# On orthogonality to uniquely ergodic systems

M. Górska, M. Lemańczyk, T. de la Rue

###### Abstract

We solve Boshernitzan’s problem of characterization (in terms of so called
Furstenberg systems) of bounded sequences that are orthogonal to all uniquely
ergodic systems. Some variations of Boshernitzan’s problem involving
characteristic classes are considered. As an application, we characterize
sequences orthogonal to all uniquely ergodic systems whose (unique) invariant
measure yields a discrete spectrum automorphism as those satisfying an
averaged Chowla property.

###### Contents

  1. 1 Introduction
    1. 1.1 Presentation of the main problem and motivations
    2. 1.2 Ergodic theory results
    3. 1.3 Solution of Boshernitzan’s problem
    4. 1.4 Applications
    5. 1.5 Auxiliary results
  2. 2 Preparatory material
    1. 2.1 Lifting lemma
    2. 2.2 Orbital uniquely ergodic models
    3. 2.3 Joinings and Markov operators
    4. 2.4 Weakly ergodic part of a dynamical system
    5. 2.5 Koopman operator and spectral measures
    6. 2.6 Furstenberg systems of arithmetic functions
    7. 2.7 Confined extensions
    8. 2.8 Joinings of a non-ergodic automorphism with an ergodic system
    9. 2.9 Characteristic classes
    10. 2.10 Measurable selectors
  3. 3 Automorphisms disjoint from all ergodic systems
    1. 3.1 Proof of (ii) $\Rightarrow$ (i)
    2. 3.2 Proof of (iii) $\Rightarrow$ (ii)
    3. 3.3 Disjointness in ${\rm Aut}(X,\mu)$
    4. 3.4 Kallman theorem and joinings
    5. 3.5 Proof of (i) $\Rightarrow$ (iii)
    6. 3.6 Relative ergodic decomposition
    7. 3.7 Example(s)
  4. 4 Orthogonality to ergodic Markov images
    1. 4.1 Preparatory observations and some motivations
    2. 4.2 Characterization result
    3. 4.3 Proof of Theorem 4.5. Sufficiency
    4. 4.4 Proof of Theorem 4.5. Necessity
    5. 4.5 A condition for being orthogonal to ergodic Markov images
  5. 5 Orthogonality to uniquely ergodic systems
    1. 5.1 Orthogonality and ergodic Markov images
    2. 5.2 Pretentious multiplicative function orthogonal to all uniquely ergodic systems
    3. 5.3 Examples of sequences orthogonal to all uniquely ergodic systems
    4. 5.4 Properties of arithmetic functions orthogonal to all u.e. systems
    5. 5.5 Mean slowly varying functions are multipliers of the UE⟂
    6. 5.6 How to recognize that $\rho$ satisfies $\rho|_{X\times X}=\mu\otimes\mu$?
    7. 5.7 Summing up
  6. 6 Characteristic classes and orthogonality to uniquely ergodic systems
    1. 6.1 Characteristic classes and disjointness
    2. 6.2 Orthogonality to zero entropy and uniquely ergodic systems. General characteristic class case
    3. 6.3 Orthogonality to systems with countably many ergodic invariant measures, all in ${\cal F}$
    4. 6.4 Examples
      1. 6.4.1 The ID class
      2. 6.4.2 Discrete spectrum case
      3. 6.4.3 When Furstenberg systems are almost ergodic
    5. 6.5 Veech and Sarnak’s conditions are not equivalent - a counterexample
    6. 6.6 Application: an averaged Chowla property
  7. 7 Proof of the second assertion in Theorem 6.20
    1. 7.1 Ergodic theory
    2. 7.2 (a) implies the second assertion
  8. 8 Empirical approximations of self-joinings of Furstenberg systems
    1. 8.1 Without dynamics
    2. 8.2 Taking dynamics into account
    3. 8.3 Proof of Proposition 8.1
    4. 8.4 Question
  9. 9 Cross-sections of ergodic components
    1. 9.1 Measure theory
      1. 9.1.1 Notation and other conventions
      2. 9.1.2 A representation of joint distributions
    2. 9.2 Ergodic theory
  10. 10 Pinsker factor in ergodic components

##  1 Introduction

###  1.1 Presentation of the main problem and motivations

We will study the following problem of orthogonality: Given a class
$\mathscr{C}$ of topological systems111By a topological system $(X,T)$, we
mean the action of a homeomorphism $T$ on a compact metric space $X$. and a
(bounded) sequence
$\boldsymbol{u}:{\mathbb{N}}\to{\mathbb{D}}:=\\{z\in{\mathbb{C}}\colon|z|\leq
1\\}$ of zero mean, i.e.
$M(\boldsymbol{u}):=\lim_{N\to\infty}\frac{1}{N}\sum_{n\leq
N}\boldsymbol{u}(n)=0$, we say that $\boldsymbol{u}$ is orthogonal to
$\mathscr{C}$, $\boldsymbol{u}\perp\mathscr{C}$, if

(1) $\lim_{N\to\infty}\frac{1}{N}\sum_{n\leq N}f(T^{n}x)\boldsymbol{u}(n)=0$

for each $(X,T)\in\mathscr{C}$, $f\in C(X)$ and all $x\in X$ (we write then
$\boldsymbol{u}\perp(X,T)$). The main motivation to study this problem is the
celebrated Sarnak’s conjecture222Sarnak’s conjecture is in turn motivated by
the Chowla conjecture from 1965 which predicts that the autocorrelation of the
Liouville function vanish, i.e. the Liouville function is a generic point for
the Bernoulli measure $B(1/2,1/2)$ for the full shift
$\\{-1,1\\}^{{\mathbb{Z}}}$; the Chowla conjecture implies Sarnak’s conjecture
[30], [33], [9]. [30] in which $\boldsymbol{u}$ is the Möbius (or Liouville)
function and $\mathscr{C}=\mathscr{C}_{\rm ZE}$ the class of (topological)
zero entropy systems. We can also study the problem of orthogonality without
the zero-mean assumption for $\boldsymbol{u}$. Namely, by
$\boldsymbol{u}\perp(X,T)$, we mean uncorrelation:

(2) $\frac{1}{N}\sum_{n\leq
N}f(T^{n}x)\boldsymbol{u}(n)=\frac{1}{N}\sum_{n\leq
N}f(T^{n}x)\cdot\frac{1}{N}\sum_{n\leq N}\boldsymbol{u}(n)+o(1)$

(when $N\to\infty$) for all $f\in C(X)$ and $x\in X$. Note that, if $(X,T)$ is
uniquely ergodic,333$(X,T)$ is uniquely ergodic if it has a unique
$T$-invariant probability measure. By unicity, such a measure has to be
ergodic. the above requirement is equivalent to
$\lim_{N\to\infty}\frac{1}{N}\sum_{n\leq N}f(T^{n}x)\boldsymbol{u}(n)=0$ for
all $f\in C(X)$ of zero mean (with respect to the unique invariant measure)
and $x\in X$. Moreover, if $M(\boldsymbol{u})=0$, we come back to (1),
although we will see later that the natural (stronger than
$M(\boldsymbol{u})=0$) assumption on $\boldsymbol{u}$ is that the first
Gowers-Host-Kra norm vanishes: $\|\boldsymbol{u}\|_{u^{1}}=0$.

As done by Tao [34], we can change the Cesàro averaging in (1), by the
logarithmic averages (the logarithmic mean of $\boldsymbol{u}$ is zero
whenever its usual mean vanishes) and study the corresponding orthogonality
problem:

(3) $\lim_{N\to\infty}\frac{1}{\log N}\sum_{n\leq
N}\frac{1}{n}f(T^{n}x)\boldsymbol{u}(n)=0$

which, in particular, leads to the logarithmic Sarnak’s conjecture. While even
the logarithmic Sarnak’s conjecture stays open, a remarkable theorem by
Franzikinakis-Host [11] asserts that the Liouville
function444$\boldsymbol{\lambda}(n)=-1$ if the number of prime divisors
(counted with multiplicity) is odd, and it is 1 otherwise.
$\boldsymbol{\lambda}$ (and many other multiplicative functions) are
logarithmically orthogonal to all uniquely ergodic zero entropy systems. A
natural question arises to characterize those sequences which are orthogonal
(or logarithmically orthogonal) to the class UE of all uniquely ergodic
systems. In [6], this particular problem is attributed to M. Boshernitzan, and
from now on we will call it Boshernitzan’s problem. Conze, Downarowicz and
Serafin showed in [6] that Boshernitzan’s problem is non-trivial by noticing
that sequences “produced” by systems which are disjoint in the Furstenberg
sense from all ergodic automorphisms are orthogonal to the UE class. This path
of reasoning, relating the fundamental problem of Furstenberg disjointness of
measure-theoretic dynamical systems in ergodic theory [14] with the problem of
orthogonality to sequences is not new and it was applied earlier, when
studying Sarnak’s conjecture, in particular, it was successfully applied in
[11] and many other papers (see the survey [9]). Let us quickly recall the
essence of this approach.

Given $\boldsymbol{u}:{\mathbb{N}}\to{\mathbb{D}}$ (which w.l.o.g. we can
treat as a two-sided sequence), we consider
$X_{\boldsymbol{u}}:=\overline{\\{S^{k}\boldsymbol{u}\colon
k\in{\mathbb{Z}}\\}}$ the subshift generated by $\boldsymbol{u}$ ($S$ stands
for the left shift). Let555Given a dynamical system $(X,T)$, we denote by
$M(X)$ the compact space of Borel probability measures on $X$ endowed with the
weak∗-topology, and by $M(X,T)$ the closed, and always non-empty, subspace of
$T$-invariant measures.

$V(\boldsymbol{u}):=\Big{\\{}\kappa\in
M(X_{\boldsymbol{u}})\colon\Big{(}\exists
N_{k}\uparrow\infty\Big{)}\;\kappa=\lim_{k\to\infty}\frac{1}{N_{k}}\sum_{n\leq
N_{k}}\delta_{S^{n}\boldsymbol{u}}\Big{\\}}\subset M(X_{\boldsymbol{u}},S)$

denote the set of $\boldsymbol{u}$-visible measures.666$V^{\rm
log}(\boldsymbol{u})\subset M(X_{\boldsymbol{u}},S)$ is defined analogously.
In general, the sets of visible and logarithmically visible measures can be
disjoint. By a Furstenberg system of $\boldsymbol{u}$, we mean the measure-
theoretic system $(X_{\boldsymbol{u}},\kappa,S)$, for each $\kappa\in
V(\boldsymbol{u})$. Suppose now that we understand (in the sense of ergodic
properties) all Furstenberg systems of $\boldsymbol{u}$. Then, if we take any
topological system $(X,T)$ for which for each $\nu\in M(X,T)$, the measure-
theoretic systems $(X,\nu,T)$ is Furstenberg disjoint777Given two
automorphisms $(X,\nu,T)$ and $(X^{\prime},\nu^{\prime},T^{\prime})$, we
consider $J(T,T^{\prime})$ the set of joinings, i.e. of all $T\times
T^{\prime}$-invariant measures on $X\times X^{\prime}$ with marginals $\nu$
and $\nu^{\prime}$, respectively. The two automorphisms are (Furstenberg)
disjoint (we write $T\perp T^{\prime}$) if
$J(T,T^{\prime})=\\{\nu\otimes\nu^{\prime}\\}$. It is not hard to see that if
$T\perp T^{\prime}$ then at least one of the two automorphisms has to be
ergodic. If $T=T^{\prime}$, we write $J_{2}(T)$ instead of $J(T,T)$. to all
$(X_{\boldsymbol{u}},\kappa,S)$ for all $\kappa\in V(\boldsymbol{u})$, then it
is not hard to see that

$\boldsymbol{u}\perp(X,T).$

In fact, to achieve this goal, as proved in [21], we need a condition
appealing only to Furstenberg systems of $\boldsymbol{u}$, namely, we need to
see that the function $\pi_{0}:X_{\boldsymbol{u}}\to{\mathbb{D}}$,
$\pi_{0}((x_{k})):=x_{0}$ is orthogonal to the $L^{2}$-space of a certain
factor ${\cal A}(\kappa)$ of $(X_{\boldsymbol{u}},\kappa,S)$ – the factor
${\cal A}(\kappa)$, called characteristic, will depend on the class
$\mathscr{C}$ to which $(X,T)$ belongs:

(4) $\pi_{0}\perp L^{2}({\cal A}(\kappa))\text{ for all }\kappa\in
V(\boldsymbol{u}).$

This is a particular instance of the Veech condition (for the definition of
the Veech condition in the general characteristic class setup, see Section
6.5). For Sarnak’s conjecture (when $\mathscr{C}=\mathscr{C}_{\rm ZE})$ the
relevant conjecture888Veech’s conjecture was proved recently in [21]. was
formulated by Veech [35] and stated that, for each Furstenberg system $\kappa$
of $\boldsymbol{\lambda}$, $\pi_{0}$ is orthogonal to the $L^{2}$-space of the
Pinsker factor $\Pi(\kappa)$.

All the above can be repeated in the logarithmic case, and in [10] it is in
fact proved that either a Furstenberg system of $\boldsymbol{\lambda}$ is
ergodic (then the Chowla conjecture holds), or it must be disjoint from all
ergodic systems with zero entropy. The latter naturally raises a question (of
independent interest in ergodic theory) how to describe the class Erg⟂ of
automorphisms disjoint from all ergodic automorphisms.999The reader can easily
check that although the disjointness theory in ergodic theory is well
developed, see e.g. the monograph [15], but it usually considers the
disjointness question between two ergodic automorphisms. Indeed, in this case,
the problem is reduced to show that each ergodic joining is product measure.
We now pass to a description of the results of the paper.

###  1.2 Ergodic theory results

Given a non-ergodic automorphism $\widetilde{T}$, we consider its ergodic
decomposition. We usually assume that the space on which $\widetilde{T}$ acts
is of the form $(X\times Y,\mu\otimes\nu)$ and

(5) $\widetilde{T}(x,y)=(x,T_{x}(y)),$

where $T_{x}\in{\rm Aut}(Y,{\cal C},\nu)$ are ergodic. Here, the map

(6) $x\mapsto T_{x}\in{\rm Aut}(Y,\nu)\text{ is Borel}.$

Moreover, the space $C$ of ergodic components can be identified with $X$.
Strictly speaking if $\widetilde{T}$ acting on
$(\widetilde{X},\widetilde{\mathcal{B}},\widetilde{\mu})$ is non-ergodic then
its ergodic decomposition is more complicated than in (5). The space $C$ of
ergodic components is represented as $C=\bigcup_{n\geq 1}C_{n}\cup C_{\infty}$
(disjoint union of measurable sets), where in what follows $C_{\infty}$ is
replaced by $X$,

$\widetilde{X}=\bigcup_{n\geq 1}C_{n}\times\\{0,\ldots,n-1\\}\cup
C_{\infty}\times Y,$

and for each $n\geq 1$, $\widetilde{T}(c,j)=(c,j+1\mod n)$ on
$C_{n}\times\\{0,\ldots,n-1\\}$. Note however that for each $2\leq n$,
assuming that $\nu(C_{n})>0$, the automorphism $\widetilde{T}$ is not disjoint
with the ergodic rotation by 1 on $n$ points $\\{0,\ldots,n-1\\}$ (considered
with the uniform measure $\lambda_{n}$).101010Indeed, if $A\subset C_{n}$ and
$B,D\subset\\{0,\ldots,n-1\\}$ then $\rho((A\times B)\times
D):=\nu(A)\lambda_{n}(B\cap D)$ yields a non-trivial joining between
$\widetilde{T}|_{C_{n}\times\\{0,\ldots,n-1\\}}$ and the rotation by 1 on
$\\{0,\ldots,n-1\\}$ and joining the rotation in an independent way with the
remaining part of $\widetilde{T}$ yields a required joining. Taking into
account that on $C_{1}$ the automorphism $\widetilde{T}$ acts as the identity
(which is disjoint with all ergodic automorphisms) and since we are interested
in the disjointness condition with the class of ergodic automorphisms, it is
natural to assume that $\widetilde{T}$ is of the form (5). In other words,
usually, we will assume that the automorphisms under our considerations are
aperiodic (the set of periodic points is of measure zero). Note that this
prevents the measure $\nu$ from having an atom (note also that if $\mu$ has an
atom at $x_{0}$ then $\widetilde{T}$ cannot be disjoint with $T_{x_{0}}$).

Theorem A. Assume that $\widetilde{T}$ is aperiodic (of the form (5)). The
following conditions are equivalent:
(i) $\widetilde{T}\in{\rm Erg}^{\perp}$.
(ii) The extension $\widetilde{T}\to{\rm Id}_{X}$ is confined.111111See
Subsection 2.7 for the relevant definition.
(iii) $T_{x}\perp T_{x^{\prime}}$ for $\mu\otimes\mu$-a.a. $(x,x^{\prime})\in
X\times X$.121212The implication (iii) $\Rightarrow$ (i), using the approach
from [21] can be thought of to be more or less implicit in [6].

(For the proof of Theorem A, see Theorem 3.1 and its proof.)

We refer to Section 2.9 for the definition of a characteristic class ${\cal
F}$ of measure-theoretic systems, and the associate largest ${\cal F}$-factor
${\cal A}_{{\cal F}}(X,\mu,T)$ in any system $(X,\mu,T)$. Recall that the
class ZE of zero-entropy systems is the largest proper characteristic class
[21], and that the largest ZE-factor ${\cal A}_{\rm ZE}(X,\mu,T)$ is the
Pinsker factor $\Pi(T)$.

Theorem B.
(i) Let ${\cal F}$ be a characteristic class. Then $\widetilde{T}\perp{\cal
F}\cap{\rm Erg}$ if and only if the extension $\widetilde{T}|_{{\cal A}_{{\cal
F}}}\to{\rm Id}_{X}$ is confined (${\cal A}_{{\cal F}}$ is the largest ${\cal
F}$-factor of $\widetilde{T}$).
(ii) $\widetilde{T}\perp{\rm ZE}\cap{\rm Erg}$ if and only if
$\Pi(T_{x})\perp\Pi(T_{x^{\prime}})$ for $\mu\otimes\mu$-a.a.
$(x,x^{\prime})\in X\times X$.

(For the proof of Theorem B, see Corollary 6.2, Corollary 6.3 and their
proofs.)

For a bi-measurable invertible transformation $T$ of a standard Borel space
$(X,{\cal B}_{X})$, we denote by $\mathcal{I}_{T}$ the sub-sigma-algebra of
$T$-invariant Borel sets. For the definition of the weakly ergodic subspace
$F_{\rm we}(\widetilde{T})$ (or of $F_{{\rm we},0}(\widetilde{T})$) of an
automorphism $\widetilde{T}$, we refer to Subsection 2.4.

Theorem C. Assume that $f\in L^{2}(X\times Y,\mu\otimes\nu)$, and
${{\mathbbm{E}}}_{\mu\otimes\nu}[f\,|\,X]=0$. The following conditions are
equivalent:
(i) $f\perp F_{\rm we}(\widetilde{T})$.
(ii) ${{\mathbbm{E}}}_{\lambda}[f\otimes\overline{f}\,|\,X\times X]=0$ mod
$\mu\otimes\mu$ for all $\lambda\in J_{2}^{\rm RelErg}(\widetilde{T})$,
$\lambda|_{X\times X}=\mu\otimes\mu$.131313$J_{2}^{\rm RelErg}(\widetilde{T})$
stands for the subset of those members $\lambda$ of $J_{2}(\widetilde{T})$
which are relatively ergodic over the factor $(X\times X,\mu\otimes\mu)$, i.e.
the only $\widetilde{T}\times\widetilde{T}$-invariant (mod $\lambda$)
functions are $X\times X$-measurable.
(iii)
${{\mathbbm{E}}}_{\lambda}[f\otimes\overline{f}\,|\,\mathcal{I}_{\widetilde{T}\times\widetilde{T}}]=0$
mod $\lambda$ for all $\lambda\in J_{2}(\widetilde{T})$, $\lambda|_{X\times
X}=\mu\otimes\mu$.

(For the proof of Theorem C, see Theorem 4.5 and Corollary 4.9 and their
proofs.)

###  1.3 Solution of Boshernitzan’s problem

Let $\boldsymbol{u}:{\mathbb{N}}\to{\mathbb{D}}$.

Proposition D. $\boldsymbol{u}\perp\,{\rm UE}$ if and only if for each
Furstenberg system $\kappa\in V(\boldsymbol{u})$, $\pi_{0}\perp F_{{\rm
we},0}(X_{\boldsymbol{u}},\kappa,S)$.

(For the proof, see Proposition 5.1 and its proof.)

