Phi$ is not one-to-one, then there exists $x_{1}\neq x_{2}\in Y$
such that $\Phi(x_{1})=\Phi(x_{2})$. As $(Y,T)$ is minimal, there exists
$p_{1},p_{2}\in E(Y,T)$ such that $x_{1}=p_{1}x$, $x_{2}=p_{2}x$. Then
$p_{1}y=\Phi(p_{1}x)=\Phi(x_{1})=\Phi(x_{2})=\Phi(p_{2}x)=p_{2}y$. Thus
$p_{1}^{-1}p_{2}\in\mathcal{G}(Y,y)$. As $p_{2}x=x_{2}\neq x_{1}=p_{1}x$, we
have

$p_{1}^{-1}p_{2}x\neq x,$

which implies that
$p_{1}^{-1}p_{2}\in\mathcal{G}(Y,y)\setminus\mathcal{G}(Y,x)$.

Let $\beta_{0}=p_{1}^{-1}p_{2}$. As $(Y,T)$ is minimal, there exists $u\in
E(Y,T)$ such that $ux=y$. Then $\mathcal{G}(Y,x)=u^{-1}\mathcal{G}(Y,y)u$. Let
$\beta_{1}=(u^{-1}\beta_{0}^{-1}u)\beta_{0}$. As
$\beta_{0}\in\mathcal{G}(Y,y)\setminus\mathcal{G}(Y,x)$, one has that

(7) $\beta_{0}\notin\mathcal{G}(Y,x),\beta_{0}\in\mathcal{G}(Y,y)\text{ and
}(u^{-1}\beta_{0}^{-1}u)\in
u^{-1}\mathcal{G}(Y,y)u=\mathcal{G}(Y,x)\subset\mathcal{G}(Y,y).$

Thus we can show that $\beta_{1}\in\mathcal{G}(Y,y)\setminus\mathcal{G}(Y,x)$.
Indeed, by (7) we know that
$\beta_{1}=(u^{-1}\beta_{0}^{-1}u)\beta_{0}\in\mathcal{G}(Y,y)$ as
$\mathcal{G}(Y,y)$ is a group. If $\beta_{1}\in\mathcal{G}(Y,x)$, then
$\beta_{0}=(u^{-1}\beta_{0}^{-1}u)^{-1}\beta_{1}\in\mathcal{G}(Y,x)$, which
constitutes a contradiction. Therefore
$\beta_{1}\in\mathcal{G}(Y,y)\setminus\mathcal{G}(Y,x)$ and
$(u^{-1}\beta_{1}^{-1}u)\in u^{-1}\mathcal{G}(Y,y)u=\mathcal{G}(Y,x)$.

Similarly, we define $\beta_{i+1}=(u^{-1}\beta_{i}^{-1}u)\beta_{i}$ for $i\geq
1$. By the same argument, one has that
$\beta_{i+1}\in\mathcal{G}(Y,y)\setminus\mathcal{G}(Y,x)$. But notice that
$\beta_{i}\in E_{i+1}$ and $E_{k+1}=\\{\operatorname{Id}\\}$, therefore
$\operatorname{Id}=\beta_{k}\in\mathcal{G}(Y,y)\setminus\mathcal{G}(Y,x)$.
Contradiction.

Thus $\Phi$ is a one-to-one topological factor map, which implies it is a
topological isomorphism. ∎

######  Proposition 3.2.

[HK18, Chapter 13, Proposition 15] Let $(Y,\nu,T)$,
$(Y^{\prime},\nu^{\prime},T)$ be minimal (uniquely ergodic) $k$-step
pronilsystems. Let $\pi:(Y,\nu,T)\rightarrow(Y^{\prime},\nu^{\prime},T)$ be a
measurable factor map. Then there exists a topological factor map
$\hat{\pi}:(Y,T)\rightarrow(Y^{\prime},T)$ such that $\pi(y)=\hat{\pi}(y)$ for
$\nu$-a.e. $y$.

Combining Theorem 3.1 and Proposition 3.2 we immediately have the following
theorem.

######  Theorem 3.3.

(Measurable coalescence for minimal pronilsystems) Let $(Y,\nu,T)$ be a
minimal (uniquely ergodic) $k$-step pronilsystem. Then $(Y,\nu,T)$ is
measurably coalescent, i.e. if $\pi:(Y,\nu,T)\rightarrow(Y,\nu,T)$ is a
measurable factor map, then $\pi$ is a measurable isomorphism (which equals
a.s. a topological isomorphism).

###### Proof.

By Proposition 3.2, there exists a topological factor map
$\hat{\pi}:(Y,\nu,T)\rightarrow(Y,\nu,T)$ such that $\pi(y)=\hat{\pi}(y)$ for
$\nu$-a.e. $y\in Y$. By Theorem 3.1, $\hat{\pi}$ is a topological isomorphism.
As $\pi$ equals a.s. $\hat{\pi}$, one may find a $T$-invariant Borel set
$Y_{0}\subset Y$ with $\nu(Y_{0})=1$, $\pi_{|Y_{0}}=\hat{\pi}_{|Y_{0}}$. As
$\hat{\pi}$ is one-to-one, $\pi_{|Y_{0}}^{-1}(\pi_{|Y_{0}}(Y_{0}))=Y_{0}$ and
therefore $\nu(\pi_{|Y_{0}}(Y_{0}))=1$. Thus
$\pi_{|Y_{0}}:Y_{0}\rightarrow\hat{\pi}(Y_{0})$ is a Borel measurable one-to-
one map between two $T$-invariant sets of full measure, which implies that
$\pi$ is a measurable isomorphism. ∎

######  Corollary 3.4.

Let $(X,\mathcal{X},\mu,T)$ be an ergodic m.p.s. and $k\in\mathbb{N}$. Let
$(Y,\mathcal{Y},\nu,S)$ be a minimal $k$-step pronilsystem isomorphic to
$(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$. Let
$\pi:(X,\mathcal{X},\mu,T)\rightarrow(Y,\mathcal{Y},\nu,S)$ be a factor map.
The following holds:

  1. (1)

There is a (topological) isomorphism $p\leavevmode\nobreak\
:\leavevmode\nobreak\
(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)\rightarrow(Y,\mathcal{Y},\nu,S)$ such
that $\pi=p\circ\pi_{k}$ a.s.

  2. (2)

For every measurable factor map
$\phi:(X,\mathcal{X},\mu,T)\rightarrow(Y^{\prime},\mathcal{Y}^{\prime},\nu^{\prime},S^{\prime})$
where $(Y^{\prime},\mathcal{Y}^{\prime},\nu^{\prime},S^{\prime})$ is a minimal
$k$-step pronilfactor, factors through $\pi$, i.e., there exists a unique (up
to measure zero)
$\psi:(Y,\mathcal{Y},\nu,S)\rightarrow(Y^{\prime},\mathcal{Y}^{\prime},\nu^{\prime},S^{\prime})$
such that $\phi=\psi\circ\pi$ a.s.