In Proposition 8.1, we describe combinatorially all self-joinings of
Furstenberg systems of $\boldsymbol{u}$. Hence, Proposition D, Theorem C
together with the von Neumann theorem describe the full solution of
Boshernitzan’s problem in terms of Furstenberg systems of $\boldsymbol{u}$,
see Remark 4.10. Proposition D has its characteristic class version
counterpart:

Corollary E. For any characteristic class ${\cal F}$, the following conditions
are equivalent:
(a) $\boldsymbol{u}\perp{\cal F}\cap{\rm UE}$.
(b) For each Furstenberg system $\kappa$ of $\boldsymbol{u}$, for each
$\lambda\in J^{\rm RelErg}_{2}(S|_{{\cal A}_{{\cal F}}},\kappa|_{{\cal
A}_{{\cal F}}})$ such that
$\lambda|_{\mathcal{I}_{S}\otimes\mathcal{I}_{S}}=\kappa|_{\mathcal{I}_{S}}\otimes\kappa|_{\mathcal{I}_{S}}$,
we have

${{\mathbbm{E}}}_{\lambda}\Bigl{[}{{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,{\cal
A}_{{\cal F}}]\otimes{{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,{\cal A}_{{\cal
F}}]\,|\,\mathcal{I}_{S}\otimes\mathcal{I}_{S}\Bigr{]}={{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,\mathcal{I}_{S}]\otimes{{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,\mathcal{I}_{S}].$

(c) For each Furstenberg system $\kappa$ of $\boldsymbol{u}$, for each
$\lambda\in J_{2}(X_{\boldsymbol{u}},\kappa,S)$ such that
$\lambda|_{\mathcal{I}_{S}\otimes\mathcal{I}_{S}}=\kappa|_{\mathcal{I}_{S}}\otimes\kappa|_{\mathcal{I}_{S}}$,
we have

${{\mathbbm{E}}}_{\lambda}\Bigl{[}{{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,{\cal
A}_{{\cal F}}]\otimes{{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,{\cal A}_{{\cal
F}}]\,|\,\mathcal{I}_{S\times
S}\Bigr{]}={{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,\mathcal{I}_{S}]\otimes{{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,\mathcal{I}_{S}].$

(For the proof, see Corollary 6.7 and its proof.)

Given a characteristic class ${\cal F}$, we denote by $\mathscr{C}_{{\cal F}}$
the class of topological systems for which all visible invariant measures
yield automorphisms in ${\cal F}$. (An invariant measure $\mu$ in $(X,T)$ is
visible if there exists $x\in X$ which is quasi-generic for $\mu$.) The main
result of [21] stated that the Veech condition (see Section 6.5) is equivalent
to the Sarnak condition: $\boldsymbol{u}\perp\mathscr{C}_{{\cal F}}$, provided
that ${\cal F}$ is a so called ec-class. It was left as an open problem
whether the equivalence of these two conditions holds for any characteristic
class. It turns out that this is not the case as the following result shows:

Proposition F. There exist a characteristic class ${\cal F}$ and
$\boldsymbol{u}:{\mathbb{N}}\to\\{-1,1\\}$ which is generic for a measure
$\kappa$ on $\\{-1,1\\}^{{\mathbb{Z}}}$ such that
$\boldsymbol{u}\perp\mathscr{C}_{{\cal F}}$ but the Veech condition fails.

(For the proof, see Section 6.5.)

###  1.4 Applications

We refer to Subsection 4.1 and Subsection 6.6 for the definitions of Gowers-
Host-Kra (GHK) norms $\|\cdot\|_{u^{s}}$ and of an averaged Chowla
property141414An averaged Chowla property (even in a quantitive version) has
been proved to hold for the Liouville function (and many other multiplicative
functions) by Matomäki-Radziwiłł-Tao [27].. Denote by DISP the
(characteristic) class of automorphisms having discrete spectrum. Our main
application is the following result:

Corollary G. Let $\boldsymbol{u}:{\mathbb{N}}\to{\mathbb{D}}$,
$\|\boldsymbol{u}\|_{u^{1}}=0$.151515This condition is equivalent to so called
“zero mean property on typical short interval”, cf. Proposition 4.1. The
following conditions are equivalent:
(i) $\boldsymbol{u}$ satisfies an averaged Chowla property.
(ii) $\boldsymbol{u}\perp{\rm UE}\cap{\rm DISP}$.
(iii) $\boldsymbol{u}\perp\mathscr{C}_{\rm DISP}$.

(For the proof, see Theorem 6.20 and its proof.)

The equivalence between (ii) and (iii) seems to be exceptional for the class
DISP, we do not expect it to hold for a general characteristic class. It
obviously does not hold for the class of all automorphisms but more
interestingly, it does not hold for the class ZE. Indeed, for all sequences
$\boldsymbol{u}$ of the form (37), we have $\|\boldsymbol{u}\|_{u^{1}}=0$ by
Remark 5.11, and they satisfy $\boldsymbol{u}\perp{\rm UE}\cap{\rm ZE}$.
However, by Remark 5.12, they do not satisfy (iii), as their own topological
entropy is zero. Denote by CE the class of topological systems whose set of
ergodic invariant measures is countable. In light of what was said, the
following result looks somewhat surprising.

Corollary H. Let ${\cal F}$ be an arbitrary characteristic class. Let
$\boldsymbol{u}:{\mathbb{N}}\to{\mathbb{D}}$, $\|\boldsymbol{u}\|_{u^{1}}=0$.
Then $\boldsymbol{u}\perp{\rm UE}\cap{\cal F}$ if and only if
$\boldsymbol{u}\perp{\rm CE}\cap{\cal F}$.161616The class ${\rm CE}\cap{\cal
F}$ is defined unambiguously due to Proposition 2.15

(For the proof, see Proposition 6.8 and its proof.)

It follows that in the theorem of Frantzikinakis-Host [11], the final
statement of orthogonality (in the logarithmic sense) of the Liouville
function to all zero entropy systems from CE, is in fact equivalent to the
orthogonality to all zero entropy uniquely ergodic systems.

Corollary I. The mean slowly varying functions171717$\boldsymbol{u}$ is called
mean slowly varying if $\lim_{N\to\infty}\frac{1}{N}\sum_{n\leq
N}|\boldsymbol{u}(n)-\boldsymbol{u}(n+1)|=0$. are multipliers of UE⟂. That is,
if $\boldsymbol{u}:{\mathbb{N}}\to{\mathbb{D}}$ is mean slowly varying and
$\boldsymbol{v}\perp{\rm UE}$ then $\boldsymbol{u}\cdot\boldsymbol{v}\perp{\rm
UE}$.

(For the proof, see Proposition 5.18 and its proof.)

Here, it seems that the class of mean slowly varying function should precisely
be the class of multipliers of UE⟂.

The next fact is for multiplicative functions.

Corollary J. The only pretentious multiplicative functions
$\boldsymbol{u}:{\mathbb{N}}\to{\mathbb{D}}$ orthogonal to UE are Archimedean
characters $n\mapsto n^{it}$.

(For the proof, see Corollary 5.6.)

It is reasonable to expect that the result holds in the class of all
multiplicative functions.

###  1.5 Auxiliary results

In the last three sections we prove some new ergodic results which are used to
obtain the aforementioned theorems, but which are also of an independent
interest. Namely, in Section 8, we provide a combinatorial description of all
self-joinings of all Furstenberg systems of a bounded arithmetic function
$\boldsymbol{u}$. In Section 9181818The proof has been written by Tim Austin.,
we prove the existence of a so-called _relative ergodic decomposition_ , with
respect to a sub-sigma-algebra of $\mathcal{I}_{T}$. Finally in Section 10, we
show that the “trace” of the Pinsker factor of an automorphism on almost all
ergodic component is equal to the Pinsker factor of the ergodic component.
This result seems to be part of folklore, but we could not find its proof in
the literature.

##  2 Preparatory material

###  2.1 Lifting lemma

We will need the following lemma about lifting of generic point (along a
subsequence) to generic sequences:

######  Theorem 2.1 ([21]).

Assume that $(Y,S)$ and $(X,T)$ are topological systems. Let $\nu\in M(X,T)$,
$u\in Y$ be generic191919This means that $\frac{1}{N_{m}}\sum_{n\leq
N_{m}}g(S^{n}u)\to\int_{Y}g\,d\nu$ for all $g\in C(Y)$. along an increasing
sequence $(N_{m})$ for $\kappa\in M(Y,S)$ and $\rho\in J(\kappa,\nu)$. Then
there exist a sequence $(x_{n})\subset X$ and a subsequence $(N_{m_{\ell}})$
such that $(S^{n}u,x_{n})$ is generic along $(N_{m_{\ell}})$ for $\rho$ and
the set $\\{n\geq 0:\>x_{n+1}\neq Tx_{n}\\}$ is of the form $(b_{k})$ with
$b_{k+1}-b_{k}\to\infty$ when $k\to\infty$.

######  Remark 2.2.

As proved in [21], this result is also true for the logarithmic averages.

######  Remark 2.3.

First lifting lemmas of that kind can be found in [20], [6].

###  2.2 Orbital uniquely ergodic models

We will also need the following:

######  Theorem 2.4 ([1]).

Assume that $(X,T)$ is uniquely ergodic and let $(x_{n})\subset X$ satisfy
$\\{n\geq 0:\>x_{n+1}\neq Tx_{n}\\}$ is of the form $(b_{k})$ with
$b_{k+1}-b_{k}\to\infty$ when $k\to\infty$. Consider
$\underline{x}:=(x_{n})\in X^{{\mathbb{N}}}$, the latter space with the one-
sided shift $S$. Set $Y:=\overline{\\{S^{n}\underline{x}:\>n\geq 0\\}}\subset
X^{{\mathbb{N}}}$. Then the subshift $(Y,S)$ is uniquely ergodic and (measure-
theoretically) isomorphic to the original system.

###  2.3 Joinings and Markov operators

Given dynamical systems $(Z,{\cal D},\nu,R)$ $(Z^{\prime},{\cal
D}^{\prime},\nu^{\prime},R^{\prime})$ and a joining $\rho\in J(R^{\prime},R)$,
we let $\Phi_{\rho}$ denote the corresponding operator
$\Phi_{\rho}:L^{2}(Z^{\prime},\nu^{\prime})\to L^{2}(Z,\nu)$ defined by

(7) $\int_{Z}\Phi_{\rho}(f^{\prime})\overline{f}\,d\nu=\int_{Z^{\prime}\times
Z}f^{\prime}(z^{\prime})\overline{f(z)}\,d\rho(z^{\prime},z)$

for all $f^{\prime}\in L^{2}(Z^{\prime},\nu^{\prime})$ and $f\in
L^{2}(Z,\nu)$. Then $\Phi_{\rho}$ is Markov:
$\Phi_{\rho}\mathbbm{1}=\Phi_{\rho}^{\ast}\mathbbm{1}=\mathbbm{1}$ and
$\Phi_{\rho}(g^{\prime})\geq 0$ whenever $g^{\prime}\geq 0$. Moreover,
$\Phi_{\rho}$ is intertwining $R$ and $R^{\prime}$: $\Phi_{\rho}\circ
R^{\prime}=R\circ\Phi_{\rho}$. And vice versa: if
$\Phi:L^{2}(Z^{\prime},\nu^{\prime})\to L^{2}(Z,\nu)$ is a Markov operator
intertwining $R$ and $R^{\prime}$, then $\Phi=\Phi_{\rho}$ for a unique
joining $\rho\in J(R^{\prime},R)$, see e.g. [15].

###  2.4 Weakly ergodic part of a dynamical system

Given a dynamical system $(Z,{\cal D},\nu,R)$ we call $f\in L^{2}(Z,{\cal
D},\nu)$ weakly ergodic if there exists an ergodic $(Z^{\prime},{\cal
D}^{\prime},\nu^{\prime},R^{\prime})$ and a joining $\rho\in J(R^{\prime},R)$
such that $f\in{\rm Im}(\Phi_{\rho})$, where
$\Phi_{\rho}:L^{2}(Z^{\prime},\nu^{\prime})\to L^{2}(Z,\nu)$ is the Markov
operator corresponding to $\rho$. The closed subspace $F_{\rm we}(R)$ spanned
by the weakly ergodic elements is called the weakly ergodic part of
$L^{2}(Z,\nu)$ for $R$. Note that

(8) If $f\perp F_{\rm we}(R)$ then $f\in L^{2}_{0}(Z,\nu)$, i.e.
$\int_{Z}f\,d\nu=0$.

In what follows when considering the problem of orthogonality, unless stated
otherwise, we will only consider the zero mean functions. In order to treat
the non-zero mean case, we consider the space $F_{{\rm we},0}(R)$ generated by
${\rm Im}(\Phi_{\rho}|_{L^{2}_{0}(Z^{\prime},\nu^{\prime})})$ for all ergodic
$R^{\prime}$.

######  Remark 2.5.

If a system $(Z,{\cal D},\nu,R)$ is non-ergodic then $F_{\rm we}(R)$ is never
dense in it. Indeed, take any non-trivial factor ${\cal A}\subset{\cal D}$ of
$R$ which belongs to ${\rm Erg}^{\perp}$ (we can take for ${\cal A}$ the
sigma-algebra of invariant sets). If $g\in L^{2}_{0}({\cal A})$ then
$\int\Phi_{\rho}(f^{\prime})\overline{g}\,d\nu=\int
f^{\prime}\otimes\overline{g}\,d\rho=0$, the latter because $R^{\prime}\perp
R|_{{\cal A}}$.

###  2.5 Koopman operator and spectral measures

Given a dynamical system $(Z,{\cal D},\nu,R)$, we have the corresponding
Koopman operator (denoted also by $R$) acting on $L^{2}(Z,\mathcal{D},\nu)$ by

$f\mapsto Rf:=f\circ R.$

This operator is unitary. Given $f\in L^{2}(Z,\mathcal{D},\nu)$, by
$\sigma_{f}$ (or $\sigma_{f,R}$ if an ambiguity can arise) we denote the
spectral measure of $f$ which is a finite positive (Borel) measure on
$\mathbb{S}^{1}$ determined by

$\widehat{\sigma}(n):=\int_{\mathbb{S}^{1}}z^{n}\,d\sigma_{f}(z)=\int_{Z}R^{n}(f)\cdot\overline{f}\,d\nu\text{
for all }n\in{\mathbb{Z}}.$

Then, $f$ is an eigenfunction corresponding to an eigenvalue $e^{2\pi
i\alpha}$ of $R$ if and only if its spectral measure is atomic concentrated at
$e^{2\pi i\alpha}$. $R$ is said to have discrete spectrum if the space
$L^{2}(Z,\mathcal{D},\nu)$ is spanned by the eigenfunctions of $R$. If $R$ is
ergodic then its eigenvalues form a (countable) subgroup of $\mathbb{S}^{1}$.

###  2.6 Furstenberg systems of arithmetic functions

Given $\boldsymbol{u}:{\mathbb{N}}\to{\mathbb{D}}$ (${\mathbb{D}}$ stands for
the unit disc; we freely treat $\boldsymbol{u}$ as a two-sided sequence, e.g.
$\boldsymbol{u}(-n)=\boldsymbol{u}(n)$), by $V(\boldsymbol{u})$, we denote the
set of shift-invariant measures on $(X_{\boldsymbol{u}},{\cal
B}(X_{\boldsymbol{u}}))$ for which $\boldsymbol{u}\in X_{\boldsymbol{u}}$ is
quasi-generic. Recall that $(X_{\boldsymbol{u}},{\cal
B}(X_{\boldsymbol{u}}),\kappa,S)$ is a Furstenberg system of $\boldsymbol{u}$.
Let $\pi_{0}(y)=y_{0}$ for $y\in X_{\boldsymbol{u}}$; then
$\pi_{n}=\pi_{0}\circ S^{n}$, $n\in{\mathbb{Z}}$, is a stationary process
(once, we fix $\kappa\in V(\boldsymbol{u})$).

###  2.7 Confined extensions

Confined extensions (see the definition below) are systematically studied in
[4] (we recall some known results for completeness). In the present paper, we
consider them only in the context of extensions of identity. Let $R\in{\rm
Aut}(Z,\mathcal{D},\kappa)$ and $T\in{\rm Aut}(X,{\cal B},\mu)$.

######  Lemma 2.6.

If $\Phi_{\rho}:L^{2}(Z,\kappa)\to L^{2}(X,\mu)$ and $\pi_{0}\in L^{2}(X,\mu)$
then

$\pi_{0}\perp{\rm Im}\,\Phi_{\rho}\text{ if and only if }\pi_{0}\perp{\rm
Im}(\Phi_{\rho}\circ\Phi^{\ast}_{\rho}).$

###### Proof.

Assume that $\pi_{0}\perp{\rm Im}(\Phi_{\rho}\circ\Phi^{\ast}_{\rho})$. Then
for each $g\in L^{2}(Z,\kappa)$, we have $g=g_{1}+g_{2}$, where
$g_{1}\in\overline{{\rm Im}\,\Phi^{\ast}_{\rho}}$ and $g_{2}\perp{\rm
Im}\,\Phi^{\ast}_{\rho}$. Hence

$\int\pi_{0}\Phi_{\rho}(g)\,d\mu=\int\pi_{0}\Phi_{\rho}(g_{1})\,d\mu+\int\pi_{0}\Phi_{\rho}(g_{2})\,d\mu=$
$\int\pi_{0}\Phi_{\rho}(g_{1})\,d\mu+\int\Phi^{\ast}_{\rho}(\pi_{0})g_{2}\,d\kappa=\int\pi_{0}\Phi_{\rho}(g_{1})\,d\mu$

and since $g_{1}$ can be approximated by $\Phi_{\rho}^{\ast}(h)$ with $h\in
L^{2}(X,\mu)$, the claim follows.

Below, we restrict to $L^{2}_{0}$-spaces. By Lemma 2.6, we obtain the
following:

######  Lemma 2.7 (del Junco-Rudolph).

Under the above assumption, $\Phi_{\rho}=0$ if and only if
$\Phi_{\rho}\circ\Phi^{\ast}_{\rho}=0$.

Assume now that we have an extension $\widetilde{R}\in{\rm
Aut}(\widetilde{Z},\widetilde{\mathcal{D}},\widetilde{\kappa})$ of $R$. If
$\widetilde{\lambda}$ is any self-joining of $\widetilde{\kappa}$ and
$\lambda$ is its restriction to $Z\times Z$ then we easily check that

(9)
$\Phi_{\lambda}=\mathrm{proj}_{L^{2}(Z)}\circ\Phi_{\widetilde{\lambda}}|_{L^{2}(Z)}.$

######  Lemma 2.8.

Assume that $\widetilde{\rho}$ is a joining of $\widetilde{R}$ and $T$ (and
$\rho$ stands for its restriction to $Z\times X$). Then

$\Phi_{\rho}^{\ast}\circ\Phi_{\rho}=\mathrm{proj}_{L^{2}(Z)}\circ\Phi^{\ast}_{\widetilde{\rho}}\circ\Phi_{\widetilde{\rho}}|_{L^{2}(Z)}.$

###### Proof.

Similarly as in (9), we have
$\Phi_{\rho}=\Phi_{\widetilde{\rho}}|_{L^{2}(Z)}$. Then, for $f,g\in
L^{2}(Z)$,

$\langle\Phi^{\ast}_{\rho}\circ\Phi_{\rho}f,g\rangle=\langle\Phi_{\rho}f,\Phi_{\rho}g\rangle=$
$\langle\Phi_{\widetilde{\rho}}f,\Phi_{\widetilde{\rho}}g\rangle=\langle\Phi^{\ast}_{\widetilde{\rho}}\circ\Phi_{\widetilde{\rho}}f,g\rangle=\langle\mathrm{proj}_{L^{2}(Z)}\circ\Phi^{\ast}_{\widetilde{\rho}}\circ\Phi_{\widetilde{\rho}}f,g\rangle.$

Following [4], an extension $\widetilde{Z}\to Z$ is called confined if for
each self-joining $\widetilde{\lambda}\in J_{2}(\widetilde{Z})$, we have

$\widetilde{\lambda}|_{Z\times
Z}=\kappa\otimes\kappa\;\Rightarrow\widetilde{\lambda}=\widetilde{\kappa}\otimes\widetilde{\kappa}.$

We have now the following lifting disjointness result (first proved in [4]).

######  Proposition 2.9.

Assume that $\widetilde{Z}\to Z$ is a confined extension and let $Z\perp X$.
Then $\widetilde{Z}\perp X$.

###### Proof.

Define $\widetilde{\lambda}\in J_{2}(\widetilde{R})$ so that
$\Phi_{\widetilde{\lambda}}=\Phi^{\ast}_{\widetilde{\rho}}\circ\Phi_{\widetilde{\rho}}$.
Then by this and (9),

$\Phi_{\lambda}=\mathrm{proj}_{L^{2}(Z)}\circ\Phi^{\ast}_{\widetilde{\rho}}\circ\Phi_{\widetilde{\rho}}|_{L^{2}(Z)}.$

But now, by Lemma 2.8, we obtain that
$\Phi_{\lambda}=\Phi_{\rho}^{\ast}\circ\Phi_{\rho}$. Since $\Phi_{\rho}=0$,
also $\Phi_{\lambda}=0$ and therefore $\Phi_{\widetilde{\lambda}}=0$ by our
assumption. The result follows from Lemma 2.7.

It follows easily that each confined extension of an identity is disjoint from
all ergodic automorphisms. In fact, in Theorem 3.1, we will show that an
automorphism $R$ is disjoint from all ergodic automorphisms if and only if it
is a confined extension of an identity.

######  Lemma 2.10.

Assume that $S\in{\rm Aut}(Y,{\cal C},\nu)$. Then the set

$S^{\not\perp}:=\\{R\in{\rm Aut}(Z,\mathcal{D},\kappa):\>R\not\perp S\\}$

cannot contain an uncountable family of pairwise disjoint automorphisms.

###### Proof.

Assume that we have $S^{\not\perp}\ni R_{i}\in{\rm
Aut}(Z,\mathcal{D},\kappa)$, $i\in I$ (uncountable) are pairwise disjoint.
Now, suppose that $\Phi_{i}:L^{2}(Z,\mathcal{D},\kappa)\to L^{2}(Y,{\cal
C},\nu)$ is a non-trivial intertwining Markov operator. Take $f,g\in
L^{2}_{0}(Z,\mathcal{D},\kappa)$, then, for each $i\neq j$, we have

$\langle\Phi_{i}f,\Phi_{j}g\rangle=\langle\Phi_{j}^{\ast}\Phi_{i}f,g\rangle=0$

as $\Phi^{\ast}_{j}\Phi_{i}=0$ is an intertwining operator of $R_{i}$ and
$R_{j}$, so on $L_{0}^{2}$ it is zero. So the images via $\Phi_{i}$’s are
pairwise orthogonal, and we obtain a contradiction by separability.

###  2.8 Joinings of a non-ergodic automorphism with an ergodic system

Assume that $\widetilde{T}(x,y)=(x,T_{x}(y))$ is an automorphism of the form
(5), acting on the space $(X\times Y,\mu\otimes\nu)$. Let us fix $R$ an
ergodic automorphism of $(Z,\mathcal{D},\kappa)$. Let $\rho\in
J(\widetilde{T},R)$. Then we have factors:

$((X\times Y)\times Z,\rho)\to(X\times Y,\mu\otimes\nu)\to(X,\mu),$

so let us disintegrate $\rho$ over $(X,\mu)$:

(10) $\rho=\int_{X}\rho_{x}\,d\mu(x).$

Note that since we consider the factor on which the action is the identity,

(11) the measures $\rho_{x}$ are $\widetilde{T}\times R$-invariant.

Moreover, for (measurable) $A\subset X\times Y$ and $B\subset Z$, we have

$\rho(A\times B)=\int_{X}\rho_{x}(A\times B)\,d\mu(x),$

so

$(\mu\otimes\nu)(A)=\rho(A\times Z)=\int_{X}\rho_{x}(A\times Z)\,d\mu(x)$

and

$\kappa(B)=\rho((X\times Y)\times B)=\int_{X}\rho_{x}((X\times Y)\times
B)\,d\mu(x),$

so by that (and in view of (11)):

(12) $\rho_{x}\in J((T_{x},\nu),(R,\kappa))$

by the uniqueness of disintegration of $\mu\otimes\nu$ (over $\mu$) and the
ergodicity of $\kappa$.

And vice versa: if (12) holds and the map $x\mapsto\rho_{x}$ is measurable
then the formula (10) defines a $\rho\in J(\widetilde{T},R)$. This observation
is quite meaningful and, given $\Phi_{\rho}$, allows us to produce more Markov
operators $\Phi_{\rho_{D}}$ with $\rho_{D}\in J(\widetilde{T},R)$, indexed by
measurable subsets $D\subset X$, by:

(13) $\begin{array}[]{c}(\rho_{D})_{x}=\rho_{x},\text{ for }x\in D\text{
and}\\\ (\rho_{D})_{x}=\nu\otimes\kappa\text{ otherwise.}\end{array}$

In what follows, in most cases we will only consider real-valued functions.
Note that Markov operators send real-valued functions to real-valued
functions. Moreover, the mean of a function is zero if and only if the mean of
its real part and of imaginary part are zero.

######  Lemma 2.11.

Assume that $\rho\in J(R,\widetilde{T})$ with $R$ ergodic. Let $f\in
L^{2}(X\times Y,\mu\otimes\nu)$ and $g\in L^{2}_{0}(Z,\mathcal{D},\kappa)$ be
real-valued functions and assume that $f\perp F_{{\rm we},0}(\widetilde{T})$
(in particular, $\int f\Phi_{\rho}(g)\,d(\mu\otimes\nu)=0$). Then, for a.a.
$x\in X$, $\int_{Y}f\Phi_{\rho_{x}}(g)\,d\nu=0$.

###### Proof.

Suppose that for some $\varepsilon_{0}>0$,

$D:=\Big{\\{}x\in
X:\;\int_{Y}f\Phi_{\rho_{x}}(g)\,d\nu\geq\varepsilon_{0}\Big{\\}}=\Big{\\{}x\in
X\colon\int_{Y\times Z}f\otimes g\,d\rho_{x}\geq\varepsilon_{0}\Big{\\}}$

has positive $\mu$-measure. Note that since $\int g\,d\kappa=0$, we have
$\int\Phi_{\rho_{x}}(g)\,d\nu=\int g\,d\kappa=0$. In particular, if
$\rho_{x}=\nu\otimes\kappa$ then
$\Phi_{\rho_{x}}(g)=\int\Phi_{\rho_{x}}(g)\,d\kappa=0$. By our assumption,
$f\perp\Phi_{\rho_{D}}(g)$. But

$\int f\Phi_{\rho_{D}}(g)\,d(\mu\otimes\nu)=\int
f(x,y)g(z)\,d\rho_{D}((x,y),z)=$ $\int\Big{(}\int
f(x,y)g(z)\,d(\rho_{D})_{x}((x,y)z)\Big{)}d\mu(x)=$ $\int_{D}\Big{(}\int
f(x,y)\Phi_{\rho_{x}}(g)(y)\,d\nu(y)\Big{)}d\mu(x)\geq\varepsilon_{0}\mu(D)>0,$

a contradiction.

In the above sense, being orthogonal to Markov images of ergodic systems is
shifted to examine Markov images of ergodic systems on the (ergodic) fibers.

###  2.9 Characteristic classes

A class ${\cal F}$ of measure-theoretic dynamical systems (implicitly closed
under isomorphism) is called characteristic if it is closed under factors and
countable joinings (the latter implies that it is closed under inverse
limits). An _${\cal F}$ -factor_ of $(Z,{\cal D},\kappa,R)$ is an
$R$-invariant sub-sigma-algebra ${\cal A}$ of ${\cal D}$ such that the action
of $R$ restricted to ${\cal A}$ defines a system in ${\cal F}$. One of the
reasons to consider characteristic classes is the following (for the proof,
see e.g. [29]):

######  Theorem 2.12.

If $\mathcal{F}$ is a characteristic class and $R\in{\rm
Aut}(Z,\mathcal{D},\kappa)$ then the largest ${\cal F}$-factor ${\cal
A}_{{\cal F}}={\cal A}_{{\cal F}}(Z,{\cal D},\kappa,R)\subset{\cal D}$ of
$(Z,{\cal D},\kappa,R)$ always exists. Moreover, any joining $\rho$ of $R$
with some $(X,\mu,T)\in{\cal F}$ is the relatively independent extension of
the relevant restriction of the joining to an element of $J(R|_{{\cal
A}_{{\cal F}}},T)$. Equivalently, for all $f\in L^{2}(\kappa)$ and all $g\in
L^{2}(\mu)$,

$\int_{Z\times X}f(z)\,g(x)\,d\rho=\int_{Z\times
X}{{\mathbbm{E}}}_{\rho}\bigl{[}f\,|\,{\cal A}_{{\cal
F}}\bigr{]}(z)\,g(x)\,d\rho(z,x)$

######  Corollary 2.13.

Assume that $\mathcal{F}$ is a characteristic class and $R\in{\rm
Aut}(Z,\mathcal{D},\kappa)$. Let $S\in\mathcal{F}$. Then $S\perp R$ if and
only if $S\perp R|_{{\cal A}_{{\cal F}}()}$.

Of course the class ZE of zero entropy automorphisms is a characteristic
class.

######  Proposition 2.14 ([21]).

If ${\cal F}$ is a proper characteristic class then ${\cal F}\subset$ZE.

It has been already noticed in [21] that if ${\cal F}$ is a characteristic
class and $T$ is automorphism on $(X,{\cal B},\mu)$ such that a.a. its ergodic
components are in ${\cal F}$ then $T$ need not belong to ${\cal F}$. However,
the following general property holds.

######  Proposition 2.15.

Let ${\cal F}$ be a characteristic class and let $T$ be an automorphism of
$(X,{\cal B},\mu)$. If $\mu$ has a countable ergodic decomposition with
respect to $T$, then $(X,{\cal B},\mu,T)$ is in ${\cal F}$ if and only if
$(X,{\cal B},\mu_{i},T)$ is in ${\cal F}$ for every ergodic component
$\mu_{i}$.

We postpone the proof of this proposition to Section 6.3.

###  2.10 Measurable selectors

We recall:

######  Theorem 2.16 (Kallman [19]).

Suppose that $X$ is a standard Borel space and $Y$ is a separable topological
space metrizable by a complete metric. Suppose that $A\subset X\times Y$ is
Borel and that for each $x\in X$, $A_{x}:=\\{y\in Y:\>(x,y)\in A\\}$ is a
countable union of compact sets ($\sigma$-compactness of $A_{x}$). Let
$f:X\times Y\to X$ be projection. Then $f(A)$ is Borel and there is a Borel
section $f(A)\to A$ of the map $f|_{A}$.

We also recall the following useful result (see Theorem 4.5.2 in [32]):

######  Theorem 2.17.

Assume that $X,Y$ are Polish spaces and let $A\subset X$ be Borel. Let $f:A\to
Y$. Then $f$ is Borel if and only if ${\rm graph}(f)$ is a Borel subset of
$A\times Y$.

For example, by (6), the set $\\{(x,T_{x}):\>x\in X\\}$ is a Borel subset of
$X\times{\rm Aut}(Y,\nu)$.

PART I

##  3 Automorphisms disjoint from all ergodic systems

Let $\widetilde{T}(x,y)=(x,T_{x}y)$ be an automorphism of $(X\times
Y,\mu\otimes\nu)$, with $x\mapsto T_{x}$ being the ergodic decomposition of
$\widetilde{T}$. We aim at proving the following theorem:

######  Theorem 3.1.

The following conditions are equivalent:
(i) $\widetilde{T}\in{\rm Erg}^{\perp}$.
(ii) The extension $\widetilde{T}\to{\rm Id}_{X}$ is confined.
(iii) For $\mu\otimes\mu$-a.e. $(x,x^{\prime})\in X\times X$, we have
$T_{x}\perp T_{x^{\prime}}$.

######  Corollary 3.2.

If $\widetilde{T}\in{\rm Erg}^{\perp}$ then $h(\widetilde{T})=0$.

###### Proof.

Ergodic positive entropy transformations are not disjoint because of Sinai’s
theorem.

######  Corollary 3.3.

If there exists $S\in{\rm Erg}$ such that $\\{T_{x}\colon x\in X\\}\subset
S^{\not\perp}$ then $\widetilde{T}\not\perp{\rm Erg}$.

###### Proof.

If $\mu$ has an atom, say $\mu(\\{x_{0}\\})>0$, then also
$\mu\otimes\mu(\\{(x_{0},x_{0})\\})>0$ and (iii) of Theorem 3.1 applies.
Otherwise, use Lemma 2.10.

###  3.1 Proof of (ii) $\Rightarrow$ (i)

The proof follows from Proposition 2.9 since classically ${\rm Id}\in{\rm
Erg}^{\perp}$.

###  3.2 Proof of (iii) $\Rightarrow$ (ii)

Assume that $\rho\in J_{2}(\widetilde{T})$, $\rho|_{X\times X}=\mu\otimes\mu$.
Then $\rho=\int_{X\times X}\rho_{x,x^{\prime}}\,d\mu(x)d\mu(x^{\prime})$,
where $\rho_{x,x^{\prime}}$ is $T_{x}\times T_{x^{\prime}}$-invariant.
Moreover,

(14) $\mu\otimes\nu(A\times B)=\rho(A\times B\times X\times Y)=\int_{A\times
X}\rho_{x,x^{\prime}}(B\times Y)\,d\mu(x)d\mu(x^{\prime}).$

By substituting $A=X$, we first obtain that the measure
$B\mapsto\int\rho_{x,x^{\prime}}(B\times Y)\,d\mu(x)d\mu(x^{\prime})$ is
$\nu$, and since $\nu$ is ergodic and the integrand measures are
$T_{x}$-invariant, we must have $\rho_{x,x^{\prime}}\in
J(T_{x},T_{x^{\prime}})$. 202020 Returning to (14), we obtain that if
$(x,x^{\prime})\mapsto\rho_{x,x^{\prime}}\in J_{2}(T_{x},T_{x^{\prime}})$ is
measurable then $\rho\in J_{2}(\widetilde{T})$:  $\int_{A\times
X}\rho_{x,x^{\prime}}(B\times Y)\,d\mu(x)d\mu(x^{\prime})=\int_{A\times
X}\nu(B)\,d\mu(x)d\mu(x^{\prime})=\mu\otimes\nu(A\times B).$ But $T_{x}\perp
T_{x^{\prime}}$ for $\mu\otimes\mu$-a.e. $x,x^{\prime}$, so
$\rho_{x,x^{\prime}}=\nu\otimes\nu$ and $\rho$ is the product measure.

###  3.3 Disjointness in ${\rm Aut}(X,\mu)$

We assume that $X$ is a compact metric space, and $\mu$ is a (probability)
Borel measure. Then ${\rm Aut}(X,\mu)$ is a Polish group with the strong
topology. Furthermore, the space $M(X\times X)$ of (Borel, probability)
measures on $X\times X$ is a compact metric space with the weak∗-topology.
Note that $C_{2}(\mu)$ which is the set of couplings of $\mu$ with itself,
i.e. of members in $M(X\times X)$ with both marginals $\mu$, is a closed
subset of $M(X\times X)$. We aim at proving the following:

######  Proposition 3.4.

The set $\\{(S,T):\>S,T\in{\rm Aut}(X,\mu),S\perp T\\}$ is a Borel subset of
${\rm Aut}(X,\mu)\times{\rm Aut}(X,\mu)$ (in fact it is a $G_{\delta}$-set).

We will adapt del Junco’s proof of Theorem 1 [17] to our needs. Given two
finite (measurable) partitions $P,Q$ of $X$ and $\delta,\varepsilon>0$, we let

$\mathscr{O}(P,\varepsilon,Q,\delta)$

denote the set of pairs $(S,T)$ of automorphisms of $(X,\mu)$ satisfying: if
$\rho\in C_{2}(\mu)$ then212121We consider the following pseudo-metrics
“responsible” for the topologies: $d(T,T^{\prime};Q):=\sum_{q\in
Q}\mu(Tq\triangle T^{\prime}q)$ and
$d(\rho,\rho^{\prime};Q):=\sum_{q_{1},q_{2}\in Q}|\rho(q_{1}\times
q_{2})-\rho^{\prime}(q_{1}\times q_{2})|$. (notation $(\cdot)_{\ast}(\rho)$
stands for the image of the measure $\rho$ under the map “$\cdot$”)

$d(\rho,(S\times
T)_{\ast}\rho;Q)<\delta\;\Longrightarrow\;d(\rho,\mu\otimes\mu;P)<\varepsilon.$

Obviously, if $0<\delta^{\prime}<\delta$ then

$\mathscr{O}(P,\varepsilon,Q,\delta)\subset\mathscr{O}(P,\varepsilon,Q,\delta^{\prime}).$

In fact, we have the following:

######  Lemma 3.5.

$\mathscr{O}(P,\varepsilon,Q,\delta)\subset{\rm
Int}(\mathscr{O}(P,\varepsilon,Q,\delta/3)).$

###### Proof.

Assume that $(S,T)\in\mathscr{O}(P,\varepsilon,Q,\delta)$. Suppose that (the
pseudo-metrics in the product space giving the relevant topology are given by
the relevant “max”)

$d((S,T),(S^{\prime},T^{\prime});Q)<\delta/(3|Q|).$

We want to show that
$(S^{\prime},T^{\prime})\in\mathscr{O}(P,\varepsilon,Q,\delta/3)$. So take
$\rho\in C_{2}(\mu)$ and assume that $d(\rho,(S^{\prime}\times
T^{\prime})_{\ast}\rho;Q)<\delta/3$. Then

$d((S^{\prime}\times T^{\prime})_{\ast}\rho,(S\times T)_{\ast}\rho;Q)\leq
d((S^{\prime}\times T^{\prime})_{\ast}\rho,(S\times
T^{\prime})_{\ast}\rho;Q)+$ $d((S\times T^{\prime})_{\ast}\rho,(S\times
T)_{\ast}\rho;Q)=$ $\sum_{q_{1},q_{2}\in Q}|\rho(S^{\prime}q_{1}\times
T^{\prime}q_{2})-\rho(Sq_{1}\times T^{\prime}q_{2})|+\sum_{q_{1},q_{2}\in
Q}|\rho(Sq_{1}\times T^{\prime}q_{2})-\rho(Sq_{1}\times Tq_{2})|\leq$
$\sum_{q_{1},q_{2}\in Q}\rho((S^{\prime}q_{1}\triangle Sq_{1})\times
X)+\sum_{q_{1},q_{2}\in Q}\rho(X\times(T^{\prime}q_{2}\triangle Tq_{2})\leq$
$\sum_{q_{1},q_{2}\in Q}\mu((S^{\prime}q_{1}\triangle
Sq_{1})+\sum_{q_{1},q_{2}\in Q}\mu(T^{\prime}q_{2}\triangle Tq_{2})\leq
2|Q|\delta/(3|Q|)=\frac{2}{3}\delta.$

It follows that

$d(\rho,(S\times T)_{\ast}(\rho);Q)\leq\frac{2}{3}\delta+\delta/3=\delta$

and since $(S,T)\in\mathscr{O}(P,\varepsilon,Q,\delta)$,
$d(\rho,\mu\otimes\mu;P)<\varepsilon$.

We have obtained that the set

$\mathscr{O}(P,\varepsilon,Q):=\bigcup_{\delta>0}\mathscr{O}(P,\varepsilon,Q,\delta)$

is open, so the set

$\mathscr{O}:=\bigcap_{m,n\in{\mathbb{N}}}\bigcup_{\ell\in{\mathbb{N}}}\mathscr{O}(P_{m},\frac{1}{n},P_{\ell})$

is $G_{\delta}$, where $P_{n}$ is going to the partition into points and it
consists of clopen sets (we assume additionally that $X$ is a zero-dimensional
space).

######  Lemma 3.6.

If $(S,T)\notin\mathscr{O}$ then $S\not\perp T$.

###### Proof.

By assumption, there exist $m,n\geq 1$ such that for all $\ell\geq 1$,
$(S,T)\notin\mathscr{O}(P_{m},\frac{1}{n},P_{\ell},\frac{1}{\ell})$. Hence,
for some $\rho_{\ell}\in C_{2}(\mu)$, we have

(15) $d(\rho_{\ell},(S\times T)_{\ast}\rho_{\ell};P_{\ell})<\frac{1}{\ell},$
(16) $d(\rho_{\ell},\mu\otimes\mu;P_{m})\geq\frac{1}{n}.$

Without loss of generality, we can assume that $\rho_{\ell}\to\rho$. Then
$\rho\in C_{2}(\mu)$ and, by (15), $(S\times T)_{\ast}\rho=\rho$, so $\rho$ is
a joining. While (16) tells us that $\rho\neq\mu\otimes\mu$, so $T$ and $S$
are not disjoint.

Proof of Proposition 3.4 All we need to prove is that
$\mathscr{O}=\\{(S,T):\>S\perp T\\}$. Assume that $(S,T)\in\mathscr{O}$. Then,
for each $m,n\geq 1$ there are $\ell\geq 1$ and $\delta>0$ such that
$(S,T)\in\mathscr{O}(P_{m},\frac{1}{n},P_{\ell},\delta)$. But if $\rho\in
C_{2}(\mu)$ is a joining then this measure is $S\times T$-invariant, so
$d(\rho,(S\times T)_{\ast}\rho;P_{\ell})=0<\delta$. Hence,
$d(\rho,\mu\otimes\mu;P_{m})<\frac{1}{n}$ and therefore $\rho=\mu\otimes\mu$.
The result follows now from Lemma 3.6.

######  Remark 3.7.

In fact, del Junco proved that the set of automorphisms disjoint with a fixed
one is Borel.

###  3.4 Kallman theorem and joinings

We come back to our aperiodic automorphism $\widetilde{T}$ written in the form
(5). From (6), Proposition 3.4 and Remark 3.7, we immediately get the
following

######  Corollary 3.8.

(i) The set $\\{(x,x^{\prime})\in X\times X\colon T_{x}\perp
T_{x^{\prime}}\\}$ is Borel.
(ii) If $S_{0}\in{\rm Aut}(Y,\nu)$ then the set $\\{x\in X\colon T_{x}\perp
S_{0}\\}$ is Borel.

Now we can prove the following result as an application of Kallman theorem.

######  Lemma 3.9.

Fix $S_{0}\in{\rm Aut}(Y,\nu)$ and suppose that $\mu(\\{x\in X\colon
T_{x}\not\perp S_{0}\\})>0$. Then $\widetilde{T}\not\perp S_{0}$.

###### Proof.

Let (by some abuse of notation)

$C_{2}(Y,\nu):=\\{J\colon L^{2}(Y,\nu)\to L^{2}(Y,\nu)\colon J\text{ is
Markov}\\}$

be the space of Markov operators corresponding to couplings of $\nu$ with
itself. Then $C_{2}(Y,\nu)$ is a compact metric space (with the weak operator
topology).

Let $\Pi$ denote the Markov operator corresponding to product measure
$\nu\otimes\nu$, and consider the set

(17) $\\{(x,T_{x},J)\colon x\in X,\Pi\neq J\in C_{2}(Y,\nu),JT_{x}=S_{0}J\\}$

which is a Borel subset of $\bigl{(}X\times{\rm Aut}(Y,\nu)\bigr{)}\times
C_{2}(Y,\nu)$ as the intersection of

$\\{(x,T_{x})\colon x\in X\\}\times(C_{2}(Y,\nu)\setminus\\{\Pi\\})$

and

$X\times\\{(R,J)\in{\rm Aut}(Y,\nu)\times C_{2}(Y,\nu)\colon JR=S_{0}J\\}.$

Let $f$ denote the projection of the set (17) on the first two coordinates.
Note that this image is precisely $\\{(x,T_{x})\colon T_{x}\not\perp
S_{0}\\}$. For each such $(x,T_{x})$ the fiber consists of all Markov
operators $J\in C_{2}(Y,\nu)\setminus\\{\Pi\\}$ such that $JT_{x}=S_{0}J$, so
it is a compact set up to the one-element set $\\{\Pi\\}$, and therefore it is
$\sigma$-compact. We can now use Theorem 2.16, to obtain a Borel map

$\\{y\in X\colon T_{y}\not\perp S_{0}\\}\ni x\to J_{x}\in
C_{2}(Y,\nu)\setminus\\{\Pi\\}$

satisfying $J_{x}T_{x}=S_{0}J_{x}$ which is precisely the fact that $J_{x}$ is
a joining of $T_{x}$ and $S_{0}$. Since we assume that $\mu(\\{x\in X\colon
T_{x}\not\perp S_{0}\\})>0$, we obtain a nontrivial joining between
$\widetilde{T}$ and $S_{0}$ as in Section 2.8, see (13).