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{k}}$$\scriptstyle{\pi}$$\scriptstyle{\phi}$$\textstyle{Z_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{Y^{\prime}}$

###### Proof.

By the maximality of $\pi_{k}$ (see Subsection 2.6) there is a measurable
factor map
$p:(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)\rightarrow(Y,\mathcal{Y},\nu,S)$
such that $\pi=p\circ\pi_{k}$ a.s. By assumption there is a measurable
isomorphism
$i:(Y,\mathcal{Y},\nu,S)\rightarrow(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$
(which equals a.s. a topological isomorphism). By Theorem 3.3, $i\circ p$ is a
measurable isomorphism and therefore $p$ is a measurable isomorphism. This
establishes (1). Thus $\pi$ inherits the maximality property of $\pi_{k}$.
This establishes (2). ∎

######  Remark 3.5.

Bernard Host has pointed out to us that it is possible to prove Theorem B
using results from [HK18, Chapter 13].

###  3.2. Universality

######  Definition 3.6.

Let $(X,\mu,T)$ be a strictly ergodic t.d.s. Denote by
$\operatorname{C_{k}^{top}}$ the collection of (topological) isomorphism
equivalence classes of topological $k$-step pronilfactors of $(X,T)$. Denote
by $\operatorname{C_{k}^{meas}}$ the collection of (measurable) isomorphism
equivalence classes of measurable $k$-step pronilfactors of $(X,T)$. An
(equivalence class of) t.d.s. $(M,T)\in\operatorname{C_{k}^{top}}$ is called
$\operatorname{C_{k}^{top}}$-universal111111This terminology is frequently
used in the literature, see [dV93, GL13]. if every
$(N,S)\in\operatorname{C_{k}^{top}}$ is a topological factor of $(M,T)$. An
(equivalence class of) m.p.s.
$(M,\mathcal{M},\mu,T)\in\operatorname{C_{k}^{meas}}$ is called
$\operatorname{C_{k}^{meas}}$-universal if every
$(N,\mathcal{N},v,S)\in\operatorname{C_{k}^{meas}}$ is a measurable factor of
$(M,\mathcal{M},\mu,T)$.

The following theorem establishes a complementary property to maximality as
described in Remark 2.19 and Remark 2.27.

######  Theorem 3.7.

Let $(X,\mu,T)$ be a strictly ergodic t.d.s., then $(W_{k}(X),T)$ is the
unique $\operatorname{C_{k}^{top}}$-universal topological $k$-step
pronilfactor of $(X,T)$ and $(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$ is the
unique $\operatorname{C_{k}^{meas}}$-universal measurable $k$-step
pronilfactor of $(X,T)$.

###### Proof.

By Remark 2.19 $(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$ is a
$\operatorname{C_{k}^{meas}}$-universal measurable $k$-step pronilfactor of
$(X,T)$. Assume
$(Z^{\prime}_{k}(X),\mathcal{Z}^{\prime}_{k}(X),\mu^{\prime}_{k},T)$ is
another $\operatorname{C_{k}^{meas}}$-universal measurable $k$-step
pronilfactor of $(X,T)$. By universality one has measurable factor maps
$Z^{\prime}_{k}(X)\rightarrow\mathcal{Z}^{\prime}_{k}(X)$ and
$Z_{k}(X)\rightarrow\mathcal{Z}^{\prime}_{k}(X)$. By Theorem 3.3, $Z_{k}(X)$
and $\mathcal{Z}^{\prime}_{k}(X)$ are isomorphic.

By Remark 2.27 $(W_{k}(X),T)$ is a $\operatorname{C_{k}^{top}}$-universal
topological $k$-step pronilfactor of $(X,T)$. By Theorem 3.1 it is unique.

∎

###### Proof of Proposition 2.31.

By Remark 2.27, one can find a topological factor map
$q:(W_{k}(X),T)\rightarrow(\hat{Z}_{k},T)$. Let $\omega_{k}$ be the unique
ergodic measure of $(W_{k}(X),T)$. By Remark 2.19, one can find a measurable
factor map
$\psi:(\hat{Z}_{k},\gamma_{k},T)\rightarrow(W_{k}(X),\omega_{k},T)$.

$\textstyle{\hat{Z}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{W_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$

By Proposition 3.2, there exists a topological factor map
$\hat{\psi}:(\hat{Z}_{k},\gamma_{k},T)\rightarrow(W_{k}(X),\omega_{k},T)$ such
that $\hat{\psi}=\psi$ a.s. In particular, $\hat{\psi}\circ
q:(W_{k}(X),\omega_{k},T)\rightarrow(W_{k}(X),\omega_{k},T)$ is a topological
factor map. By Theorem 3.1, $\hat{\psi}\circ q$ is a topological isomorphism.
Thus $q$ is a topological isomorphism. As $(\hat{Z}_{k},T)$ and $(W_{k},T)$
are uniquely ergodic, $q$ is also a measurable isomorphism. In particular
$(W_{k}(X),\mathcal{W}_{k}(X),\omega_{k},T)$ and
$(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$ are isomorphic as m.p.s. and
$(X,\mu,T)$ is CF-Nil($k$).∎

##  4\. Cubespace characterization of CF-Nil($k$).

In this section, we prove Theorem C. We need some lemmas.

######  Lemma 4.1.

[HKM10, Lemma 5.6] Let $(X,T)$ be a minimal topological dynamical system and
$\mu$ be an invariant ergodic measure on $X$. Then the measure $\mu^{[k]}$ is
supported on $\operatorname{C}_{\operatorname{T}}^{k}(X)$ for any $k\geq 1$.

###### Proof.

The Lemma is proven in [HKM10, Lemma 5.6] with the help of the so called
$L^{2}$-convergence of cubical averages theorem [HK05, Theorem 1.2]. This is a
deep theorem with a highly non-trivial proof. We note that we are able to give
a direct proof of this Lemma which we hope to publish elsewhere. ∎

######  Definition 4.2.

Let $G$ be a countable amenable group. A Følner sequence
$\\{F_{N}\\}_{N\in\mathbb{N}}$ is a sequence of finite subsets of $G$ such
that for any $g\in G$, $\lim_{n\rightarrow\infty}|gF_{N}\cap
F_{N}|/|F_{N}|=1$.