###  3.5 Proof of (i) $\Rightarrow$ (iii)

(a.a.) Suppose that $\mu\otimes\mu(\\{(x,x^{\prime}):\>T_{x}\not\perp
T_{x^{\prime}}\\})>0$.222222The set we are considering is measurable by the
same token as (i) in Corollary 3.8.. By Fubini’s theorem, there is $x_{0}\in
X$ such that the set $D$ of $x^{\prime}$ such that $T_{x_{0}}\not\perp
T_{x^{\prime}}$ has positive $\mu$-measure. By Lemma 3.9, it follows that
$\widetilde{T}\not\perp T_{x_{0}}$, a contradiction.

###  3.6 Relative ergodic decomposition

We consider an invertible, bi-measurable transformation $T$ of a standard
Borel space $(X,{\cal B}_{X})$. Let $\mathcal{I}_{T}$ be the sub-sigma-algebra
of $T$-invariant Borel sets (which is defined without reference to any
measure: no completion here). Recall that $M(X)$ stands for the space of
Borel, probability measures on $X$, and $M(X,T)\subset M(X)$ for the subset of
$T$-invariant measures. Let $(Y,{\cal A})$ be another measurable space (not
necessarily standard). Let $\varphi:X\to Y$ be a measurable map which
satisfies the property

(18) $\varphi^{-1}({\cal A})\subset\mathcal{I}_{T}.$

We also fix a $T$-invariant probability measure $\mu\in M(X,T)$.

######  Proposition 3.10.

There exists a probability kernel $[0,1]\ni t\mapsto\nu_{t}\in M(X)$ such
that:
(a) for all $t\in[0,1]$, $\nu_{t}\in M(X,T)$,
(b) we have

$\int_{0}^{1}\nu_{t}\,dt=\mu$

and
(c) each $\nu_{t}$ satisfies $\varphi_{\ast}(\nu_{t})=\varphi_{\ast}(\mu)$ and

$\mathcal{I}_{T}=\varphi^{-1}({\cal A})\text{ mod }\nu_{t}\;(cf.\
\eqref{martim1}).$

Moreover, if $\varphi^{-1}({\cal A})=\mathcal{I}_{T}\text{ mod }\mu$, then for
each decomposition satisfying (a), (b) and (c), we have $\nu_{t}=\mu$ for all
$t$.

We postpone the proof of Proposition 3.10 to Section 9. Let us consider now
the above result in the context of self-joinings of $\widetilde{T}$ acting on
$(X\times Y,\mu\otimes\nu)$ by

$\widetilde{T}(x,y)=(x,T_{x}(y)),$

where $T_{x}$ are ergodic for $\nu$. Let $\lambda\in J_{2}(\widetilde{T})$,
$\lambda|_{X\times X}=\mu\otimes\mu$. The disintegration of $\lambda$ with
respect to the projection on $X\times X$ writes

(19) $\lambda=\int_{X\times
X}\lambda_{x,x^{\prime}}\,d\mu(x)d\mu(x^{\prime}),$

where $\lambda_{x,x^{\prime}}\in J(T_{x},T_{x^{\prime}})$ for
$\mu\otimes\mu$-a.a. $(x,x^{\prime})\in X\times X$. We can apply Proposition
3.10 to $(\widetilde{T}\times\widetilde{T},\lambda)$, when $\phi:(X\times
Y)\times(X\times Y)\to X\times X$ is the projection onto $X\times X$. We
obtain that

$\lambda=\int_{0}^{1}\lambda_{t}\,dt,$

where, from (a), each $\lambda_{t}$ is
$\widetilde{T}\times\widetilde{T}$-invariant, and from (c) the marginal on
$X\times X$ of each $\lambda_{t}$ is the same as the marginal of $\lambda$,
which is $\mu\otimes\mu$, and moreover the system $\bigl{(}(X\times
Y)\times(X\times Y),\lambda_{t},\widetilde{T}\times\widetilde{T}\bigr{)}$ is a
relatively ergodic extension of $(X\times X,\mu\otimes\mu,{\rm Id})$. It
follows that the ergodic decomposition of $\lambda_{t}$ takes the form

$\lambda_{t}=\int_{X\times
X}\lambda_{t,x,x^{\prime}}\,d\mu(x)d\mu(x^{\prime}),$

where the measures $\lambda_{t,x,x^{\prime}}$ are ergodic (for $T_{x}\times
T_{x^{\prime}})$. Integrating with respect to $t$, and comparing with (19), we
get that for $\mu\otimes\mu$-almost all $(x,x^{\prime})$, we have

$\lambda_{x,x^{\prime}}=\int_{0}^{1}\lambda_{t,x,x^{\prime}}\,dt.$

Projecting on each $Y$-coordinate the above relation, and remembering that the
systems $(Y,\nu,T_{x})$ are ergodic, we get that for Lebesgue-almost all
$t\in[0,1)$ (depending on $(x,x^{\prime})$), $\lambda_{t,x,x^{\prime}}$ is an
ergodic joining of $(Y,\nu,T_{x})$ and $(Y,\nu,T_{x^{\prime}})$.

Finally, defining

(20) $J_{2}^{\rm RelErg}(\widetilde{T}):=\\{\lambda\in
J_{2}(\widetilde{T})\colon\lambda|_{X\times X}=\mu\otimes\mu,\\\
\lambda_{x,x^{\prime}}\in J^{e}(T_{x},T_{x^{\prime}})\text{ for
}\mu\otimes\mu\text{-a.a. }x,x^{\prime}\in X\\},$

we obtain the following:

######  Corollary 3.11.

Each self-joining $\lambda$ of $\widetilde{T}$ projecting into $\mu\otimes\mu$
(on $X\times X$) is of the form $\lambda=\int_{0}^{1}\lambda_{t}\,dt$, where
$\lambda_{t}\in J_{2}^{\rm RelErg}(\widetilde{T})$ for all $t\in[0,1]$.

######  Remark 3.12.

This proves in particular that $J_{2}^{\rm RelErg}(\widetilde{T})$ is not
empty, since the corollary applies to
$\lambda=(\mu\otimes\nu)\otimes(\mu\otimes\nu)$.

###  3.7 Example(s)

Let $M=G/\Gamma$, where $G=H_{3}({\mathbb{R}})$ and
$\Gamma=H_{3}({\mathbb{Z}})$ ($\Gamma$ is an example of a lattice in the
Heisenberg group; $M$ is an example of a nil-manifold). On $M$ we consider the
(normalized) measure $\kappa$ which is the image of Haar measure on $G$. Given
$g\in G$, let $T_{g}:M\to M$: $T_{g}(x\Gamma)=gx\Gamma$. Denote by
$\pi:M\to{\mathbb{T}}^{2}$ the map

$\left[\begin{array}[]{ccc}1&a&c\\\ 0&1&b\\\
0&0&1\end{array}\right]/\Gamma\mapsto(a~{}{\rm mod}~{}1,b~{}{\rm
mod}~{}1)\in{\mathbb{T}}^{2}.$

Note that $\pi$ is well defined. Moreover, since

$\left[\begin{array}[]{ccc}1&a&c\\\ 0&1&b\\\
0&0&1\end{array}\right]\cdot\left[\begin{array}[]{ccc}1&a^{\prime}&c^{\prime}\\\
0&1&b^{\prime}\\\
0&0&1\end{array}\right]=\left[\begin{array}[]{ccc}1&a+a^{\prime}&c^{\prime}+ab^{\prime}+c\\\
0&1&b+b^{\prime}\\\ 0&0&1\end{array}\right],$

the fibers of $\pi$ are the circle ${\mathbb{T}}$. Moreover, the rotation
$T_{g}$ on $M$ is sent to the rotation $R_{(a,b)}$ on ${\mathbb{T}}^{2}$. By a
theorem of Auslander-Green-Hahn [3] if $1,a,b$ are rationally independent
(i.e. when $R_{(a,b)}$ is ergodic) $T_{g}$ is ergodic and it is a compact
group (circle) extension of the rotation (from the measure-theoretic point of
view $M$ is ${\mathbb{T}}^{2}\times{\mathbb{T}}$ and in these “coordinates”
$\kappa$ is Leb${}_{{\mathbb{T}}^{2}}\otimes$LebT) such that in the
orthocomplement of $L^{2}({\mathbb{T}}^{2},{\rm Leb}_{{\mathbb{T}}^{2}})$ the
spectrum is countable Lebesgue. It follows that when $T_{g}$ and
$T_{g^{\prime}}$ are ergodic then

(21) $T_{g}\perp T_{g^{\prime}}\Leftrightarrow ka+\ell
b=k^{\prime}a^{\prime}+\ell^{\prime}b^{\prime}\neq 0$

for some $k,\ell,k^{\prime}\ell^{\prime}\in{\mathbb{Z}}$ (i.e. $R_{(a,b)}$ and
$R_{(a^{\prime},b^{\prime})}$ have a non-trivial common factor).

Denote by $q$ the natural quotient map $q:G\to G/\Gamma$ and set
$N_{k,\ell,m}:=\\{(a,b)\in{\mathbb{T}}^{2}\colon k+\ell a+mb=0\\}$. Let

$A:=(\pi\circ q)^{-1}\big{(}\bigcup_{k,\ell,m}({\mathbb{T}}^{2}\setminus
N_{k,\ell,m})\big{)}.$

Then $A\subset G$ is a subset of full HaarG-measure. Now, given $g=(a,b,c)\in
G$, we can find only a subset of $(a^{\prime},b^{\prime})\in{\mathbb{T}}^{2}$
of zero Leb${}_{{\mathbb{T}}^{2}}$ measure to see that $R_{(a,b)}\not\perp
R_{(a^{\prime},b^{\prime})}$. It follows that the condition $T_{g}\perp
T_{g^{\prime}}$ is satisfied Haar${}_{G}\otimes$HaarG-a.e.

It follows that, by considering any probability measure $\mu$ on $G$
equivalent to HaarG, the automorphism:

$\widetilde{T}(g,x\Gamma):=(g,gx\Gamma),\;\widetilde{T}:G\times M\to G\times
M,$

preserves the measure $\mu\otimes\kappa$, and $(G,\mu)$ is the space of
ergodic components. By Theorem 3.1, $\widetilde{T}\perp$ Erg.

The above reasoning has its natural generalization for any connected nilpotent
$d$-step group $G$ and the corresponding rotations on nil-manifolds. Indeed,
as before, we consider the quotient map $G/\Gamma\to G/([G,G]\Gamma)$, where
the latter space is the torus ${\mathbb{T}}^{d}$ and a nil-rotation $T_{g}$ is
sent to a rotation on ${\mathbb{T}}^{d}$. Moreover, by Auslander-Green-Hahn
theorem [3], $T_{g}$ is ergodic iff the corresponding rotation on
${\mathbb{T}}^{d}$ is ergodic. As the set of ergodic rotations on
${\mathbb{T}}^{d}$ is of full Lebesgue measure, also the set of ergodic nil-
rotations on $G/\Gamma$ is of full Haar measure. We now, apply the same
reasoning to the product space $(G\times G)/(\Gamma\times\Gamma)$ to obtain
that the set of ergodic nil-rotations $T_{g,g^{\prime}}$ is of full (product)
Haar measure. As $T_{g,g^{\prime}}=T_{g}\times T_{g^{\prime}}$, we use now the
following result: assuming $T_{g}$, $T_{g^{\prime}}$ ergodic,

$T_{g}\perp T_{g^{\prime}}\text{ iff Cartesian product }T_{g}\times
T_{g^{\prime}}\text{ is ergodic}$

(see Corollary 1.5 in [28]) to conclude that on the set of $(g,g^{\prime})$ of
full (product) Haar measure, we have $T_{g}\perp T_{g^{\prime}}$. Hence the
corresponding $\widetilde{T}$ (see (21)) is disjoint from all ergodic systems.

######  Remark 3.13.

Automorphisms of that type considered for connected and simply connected
nilpotent Lie groups are fundamental to prove Sarnak’s conjecture: they
satisfy Sarnak’s conjecture232323We can consider other measures $\mu$ on $G$,
including those with compact support; but we should consider only those for
which $T_{g}\perp T_{g^{\prime}}$ for $\mu\otimes\mu$-a.a. $(g,g^{\prime})$;
Then $\widetilde{T}$ will be considered on the compact space ${\rm
supp}(\mu)\times M$ and it will be a homeomorphism of zero entropy. but
whether the strong MOMO property holds for them is a big open problem, see
[21], [22], [34].

##  4 Orthogonality to ergodic Markov images

###  4.1 Preparatory observations and some motivations

Before we formulate the main result of this section, we need some preparatory
observations. We recall that we consider $\widetilde{T}(x,y)=(x,T_{x}y)$ on
$(X\times Y,\mu\otimes\nu)$ with the fiber automorphisms ergodic. It follows
that the extension

(22) $\widetilde{T}\to\widetilde{T}|_{X}={\rm Id}_{X}\text{ is relatively
ergodic.}$

Let us consider now $\lambda\in J_{2}(\widetilde{T})$, $\lambda|_{X\times
X}=\mu\otimes\mu$. Then clearly

(23) $({\rm Id}_{X\times X},\mu\otimes\mu)\to{\rm Id}_{X}\text{ is relatively
identity.}$

Hence, using the relative version of the classical fact of disjointness of
ergodic systems with any identity system, we obtain that

(24) $\lambda|_{(X\times Y)\times X}=(\mu\otimes\nu)\otimes\mu\text{ i.e., it
is the relative product over }\mu,~{}$

see also Section 5.6. If now $f=f(x,y)\in L_{0}^{2}(X\times Y,\mu\otimes\nu)$
and if $\widetilde{f}:={{\mathbbm{E}}}_{\mu\otimes\nu}[f\,|\,X]$, then (24)
implies that

(25)
${{\mathbbm{E}}}_{\lambda}[(f-\widetilde{f})\otimes(f-\widetilde{f})\,|\,X\times
X]={{\mathbbm{E}}}_{\lambda}[f\otimes f\,|\,X\times X]\\\
-{{\mathbbm{E}}}_{\mu\otimes\nu}[f\,|\,X]\otimes{{\mathbbm{E}}}_{\mu\otimes\nu}[f\,|\,X]$

as

(26) ${{\mathbbm{E}}}_{\lambda}[f\otimes\widetilde{f}\,|\,X\times
X]=(1\otimes\widetilde{f})\,{{\mathbbm{E}}}_{\lambda}\bigl{[}(f\otimes
1)\,|\,X\times X\bigr{]}=\widetilde{f}\otimes\widetilde{f}$

the latter equality due to (24). Moreover, notice that252525Note that
$\widetilde{f}\perp F_{\rm we}(\widetilde{T})$.

(27) $f\perp F_{\rm we}(\widetilde{T})$ if and only if $f-\widetilde{f}\perp
F_{\rm we}(\widetilde{T})$.

Therefore, in what follows, we will constantly assume that we consider $f$
such that

(28) ${{\mathbbm{E}}}[f\,|\,X]=0.$

In order to understand how natural this assumption is, recall first the notion
of Gowers-Host-Kra norms of $f$:

(29) $\|f\|^{2}_{u^{1}}:=\lim_{H\to\infty}\frac{1}{H}\sum_{h\leq H}\int
f\circ\widetilde{T}^{h}\cdot\overline{f}\,d(\mu\otimes\nu),$

hence,

(30) $\|f\|^{2}_{u^{1}}=\Big{|}\int f\,d(\mu\otimes\nu)\Big{|}^{2}\text{ if
}\widetilde{T}\text{ is ergodic}$

and $\|f\|^{2^{s+1}}_{u^{s+1}}:=\lim_{H\to\infty}\frac{1}{H}\sum_{h\leq
H}\|f\circ\widetilde{T}^{h}\cdot\overline{f}\|_{u^{s}}^{2^{s}}$ for $s\geq 1$.
Note that, by the von Neumann theorem,
$\|f\|_{u^{1}}=\int|{{\mathbbm{E}}}[f\,|\,X]|^{2}\,d\mu$. Remember that we aim
at a study of (bounded) arithmetic functions $\boldsymbol{u}$ through the
associated Furstenberg systems $\kappa\in V(\boldsymbol{u})$ and we consider
$f=\pi_{0}$. In particular, we say that $\|\boldsymbol{u}\|_{u^{1}}=0$ if
$\|\pi_{0}\|_{u^{1}}=0$ for all $\kappa\in V(\boldsymbol{u})$. Hence, in the
problem of classifying arithmetic functions $\boldsymbol{u}$ which are
orthogonal to uniquely ergodic systems, the assumption (28) is now explained
by the following classical result (see also Section 3.4.1 in [9]):

######  Proposition 4.1.

The following conditions are equivalent:
(a) $\|\boldsymbol{u}\|_{u^{1}}=0$.
(b) For each $(N_{k})$ defining a Furstenberg system of $\boldsymbol{u}$, we
have

$\limsup_{H\to\infty}\lim_{k\to\infty}\frac{1}{N_{k}}\sum_{n\leq
N}\Big{|}\frac{1}{H}\sum_{h\leq H}\boldsymbol{u}(n+h)\Big{|}^{2}=0$

(i.e. the property of “zero mean on typical short interval”).
(c) ${{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,\mathcal{I}_{S}]=0$ for each
$\kappa\in V(\boldsymbol{u})$.
(d) $\lim_{H\to\infty}\frac{1}{H}\sum_{h\leq H}\pi_{0}\circ S^{h}=0$ in
$L^{2}(X_{\boldsymbol{u}},\kappa)$ for each $\kappa\in V(\boldsymbol{u})$.

######  Remark 4.2.

(For some basics about multiplicative, bounded by 1 functions and their
multiplicative distance $D$, see Section 5.2.) A break-through theorem by
Matomäki and Radziwiłł [26] established zero value for the first GHK-norm for
many classical multiplicative (unpretensious) functions like the Liouville or
Möbius functions. But there are also pretentious multiplicative functions
$\boldsymbol{u}$ for which $\|\boldsymbol{u}\|_{u^{1}}=0$. In fact, all
multiplicative functions pretending being non-principal Dirichlet character
$\chi$ enjoy this property. Indeed, as shown in [13], Furstenberg systems of
all such arithmetic functions are ergodic, so (in view of (30)) our claim
follows from the fact that they have mean equal to zero.262626As for the non-
principal Dirichlet character, we have $D(\chi,n^{it})=+\infty$ for all
$t\in{\mathbb{R}}$, and $D(\boldsymbol{u},\chi)<+\infty$. It follows that
$D(\boldsymbol{u},n^{it})=+\infty$ as the multiplicative “distance” $D$
satisfies the triangle inequality. The claim now follows from the Halás
theorem.

######  Remark 4.3.

Note that if all Furstenberg systems of $\boldsymbol{u}$ are identities, then
$\|\boldsymbol{u}\|_{u^{1}}>0$ unless the Besicovitch norm vanishes
$\|\boldsymbol{u}\|_{B}=0$ (see Section 5.2). Indeed, if in the definition of
$u^{1}$ of $f$, the automorphism $T$ is an identity then
$\|f\|_{u^{1}}=\|f\|_{L^{2}}$. Hence, by an observation in [16], all non-
trivial slowly varying functions have positive GHK first norm. In particular,
for all Archimedean characters, we have $\|n^{it}\|_{u^{1}}>0$.

Let us consider some examples of orthogonality to ergodic Markov images.

  1. A.

If $\widetilde{T}$ is ergodic (that is, $\widetilde{X}=\\{\ast\\}\times Y$)
then all $f\in L^{2}(\widetilde{X})$ are in the Markov image of
$\Phi_{\Delta_{\nu}}$, where $\Delta_{\nu}$ stands for the diagonal joining on
$(Y,\nu)$.

  2. B.

If for $\mu\otimes\mu$-a.a. $x,x^{\prime}\in X$, we have $T_{x}\perp
T_{x^{\prime}}$, then, by Theorem 3.1, all zero mean functions are orthogonal
to all ergodic Markov images.

######  Proposition 4.4.

Assume that for all $x\in X$, $T_{x}=T$ (in other words, $\widetilde{T}={\rm
Id}_{X}\times T$ with $T$ ergodic). Then for all $f\in L^{2}(\mu\otimes\nu)$,
$f\perp F_{{\rm we},0}(\widetilde{T})$ if and only if $f$ is measurable with
respect to $x$.

###### Proof.

Suppose that $f\perp F_{{\rm we},0}(\widetilde{T})$. Then, by considering the
joining $\rho$ of $\widetilde{T}$ with $T$ in which we join diagonally $T$
with itself and the $X$ part is independent of $Y$ and using Lemma 2.11, for
every $g\in L_{0}^{2}(Y,\nu)$, we have

(31) $\int_{X\times Y}f(x,y)g(y)\,d\nu(y)=0$

for $\mu$-a.a. $x\in X$. This $\mu$-a.e. condition persists if we consider a
countable linearly dense set of $g\in L_{0}^{2}(Y)$. It follows that for
$\mu$-a.a. $x\in X$, $y\mapsto f(x,y)$ is $\nu$-almost everywhere equal to
$\int_{Y}f(x,y)\,d\nu(y)$, _i.e._ $f$ is $\mu\otimes\nu$-almost surely equal
to a function of $x$.