######  Theorem 4.3.

(Lindenstrauss) Let $G$ be an amenable group acting on a measure space
$(X,\mathcal{X},\mu)$ by measure preserving transformations. Let
$\mathcal{I}_{G}$ be the $G$-invariant $\sigma$-algebra of
$(X,\mathcal{X},\mu)$. There is a Følner sequence
$\\{F_{N}\\}_{N\in\mathbb{N}}$ such that for any $f\in L^{\infty}(\mu)$, for
$\mu$-a.e. $x\in X$,

${\displaystyle\lim_{N\rightarrow\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}}f(gx)=\mathbb{E}(f|\mathcal{I}_{G})(x),$

In particular, if the $G$ action is ergodic, for $\mu$-a.e. $x\in X$,

${\displaystyle\lim_{N\rightarrow\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}}f(gx)=\int f(x)d\mu\text{ a.e.}$

###### Proof.

The theorem follows from [Lin01, Theorem 1.2] and [Lin01, Proposition 1.4]. In
[Lin01, Theorem 1.2] the statement reads

(8) $\lim_{N\rightarrow\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}f(gx)=\overline{f}(x)\text{ a.e.}$

for some $G$-invariant $\overline{f}\in L^{\infty}(\mu)$.

Note that if we replace $f$ by $\mathbb{E}(f|\mathcal{I}_{G})$ in (8), we have
trivially as $\mathbb{E}(f|\mathcal{I}_{G})$ is $G$-invariant:

$\mathbb{E}(f|\mathcal{I}_{G})(x)=\lim_{N\rightarrow\infty}\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}\mathbb{E}(f|\mathcal{I}_{G})(gx)$

Using the Lebesgue dominated convergence theorem for conditional
expectation121212It follows easily from applying the Lebesgue dominated
convergence theorem in Equation (4). one has:

$\mathbb{E}(f|\mathcal{I}_{G})(x)=\lim_{N\rightarrow\infty}\mathbb{E}(\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}f(g\cdot)|\mathcal{I}_{G})(x)=\mathbb{E}(\overline{f}|\mathcal{I}_{G})(x)=\overline{f}(x)\text{
a.e.}$

Thus $\overline{f}(x)=\mathbb{E}(f|\mathcal{I}_{G})(x)$, which gives the
statement above.

∎

###### Proof of Theorem C.

(I) $\Rightarrow$ (II): This follows from the proof in [HSY17, Section 4.4.3],
where it is shown that if one has a commutative diagram of the following form:

$\begin{CD}(X,\mathcal{X},\mu,T)@>{\phi}>{}>(\hat{X},T)\\\
@V{\pi_{k}}V{}V@V{}V{\hat{\pi}_{k}}V\\\
(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)@>{\operatorname{Id}}>{}>(Z_{k}(X),T),\end{CD}$

then $(C^{k+1}_{T}(\hat{X}),\mathcal{HK}^{k+1}(T))$ is uniquely ergodic. Here
$(X,\mathcal{X},\mu,T)$ is an ergodic system, $(\hat{X},T)$ is strictly
ergodic, $\phi$ is a measurable isomorphism w.r.t. the uniquely ergodic
measure of $(\hat{X},T)$ and $\hat{\pi}_{k}$ is a topological factor map.
Indeed, it is easy to obtain such a diagram for a CF-Nil$(k)$ system using
Proposition 2.31.

(II) $\Rightarrow$ (I): We assume that
$(\operatorname{C}_{\operatorname{T}}^{k+1}(X),\mathcal{HK}^{k+1}(T))$ is
uniquely ergodic. By Lemma 4.1, the unique invariant measure is $\mu^{[k+1]}$.
As $(X,T)$ is a topological factor of
$(\operatorname{C}_{\operatorname{T}}^{k+1}(X),\mathcal{HK}^{k+1}(T))$ w.r.t.
the projection to the first coordinate, $(X,T)$ is uniquely ergodic.

Let $p_{k}:(X,T)\rightarrow(W_{k}(X),T)$ be the topological canonical $k$-th
projection. By Proposition 2.1, as $(X,T)$ is uniquely ergodic so is
$(W_{k}(X),T)$. Let us denote by $\omega_{k}$ the unique invariant measure of
$(W_{k}(X),T)$. Obviously $(p_{k})_{*}\mu=\omega_{k}$. Thus
$p_{k}:(X,\mu,T)\rightarrow(W_{k}(X),\omega_{k},T)$ is a measurable factor
map. Let $\mathcal{W}_{k}$ be the $\sigma$-algebra corresponding to the map
$p_{k}$. Let $\mathcal{Z}_{k}$ be the $\sigma$-algebra corresponding to the
measurable canonical $k$-th projection
$\pi_{k}:(X,\mu,T)\rightarrow(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$. We will
show that $\mathcal{W}_{k}=\mathcal{Z}_{k}$, which implies that
$(W_{k}(X),\omega_{k},T)$ is isomorphic to
$(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$ as m.p.s. The map
$p_{k}:(X,T)\rightarrow(W_{k}(X),T)$ induces a factor map

$(\operatorname{C}_{\operatorname{T}}^{k+1}(X),\mathcal{HK}^{k+1}(T))\rightarrow(\operatorname{C}_{\operatorname{T}}^{k+1}(W_{k}(X)),\mathcal{HK}^{k+1}(T)).$

By Proposition 2.1, as
$(\operatorname{C}_{\operatorname{T}}^{k+1}(X),\mathcal{HK}^{k+1}(T))$ is
uniquely ergodic so is
$(\operatorname{C}_{\operatorname{T}}^{k+1}(W_{k}(X)),\mathcal{HK}^{k+1}(T))$.
By Lemma 4.1 the unique invariant measure on
$(\operatorname{C}_{\operatorname{T}}^{k+1}(W_{k}(X)),\mathcal{HK}^{k+1}(T))$
is $\omega_{k}^{[k+1]}$. Let $\gamma_{k+1}$ be the conditional product measure
relative to $(W_{k}(X)^{[k+1]},\omega_{k}^{[k+1]})$ on $X^{[k+1]}$ ([Fur77,
Definition 9.1]). This is the unique measure on $X^{[k+1]}$ such that for all
$f_{v}\in L^{\infty}(X,\mu)$, $v\in\\{0,1\\}^{k+1}$ ([Fur77, Lemma 9.1]):

(9)
$\int_{X^{[k+1]}}\prod_{v\in\\{0,1\\}^{k+1}}f_{v}(c_{v})d\gamma_{k+1}(c)=\\\
\int_{W_{k}(X)^{[k+1]}}\prod_{v\in\\{0,1\\}^{k+1}}\mathbb{E}(f_{v}|W_{k}(X))(c_{v})d\omega_{k}^{[k+1]}(c).$

As $\mathbb{E}(\cdot|W_{k}(X))$ commutes with $T$ and $\omega_{k}^{[k+1]}$ is
$\mathcal{HK}^{k+1}(T)$-invariant, one has that $\gamma_{k+1}$ is
$\mathcal{HK}^{k+1}(T)$-invariant. It is natural to introduce the measure
$\gamma_{k+1}$ as by [HK18, Chapter 9, Theorem 14], $\mu^{[k+1]}$ is the
conditional product measure relative to $\mu_{k}^{[k+1]}$. Thus if
$\mu_{k}=\omega_{k}$ then $\gamma_{k+1}=\mu^{[k+1]}$. It turns out one can
reverse the direction of implications. Indeed we claim that
$\gamma_{k+1}(\operatorname{C}_{\operatorname{T}}^{k+1}(X))=1$. Assuming this
claim and recalling the assumption that
$(\operatorname{C}_{\operatorname{T}}^{k+1}(X),\mathcal{HK}^{k+1}(T))$ is
uniquely ergodic, one has by Lemma 4.1 that $\gamma_{k+1}=\mu^{[k+1]}$. With
the end goal of showing $\mathcal{Z}_{k}=\mathcal{W}_{k}$ we start by showing
$\mathcal{Z}_{k}\subset\mathcal{W}_{k}$. It is enough to show
$L^{\infty}(\mu)\cap L^{2}(\mathcal{W}_{k})^{\perp}\subset L^{\infty}(\mu)\cap
L^{2}(\mathcal{Z}_{k})^{\perp}$. To this end we will show that for any
function $f\in L^{\infty}(\mu)$ such that $\mathbb{E}(f|\mathcal{W}_{k})=0$,
it holds that $\mathbb{E}(f|\mathcal{Z}_{k})=0$. By Definition 2.15, as
$\gamma_{k+1}=\mu^{[k+1]}$,

$|||f|||_{k+1}^{2^{k+1}}=\int\prod_{v\in\\{0,1\\}^{k+1}}\mathcal{C}^{|v|}f(c_{v})d\gamma_{k+1}(c)=\\\
\int\prod_{v\in\\{0,1\\}^{k+1}}\mathbb{E}(\mathcal{C}^{|v|}f|W_{k}(X))(c_{v})d\omega_{k}^{[k+1]}(c).$

As $\mathbb{E}(f|\mathcal{W}_{k})\equiv 0$, it holds that
$\mathbb{E}(\mathcal{C}^{|v|}f|W_{k}(X))\equiv 0$ for any
$v\in\\{0,1\\}^{k+1}$. Therefore $|||f|||_{k+1}=0$. This implies by Lemma 2.16
that $\mathbb{E}(f|\mathcal{Z}_{k})=0$ as desired. By Remark 2.19, $Z_{k}(X)$
is the maximal measurable $k$-step pronilfactor of $(X,\mu,T)$. As
$(W_{k}(X),\omega_{k},T)$ is a $k$-step pronilfactor of $(X,T)$, one has that
$\mathcal{W}_{k}\subset\mathcal{Z}_{k}$. Thus
$\mathcal{W}_{k}=\mathcal{Z}_{k}$, which implies that
$(W_{k}(X),\omega_{k},T)$ is isomorphic to
$(Z_{k}(X),\mathcal{Z}_{k}(X),\mu_{k},T)$ as m.p.s.

As a final step, we will now show that
$\gamma_{k+1}(\operatorname{C}_{\operatorname{T}}^{k+1}(X))=1$. Let $f_{v}\in
L^{\infty}(X,\mu)$, $v\in\\{0,1\\}^{k+1}$ and set
$H_{0}=\prod_{v\in\\{0\\}\times\\{0,1\\}^{k}}f_{v}$ and
$H_{1}=\prod_{v\in\\{1\\}\times\\{0,1\\}^{k}}f_{v}$ as well as
$\hat{H}_{0}=\prod_{v\in\\{0\\}\times\\{0,1\\}^{k}}\mathbb{E}(f_{v}|W_{k}(X))$,
$\hat{H}_{1}=\prod_{v\in\\{1\\}\times\\{0,1\\}^{k}}\mathbb{E}(f_{v}|W_{k}(X))$.
By Equation (9), we have

(10)
$\int_{X^{[k+1]}}H_{0}(c)H_{1}(c^{\prime})d\gamma_{k+1}(c,c^{\prime})=\int_{W_{k}(X)^{[k+1]}}\hat{H}_{0}(c)\hat{H}_{1}(c^{\prime})d\omega_{k}^{[k+1]}(c,c^{\prime}).$

By Equation (6) in Definition 2.12,

(11)
$\int_{W_{k}(X)^{[k+1]}}\hat{H}_{0}(c)\hat{H}_{1}(c^{\prime})d\omega_{k}^{[k+1]}(c,c^{\prime})=\int_{W_{k}(X)^{[k]}}\mathbb{E}(\hat{H}_{0}|\mathcal{I}_{T^{[k]}})(c)\hat{H}_{1}(c)d\omega_{k}^{[k]}(c).$

By Birkhoff’s ergodic theorem (see also Theorem 4.3), one has that

(12)
$\begin{array}[]{ll}\int_{W_{k}(X)^{[k]}}\mathbb{E}(\hat{H}_{0}|\mathcal{I}_{T^{[k]}})(c)\hat{H}_{1}(c)d\omega_{k}^{[k]}(c)\\\
{\displaystyle=\int\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\hat{H}_{0}((T^{[k]})^{n}c)\hat{H}_{1}(c)d\omega_{k}^{[k]}(c)}\\\
={\displaystyle\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int\hat{H}_{0}((T^{[k]})^{n}c)\hat{H}_{1}(c)d\omega_{k}^{[k]}(c)},\end{array}$

here we used the Lebesgue dominated convergence theorem.

Abusing notation one may consider $\hat{H}_{0}$ and $\hat{H}_{1}$ as defined
on $X^{[k]}$ (see Subsection 2.3). As
$p_{k}:(X,\mu,T)\rightarrow(W_{k}(X),\omega_{k},T)$ is a measurable factor
map, one has

$\int\hat{H}_{0}((T^{[k]})^{n}c)\hat{H}_{1}(c)d\omega_{k}^{[k]}(c)=\int\hat{H}_{0}((T^{[k]})^{n}c)\hat{H}_{1}(c)d\mu^{[k]}(c).$

As $(C_{T}^{k}(X),\mathcal{HK}^{k}(T))$ is a topological factor of
$(\operatorname{C}_{\operatorname{T}}^{k+1}(X),\mathcal{HK}^{k+1}(T))$ w.r.t.
the “lower” $2^{k}$ coordinates, $(C_{T}^{k}(X),\mathcal{HK}^{k}(T))$ is
uniquely ergodic. By Lemma 4.1, the unique ergodic measure is $\mu^{[k]}$. By
Theorem 4.3 applied to $(C_{T}^{k}(X),\mu^{[k]},\mathcal{HK}^{k}(T))$, there
is a Følner sequence $\\{F_{M}\subset\mathcal{HK}^{k}(T)\\}_{M\in\mathbb{N}}$
such that