Conversely, if $f$ is $x$-measurable, the fact that $f\perp F_{{\rm
we},0}(\widetilde{T})$ is a consequence of the disjointness between identities
and ergodic systems.

###  4.2 Characterization result

We aim at proving the following result:272727 If we do not want to assume (28)
then the assertion ${{\mathbbm{E}}}_{\lambda}[f\otimes f\,|\,X\times
X](x,x^{\prime})=0$, in view of (25), should be replaced by
$(\ast)\;{{\mathbbm{E}}}_{\lambda}[f\otimes f\,|\,X\times
X](x,x^{\prime})={{\mathbbm{E}}}_{\kappa}(f\,|\,X)(x){{\mathbbm{E}}}_{\kappa}(f\,|\,X)(x^{\prime})$
for $\mu\otimes\mu$-a.e. $(x,x^{\prime})\in X\times X$. For example, if we
take $\boldsymbol{u}$ such that all its Furstenberg systems are identity, then
the only self-joining that we have to consider is the product measure, and the
assumption $(\ast)$ holds. We will see later (see Proposition 5.1) that in
this case $\boldsymbol{u}$ will be orthogonal to all u.e. systems even though
the function $\pi_{0}$ does not satisfy (28).

######  Theorem 4.5.

Assume that $f\in L^{2}_{0}(X\times Y,\mu\otimes\nu)$ satisfies (28). Then
$f\perp F_{\rm we}(\widetilde{T})$ if and only if for all $\lambda\in J^{\rm
RelErg}_{2}(\widetilde{T})$, we have ${{\mathbbm{E}}}_{\lambda}[f\otimes
f\,|\,X\times X](x,x^{\prime})=0$ for $\mu\otimes\mu$-a.e. $(x,x^{\prime})\in
X\times X$.

  * •

Note that $\lambda\in J_{2}^{\rm RelErg}(\widetilde{T})$ if and only if
$\mathcal{I}_{\widetilde{T}\times\widetilde{T}}$ is the sigma-algebra
generated by the projection to $X\times X$ modulo $\lambda$.

  * •

Under the assumption on $\lambda$, ${{\mathbbm{E}}}_{\lambda}[f\otimes
f\,|\,X\times X](x,x^{\prime})=0$ for $\mu\otimes\mu$-a.e. $(x,x^{\prime})\in
X\times X$ if and only if $\lim_{N\to\infty}\frac{1}{N}\sum_{n\leq N}(f\otimes
f)\circ(\widetilde{T}\times\widetilde{T})^{n}=0$ (by the von Neumann theorem).

  * •

Note also that (assuming the validity of Theorem 4.5), due to Corollary 3.11:
$f\perp F_{\rm we}(\widetilde{T})$ if and only if for all $\lambda\in
J_{2}(\widetilde{T})$, we have ${{\mathbbm{E}}}_{\lambda}[f\otimes
f\,|\,X\times X](x,x^{\prime})=0$ for $\mu\otimes\mu$-a.e. $(x,x^{\prime})\in
X\times X$.

  * •

For another equivalent condition (important for applications), see Corollary
4.9.

######  Remark 4.6.

It follows directly from Theorem 4.5 that if $f\in L^{2}_{0}(X\times
Y,\mu\otimes\nu)$ satisfies (28) and $f\perp F_{\rm we}(\widetilde{T})$ then,
for each $g\in L^{\infty}(X,\mu)$, we have $gf\perp F_{\rm
we}(\widetilde{T})$.

We need the following:

######  Lemma 4.7.

Assume that $\rho\in J(\widetilde{T},R)$ with $R$ ergodic. Set, by some abuse
of notation,282828i.e. $\lambda$ is the only joining for which
$\Phi_{\lambda}=\Phi_{\rho}^{\ast}\circ\Phi_{\rho}$.
$\lambda:=\rho^{\ast}\circ\rho\in J_{2}(\widetilde{T})$. Then
$\lambda|_{X\times X}=\mu\otimes\mu$.

###### Proof.

Since $R$ is ergodic, $\rho|_{X\times Z}=\mu\otimes\kappa$. That is,
$\Phi_{\rho}|_{L_{0}^{2}(X,\mu)}=0$. By the definition of $\lambda$,

$\Phi_{\lambda}|_{L_{0}^{2}(X)}=\Phi^{\ast}_{\rho}\circ\Phi_{\rho}|_{L_{0}^{2}(X)}=0,$

whence, $\lambda|_{X\times X}=\mu\otimes\mu$.

We have $\rho=\int_{X}\rho_{x}\,d\mu(x)$, where $\rho_{x}\in J(T_{x},R)$ for
$\mu$-a.e. $x\in X$. Moreover, by disintegrating

$\lambda=\int_{X\times X}\lambda_{x,x^{\prime}}\,d\mu(x)d\mu(x^{\prime}),$

we obtain that

(32) $\lambda_{x,x^{\prime}}=\rho^{\ast}_{x^{\prime}}\circ\rho_{x}.$

Indeed,

$\Phi_{\lambda}=\Phi^{\ast}_{\rho}\circ\Phi_{\rho}=\int_{X}\Phi^{\ast}_{\rho_{x}}\,d\mu(x)\circ\int_{X}\Phi_{\rho_{x^{\prime}}}\,d\mu(x^{\prime})=$
$\int_{X\times
X}\Phi^{\ast}_{\rho_{x}}\circ\Phi_{\rho_{x^{\prime}}}\,d\mu(x)d\mu(x^{\prime}).$

###  4.3 Proof of Theorem 4.5. Sufficiency

We first notice that if $\mu$ has an atom, say at $x_{0}$, then by joining
$\widetilde{T}$ with $R=T_{x_{0}}$, diagonally between $T_{x_{0}}$ with $R$,
and as direct product between remaining $T_{x}$ with $R$, we get automatically
that $f|_{\\{x_{0}\\}\times Y}=0$. Therefore, we may assume that $\mu$ has no
atoms.

Take any ergodic $R$ on $(Z,\mathcal{D},\kappa)$ and
$\Phi_{\rho}:L^{2}(X\times Y,\mu\otimes\nu)\to L^{2}(Z,\kappa)$. Set
$\lambda=\rho^{\ast}\circ\rho\in J_{2}(\widetilde{T})$. In view of Lemma 4.7,
$\lambda|_{X\times X}=\mu\otimes\mu$. Although $\lambda$ need not belong to
$J_{2}^{\rm RelErg}(\widetilde{T})$, by Corollary 3.11, we can assume that
${{\mathbbm{E}}}_{\lambda}[f\otimes f\,|\,X\times X]=0$ $\mu\otimes\mu$-a.e.
Hence, by (32),

$0={{\mathbbm{E}}}_{\lambda}[f\otimes f](x,x^{\prime})=\int f\otimes
f\,d\lambda_{x,x^{\prime}}=$
$\int\Phi_{\lambda_{x,x^{\prime}}}(f)f\,d\nu=\int_{Z}\Phi_{\rho_{x}}(f)\cdot\Phi_{\rho_{x^{\prime}}}(f)\,d\kappa$

for $\mu\otimes\mu$-a.e. $(x,x^{\prime})\in X\times X$. A point $x\in X$ is
called “good” if

$\mu(\\{x^{\prime}\in
X:\>\Phi_{\rho_{x}}(f)\perp\Phi_{\rho_{x^{\prime}}}(f)\\})=1.$

Since the set of $(x,x^{\prime})$ such that
$\Phi_{\rho_{x}}(f)\perp\Phi_{\rho_{x^{\prime}}}(f)$ is Borel, by Fubini’s
theorem the set $G$ of “good” points has full measure. Set

$C:=\\{x\in X:\>\Phi_{\rho_{x}}(f)\neq 0\\}.$

Suppose that $\mu(C)>0$. Then $\mu(G\cap C)=\mu(C)>0$. In the set $C\cap G$
select a maximal subset $\\{x_{n}:\>n\geq 1\\}$, so that
$\Phi_{\rho_{x_{n}}}(f)\perp\Phi_{\rho_{x_{m}}}(f)$ for $n\neq m$ (this set is
countable because $L^{2}(Z,\kappa)$ is separable, and we can find a maximal
set by Zorn’s lemma). Let

$B_{n}:=\\{x\in X:\>\Phi_{\rho_{x}}(f)\perp\Phi_{\rho_{x_{n}}}(f)\\}.$

Then $\mu(B_{n})=1$ and therefore

$\mu(\bigcap_{n\geq 1}B_{n}\cap G\cap C)>0$

and adding any $x\in\bigcap_{n\geq 1}B_{n}\cap G\cap C$ makes a larger family
than $\\{x_{n}:\>n\geq 1\\}$, a contradiction.

It follows that the set $C$ has measure zero, and this yields
$\Phi_{\rho}(f)=0$.

###  4.4 Proof of Theorem 4.5. Necessity

(a.a.) Suppose that there exists $\lambda\in J_{2}^{\rm
RelErg}(\widetilde{T})$,
$\lambda=\int\lambda_{x,x^{\prime}}\,d\mu(x)d\mu(x^{\prime})$, for which the
claim does not hold. Remembering that ${{\mathbbm{E}}}_{\lambda}[f\otimes
f\,|\,X\times X](x,x^{\prime})=\int f\otimes f\,d\lambda_{x,x^{\prime}}$ and
using Fubini’s theorem, we can assume that there exists a point $x_{0}$ such
that for a set of $x^{\prime}$ from a set $A\subset X$ of positive measure (we
assume that $f$ is real),

$\int\Phi_{\lambda_{x_{0},x^{\prime}}}(f|_{\\{x_{0}\\}\times Y})\cdot
f|_{\\{x^{\prime}\\}\times Y}\,d\nu>0.$

We set $R=T_{x_{0}}$ (which is ergodic) and define $\rho\in
J(\widetilde{T},R)$, by setting $\rho=\int\rho_{x}\,d\mu(x)$, where
$\rho_{x}=\lambda_{x_{0},x}$ for $x\in A$, and $\rho_{x}$ equals the product
measure otherwise. We easily obtain that $\rho\in J(T_{x_{0}},\widetilde{T})$
and $\int\Phi_{\rho}(f|_{\\{x_{0}\\}\times Y})\cdot f\,d(\mu\otimes\nu)>0$.

###  4.5 A condition for being orthogonal to ergodic Markov images

We need the following lemma (relativizing the case of ergodic decomposition).

######  Lemma 4.8.

Let $T$ be an automorphism of $(X,{\cal B},\mu)$ and let ${\cal
A}\subset\mathcal{I}_{T}$ be a factor. Assume that
$\mu=\int_{0}^{1}\mu_{t}\,dQ(t)$, where $\mu_{t}$ are $T$-invariant and ${\cal
A}=\mathcal{I}_{T}\mod\mu_{t}$ for $Q$-a.a. $t\in[0,1]$. Let $f\in
L^{\infty}(X,\mu)$ and ${{\mathbbm{E}}}_{\mu_{t}}[f|{\cal A}]=0$ for $Q$-a.a.
$t\in[0,1]$. Then ${{\mathbbm{E}}}_{\mu}[f\,|\,\mathcal{I}_{T}]=0$.

###### Proof.

Denote $F_{N,t}:=\int_{X}|\frac{1}{N}\sum_{n\leq N}f\circ
T^{n}|^{2}\,d\mu_{t}$. Then, for $Q$-a.e. $t\in[0,1]$, we have
$\lim_{N\to\infty}F_{N,t}=0$ by the von Neumann theorem (in
$L^{2}(X,\mu_{t})$) and our assumptions. By the Lebesgue dominated theorem:
$\lim_{N\to\infty}\int_{0}^{1}F_{N,t}\,dQ(t)=0$. That is,

$0=\lim_{N\to\infty}\int_{0}^{1}\Big{(}\int_{X}\Big{|}\frac{1}{N}\sum_{n\leq
N}f\circ
T^{n}\Big{|}^{2}\,d\mu_{t}\Big{)}dQ(t)=\lim_{N\to\infty}\int_{X}\Big{|}\frac{1}{N}\sum_{n\leq
N}f\circ T^{n}\Big{|}^{2}\,d\mu.$

It follows again by the von Neumann theorem that
${{\mathbbm{E}}}_{\mu}[f\,|\,\mathcal{I}_{T}]=0$.

Returning to our context, by Corollary 3.11, Theorem 4.5 and Lemma 4.8, we
obtain the following:

######  Corollary 4.9.

Assume that $f\in L^{2}_{0}(X\times Y,\mu\otimes\nu)$ satisfies (28). Then
$f\perp F_{\rm we}(\widetilde{T})$ if and only if for all $\lambda\in
J_{2}(\widetilde{T})$ satisfying $\lambda|_{X\times X}=\mu\otimes\mu$, we have
${{\mathbbm{E}}}_{\lambda}[f\otimes
f\,|\,\mathcal{I}_{\widetilde{T}\otimes\widetilde{T}}]=0$.

######  Remark 4.10.

The above corollary yields the full solution of Boshernitzan’s problem (see
next section). To test whether $f\perp F_{\rm we}(\widetilde{T})$ and assuming
that ${\rm Id}_{=}0$, we only need (and it is sufficient) to show that

$(\ast)\;\;\;\;\lim_{N\to\infty}\int_{\widetilde{X}\times\widetilde{X}}\Big{|}\frac{1}{N}\sum_{n\leq
N}(f\otimes
f)\circ(\widetilde{T}\times\widetilde{T})^{n}\,d\lambda\Big{|}^{2}=0$

for all self-joinings $\lambda$ of $\widetilde{T}$ for which
$\lambda|_{X\times X}=\mu\otimes\mu$. When we consider Furstenberg systems of
$\boldsymbol{u}$ (satisfying the condition: “zero mean on typical short
interval”), then $f=\pi_{0}$ and the condition $(\ast)$ is expressed
combinatorially (i.e. expressed in terms of properties of $\boldsymbol{u}$),
see Proposition 4.1 and next sections.

######  Remark 4.11.

If $\widetilde{T}\perp{\rm Erg}$ then of course each $f$ as above is
orthogonal to all ergodic images, but we can also see that the condition
$(\ast)$ is trivially satisfied because each $\lambda$ as above will simply be
product measure (as the extension $\widetilde{T}\to\widetilde{T}|_{X}$ is
confined).

######  Remark 4.12.

Note that if $\lambda|_{X\times X}=\mu\otimes\mu$ then (unless $\widetilde{T}$
is ergodic) $\lambda$ cannot be the graph measure
$\widetilde{\mu}_{\widetilde{S}}$ for an element $\widetilde{S}$of the
centralizer $C(\widetilde{T})$ of $\widetilde{T}$. Indeed, if
$\widetilde{S}\in C(\widetilde{T})$ then $\widetilde{S}$ must preserve the
sigma-algebra of invariant sets. We obtain

$\widetilde{\mu}_{\widetilde{S}}|_{X\times X}=\mu_{S},$

where $S=\widetilde{S}|_{X}$. But $\mu_{S}\neq\mu\otimes\mu$ unless $X$
reduces to one point.

##  5 Orthogonality to uniquely ergodic systems

###  5.1 Orthogonality and ergodic Markov images

######  Proposition 5.1.

Let $\boldsymbol{u}:{\mathbb{N}}\to{\mathbb{D}}$. Then
$\boldsymbol{u}\perp\,{\rm UE}$292929We recall that (cf. (2)) the meaning of
orthogonality is that $\lim_{N\to\infty}\frac{1}{N}\sum_{n\leq
N}f(T^{n}x)\boldsymbol{u}(n)=0$ for each $(X,T)$ uniquely ergodic, $f\in C(X)$
of zero mean (with respect to the unique invariant measure), $x\in X$. if and
only if for each Furstenberg system $\kappa\in V(\boldsymbol{u})$,
$\pi_{0}\perp F_{{\rm we},0}(X_{\boldsymbol{u}},\kappa,S)$.

###### Proof.

$\Rightarrow$ (a.a.) Suppose that for some $\kappa\in V(\boldsymbol{u})$ there
is an ergodic system $(Z^{\prime},\nu^{\prime},R^{\prime})$ and a joining
$\rho^{\prime}\in J(\nu^{\prime},\kappa)$ for which $\pi_{0}\not\perp{\rm
Im}(\Phi_{\rho^{\prime}}|_{L^{2}_{0}(Z^{\prime},\nu^{\prime})})$. Using the
Jewett-Krieger theorem, we obtain that there exists a uniquely ergodic system
$(Z,\nu,R)$ and a joining $\rho\in J(\nu,\kappa)$ such that
$\pi_{0}\not\perp{\rm Im}(\Phi_{\rho}|_{L^{2}_{0}(Z,\nu)})$ (in
$L^{2}(\kappa)$). It follows that there exists $f\in C(Z)$ of zero mean such
that
$\int{{\mathbbm{E}}}_{\kappa}[f\,|\,X_{\boldsymbol{u}}]\cdot\pi_{0}\,d\kappa\neq
0$, or else $\int_{Z\times X_{\boldsymbol{u}}}f\otimes\pi_{0}\,d\rho\neq 0$.
Let $(N_{n})$ satisfy $\frac{1}{N_{n}}\sum_{j\leq
N_{n}}\delta_{S^{j}\boldsymbol{u}}\to\kappa$. In view of Theorem 2.1, there
exist a sequence $(z_{\ell})\subset Z$ and a subsequence $(N_{n_{k}})$ such
that

$\frac{1}{N_{n_{k}}}\sum_{j\leq
N_{n_{k}}}\delta_{(z_{j},S^{j}\boldsymbol{u})}\to\rho$

and the set

$\\{\ell\geq 0:\>z_{\ell+1}\neq Rz_{\ell}\\}=:\\{b_{1}<b_{2}<\ldots\\}$

satisfies $b_{\ell+1}-b_{\ell}\to\infty$. It follows that

$0\neq\int
f\otimes\pi_{0}\,d\rho=\lim_{k\to\infty}\frac{1}{N_{n_{k}}}\sum_{j\leq
N_{n_{k}}}f(z_{j})\boldsymbol{u}(j)=$
$\lim_{m\to\infty}\frac{1}{b_{K_{m}}}\sum_{r<K_{m}}\Big{(}\sum_{b_{r}\leq
j<b_{r+1}}f(R^{j-b_{r}}z_{b_{r}})\boldsymbol{u}(j)\Big{)}.$

However, the latter limit is $0$ because of Theorem 2.4 (after a slight
modification of $(b_{k})$ if necessary) and our assumption of orthogonality on
$\boldsymbol{u}$.

$\Leftarrow$ (a.a.) Let $(X,T)$ be uniquely ergodic. Suppose that for some
subsequence $(N_{k})$ we have $\frac{1}{N_{k}}\sum_{n\leq
N_{k}}f(T^{n}x)\boldsymbol{u}(n):=c\neq 0$ (for some zero mean $f\in C(X)$ and
$x\in X$) and $\frac{1}{N_{k}}\sum_{n\leq
N_{k}}\delta_{(T^{n}x,S^{n}\boldsymbol{u})}\to\rho$. Then $\rho$ is a joining,
$\rho\in J(\nu,\kappa)$, where $\nu$ is the unique invariant measure for $T$,
and $\kappa\in V(\boldsymbol{u})$. Since now

$c=\int
f\otimes\pi_{0}\,d\rho=\int{{\mathbbm{E}}}_{\kappa}[f\,|\,X_{\boldsymbol{u}}]\pi_{0}\,d\kappa,$

we obtain a contradiction as $\nu$ is ergodic.

######  Remark 5.2.

If we assume that $M(\boldsymbol{u})=0$ (which is equivalent to
${{\mathbbm{E}}}_{\kappa}[\pi_{0}]=0$ for each Furstenberg system $\kappa$),
then in the proposition above we can replace $F_{{\rm
we},0}(X_{\boldsymbol{u}},\kappa,S)$ with $F_{{\rm
we}}(X_{\boldsymbol{u}},\kappa,S)$.

######  Remark 5.3.

Note that in the proof we used unique ergodicity twice: each ergodic system
has a uniquely ergodic model and then the use of Theorem 1.4 for orbital
uniquely ergodic models. As the latter models are not (in general) minimal,
the question arises whether orthogonality with respect to all uniquely ergodic
systems is the same as orthogonality to all strictly ergodic systems.

######  Remark 5.4.

If all Furstenberg systems of $\boldsymbol{u}$ are identities then
$\boldsymbol{u}$ is orthogonal to all u.e. systems. Hence (see [16]) all mean
slowly varying functions are orthogonal to all u.e. systems. In the next
section we will consider Boshernitzan’s problem in the class of multiplicative
functions.

######  Remark 5.5.

If orthogonality is given by the logarithmic way of averaging then all the
results in the paper persist, with the only change that instead of considering
Furstenberg systems of $\boldsymbol{u}$ we need to consider all logarithmic
Furstenberg systems of $\boldsymbol{u}$. For example,
$\boldsymbol{u}\perp_{\rm log}{\rm UE}$ if and only if for each $\kappa\in
V^{\rm log}(\boldsymbol{u})$, we have $\pi_{0}\perp F_{{\rm
we},0}(X_{\boldsymbol{u}},\kappa,S)$.