(13)
$\int\hat{H}_{0}\big{(}(T^{[k]})^{n}c\big{)}\hat{H}_{1}(c)d\mu^{[k]}(c)=\lim_{M\rightarrow\infty}\frac{1}{|F_{M}|}\sum_{h\in
F_{M}}\hat{H}_{0}\big{(}(T^{[k]})^{n}hs\big{)}\hat{H}_{1}(hs)$

for $\mu^{[k]}$-a.e. $s\in C_{T}^{k}(X)$. Thus from Equations (10), (11), (12)
and (13), it holds for arbitrary $f_{v}\in L^{\infty}(X,\mu)$,
$v\in\\{0,1\\}^{k+1}$, $H_{0}=\prod_{v\in\\{0\\}\times\\{0,1\\}^{k}}f_{v}$ and
$H_{1}=\prod_{v\in\\{1\\}\times\\{0,1\\}^{k}}f_{v}$, for $\mu^{[k]}$-a.e.
$s\in C_{T}^{k}(X)$,

(14) $\int_{X^{[k+1]}}H_{0}(c)H_{1}(c^{\prime})d\gamma_{k+1}(c,c^{\prime})=\\\
\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\lim_{M\rightarrow\infty}\frac{1}{|F_{M}|}\sum_{h\in
F_{M}}\hat{H}_{0}\big{(}(T^{[k]})^{n}hs\big{)}\hat{H}_{1}(hs)$

Let $R\in C(X^{[k+1]},\mathbb{R})$ be a continuous function. We claim for
$\mu^{[k]}$-a.e. $s\in\operatorname{C}_{\operatorname{T}}^{k}(X)$,

(15) $\int
R(c)d\gamma_{k+1}(c)=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\lim_{M\rightarrow\infty}\frac{1}{|F_{M}|}\sum_{h\in
F_{M}}R\big{(}(T^{[k]})^{n}hs,hs\big{)}$

Notice that it follows from Definitions 2.11 and 2.21 that if $s\in
C_{T}^{k}(X)$, then $((T^{[k]})^{n}hs,hs)\in C_{T}^{k+1}(X)$ for arbitrary
$h\in\mathcal{HK}^{k}(T)$ and $n\in\mathbb{Z}^{+}$ (see also [GGY18,
Subsection A.2]). Thus using Equation (15) with functions $R_{\delta}\in
C(X^{[k+1]},[0,1])$ such that
$R_{\delta}|_{\operatorname{C}_{\operatorname{T}}^{k+1}(X)}\equiv 1$ and
$R|_{X^{[k+1]}\setminus
B_{\delta}(\operatorname{C}_{\operatorname{T}}^{k+1}(X))}\equiv 0$, (taking
$\delta$ to zero) one obtains:

$\gamma_{k+1}(C_{T}^{k+1}(X))=1.$

We now prove (15). For $d\in\mathbb{N}$, let $H_{d}^{(i)}$ be functions of the
form $\prod_{v\in\\{0,1\\}^{k+1}}h^{(i)}_{v}$, $i\in I_{d}$ for some finite
set $I_{d}$, such that $|R(z)-\sum_{i\in I_{d}}H_{d}^{(i)}(z)|<\frac{1}{2d}$
for all $z\in\operatorname{C}_{\operatorname{T}}^{k+1}(X)$. Denote by
$C(R)=\int R(c)d\gamma_{k+1}(c)$ the (LHS) of (15). Denote by $D(R)(z)$ be the
(RHS) of Equation (15). By Equation (14), $C(H_{d}^{(i)})=D(H_{d}^{(i)})(z)$
for $\mu^{[k]}$-a.e. $z\in\operatorname{C}_{\operatorname{T}}^{k}(X)$. Note
that $|C(R)-\sum_{i\in I_{d}}C(H_{d}^{(i)})|<\frac{1}{2d}$ and
$|D(R)(y)-\sum_{i\in I_{d}}D(H_{d}^{(i)})(y)|<\frac{1}{2d}$ for all
$y\in\operatorname{C}_{\operatorname{T}}^{k}(X)$. Thus for any $d$,
$E_{d}:=\\{y\in\operatorname{C}_{\operatorname{T}}^{k}(X):|C(R)(y)-D(R)(y)|<\frac{1}{d}\\}$
has full $\mu^{[k]}$ measure. Let $E=\bigcap_{d\in\mathbb{N}}E_{d}$, then
$\mu^{[k]}(E)=1$ and for any $y\in E$, Equation (15) holds. ∎

The following remark may be of interest:

######  Remark 4.4.

In [GHSY20, Section 6] an example is given showing there exists a strictly
ergodic distal system which is not CF-Nil($1$).

##  5\. A topological Wiener-Wintner theorem.

In this section, we prove Theorem A.

######  Definition 5.1.

Let $(X,T)$ be a t.d.s. and $\mu\in\operatorname{P_{T}}(X)$. A point $x\in X$
is generic (for $\mu$) if for all $f\in C(X)$

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N}f(T^{n}x)=\int fd\mu$

######  Lemma 5.2.

Let $(X,T)$ be a t.d.s. and $x_{0}\in X$. Assume that for all $f\in C(X)$,
there exists $c_{f}\in\mathbb{R}$, a constant depending on $f$, so that :

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N}f(T^{n}x_{0})=c_{f}$

Then $x_{0}$ is generic for some $\mu\in\operatorname{P_{T}}(X)$.

###### Proof.

Define the functional $\phi:C(X)\rightarrow\mathbb{R}$ by $\phi(f)=c_{f}$. It
is easy to see that $\phi$ is a bounded linear positive functional of supremum
norm $1$. By the Riesz representation theorem $c_{f}=\int fd\mu$ for some
Borel probability measure $\mu$ on $X$ ([Rud06, Theorem 2.14]). As
$c_{f}=c_{Tf}$ for all $f\in C(X)$, it follows that
$\mu\in\operatorname{P_{T}}(X)$. Thus $x_{0}$ is generic by Definition 5.1. ∎

######  Theorem 5.3.

([Gla03, Theorem 4.10]) Let $(X,T)$ be a minimal t.d.s., then $(X,T)$ is
uniquely ergodic iff every $x\in X$ is generic for some
$\mu\in\operatorname{P_{T}}(X)$ (depending on $x$).

######  Lemma 5.4.

Let $(X,T)$ be a t.d.s. and $\mu\in\operatorname{P_{T}}(X)$. If a point $x\in
X$ is generic for $\mu$, then $\mu$ is supported on
$\operatorname{\overline{\mathcal{O}}}(x)$.

###### Proof.

Let $f$ be a non-negative function supported outside
$\operatorname{\overline{\mathcal{O}}}(x)$. Then $\int
fd\mu=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}f(T^{n}x)=0$. Q.E.D. ∎

###### Proof of Theorem A.

$(I)\Rightarrow(II)$. It follows from [HK09, Theorem 2.19 and Proposition
7.1].

We will show $(II)\Rightarrow(I)$ inductively. For $k=0$ note that Condition
$(II)$ with the constant nilsequence $a(n)\equiv 1$ implies that for a fixed
arbitrary $x\in X$ and every $f\in C(X)$,
$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}a(n)f(T^{n}x)=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}f(T^{n}x)$
exists. From Lemma 5.2, $x\in X$ is generic for some $\mu_{x}\in P_{T}(X)$. By
Theorem 5.3, $(X,T)$ is uniquely ergodic. By assumption $(X,T)$ is minimal and
thus $(X,T)$ is a CF-Nil$(k)$ system.

Assume the $(II)\Rightarrow(I)$ holds for $k-1$. We will now show
$\sim(I)\Rightarrow\,\,\,\sim(II)$ for $k$. Thus we assume that $(X,T)$ is not
CF-Nil($k$). If $(X,T)$ is not CF-Nil($k-1$) then the result follows from the
inductive assumption. Thus we may assume $(X,T)$ is CF-Nil($k-1$) and in
particular uniquely ergodic. Denote the unique probability measure of $(X,T)$
by $\mu$. By definition one has that
$(Z_{k-1}(X),\mathcal{Z}_{k-1}(X),\mu_{k-1},T)$ is isomorphic as an m.p.s. to
$(W_{k-1}(X),\omega_{k-1},T)$, where $\omega_{k-1}$ is the unique ergodic
measure of $(W_{k-1}(X),T)$.

An important result of the Host-Kra structure theory is that
$\pi:Z_{k}(X)\rightarrow Z_{k-1}(X)$, determined by
$\pi_{k-1}=\pi\circ\pi_{k}$ (as defined in Definition 2.14), is a measurable
group extension w.r.t. some abelian group $A$ (See [HK05, Section 6.2], [HK18,
Chapter 9, Section 2.3]). By [GL19, Theorem 1.1, proof of Theorem 5.3], we can
find a topological model
$\hat{\pi}:(\hat{Z}_{k},T)\rightarrow(\hat{Z}_{k-1},T)$ of $\pi$ which is an
abelian topological group extension w.r.t. the abelian group $A$ such that
$(\hat{Z}_{k},T)$ is a minimal $k$-step pronilsystem and $(\hat{Z}_{k-1},T)$
is a minimal $(k-1)$-step pronilsystem. Denote by $\phi$ and $\psi$ the
measurable isomorphisms between $Z_{k}(X)$ and $\hat{Z}_{k}(X)$ and
$Z_{k-1}(X)$ and $\hat{Z}_{k-1}(X)$ respectively.

$\begin{CD}Z_{k}(X)@>{\phi}>{}>\hat{Z}_{k}(X)\\\
@V{\pi}V{}V@V{}V{\hat{\pi}}V\\\
Z_{k-1}(X)@>{\psi}>{}>\hat{Z}_{k-1}(X)\end{CD}$

For clarity denote $\pi_{Z_{k}}:=\pi_{k}$ from the previous paragraph.

Define $\pi_{\hat{Z}_{k}}=\phi\circ\pi_{Z_{k}}$. Let $p_{k-1}:X\rightarrow
W_{k-1}(X)$ be the topological canonical $(k-1)$-th projection. Let
$\pi_{\hat{Z}_{k-1}}=\hat{\pi}\circ\pi_{\hat{Z}_{k}}$. By Corollary 3.4(2),
$\hat{\pi}\circ\pi_{\hat{Z}_{k}}$ inherits the maximality property of
$\pi_{k-1}=\pi\circ\pi_{Z_{k}}$. By Corollary 3.4(1), there exists a
measurable factor map $p:\hat{Z}_{k-1}(X)\rightarrow W_{k-1}(X)$ such that
$p_{k-1}=p\circ\hat{\pi}\circ\pi_{\hat{Z}_{k}(X)}$ a.s. As $\hat{Z}_{k-1}(X)$
is isomorphic to both $Z_{k-1}(X)$ and $W_{k-1}(X)$ as m.p.s.131313Here we use
that $(X,T)$ is CF-Nil($k-1$)., by Theorem 3.3, $p$ may be chosen to be a
topological isomorphism. W.l.o.g. we will assume $p=\operatorname{Id}$. Thus
we have:

(16) $p_{k-1}(x)=\hat{\pi}\circ\pi_{\hat{Z}_{k}(X)}(x)$ for $\mu$-a.e. $x\in
X$.

---
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{Z_{k}}}$$\scriptstyle{\operatorname{Id}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}}$$\scriptstyle{\pi_{\hat{Z}_{k}}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{k-1}}$$\textstyle{Z_{k}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\pi}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hat{Z}_{k}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\pi}}$$\textstyle{Z_{k-1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hat{Z}_{k-1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
W_{k-1}(X)}$

We claim that there exists a minimal subsystem $(Y,T\times
T)\subset(X\times\hat{Z}_{k},T\times T)$ such that $(Y,T\times T)$ is not
uniquely ergodic. Assuming this, as by Theorem 5.3 a minimal system is
uniquely ergodic if and only if every point is generic, there exists
$(x_{3},u_{3})\in Y$ such that $(x_{3},u_{3})$ is not a generic point for any
measure. By Lemma 5.2, there exist continuous functions $h\in C(\hat{Z}_{k})$,
$f\in C(X)$ such that

(17)
$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}h(T^{n}u_{3})f(T^{n}x_{3})$

does not exist. As $(\hat{Z}_{k},T)$ is a $k$-step pronilsystem,
$h(T^{n}u_{3})$ is a $k$-step nilsequence (Definition 2.7). Thus $(II)$ does
not hold as required.

Our strategy in proving the claim is finding a minimal subsystem $(Y,T\times
T)$ of $(X\times\hat{Z}_{k},T\times T)$ which supports an invariant measure
$\nu$, w.r.t which $(Y,T\times T)$ is isomorphic to $(X,\mu,T)$ as an m.p.s.
We then assume for a contradiction that $(Y,T\times T)$ is uniquely ergodic.
Next we notice that the strictly ergodic system $(Y,T\times T)$, being
measurably isomorphic to $(X,\mu,T)$, has $Z_{k}(Y)\simeq Z_{k}(X)$. Moreover
as $(Y,T\times T)$ is a minimal subsystem of a product of the two minimal
systems, $(X,T)$ and $(\hat{Z}_{k},T)$, it maps onto each of them through the
first, respectively second coordinate projections. From the projection on
$(\hat{Z}_{k},T)$, we conclude that $(Y,T)$ has a topological $k$-step
pronilfactor $\hat{Z}_{k}$ which is measurably isomorphic to $Z_{k}(Y)$. By
Proposition 2.31, one has that $(Y,T)$ is CF-Nil($k$).From the projection on
$(X,T)$, we conclude by Proposition 2.32, that $(X,T)$ is CF-Nil($k$).This
constitutes a contradiction implying that $(Y,T)$ is not uniquely ergodic as
desired.