###  5.2 Pretentious multiplicative function orthogonal to all uniquely
ergodic systems

We consider arithmetic functions $\boldsymbol{u}:{\mathbb{N}}\to{\mathbb{D}}$
which are multiplicative (i.e.
$\boldsymbol{u}(mn)=\boldsymbol{u}(m)\boldsymbol{u}(n)$ whenever $m,n$ are
coprime), where ${\mathbb{D}}$ denotes the unit disc. Let

$\|\boldsymbol{u}\|_{B}:=\limsup_{N\to\infty}\frac{1}{N}\sum_{n\leq
N}|\boldsymbol{u}(n)|$

stand for the Besicovitch pseudo-norm of $\boldsymbol{u}$. We recall that (see
e.g. [7], Lemma 2.9)

(33)
$\|\boldsymbol{u}\|_{B}=0\Leftrightarrow\sum_{p\in{\mathbb{P}}}\frac{1}{p}(1-|\boldsymbol{u}(p)|)=\infty$

(e.g. $\boldsymbol{u}(n)=n^{-r}$, where $r>0$ is a completely multiplicative
function of zero Besicovitch pseudo-norm; the RHS in (33) yields many other
possibilities to create multiplicative functions of zero Besicovitch pseudo-
norm). When $\|\boldsymbol{u}\|_{B}=0$ then $\boldsymbol{u}(n)\to 0$ along a
subsequence of full density in ${\mathbb{N}}$. It is easy to see that in this
case $\pi_{0}=0$ $\kappa$-a.e., where $\kappa=\delta_{\underline{0}}$ is the
only Furstenberg system of $\boldsymbol{u}$. It follows that multiplicative
functions of Besicovitch pseudo-norm zero are orthogonal to all topological
systems,303030This fact does not use multiplicativity of $\boldsymbol{u}$. in
particular, to all uniquely ergodic systems. Another easy case arises when
$\boldsymbol{u}(n)=n^{it}$ ($t\in{\mathbb{R}}$) is an Archimedean character.
In this case, $\boldsymbol{u}$ is so called mean slowly varying function:
$\frac{1}{N}\sum_{n\leq N}|\boldsymbol{u}(n+1)-\boldsymbol{u}(n)|\to 0$ when
$N\to\infty$. 313131In fact, $(n+1)^{it}-n^{it}\to 0$ when $n\to\infty$. Note
also that, in general, if $\|\boldsymbol{u}\|_{B}=0$ then $\boldsymbol{u}$ is
mean slowly varying. It is already remarked in [16] that mean slowly varying
functions (without the multiplicativity restriction) are precisely those whose
all Furstenberg systems yield identities (so we obtain functions orthogonal to
all uniquely ergodic systems). Klurman [23] proved that if $\boldsymbol{u}$ is
multiplicative ($|\boldsymbol{u}|\leq 1$) and mean slowly varying then either
$\|\boldsymbol{u}\|_{B}=0$ or $\boldsymbol{u}$ is an Archimedean
character.323232Klurman’s theorem contains also the unbounded case: $n\mapsto
n^{z}$, where $0<{\rm Re}\,z<1$ which we do not consider as we assume
$|\boldsymbol{u}(n)|\leq 1$. Recall also that $\boldsymbol{u}$ is called
pretentious if for some $t\in{\mathbb{R}}$ and a Dirichlet character $\chi$,
we have333333The multiplicative “distance” between two multiplicative
functions $\boldsymbol{u},\boldsymbol{v}:{\mathbb{N}}\to{\mathbb{D}}$ is
defined as
$D(\boldsymbol{u},\boldsymbol{v})^{2}:=\sum_{p\in{\mathbb{P}}}\frac{1}{p}(1-{\rm
Re}(\boldsymbol{u}(p)\overline{\boldsymbol{v}(p)})).$

$D(\boldsymbol{u},\chi\cdot
n^{it})^{2}:=\sum_{p\in{\mathbb{P}}}\frac{1}{p}(1-{\rm
Re}(\boldsymbol{u}(p)\chi(p)p^{it}))<\infty.$

Note that, in view of (33), $\boldsymbol{u}$ is not pretentious whenever
$\|u\|_{B}=0$. We aim at proving the following result.

######  Corollary 5.6.

The only pretentious multiplicative functions
$\boldsymbol{u}:{\mathbb{N}}\to{\mathbb{D}}$ orthogonal to all uniquely
ergodic systems are Archimedean characters $n\mapsto n^{it}$,
$t\in{\mathbb{R}}$.

###### Proof.

Suppose that $\boldsymbol{u}$ is not an Archimedean character. Then
$D(\boldsymbol{u},\chi\cdot n^{it})<\infty$, where $\chi$ is a (non-trivial)
primitive Dirichlet character. If $t=0$ then by Theorem 2.8(i) [13] all
Furstenberg systems of $\boldsymbol{u}$ are ergodic, and each of them is non-
trivial (unless $\boldsymbol{u}=1$). If $t\neq 0$, then since $\boldsymbol{u}$
is (by assumption) not an Archimedean character, its spectrum cannot be
trivial by Theorem 2.8(ii) [13], and it follows by the same theorem that each
Furstenberg system of it is of the form “identity$\times T$” with $T$ ergodic,
and the ergodic part is non-trivial. Now, since $\boldsymbol{u}\perp{\rm UE}$,
$\pi_{0}\perp F_{{\rm we},0}(X_{\boldsymbol{u}},\kappa,S)$ for each $\kappa\in
V(\boldsymbol{u})$ in view of Proposition 5.1. By Proposition 4.4, it follows
that for each Furstenberg system $\kappa$, $\pi_{0}$ is $\kappa$-almost-surely
measurable with respect to $\mathcal{I}_{S}$. But then the sigma-algebra
generated by the process $(\pi_{0}\circ S^{n})$ is $\mathcal{I}_{S}$, which
yields an identity action, contradicting non-triviality of the ergodic part of
the Furstenberg systems.

######  Remark 5.7.

In view of the recent development around the Chowla and Sarnak conjectures, it
is reasonable to expect that Archimedean characters and zero Besicovitch
pseudo-norm multiplicative functions are the only (bounded by 1)
multiplicative functions which are orthogonal to all uniquely ergodic
sequences.

###  5.3 Examples of sequences orthogonal to all uniquely ergodic systems

######  Lemma 5.8.

Let $(X,T)$ be a topological dynamical system and let $(x_{n})\subset X$
satisfy:
(i) the density of $\\{n\in{\mathbb{N}}\colon x_{n+1}\neq Tx_{n}\\}$ is zero,
(ii) $(x_{n})$ is generic along $(N_{k})$343434It means that
$\lim_{k\to\infty}\frac{1}{N_{k}}\sum_{n\leq N_{k}}f(x_{n})=\int f\,d\nu$ for
all $f\in C(X)$. A general problem arises whether generic sequences satisfying
(i) do exist. The lifting lemma (when applied directly) from [21] yields the
existence of such a sequence $(x_{n})$ satisfying (i) and also (ii) along a
subsequence $(N_{k})$; however we do not control other subsequences. for
$\nu\in M(X)$.
Then $\nu$ is $T$-invariant.

###### Proof.

Obvious by considering the integrals of $f,f\circ T\in C(X)$.

######  Lemma 5.9.

Assume that $(Z,R)$ is another topological system in which $z$ is a generic
point for a measure $\kappa$ (which must be in $M(Z,R)$). Suppose that

$\frac{1}{N_{k}}\sum_{n\leq N_{k}}\delta_{(x_{n},R^{n}z)}\to\rho\in M(X\times
Z).$

Then $\rho$ is $T\times R$-invariant.

###### Proof.

The sequence $(x_{n},R^{n}z)$ is generic for $\rho$ along a subsequence and
satisfies (i) from Lemma 5.8 for $T\times R$.

######  Lemma 5.10.

Assume that in Lemma 5.8 additionally $(X,\nu,T)\in{\rm Erg}^{\perp}$ for any
convergent $(N_{k})$. Then, for each $f\in C(X)$ (for each $\nu\in
V((x_{n}))$), we have

$(\boldsymbol{u}(n):=f(x_{n}))\perp\,{\rm UE}.$

###### Proof.

Take any $(Z,R)$ u.e. (with a unique $R$-invariant measure $\kappa$). Suppose
that for some $(N_{k})$ and $z\in Z$, we have

$\frac{1}{N_{k}}\sum_{n\leq N_{k}}\delta_{(x_{n},R^{n}z)}\to\rho.$

By Lemma 5.9 it follows that $\rho\in M(X\times Z,T\times R)$ and moreover the
marginals of $\rho$ are $\nu$ and $\kappa$, respectively. It follows that for
each $\kappa$-zero mean $g\in C(Z)$, we have

$\lim_{k\to\infty}\frac{1}{N_{k}}\sum_{n\leq
N_{k}}\boldsymbol{u}(n)g(R^{n}z)=\int f\otimes g\,d\rho=\int f\,d\nu\cdot\int
g\,d\kappa=0$

by the disjointness of $(X,\nu,T)$ and $(Z,\kappa,R)$.

######  Remark 5.11.

Assume for simplicity that $f:X\to{\mathbb{C}}$ has modulus 1. Note that for
the sequence $\boldsymbol{u}$ in the lemma if $\frac{1}{N_{k}}\sum_{n\leq
N_{k}}\delta_{S^{n}\boldsymbol{u}}\to\kappa$, and $\frac{1}{N_{k}}\sum_{n\leq
N_{k}}\delta_{x_{n}}\to\nu$ then for all $m_{i},\ell_{i}\in{\mathbb{Z}}$
($i=1,\ldots,r$), if we set $g:=\prod_{i\leq r}\pi_{m_{i}}^{\ell_{i}}$, then
using (ii) from Lemma 5.8, we have

$\displaystyle\int_{X_{\boldsymbol{u}}}g\,d\kappa$
$\displaystyle=\lim_{k\to\infty}\frac{1}{N_{k}}\sum_{n\leq
N_{k}}g(S^{n}\boldsymbol{u})$
$\displaystyle=\lim_{k\to\infty}\frac{1}{N_{k}}\sum_{n\leq N_{k}}\prod_{i\leq
r}\pi_{m_{i}}^{\ell_{i}}(S^{n}\boldsymbol{u})$
$\displaystyle=\lim_{k\to\infty}\frac{1}{N_{k}}\sum_{n\leq N_{k}}\prod_{i\leq
r}\boldsymbol{u}_{n+m_{i}}^{\ell_{i}}$
$\displaystyle=\lim_{k\to\infty}\frac{1}{N_{k}}\sum_{n\leq N_{k}}\prod_{i\leq
r}\bigl{(}f(x_{n+m_{i}})\bigr{)}^{\ell_{i}}$
$\displaystyle=\lim_{k\to\infty}\frac{1}{N_{k}}\sum_{n\leq N_{k}}\prod_{i\leq
r}\bigl{(}f(T^{m_{i}}x_{n})\bigr{)}^{\ell_{i}}=\int_{X}\prod_{i\leq r}(f\circ
T^{m_{i}})^{\ell_{i}}\,d\nu.$

It follows that, for any $H\geq 1$,

$\int_{X_{\boldsymbol{u}}}\Big{|}\frac{1}{H}\sum_{h\leq H}\pi_{0}\circ
S^{h}\Big{|}^{2}\,d\kappa=\int_{X}\Big{|}\frac{1}{H}\sum_{h\leq H}f\circ
T^{h}\Big{|}^{2}\,d\nu.$

Hence, using the von Neumann theorem, we obtain that

(34) ${{\mathbbm{E}}}_{\nu}[f\,|\,\mathcal{I}_{T}]=0\text{ iff
}{{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,\mathcal{I}_{S}]=0.$

In view of Proposition 4.1, with the additional assumption that
${{\mathbbm{E}}}_{\nu}[f\,|\,\mathcal{I}_{T}]=0$ for each $\nu$ for which
$(x_{n})$ is quasi-generic, the sequence $\boldsymbol{u}$ which we obtain
satisfies $\|\boldsymbol{u}\|_{u^{1}}=0$.

######  Remark 5.12.

If we also assume that the topological system $(X,T)$ has zero topological
entropy, then it follows from [2] that the sequence
$\boldsymbol{u}=(f(x_{n}))$ has zero topological entropy as well.

Now we provide examples of sequences which are orthogonal to all uniquely
ergodic systems. Fix an irrational $\alpha\in[0,1)$. We define the sequence
$(w_{n})\subset{\mathbb{T}}^{2}$ in the following way:

(35)
$\begin{array}[]{c}(\alpha,0),(2\alpha,0),(2\alpha,2\alpha),(2\alpha,4\alpha),(2\alpha,6\alpha),\ldots\\\
\ldots,(k\alpha,0),(k\alpha,k\alpha),\ldots,\bigl{(}k\alpha,(k^{2}-1)k\alpha\bigr{)},\ldots\end{array}$

This sequence is the concatenation of pieces, indexed by $k\geq 1$: at stage
$k$ we take $(k\alpha,0)$ and the initial $k^{2}$-long piece of its orbit via
the map $T:(x,y)\mapsto(x,x+y)$ on ${\mathbb{T}}^{2}$, cf. Section 3.7. We
claim that the sequence we defined is generic for $\nu={\rm
Leb}_{{\mathbb{T}}}\otimes{\rm Leb}_{{\mathbb{T}}}$. Indeed, to see this,
first notice that we only need to check the case $N=1^{2}+2^{2}+\ldots+L^{2}$
as the RHS is of order $L^{3}$ while the next group in the definition of
$(w_{n})$ has length $(L+1)^{2}$ which is $o(1)$ with respect to $N$. For
$r,s\in{\mathbb{Z}}$, consider $F(x,y)=e^{2\pi i(rx+sy)}$ (we check the weak
convergence of our sequence testing on characters of ${\mathbb{T}}^{2}$).
Assume first that $s\neq 0$, then we have

$\frac{1}{N}\sum_{n\leq N}F(w_{n})=\frac{1}{N}\sum_{k\leq L}e^{2\pi
irk\alpha}\sum_{j=0}^{k^{2}-1}e^{2\pi isjk\alpha}=\frac{1}{N}\sum_{k\leq
L}e^{2\pi irk\alpha}\frac{e^{2\pi ik^{2}sk\alpha}-1}{e^{2\pi isk\alpha}-1}.$

Fix $\varepsilon_{0}>0$. Then, if $L$ is large enough,

$\Big{|}\\{k\leq L\colon\Big{|}e^{2\pi
ik(s\alpha)}-1\Big{|}\geq\varepsilon_{0}\\}\Big{|}\geq(1-3\varepsilon_{0})L$

and for each $k$ in this large set, we have

$\sum_{j=0}^{k^{2}-1}e^{2\pi isjk\alpha}=O(1/\varepsilon_{0}).$

For the remaining $k\leq L$ the sums are bounded by $k^{2}$. Thus353535We
recall that $\sum_{m\leq M}\frac{1}{m^{2}}=\frac{1}{3}M^{3}+O(M^{2})$.

$\displaystyle\Big{|}\frac{1}{N}\sum_{n\leq N}F(w_{n})\Big{|}$
$\displaystyle\leq\frac{1}{N}\sum_{k\leq
L}O(1/\varepsilon_{0})+\frac{1}{N}\sum_{k=L(1-3\varepsilon_{0})}^{L}k^{2}$
$\displaystyle\leq
O(1/\varepsilon_{0})\frac{L}{N}+\frac{1}{N}\Big{(}\frac{1}{3}L^{3}-\frac{1}{3}(L(1-3\varepsilon_{0}))^{3}+O(L^{2})\Big{)}$
$\displaystyle=o(1)+O(\varepsilon_{0})+\frac{O(L^{2})}{N}=o(1)+O(\varepsilon_{0})$

when $L\to\infty$.

Now let us deal with the case $(s=0,r\neq 0)$. We then write, using Abel’s
summation formula

(36) $\frac{1}{N}\sum_{n\leq N}F(w_{n})=\\\ \frac{1}{N}\sum_{1\leq k\leq
L}k^{2}e^{2\pi ikr\alpha}=\frac{1}{N}\sum_{1\leq j\leq
L}\bigl{(}j^{2}-(j-1)^{2}\bigr{)}\sum_{j\leq k\leq L}e^{2\pi ikr\alpha}.$

For a fixed $\varepsilon_{0}>0$, we consider the contribution of $j$’s which
are less than $(1-\varepsilon_{0})L$: for such a $j$, we have

$\left|\frac{1}{L-j+1}\sum_{j\leq k\leq L}e^{2\pi
ikr\alpha}\right|\leq\frac{1}{\varepsilon_{0}L}\left|\frac{2}{1-e^{2\pi
ir\alpha}}\right|=\frac{1}{L}O(1/\varepsilon_{0}),$

and since

$\frac{1}{N}\sum_{1\leq j\leq L}\bigl{(}j^{2}-(j-1)^{2}\bigr{)}(L-j+1)=1$

(this corresponds to (36) when $r=0$), we get

$\left|\frac{1}{N}\sum_{1\leq
j\leq(1-\varepsilon_{0})L}\bigl{(}j^{2}-(j-1)^{2}\bigr{)}\sum_{j\leq k\leq
L}e^{2\pi ikr\alpha}\right|=\frac{1}{L}O(1/\varepsilon_{0}).$

In the remaining terms, corresponding to $j>(1-\varepsilon_{0}L)$, we just use

$\left|\sum_{j\leq k\leq L}e^{2\pi ikr\alpha}\right|=O(\varepsilon_{0}L)$

to get

$\left|\frac{1}{N}\sum_{(1-\varepsilon_{0})L<j\leq
L}\bigl{(}j^{2}-(j-1)^{2}\bigr{)}\sum_{j\leq k\leq L}e^{2\pi
ikr\alpha}\right|=\frac{1}{N}O(L^{3}\varepsilon_{0})=O(\varepsilon_{0}).$

We finally have for $(s=0,r\neq 0)$

$\Big{|}\frac{1}{N}\sum_{n\leq
N}F(w_{n})\Big{|}=\frac{1}{L}O(1/\varepsilon_{0})+O(\varepsilon_{0}),$

which concludes the proof of the genericity of the sequence $(w_{n})$.

We now apply Lemma 5.10 to $T:{\mathbb{T}}^{2}\to{\mathbb{T}}^{2}$,
$T(x,y)=(x,x+y)$ and $f(x,y)=e^{2\pi iy}$ and the sequence (35) to obtain that
the sequence

(37) $\begin{array}[]{c}1,1,e^{2\pi i2\alpha},e^{2\pi i4\alpha},e^{2\pi
i6\alpha},\ldots\\\ \ldots,1,e^{2\pi ik\alpha},\ldots,e^{2\pi
i(k^{2}-1)k\alpha},\ldots\end{array}$

is orthogonal to all uniquely ergodic systems.

###  5.4 Properties of arithmetic functions orthogonal to all u.e. systems

We start by the following consequence of Theorem 4.5.

######  Corollary 5.13.

Let $(\widetilde{X},\widetilde{\mu},\widetilde{T})$ be a measure-preserving
system and let $f\perp F_{\rm
we}(\widetilde{X},\widetilde{\mu},\widetilde{T})$. Let $\eta$ be a
$\widetilde{T}$-invariant probability measure with $\eta\ll\widetilde{\mu}$.
Assume that $d\eta/d\widetilde{\mu}$ is bounded so that $f$ can also be
considered in $L^{2}(\nu)$. Then, we have $f\perp F_{\rm
we}(\widetilde{X},\eta,\widetilde{T})$.

###### Proof.

As usual, we can represent $(\widetilde{X},\widetilde{\mu},\widetilde{T})$ in
the form $(X\times Y,\mu\otimes\nu,(x,y)\mapsto(x,T_{x}y))$ (with $x\mapsto
T_{x}$ being the ergodic decomposition). Then $h:=d\eta/d(\mu\otimes\nu)$ is
$X$-measurable (because $\eta$ is $\widetilde{T}$-invariant and so is $h$). We
consider the probability measure $\mu_{h}$ on $X$ defined by
$d\mu_{h}/d\mu=h$, so that $\eta=\mu_{h}\otimes\nu$. Since $f\perp F_{\rm
we}(\widetilde{X},\widetilde{\mu},\widetilde{T})$, the condition in the
statement of Theorem 4.5 (see also Footnote 27) is satisfied, and we rewrite
it as follows: for each measurable map
$\lambda:(x,x^{\prime})\mapsto\lambda_{x.x^{\prime}}\in
J(T_{x},T_{x^{\prime}})$ such that $\lambda_{x,x^{\prime}}$ is ergodic
$\mu\otimes\mu$\- a.s, we have

$\int_{Y\times
Y}f(x,y)f(x^{\prime},y^{\prime})\,d\lambda_{x.x^{\prime}}(y,y^{\prime})=\Big{(}\int_{Y}f(x,y)\,d\nu(y)\Big{)}\Big{(}\int_{Y}f(x^{\prime},y)\,d\nu(y)\Big{)}.$

But the same condition remains satisfied if we replace everywhere $\mu$ with
$\mu_{h}$. Indeed, we know (see Remark 3.12) that there exists a map
$\rho:(x,x^{\prime})\mapsto\rho_{x.x^{\prime}}\in J(T_{x},T_{x^{\prime}})$
such that $\rho_{x,x^{\prime}}$ is ergodic $\mu\otimes\mu$\- a.s. Therefore,
if $\lambda:(x,x^{\prime})\mapsto\lambda_{x.x^{\prime}}\in
J(T_{x},T_{x^{\prime}})$ satisfies $\lambda_{x,x^{\prime}}$ is ergodic
$\mu_{h}\otimes\mu_{h}$\- a.s, we can always modify this map by setting

$\bar{\lambda}_{x,x^{\prime}}:=\begin{cases}\lambda_{x,x^{\prime}}&\text{ if
}h(x)h(x^{\prime})>0,\\\ \rho_{x,x^{\prime}}&\text{ otherwise.}\end{cases}$

Then $\bar{\lambda}_{x,x^{\prime}}$ is ergodic $\mu\otimes\mu$-a.s, and
therefore we have $\mu\otimes\mu$-a.s.