A natural copy of $(X,\mu,T)$ inside $(X\times\hat{Z}_{k},T\times T)$ is given
by the graph joining of $\pi_{\hat{Z}_{k}(X)}$, defined by the measure
$\mu^{(k)}=(\operatorname{Id}\times\pi_{\hat{Z}_{k}(X)})_{*}\mu:=\int\delta_{x}\times\delta_{\pi_{\hat{Z}_{k}(X)}(x)}d\mu(x)$
on $(X\times\hat{Z}_{k},T)$ (see [Gla03, Chapter 6, Example 6.3]). Clearly

(18)
$\operatorname{Id}\times\pi_{\hat{Z}_{k}(X)}:(X,\mathcal{X},\mu,T)\rightarrow(X\times\hat{Z}_{k},\mathcal{X}\times\hat{\mathcal{Z}}_{k},\mu^{(k)},T\times
T),\,x\mapsto(x,\pi_{\hat{Z}_{k}(X)}(x)).$

is a measurable isomorphism and in particular $\mu^{(k)}$ is an ergodic
measure of $(X\times\hat{Z}_{k},T\times T)$. However
$(X\times\hat{Z}_{k},\mathcal{X}\times\hat{\mathcal{Z}}_{k},\mu^{(k)},T\times
T)$ is a m.p.s. and not a (minimal) t.d.s. We consider an orbit closure of a
$\mu^{(k)}$-generic point $(x_{1},\pi_{\hat{Z}_{k}(X)}(x_{1}))$ to be
determined later. By Lemma 5.4, $\mu^{(k)}$ is supported on
$\operatorname{\overline{\mathcal{O}}}(x_{1},\pi_{\hat{Z}_{k}(X)}(x_{1}))$.
However
$(\operatorname{\overline{\mathcal{O}}}(x_{1},\pi_{\hat{Z}_{k}(X)}(x_{1})),T\times
T)$ is not necessarily minimal. We thus pass to an (arbitrary) minimal
subsystem $(Y,T\times
T))\subset(\operatorname{\overline{\mathcal{O}}}(x_{1},\pi_{\hat{Z}_{k}(X)}(x_{1})),T\times
T))$. However $\mu^{(k)}$ is not necessarily supported on $Y$. As explained in
the previous paragraph, our final aim will be to find (a possibly different)
invariant measure $\nu\in\operatorname{P_{T\times T}}(Y)$ which is isomorphic
to $\mu$.

As $\hat{\pi}$ is a topological group extension w.r.t. the abelian group $A$,

(19) $\operatorname{Id}\times\hat{\pi}:(X\times\hat{Z}_{k},T\times
T)\rightarrow(X\times W_{k-1}(X),T\times T):(x,z)\mapsto(x,\hat{\pi}(z))$

is also a topological group extension w.r.t. the abelian group $A$. Thus $A$
acts on the fibers of $\operatorname{Id}\times\hat{\pi}$ transitively and
continuously by homeomorphisms. Moreover for all $a\in A$,
$(\operatorname{Id}\times a)_{*}\mu^{(k)}$ is an invariant measure on
$(X\times\hat{Z}_{k},T\times T)$ isomorphic to $\mu^{(k)}$ and thus isomorphic
to $\mu$. We will find $\nu\in\operatorname{P_{T\times T}}(Y)$ of the form
$\nu=(\operatorname{Id}\times a)_{*}\mu^{(k)}$. Indeed for $\mu$-a.e. $x\in
X$, $(x,\pi_{\hat{Z}_{k}(X)}(x))$ is a generic point of $\mu^{(k)}$. Using
(16), one may choose $x_{1}\in X$ such that

  * •

$(x_{1},\pi_{\hat{Z}_{k}(X)}(x_{1}))$ is a generic point of $\mu^{(k)}$;

  * •

$\hat{\pi}(\pi_{\hat{Z}_{k}(X)}(x_{1}))=p_{k-1}(x_{1})$.

From the second point it follows that:

$\operatorname{Id}\times\hat{\pi}:(\operatorname{\overline{\mathcal{O}}}(x_{1},\pi_{\hat{Z}_{k}(X)}(x_{1})),T\times
T)\rightarrow(\operatorname{\overline{\mathcal{O}}}(x_{1},p_{k-1}(x_{1})),T\times
T)$

is a topological factor map. As $p_{k-1}$ is a topological factor map,

(20) $\operatorname{Id}\times
p_{k-1}:(X,T)\rightarrow(\operatorname{\overline{\mathcal{O}}}(x_{1},p_{k-1}(x_{1})),T\times
T),\,x\rightarrow(x,p_{k-1}(x))$

is a topological isomorphism. Therefore
$(\operatorname{\overline{\mathcal{O}}}(x_{1},p_{k-1}(x_{1})),T\times T)$ is
minimal. Thus
$(\operatorname{Id}\times\hat{\pi})_{|Y}:(Y,T)\rightarrow(\operatorname{\overline{\mathcal{O}}}(x_{1},p_{k-1}(x_{1})),T)$
factors onto. In particular there exists $z_{1}\in\hat{Z}_{k}(X)$, such that
$(x_{1},z_{1})\in Y$ and $\hat{\pi}(z_{1})=p_{k-1}(x_{1})$. As by assumption
$\hat{\pi}(\pi_{\hat{Z}_{k}(X)}(x_{1}))=p_{k-1}(x_{1})$, we can find $a\in A$
such that $a.\pi_{\hat{Z}_{k}(X)}(x_{1})=z_{1}$. As
$(x_{1},\pi_{\hat{Z}_{k}(X)}(x_{1}))$ is a generic point of $\mu^{(k)}$, it
follows that $(x_{1},a.\hat{\pi}_{k}(x_{1}))=(x_{1},z_{1})$ is a generic point
of $\nu:=(\operatorname{Id}\times a)_{*}\mu^{(k)}$. Therefore by Lemma 5.4,
the invariant measure $\nu\simeq\mu$ is supported on the minimal subsystem
$\operatorname{\overline{\mathcal{O}}}{(x_{1},z_{1})}=Y$. This ends the proof.
∎