$\int_{Y\times
Y}f(x,y)f(x^{\prime},y^{\prime})\,d\bar{\lambda}_{x.x^{\prime}}(y,y^{\prime})=\Big{(}\int_{Y}f(x,y)\,d\nu(y)\Big{)}\Big{(}\int_{Y}f(x^{\prime},y)\,d\nu(y)\Big{)}.$

But $\bar{\lambda}_{x.x^{\prime}}=\lambda_{x,x^{\prime}}$,
$\mu_{h}\otimes\mu_{h}$-a.s, so again by Theorem 4.5, this proves $f\perp
F_{\rm we}(\widetilde{X},\eta,\widetilde{T})$.

Now, we will prove the following result.

######  Proposition 5.14.

Let $\boldsymbol{u}$, $\|\boldsymbol{u}\|_{u^{1}}=0$, be a bounded arithmetic
function orthogonal to all u.e. systems. Then $\boldsymbol{u}$ is also
orthogonal to all systems whose set of ergodic invariant measures is
countable. More precisely: if $(Z,R)$ is a topological system whose set
$\\{\rho_{i}\colon i\in I\\}$ of ergodic measures is countable, then for any
$z_{0}\in Z$ and any function $g\in C(Z)$, we have

$\frac{1}{N}\sum_{n<N}\boldsymbol{u}(n)g(R^{n}z_{0})\to 0.$

###### Proof.

Let $(N_{k})$ be an increasing sequence along which $(u,z_{0})$ is generic for
some $S\times R$-invariant probability measure $\kappa$:

$\frac{1}{N_{k}}\sum_{n<N_{k}}\delta_{(S^{n}\boldsymbol{u},R^{n}z_{0})}\to\kappa\text{
when }k\to\infty.$

Let $\rho$ be the marginal of $\kappa$ on $Z$. Then $\rho=\sum_{i\in
I}\alpha_{i}\rho_{i}$ for some non-negative real numbers $\alpha_{i}$ and
$\sum_{i\in I}\alpha_{i}=1$. For each $i\in I$, set

$Z_{i}:=\\{z\in Z\colon z\text{ is generic for }\rho_{i}\\}.$

Then, $Z_{i}$ is Borel, $R$-invariant and $Z_{i}\cap Z_{j}=\emptyset$ for
$i\neq j$. Moreover, $\rho(\bigcup_{i\in I}Z_{i})=1$. Let $\mu$ be the
marginal of $\kappa$ on $X_{\boldsymbol{u}}$: $\mu$ determines a Furstenberg
system of $\boldsymbol{u}$. Since we assume $\boldsymbol{u}\perp{\rm UE}$,
Proposition 5.1 ensures that, in $L^{2}(X_{\boldsymbol{u}},\mu)$, we have

$\pi_{0}\perp F_{\rm we}(X_{\boldsymbol{u}},\mu,S).$

For each $i\in I$ such that $\alpha_{i}>0$, we let $\kappa_{i}\ll\kappa$ be
the probability measure defined by

$\frac{d\kappa_{i}}{d\kappa}:=\frac{1}{\alpha_{i}}\mathbbm{1}_{X_{\boldsymbol{u}}\times
Z_{i}}.$

This Radon-Nikodym derivative is $S\times R$-invariant, hence, $\kappa_{i}$ is
$S\times R$-invariant. Let $\mu_{i}$ be the marginal of $\kappa_{i}$ on
$X_{\boldsymbol{u}}$. We have:

  * •

$\mu_{i}\ll\mu$,

  * •

$\mu_{i}$ is $S$-invariant,

  * •

$\frac{d\mu_{i}}{d\mu}\leq\frac{1}{\alpha_{i}}$.

Furthermore, we have

$\frac{1}{N_{k}}\sum_{n<N_{k}}\boldsymbol{u}(n)g(R^{n}z_{0})\to\int_{X_{\boldsymbol{u}}\times
Z}\pi_{0}\otimes g\,d\kappa=\sum_{i\in
I}\alpha_{i}\int_{X_{\boldsymbol{u}}\times Z}\pi_{0}\otimes g\,d\kappa_{i}.$

But $\kappa_{i}$ is a joining of $\mu_{i}$ with the ergodic measure
$\rho_{i}$. By Corollary 5.13, we have $\pi_{0}\perp F_{\rm
we}(X_{\boldsymbol{u}},\mu_{i},S)$ and finally, we get

$\frac{1}{N_{k}}\sum_{n<N_{k}}\boldsymbol{u}(n)g(R^{n}z_{0})\to 0.$

######  Remark 5.15.

If we drop the assumption $\|\boldsymbol{u}\|_{u_{1}}=0$, then we obtain that
for any function $g\in C(Z)$, satisfying $\int_{Z}gd\rho_{i}=0$ for all $i\in
I$, we have $\frac{1}{N}\sum_{n<N}\boldsymbol{u}(n)g(R^{n}z)\to 0$ when
$N\to\infty$ (for all $z\in Z$).

If we take $\boldsymbol{u}\perp{\rm UE}$ ($\|\boldsymbol{u}\|_{u^{1}}=0$), we
know that $\boldsymbol{u}$ is in fact orthogonal to all topological systems
whose set of invariant measures is countable. Using Proposition 5.14 and
proceeding now as in the proof P${}_{2}\Rightarrow{\rm P}_{3}$ in Section 2.2
[2], we obtain the following result.

######  Corollary 5.16.

Suppose that $\boldsymbol{u}\perp{\rm UE}$ ($\|\boldsymbol{u}\|_{u^{1}}=0$).
Then, for any uniquely ergodic system $(Y,S)$, for any increasing $(b_{k})$
with $b_{k+1}-b_{k}\to\infty$, for all $(y_{k})\subset Y$ and all $f\in C(Y)$
with $\int f\,d\nu=0$ ($\nu$ is the unique invariant measure on $Y$), we have

$\lim_{K\to\infty}\frac{1}{b_{K}}\sum_{k<K}\Big{|}\sum_{b_{k}\leq
n<b_{k+1}}f(S^{n-b_{k}}y_{k})\boldsymbol{u}(n)\Big{|}=0,$

i.e. $(Y,S)$ satisfies the strong MOMO property (relative to
$\boldsymbol{u}$).

###  5.5 Mean slowly varying functions are multipliers of the UE⟂

We say that an arithmetic function $\boldsymbol{v}$ is a multiplier for the
problem of orthogonality to all u.e. systems if

$\boldsymbol{u}\cdot\boldsymbol{v}\perp{\rm UE}\text{ whenever
}\boldsymbol{u}\perp{\rm UE}.$

Note that if $\boldsymbol{v}$ is a multiplier and $\widetilde{\boldsymbol{v}}$
satisfies $\|\boldsymbol{v}-\widetilde{\boldsymbol{v}}\|_{B}=0$ then also
$\widetilde{\boldsymbol{v}}$ is a multiplier. Note also that
$\boldsymbol{v}(n)=n^{it}$ is a multiplier in the above sense since taking any
u.e. sequence $(a(n))$ and denoting $s_{n}:=\sum_{k\leq
n}\boldsymbol{u}(k)a(k)$, we have (summation by parts):

$\frac{1}{N}\sum_{n\leq
N}\boldsymbol{v}(n)\boldsymbol{u}(n)a(n)=o(1)+\frac{1}{N}\sum_{n\leq
N}n(\boldsymbol{v}(n+1)-\boldsymbol{v}(n))\frac{s_{n}}{n}$

and the claim follows from the fact that $(n+1)^{it}-n^{it}=O(1/n)$
(Archimedean characters are slowly varying function with a speed). A natural
question arises whether we can replace here Archimedean characters by any
function whose all Furstenberg systems are identities, in other words, is the
above true for all mean slowly varying functions [16]? We will see that the
answer to this question is positive.

We need a general lemma on mean slowly varying functions.

######  Lemma 5.17.

If $\boldsymbol{v}$ is a (bounded) mean slowly varying function then there
exist:

  * •

another mean slowly varying function $\widetilde{\boldsymbol{v}}$ such that

$\frac{1}{N}\sum_{n\leq N}|\boldsymbol{v}(n)-\widetilde{\boldsymbol{v}}(n)|\to
0\text{ when }N\to\infty,$

  * •

an increasing sequence $(b_{k})$ satisfying $b_{k+1}-b_{k}\to\infty$,

  * •

a bounded sequence $(z_{k})\subset{\mathbb{C}}$,

  * •

a sequence $0<\varepsilon_{k}\to 0$ monotonically

such that for all $k\geq 1$ and all $n\in\\{b_{k},\ldots,b_{k+1}-1\\}$ we have
$|\widetilde{\boldsymbol{v}}(n)-z_{k}|<\varepsilon_{k}$.

###### Proof.

Since $\boldsymbol{v}$ is MSV, for any $\delta>0$, we have

$\frac{1}{N}\sum_{n<N}\mathbbm{1}_{|\boldsymbol{v}(n)-\boldsymbol{v}(n+1)|\geq\delta}\to
0\text{ when }N\to\infty.$

Let us fix a decreasing sequence $0<\delta_{j}\to 0$. Then, we get an
increasing sequence $(M_{j})$ such that for all $j\geq 1$ and all $N\geq
M_{j}$, we have

$\frac{1}{N}\sum_{n<N}\mathbbm{1}_{|\boldsymbol{v}(n)-\boldsymbol{v}(n+1)|\geq\delta_{j}}<\frac{1}{2^{j}}.$

We now define

$B:=\bigcup_{j\geq 1}\\{n\colon M_{j}\leq
n<M_{j+1},|\boldsymbol{v}(n)-\boldsymbol{v}(n+1)|\geq\delta_{j}\\}.$

If $M_{j}\leq N<M_{j+1}$ then we have

$\frac{1}{N}\sum_{n<N}\mathbbm{1}_{B}(n)\leq\frac{1}{N}\sum_{n<N}\mathbbm{1}_{|\boldsymbol{v}(n)-\boldsymbol{v}(n+1)|\geq\delta_{j}}<\frac{1}{2^{j}}.$

The $b_{k}$’s in the statement of the lemma are destined to cover this set
where the sequence has large gaps. But as we want to have
$b_{k+1}-b_{k}\to\infty$, we have to avoid integers in $B$ which are too close
to each other. Let us define more precisely what we mean: we say that two
consecutive elements $b,b^{\prime}$ of $B$ are _too close_ if

  * •

$M_{j}\leq b<M_{j+1}$ for some $j$, and

  * •

$b^{\prime}\leq b+j$.

Then we set $C:=\bigcup_{b}\\{b,b+1,\ldots,b^{\prime}-1\\}$, where the union
ranges over all pairs $(b,b^{\prime})$ in $B$ which are too close to each
other. For each $n$ in $C$, we now change $\boldsymbol{v}(n)$ by setting
$\widetilde{\boldsymbol{v}}(n):=\boldsymbol{v}(\ell(n))$, where $\ell(n)$ is
defined as the largest integer smaller than $n$ which is not in $C$. For
$n\in{\mathbb{N}}\setminus C$, we set
$\widetilde{\boldsymbol{v}}(n):=\boldsymbol{v}(n)$. Define $\widetilde{B}$
from $\widetilde{\boldsymbol{v}}$ as we defined $B$ from $\boldsymbol{v}$. We
get $\widetilde{B}\subset B$, but we have removed from $B$ all integers $b$
which are too close to the next element of $B$. Observe now that we have for
$M_{j}\leq N<M_{j+1}$

$\frac{1}{N}|C\cap\\{1,\ldots,N\\}|\leq
j\frac{1}{N}|B\cap\\{1,\ldots,N\\}|\leq\frac{j}{2^{j}}\to 0$

when $j\to\infty$. This shows that
$\frac{1}{N}\sum_{n<N}|\boldsymbol{v}(n)-\widetilde{\boldsymbol{v}}(n)|\to 0$.
Moreover, the set $\widetilde{B}$ (where
$|\widetilde{\boldsymbol{v}}(n)-\widetilde{\boldsymbol{v}}(n-1)|$ is “too
large”) now satisfies: if $b<b^{\prime}$ are two elements in $\widetilde{B}$
with $M_{j}\leq b<m_{j+1}$ then $b^{\prime}-b>j$. Then, we can choose the
increasing sequence $(b_{k})$ such that:

  * •

$\widetilde{B}\subset\\{b_{k}\colon k\geq 1\\}$,

  * •

if $M_{j}\leq b_{k}<M_{j+1}$ then $b_{k+1}-b_{k}\in\\{j,\ldots,2j-1\\}$.

Then, the inequality $b_{k+1}-b_{k}\geq j$ ensures that
$b_{k+1}-b_{k}\to\infty$, and the inequality $b_{k+1}-b_{k}\leq 2j$ together
with the fact that all $n\in\\{b_{k}+1,\ldots,b_{k+1}-1\\}$ are not in
$\widetilde{B}$ ensure that for any such $n$,

$|\widetilde{\boldsymbol{v}}(n)-\widetilde{\boldsymbol{v}}(b_{k})|\leq
2j\cdot\delta_{j}.$

We set $z_{k}:=\widetilde{\boldsymbol{v}}(b_{k})$, and we choose $\delta_{j}$
so that $2j\delta_{j}<2^{-j}$. Then we get

$\sup_{n\in\\{b_{k},\ldots,b_{k+1}-1\\}}|\widetilde{\boldsymbol{v}}(n)-z_{k}|\to
0$

when $k\to\infty$.

######  Proposition 5.18.

If $\boldsymbol{v}$ is mean slowly varying then $\boldsymbol{v}$ is a
multiplier of the class UE⟂.

###### Proof.

Using Lemma 5.17, we only need to show that $\widetilde{\boldsymbol{v}}$ is a
multiplier. Given $\boldsymbol{u}\perp{\rm UE}$ and $(X,T)$ a uniquely ergodic
system with $f\in C(X)$ and $x\in X$, we have

$\Big{|}\frac{1}{b_{K}}\sum_{k<K}\sum_{b_{k}\leq
n<b_{k+1}}\boldsymbol{u}(n)\widetilde{\boldsymbol{v}}(n)f(T^{n}x)\Big{|}\leq$
$\frac{1}{b_{K}}\sum_{k<K}|z_{k}|\Big{|}\sum_{b_{k}\leq
n<b_{k+1}}\boldsymbol{u}(n)f(T^{n}x)\Big{|}+\Big{(}\sum_{k<K}\varepsilon_{k}\frac{b_{k+1}-b_{k}}{b_{K}}\Big{)}\|\boldsymbol{u}\|_{\infty}\|f\|_{\infty}$

and when $K\to\infty$, the first summand is going to zero because of the
strong MOMO property (Corollary 5.16), while the second also goes to zero
since $\varepsilon_{k}\to 0$.

We have been unable to answer the converse: Is every multiplier of ${\rm
UE^{\perp}}$ an MSV function?

###  5.6 How to recognize that $\rho$ satisfies $\rho|_{X\times
X}=\mu\otimes\mu$?

So we have a Furstenberg system $(X_{\boldsymbol{u}},\kappa,S)$ of
$\boldsymbol{u}$ with identifications $\kappa=\mu\otimes\nu$ and
$X_{\boldsymbol{u}}=X\times Y$ and $S$ represented as $\widetilde{T}$. Then,
in view of (24),

$\rho|_{X\times X}=\mu\otimes\mu\text{ if and only if }\rho|_{(X\times
Y)\times X}=(\mu\otimes\nu)\otimes\mu$

and the latter holds if and only if for all “monomials”
$P=\prod_{j=1}^{k}\pi_{0}^{p_{j}}\circ S^{r_{j}}$ and
$Q=\prod_{j=1}^{\ell}\pi_{0}^{q_{j}}\circ S^{s_{j}}$, we have

$\int_{(X\times Y)\times(X\times
Y)}P(x,y){{\mathbbm{E}}}_{\kappa}[Q\,|\,X](x^{\prime})\,d\rho(x,y,x^{\prime})=$
$\int_{X_{\boldsymbol{u}}}P\,d\kappa\int_{X\times
Y}{{\mathbbm{E}}}_{\kappa}[Q\,|\,X](x^{\prime})\,d\mu(x^{\prime})=\int_{X_{\boldsymbol{u}}}P\,d\kappa\int_{X_{\boldsymbol{u}}}Q\,d\kappa.$

Now, in $L^{2}(\kappa)$, we have

${{\mathbbm{E}}}_{\kappa}[Q\,|\,X]=\lim_{N\to\infty}\frac{1}{N}\sum_{n\leq
N}Q\circ S^{n}.$

We can also view the equality above in $L^{2}(\rho)$. It follows that

$\int_{(X\times Y)\times(X\times
Y)}P(x,y){{\mathbbm{E}}}_{\kappa}[Q\,|\,X](x^{\prime})\,d\rho(x,y,x^{\prime})=$
$\lim_{N\to\infty}\frac{1}{N}\sum_{n\leq N}\int_{X_{\boldsymbol{u}}\times
X_{\boldsymbol{u}}}P\otimes Q\circ S^{n}\,d\rho.$

Collecting the remarks above, we have proved the following result:

######  Proposition 5.19.

Under the notation above, let $\rho\in J_{2}(S,\kappa)$. Then $\rho|_{X\times
X}=\mu\otimes\mu$ if and only if we have

(38) $\lim_{N\to\infty}\frac{1}{N}\sum_{n\leq N}\int_{X_{\boldsymbol{u}}\times
X_{\boldsymbol{u}}}P\otimes Q\circ
S^{n}\,d\rho=\int_{X_{\boldsymbol{u}}}P\,d\kappa\int_{X_{\boldsymbol{u}}}Q\,d\kappa$

for all monomials $P,Q$.

An alternative arises by using, instead of “monomials”, the characteristic
functions $\mathbbm{1}_{[B]_{0}}$: $B\in A^{k}$, where $A$ is the (finite) set
of values of $\boldsymbol{u}$ and $k\geq 1$ (here $[B]_{s}=\\{x\in
X_{\boldsymbol{u}}\colon\boldsymbol{u}[s,s+k-1]=B\\}$. As shifts of such
functions yield a linearly dense subset in $L^{2}(\kappa)$, by repeating all
arguments that led to Proposition 5.19, we obtain the following:

######  Corollary 5.20.

Under the notation above, let $\rho\in J_{2}(S,\kappa)$. Then $\rho|_{X\times
X}=\mu\otimes\mu$ if and only if we have

(39) $\lim_{N\to\infty}\frac{1}{N}\sum_{n\leq
N}\rho([B]_{0}\times[C]_{-n})=\kappa([B]_{0})\kappa([C])$

for all blocks $B\in A^{k},C\in A^{\ell}$ with $k,\ell\geq 1$.373737An
equivalent form of this corollary is that: if $\Phi_{\rho}$ stands for the
Markov operator corresponding to $\rho$ then
$\Phi_{\rho}|_{L^{2}(\mathcal{I}_{S})}=0$ if and only if
$\Phi_{\rho}\circ\frac{1}{N}\sum_{n\leq N}S^{n}\to 0$ weakly in
$L_{0}^{2}(\kappa)$. Indeed, $\frac{1}{N}\sum_{n\leq
N}S^{n}\to{{\mathbbm{E}}}_{\kappa}[\cdot\,|\,\mathcal{I}_{S}]$ (in
$L^{2}(\kappa)$) by the von Neumann theorem.

Note that the quantities we have: the LHS terms and the RHS term in (38) and
in (39) are computable through the generic sequence
$(S^{n}\boldsymbol{u},S^{\phi_{k}(n)}\boldsymbol{u})$ along $(N_{k})$, see
Proposition 8.1, and the generic point $\boldsymbol{u}$ along $(N_{k})$,
respectively.

###  5.7 Summing up

Our aim is to describe those $\boldsymbol{u}$ which are orthogonal to all u.e.
sequences (we assume that $\boldsymbol{u}$ has zero mean on typical short
interval). For that, for all Furstenberg systems $\kappa\in V(\boldsymbol{u})$
we need to check the assertion of Corollary 4.9. In this corollary, we need to
deal with some self-joinings $\lambda$ (of $\kappa$). In fact, all of them are
described combinatorially, using only $\boldsymbol{u}$, see Proposition 8.1.
We need to check the assertion of Corollary 4.9, which by Remark 4.10 (see
$(\ast)$ there) is

$\lim_{L\to\infty}\int_{X_{\boldsymbol{u}}\times
X_{\boldsymbol{u}}}\Big{|}\frac{1}{L}\sum_{\ell\leq
L}(\pi_{0}\otimes\pi_{0})(S\times S)^{\ell}\Big{|}^{2}d\lambda=0.$

That is, given $\varepsilon>0$, for $L>L_{0}$, we want to see

$\int_{X_{\boldsymbol{u}}\times
X_{\boldsymbol{u}}}\Big{|}\frac{1}{L}\sum_{\ell\leq
L}(\pi_{0}\otimes\pi_{0})(S\times S)^{\ell}\Big{|}^{2}d\lambda<\varepsilon,$

where the integral is computable along a subsequence $(N_{k})$ (in fact, a
subsequence of it), so (using Proposition 8.1), we need

$\limsup_{k\to\infty}\frac{1}{N_{k}}\sum_{n\leq
N_{k}}\Big{|}\frac{1}{L}\sum_{\ell\leq L}(\pi_{0}\otimes\pi_{0})\circ(S\times
S)^{\ell}(S^{n}\boldsymbol{u},S^{\phi_{k}(n)}\boldsymbol{u})\Big{|}^{2}=$
$\limsup_{k\to\infty}\frac{1}{N_{k}}\sum_{n\leq
N_{k}}\Big{|}\frac{1}{L}\sum_{\ell\leq
L}\boldsymbol{u}(n+\ell)\boldsymbol{u}(\phi_{k}(n)+\ell)\Big{|}^{2}<\varepsilon,$

which is, from the combinatorial point of view, a certain condition on the
behaviour on short intervals. However, it is clear that if we consider all
self-joinings, then, the above condition is not satisfied (take the diagonal
self-joining). The key here is that we only consider those $\lambda$ which
satisfy (39) (or (38)).