## References

  * [AKL14] Omer Angel, Alexander S. Kechris, and Russell Lyons.  Random orderings and unique ergodicity of automorphism groups.  Journal of the European Mathematical Society, 16(10):2059–2095, 2014.
  * [Ass92] I. Assani.  Uniform Wiener-Wintner theorems for weakly mixing dynamical systems.  Unpublished preprint, 1992.
  * [Aus63] Joseph Auslander.  Endomorphisms of minimal sets.  Duke Mathematical Journal, 30(4):605–614, 1963.
  * [BDM10] Xavier Bressaud, Fabien Durand, and Alejandro Maass.  On the eigenvalues of finite rank Bratteli-Vershik dynamical systems.  Ergodic Theory Dynam. Systems, 30:639–664, 2010.
  * [Bou90] J. Bourgain.  Double recurrence and almost sure convergence.  J. Reine Angew. Math., 404:140–161, 1990.
  * [BŚ17] Wojciech Bartoszek and Adam Śpiewak.  A note on a Wiener-Wintner theorem for mean ergodic Markov amenable semigroups.  Proceedings of the American Mathematical Society, 145(7):2997–3003, 2017.
  * [DFM19] Fabien Durand, Alexander Frank, and Alejandro Maass.  Eigenvalues of minimal cantor systems.  J. Eur. Math. Soc, 21:727–775, 2019.
  * [Don14] Sebastián Donoso.  Enveloping semigroups of systems of order $d$.  Discrete Contin. Dyn. Syst., 34(7):2729–2740, 2014.
  * [dV93] J. de Vries.  Elements of topological dynamics, volume 257 of Mathematics and its Applications.  Kluwer Academic Publishers Group, Dordrecht, 1993.
  * [Ell58] Robert Ellis.  Distal transformation groups.  Pacific J. Math., 8:401–405, 1958.
  * [EZK13] Tanja Eisner and Pavel Zorin-Kranich.  Uniformity in the Wiener-Wintner theorem for nilsequences.  Discrete Contin. Dyn. Syst., 33(8):497–516, 2013.
  * [Fan18] Ai-Hua Fan.  Topological Wiener-Wintner ergodic theorem with polynomial weights.  Chaos, Solitons and Fractals, 117:105–116, 2018.
  * [Fra06] Nikos Frantzikinakis.  Uniformity in the polynomial Wiener-Wintner theorem.  Ergodic Theory Dynam. Systems, 26(4):1061–1071, 2006.
  * [Fur77] Harry Furstenberg.  Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions.  J. Analyse Math., 31:204–256, 1977.
  * [GGY18] Eli Glasner, Yonatan Gutman, and XiangDong Ye.  Higher order regionally proximal equivalence relations for general minimal group actions.  Advances in Mathematics, 333(6), 2018.
  * [GHSY20] Eli Glasner, Wen Huang, Song Shao, and Xiangdong Ye.  Regionally proximal relation of order $d$ along arithmetic progressions and nilsystems.  Sci. China Math., 63(9):1757–1776, 2020.
  * [GL13] Yonatan Gutman and Hanfeng Li.  A new short proof for the uniqueness of the universal minimal space.  Proceedings of the American Mathematical Society, 141(1):265–267, 2013.
  * [GL19] Yonatan Gutman and Zhengxing Lian.  Strictly ergodic distal models and a new approach to the Host-Kra factors.  Preprint. https://arxiv.org/abs/1909.11349, 2019.
  * [Gla03] Eli Glasner.  Ergodic theory via joinings, volume 101 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, 2003.
  * [GMV20] Yonatan Gutman, Freddie Manners, and Péter P. Varjú.  The structure theory of nilspaces III: Inverse limit representations and topological dynamics.  Advances in Mathematics, 365(13), 2020.
  * [HK05] Bernard Host and Bryna Kra.  Nonconventional ergodic averages and nilmanifolds.  Ann. of Math. (2), 161(1):397–488, 2005.
  * [HK09] Bernard Host and Bryna Kra.  Uniformity seminorms on $\ell^{\infty}$ and applications.  J. Anal. Math., 108:219–276, 2009.
  * [HK18] Bernard Host and Bryna Kra.  Nilpotent Structures in Ergodic Theory.  American Mathematical Society, 2018.
  * [HKM10] Bernard Host, Bryna Kra, and Alejandro Maass.  Nilsequences and a structure theorem for topological dynamical systems.  Adv. Math., 224(1):103–129, 2010.
  * [HKM14] Bernard Host, Bryna Kra, and Alejandro Maass.  Complexity of nilsystems and systems lacking nilfactors.  Journal d’Analyse Mathématique, 124(1):261–295, 2014.
  * [Hos86] Bernard Host.  Valeurs propres des systèmes dynamiques définis par des substitutions de longueur variable.  Erg. Theory Dyn. Systems, 6:529–540, 1986.
  * [HP68] Frank Hahn and William Parry.  Some characteristic properties of dynamical systems with quasi-discrete spectra.  Mathematical systems theory, 2(2):179–190, 1968.
  * [HSY17] Wen Huang, Song Shao, and Xiangdong Ye.  Strictly ergodic models under face and parallelepiped group actions.  Commun. Math. Stat, 5:93–122, 2017.
  * [HSY19] W. Huang, S. Shao, and X. Ye.  Pointwise convergence of multiple ergodic averages and strictly ergodic models.  Journal d Analyse Matématique, 139(2), 2019.
  * [Leh87] Ehud Lehrer.  Topological mixing and uniquely ergodic systems.  Israel Journal of Mathematics, 57(2):239–255, 1987.
  * [Les96] Emmanuel Lesigne.  Un théorème de disjonction de systèmes dynamiques et une généralisation du théorème ergodique de Wiener-Wintner.  Ergodic Theory Dynam. Systems, 10:513–521, 1996.
  * [Lin01] Elon Lindenstrauss.  Pointwise theorems for amenable groups.  Invent. Math., 146(2):259–295, 2001.
  * [LLT92] Mariusz Lemańczyk, Pierre Liardet, and Jean-Paul Thouvenot.  Coalescence of circle extensions of measure-preserving transformations.  Ergodic Theory and Dynamical Systems, 12(4):769–789, 1992.
  * [Que10] Martine Queffélec.  Substitution dynamical systems - spectral analysis.  Lecture Notes in Mathematics, Springer, 2010.
  * [Rob94] E. Arthur Robinson.  On uniform convergence in the Wiener-Wintner theorem.  Journal of the London Mathematical Society, 49, 1994.
  * [Rud06] Walter Rudin.  Real and complex analysis.  Tata McGraw-hill education, 2006.
  * [Sch14] Marco Schreiber.  Topological Wiener–Wintner theorems for amenable operator semigroups.  Ergodic Theory and Dynamical Systems, 34(5):1674–1698, 2014.
  * [SY12] Song Shao and Xiangdong Ye.  Regionally proximal relation of order $d$ is an equivalence one for minimal systems and a combinatorial consequence.  Adv. Math., 231(3-4):1786–1817, 2012.
  * [Wei85] Benjamin Weiss.  Strictly ergodic models for dynamical systems.  Bull. Amer. Math. Soc. (N.S.), 13(2):143–146, 1985.
  * [WW41] Norbert Wiener and Aurel Wintner.  Harmonic analysis and ergodic theory.  Amer J Math, 63:415–426, 1941.
  * [Zie07] Tamar Ziegler.  Universal characteristic factors and Furstenberg averages.  J. Amer. Math. Soc., 20(1):53–97 (electronic), 2007.

Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8,
00-656 Warszawa, Poland.

Yonatan Gutman<EMAIL_ADDRESS>
School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005,
P.R. China;

Zhengxing Lian<EMAIL_ADDRESS><EMAIL_ADDRESS>