Part II

##  6 Characteristic classes and orthogonality to uniquely ergodic systems

###  6.1 Characteristic classes and disjointness

In what follows, we consider ${\cal F}$ a characteristic class. We recall that
each non-trivial characteristic class contains the class ID of all identities
[21].

######  Proposition 6.1.

Let $\widetilde{T}\in{\rm
Aut}(\widetilde{X},\widetilde{\mathcal{B}},\widetilde{\mu})$ and let ${\cal
F}$ be a characteristic class. Let ${\cal A}_{{\cal F}}$ be the largest ${\cal
F}$-factor of $\widetilde{T}$. Then the following conditions are equivalent:
(i) $\widetilde{T}\perp{\cal F}\cap{\rm Erg}$,
(ii) $\widetilde{T}|_{{\cal A}_{{\cal F}}}\perp{\cal F}\cap{\rm Erg}$,
(iii) $\widetilde{T}|_{{\cal A}_{{\cal F}}}\perp{\rm Erg}$.

###### Proof.

(ii) $\Rightarrow$ (i) Take any $R$ ergodic in ${\cal F}$. Then
$\widetilde{T}|_{{\cal A}_{{\cal F}}}$ is disjoint with $R$ by (ii), and this
disjointness lifts to $\widetilde{T}$ by Corollary 2.13, as $R\in{\cal F}$..

(ii) $\Rightarrow$ (iii) Take any $R$ ergodic. Then $\widetilde{T}|_{{\cal
A}_{{\cal F}}}$ is disjoint with the largest ${\cal F}$-factor of $R$ (by
(ii)), and then this disjointness lifts to $R$ by Corollary 2.13, as
$\widetilde{T}|_{{\cal A}_{{\cal F}}}\in{\cal F}$.

The other implications are straightforward.

Remembering that $\mathcal{I}_{\widetilde{T}}\subset{\cal A}_{{\cal
F}}\mod\widetilde{\mu}$ and using Proposition 6.1 together with Theorem 3.1,
we obtain the following.

######  Corollary 6.2.

$\widetilde{T}\perp{\cal F}\cap{\rm Erg}$ if and only if
$\widetilde{T}|_{{\cal A}_{{\cal F}}}$ is a confined extension of the sigma-
algebra of invariant sets.

Before, we formulate a result which resembles more Theorem 3.1 than the above
corollary, let us consider two examples.

######  Example 6.1.

Let ${\cal F}$=DISP be the (characteristic) class of all automorphisms with
discrete spectrum. Let $T:{\mathbb{T}}^{2}\to{\mathbb{T}}^{2}$,
$T(x,y)=(x,y+x)$ (considered with Leb${}_{{\mathbb{T}}^{2}}$). It is not hard
to see383838Note that on $L^{2}({\mathbb{T}}^{2})\ominus
L^{2}({\mathbb{T}}\otimes\\{\emptyset,{\mathbb{T}}\\})$ the spectrum is purely
Lebesgue. Of course, here $T\in{\rm Erg}^{\perp}$. In fact, if $T$ is any
automorphism from Erg⟂ then its only eigenvalue is 1, so its Kronecker factor
is always the sigma-algebra of invariant sets, while for the fiber
automorphisms we can have discrete spectrum. that ${\cal A}_{{\cal F}}$ is
precisely $\mathcal{I}_{T}$ (that is, the sigma-algebra of the first
coordinate). On the other hand, on (a.a.) fibers (ergodic components) the
spectrum is discrete, so on a.a. fibers the largest ${\cal F}$-factor is the
whole space. In other word, the restriction of the factor ${\cal A}_{{\cal
F}}$ to fibers does not give the (largest) ${\cal F}$-factors on the fibers.

######  Example 6.2.

Let ${\cal F}=$ZE be the class of automorphisms with zero entropy. According
to [21], this is the largest (proper) characteristic class. Assume that
$\widetilde{T}$ is aperiodic, so it is of the form
$\widetilde{T}(x,y)=(x,T_{x}y)$ on a space $(X\times Y,\mu\otimes\nu$), and
the space $(Y,\mathcal{C},\nu)$ has no atoms. In Proposition 10.1 below, we
will show that contrary to the phenomenon in the previous example, here, the
largest ${\cal F}$-factor (the Pinsker sigma-algebra) of $\widetilde{T}$ is
also the largest ${\cal F}$-factor for a.e. fiber automorphism. By applying
Theorem 3.1 (the equivalence of (ii) and (iii)), we hence obtain the following
result.

######  Corollary 6.3.

The following conditions are equivalent:
(i) $\widetilde{T}\perp{\rm Erg}\cap{\rm ZE}$.
(ii) The extension $\widetilde{T}|_{\Pi(\widetilde{T})}\to{\rm Id}_{X}$ is
confined.
(iii) For $\mu\otimes\mu$-a.e. $(x,x^{\prime})\in X\times X$, we have
$\Pi(T_{x})\perp\Pi(T_{x^{\prime}})$.

In particular, if the fiber automorphisms $T_{x}$ are K, then
$\widetilde{T}\perp{\rm Erg}\cap{\rm ZE}$.

######  Example 6.3.

Let $X=(0,1]$, $Y=SL_{2}({\mathbb{R}})/SL_{2}({\mathbb{Z}})$ (considered with
the corresponding Haar measure) and
$\widetilde{T}(t,x\Gamma):=(t,g_{t}x\Gamma)$, where
$\Gamma=SL_{2}({\mathbb{Z}})$, $g_{t}:=\left[\begin{array}[]{cc}e^{-t}&0\\\
0&e^{t}\end{array}\right]$. Then $\widetilde{T}\perp{\rm Erg}\cap{\rm ZE}$
while $\widetilde{T}$ is not disjoint393939On the other hand, $\widetilde{T}$
has no non-trivial ergodic factor as each such would have to be a factor of
a.e. ergodic component which is in conflict with entropy on the fibers. with
Erg (because the fiber automorphisms are not a.e. disjoint; $T_{t}$ is
Bernoulli with entropy $t$) neither with ZE (its Pinsker factor equal to the
sigma-algebra of invariant sets is non-trivial).

######  Remark 6.4.

Two further classes: ID and NIL1 404040This is the class of all automorphisms
whose a.e. ergodic component has discrete spectrum, see [21]. behave similarly
to ZE. For the ID class simply the trace of the sigma-algebra of invariant
sets is the trivial sigma-algebra on each fiber. For the NIL1 class the
largest ${\cal F}$-factor is the relative Kronecker over the sigma-algebra of
invariant sets. It is proved in [21] that this factor comes exactly from the
Kronecker factors on the fibers (and in the ergodic case, the relative
Kronecker factor over the sigma-algebra of invariant sets is exactly the
Kronecker factor.

On the base of the above, we have:414141Note that the last three classes
satisfy ${\cal F}={\cal F}_{\rm ec}$.
Conjecture: Corollary 6.3 holds for each characteristic class ${\cal F}={\cal
F}_{\rm ec}$, that is, almost every ergodic component of a member in ${\cal
F}$ also belongs to ${\cal F}$.

###  6.2 Orthogonality to zero entropy and uniquely ergodic systems. General
characteristic class case

We now merge Boshernitzan’s problem with the approach of [21] to characterize
$\boldsymbol{u}$ orthogonal to the systems whose invariant measures determine
systems belonging to a fixed characteristic class (in fact, to obtain a good
theory, in [21], so called ec-characteristic classes are considered; the ec-
problem disappears in Boshernitzan’s problem).

######  Proposition 6.5.

Let $\boldsymbol{u}:{\mathbb{N}}\to{\mathbb{D}}$. Let ${\cal F}$ be any non-
trivial characteristic class. Then $\boldsymbol{u}\perp$ uniquely ergodic
systems in ${\cal F}$ if and only if for each Furstenberg system $\kappa\in
V(\boldsymbol{u})$,

(40) ${{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,{\cal A}_{\cal
F}(X_{\boldsymbol{u}},\kappa,S)]\perp F_{\rm
we}(X_{\boldsymbol{u}},\kappa,S).$

###### Proof.

$\Rightarrow$ Assume that $\boldsymbol{u}\perp\rm{UE}\cap{\cal F}$, and
suppose that for some $\kappa\in V(\boldsymbol{u})$ there are an ergodic
system $(Z^{\prime},\nu^{\prime},R^{\prime})$, a joining $\rho^{\prime}\in
J(\nu^{\prime},\kappa)$ and a function $g\in L^{2}_{0}(\nu^{\prime})$ for
which

${{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,{\cal A}_{{\cal
F}}(X_{\boldsymbol{u}},\kappa,S)]\not\perp\Phi_{\rho^{\prime}}(g).$

Since the function on the left-hand side is measurable with respect to a
factor in the class ${\cal F}$, we can replace above $g$ by
${{\mathbbm{E}}}_{\nu^{\prime}}[g\,|\,{\cal A}_{{\cal
F}}(Z^{\prime},\nu^{\prime},R^{\prime})]$. Thus, replacing if necessary
$(Z^{\prime},\nu^{\prime},R^{\prime})$ by its largest ${\cal F}$-factor, we
can assume with no loss of generality that $R^{\prime}\in{\cal F}$. But once
we know that, we have

$0\neq{{\mathbbm{E}}}_{\kappa}\bigl{[}{{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,{\cal
A}_{{\cal
F}}(X_{\boldsymbol{u}},\kappa,S)]\,\Phi_{\rho^{\prime}}(g)\bigr{]}={{\mathbbm{E}}}_{\kappa}\bigl{[}\pi_{0}\,\Phi_{\rho^{\prime}}(g)\bigr{]}.$

Then we proceed as in the proof of Proposition 5.1: Using the Jewett-Krieger
theorem, we obtain that there exists a uniquely ergodic system (belonging to
${\cal F}$) $(Z,\nu,R)$ and a joining $\rho\in J(\nu,\kappa)$ such that
$\pi_{0}\not\perp{\rm Im}(\Phi_{\rho})$ (in $L^{2}(\kappa)$). It follows that
there exists $f\in C(Z)$ such that $\int\Phi_{\rho}(f)\,\pi_{0}\,d\kappa\neq
0$, in other words

$\int_{Z\times X_{\boldsymbol{u}}}f\otimes\pi_{0}\,d\rho\neq 0.$

Let $(N_{n})$ satisfy $\frac{1}{N_{n}}\sum_{j\leq
N_{n}}\delta_{S^{j}\boldsymbol{u}}\to\kappa$. In view of Theorem 2.1, there
exist a sequence $(z_{\ell})\subset Z$ and a subsequence $(N_{n_{k}})$ such
that

$\frac{1}{N_{n_{k}}}\sum_{j\leq
N_{n_{k}}}\delta_{(z_{j},S^{j}\boldsymbol{u})}\to\rho$

and the set

$\\{\ell\geq 0:\>z_{\ell+1}\neq Rz_{\ell}\\}=:\\{b_{1}<b_{2}<\ldots\\}$

satisfies $b_{\ell+1}-b_{\ell}\to\infty$. It follows that

$0\neq\int
f\otimes\pi_{0}\,d\rho=\lim_{k\to\infty}\frac{1}{N_{n_{k}}}\sum_{j\leq
N_{n_{k}}}f(z_{j})\boldsymbol{u}(j)=$
$\lim_{m\to\infty}\frac{1}{b_{K_{m}}}\sum_{r<K_{m}}\Big{(}\sum_{b_{r}\leq
j<b_{r+1}}f(R^{j-b_{r}}z_{b_{r}})\boldsymbol{u}(j)\Big{)}.$

However, from our assumption of orthogonality on $\boldsymbol{u}$, the latter
limit is $0$ because the sequence $\bigl{(}f(z_{j})\bigr{)}$ can be observed
in the orbital system described in Theorem 2.4, which is uniquely ergodic and
in ${\cal F}$.

$\Leftarrow$ Let $(X,T)$ be uniquely ergodic and in ${\cal F}$. Suppose that,
for some $f\in C(X)$, some $x\in X$ and some increasing sequence $(N_{k})$ we
have the existence of the limit

$c:=\lim_{k\to\infty}\frac{1}{N_{k}}\sum_{n\leq
N_{k}}f(T^{n}x)\boldsymbol{u}(n).$

Extracting a subsequence if necessary, we can also assume that

$\frac{1}{N_{k}}\sum_{n\leq
N_{k}}\delta_{(T^{n}x,S^{n}\boldsymbol{u})}\to\rho.$

Then $\rho$ is a joining, $\rho\in J(\nu,\kappa)$, where $\nu$ is the unique
invariant measure for $T$, and $\kappa\in V(\boldsymbol{u})$. Now, since
$(X,\nu,T)\in{\cal F}$, we get by Theorem 2.12

$c=\int f\otimes\pi_{0}\,d\rho=\int
f\otimes{{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,{\cal A}_{{\cal
F}}(X_{\boldsymbol{u}},\kappa,S)]\,d\rho.$

So, if we assume that

${{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,{\cal A}_{{\cal
F}}(X_{\boldsymbol{u}},\kappa,S)]\perp F_{\rm
we}(X_{\boldsymbol{u}},\kappa,S),$

we get $c=0$.

######  Remark 6.6.

The above result also holds in the logarithmic case. Now, by Frantzikinakis-
Host’s theorem [11], all zero entropy, uniquely ergodic systems are Möbius
orthogonal. By the above Proposition 6.5, the relevant Veech condition is
satisfied for $\boldsymbol{\mu}$. However, to read a combinatorial
reformulation of Frantzikinakis-Host’s theorem is unclear because we have been
unable to get a logarithmic version of Proposition 8.1 (see Section 8.4).

Note that if for ${\cal F}$ we take the class of all dynamical systems, then
we obtain the original Boshernitzan’s problem, and we recover the result
stated in Proposition 5.1.

Proposition 6.5 allows us to use directly Theorem 4.5 and Corollary 4.9 (with
$\widetilde{T}$ replaced by any Furstenberg system of $\boldsymbol{u}$).

######  Corollary 6.7.

The following conditions are equivalent:
(a) $\boldsymbol{u}\perp{\cal F}\cap{\rm UE}$.
(b) For each Furstenberg system $\kappa$ of $\boldsymbol{u}$ the following
holds: for each $\lambda\in J^{\rm RelErg}_{2}(X_{\boldsymbol{u}},\kappa,S)$
(in particular,
$\lambda|_{\mathcal{I}_{S}\otimes\mathcal{I}_{S}}=\kappa|_{\mathcal{I}_{S}}\otimes\kappa|_{\mathcal{I}_{S}}$),
we have

(41)
${{\mathbbm{E}}}_{\lambda}\Bigl{[}{{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,{\cal
A}_{{\cal F}}]\otimes{{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,{\cal A}_{{\cal
F}}]\,|\,\mathcal{I}_{S}\otimes\mathcal{I}_{S}\Bigr{]}={{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,\mathcal{I}_{S}]\otimes{{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,\mathcal{I}_{S}].$

(c) For each Furstenberg system $\kappa$ of $\boldsymbol{u}$ the following
holds: for each $\lambda\in J_{2}(X_{\boldsymbol{u}},\kappa,S)$ with
$\lambda|_{\mathcal{I}_{S}\otimes\mathcal{I}_{S}}=\kappa|_{\mathcal{I}_{S}}\otimes\kappa|_{\mathcal{I}_{S}}$,
we have

(42)
${{\mathbbm{E}}}_{\lambda}\Bigl{[}{{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,{\cal
A}_{{\cal F}}]\otimes{{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,{\cal A}_{{\cal
F}}]\,|\,\mathcal{I}_{S\times
S}\Bigr{]}={{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,\mathcal{I}_{S}]\otimes{{\mathbbm{E}}}_{\kappa}[\pi_{0}\,|\,\mathcal{I}_{S}].$

###  6.3 Orthogonality to systems with countably many ergodic invariant
measures, all in ${\cal F}$

The purpose of this section is to prove the following generalization of
Proposition 5.14.

######  Proposition 6.8.

Let $\boldsymbol{u}$, $\|\boldsymbol{u}\|_{u^{1}}=0$, be a bounded arithmetic
function orthogonal to all u.e. systems in some characteristic class ${\cal
F}$. Then $\boldsymbol{u}$ is also orthogonal to all systems whose set of
ergodic invariant measures is countable, all of them giving rise to systems in
${\cal F}$. More precisely: if $(Z,R)$ is a topological system whose set
$\\{\rho_{i}\colon i\in I\\}$ of ergodic measures is countable, and satisfy
for all $i$, $(Z,\rho_{i},R)\in{\cal F}$, then for any $z_{0}\in Z$ and any
function $g\in C(Z)$, we have

$\frac{1}{N}\sum_{n<N}\boldsymbol{u}(n)g(R^{n}z_{0})\to 0.$

The proof of the above proposition follows essentially the same lines as the
proof of Proposition 5.14, but with a further refinement to take into account
the underlying characteristic class ${\cal F}$. This is what the lemmas below
are made for. The framework of these lemmas is quite the same as in Corollary
5.13: we have a measure-preserving system $(X,\mu,T)$, and a $T$-invariant
probability measure $\eta$ such that $\eta\ll\mu$, and $\phi:=d\eta/d\mu$ is
assumed to be bounded. We identify elements of $L^{\infty}(\eta)$ with
elements of $L^{\infty}(\mu)$ which vanish on $\\{x:\phi(x)=0\\}$. We
constantly use this identification in the following lemmas.

######  Lemma 6.9.

In the framework described above, let $f\in L^{\infty}(\mu)$ satisfy $f=0$ on
$\\{x:\phi(x)=0\\}$. Then, we have the equivalence:

$f\text{ is }{\cal A}_{{\cal F}}(X,\mu,T)\text{-measurable}\Longleftrightarrow
f\text{ is }{\cal A}_{{\cal F}}(X,\eta,T)\text{-measurable}.$

###### Proof.

Let $0<\alpha<1$ be such that $\alpha\phi\leq 1$, so that $\alpha\eta\leq\mu$.
For clarity, we will use the notation $X_{\mu}$ (respectively, $X_{\eta}$) for
the space $X$ endowed with the measure $\mu$ (respectively the measure
$\eta$). We first construct a joining424242The associated Markov operator
$\Phi_{\lambda}:L^{2}(X,\eta)\to L^{2}(X,\mu)$ is given by
$\Phi_{\lambda}(f)=\alpha\phi f+(1-\alpha\phi)\int f\,d\eta$. $\lambda$ of
$(X_{\mu},\mu,T)$ and $(X_{\eta},\eta,T)$ by setting, for all measurable
subsets $A\subset X_{\mu}$, $B\subset X_{\eta}$,

$\displaystyle\lambda(A\times B)$
$\displaystyle:=\int_{A}\Bigl{(}\alpha\phi\mathbbmss{1}_{B}+\bigl{(}1-\alpha\phi\bigr{)}\eta(B)\Bigr{)}\,d\mu$
$\displaystyle=\alpha\eta(A\cap
B)+\bigl{(}\mu(A)-\alpha\eta(A)\bigr{)}\eta(B).$

In particular, the conditional distribution of the second coordinate
$x_{\eta}$ given the first one $x_{\mu}$ is

$\alpha\phi(x_{\mu})\delta_{x_{\mu}}+\bigl{(}1-\alpha\phi(x_{\mu})\bigr{)}\eta.$

We then construct the joining $\lambda_{\infty}^{\eta}$ as the relative
product of infinitely many copies of $(X_{\mu}\times X_{\eta},\lambda)$ over
$(X_{\mu},\mu)$: this is the $T\times T^{{{\mathbbm{N}}}}$\- invariant
probability measure on $X_{\mu}\times X_{\eta}^{{{\mathbbm{N}}}}$ whose
marginal on $X_{\mu}$ is $\mu$, and whose conditional distribution on
$X_{\eta}^{{\mathbbm{N}}}$ given $x_{\mu}$ is the infinite product measure

$\Bigl{(}\alpha\phi(x_{\mu})\delta_{x_{\mu}}+\bigl{(}1-\alpha\phi(x_{\mu})\bigr{)}\eta\Bigr{)}^{\otimes{{\mathbbm{N}}}}.$

By applying the law of large numbers in each fiber determined by $x_{\mu}$, we
get that for $\lambda_{\infty}^{\eta}$-almost all
$\bigl{(}x_{\mu},(x_{\eta}^{k})_{k\in{{\mathbbm{N}}}}\bigr{)}$, the limit

$\ell=\ell\bigl{(}x_{\mu},(x_{\eta}^{k})_{k\in{{\mathbbm{N}}}}\bigr{)}:=\lim_{K\to\infty}\frac{1}{K}\sum_{k=1}^{K}f(x_{\eta}^{k})$

exists, and satisfy

$\ell=\alpha\phi(x_{\mu})f(x_{\mu})+\bigl{(}1-\alpha\phi(x_{\mu})\bigr{)}\int
f\,d\eta.$

It follows that, $\lambda_{\infty}^{\eta}$-almost surely, we can get the value
$f(x_{\mu})$ by the formula

$f(x_{\mu})=\mathbbmss{1}_{\phi(x_{\mu})>0}\dfrac{\ell-\bigl{(}1-\alpha\phi(x_{\mu})\bigr{)}\int
f\,d\eta}{\alpha\phi(x_{\mu})}.$

Now, if $f$ is ${\cal A}_{{\cal F}}(X_{\eta},\eta,T)$-measurable, the limit
$\ell$ is measurable with respect to an infinite joining of ${\cal F}$-systems
(namely the infinite self-joining of ${\cal A}_{{\cal F}}(X_{\eta},\eta,T)$
arising from $\lambda_{\infty}^{\eta}$). Moreover, as $\phi$ is $T$-invariant,
and since the class ${\cal F}$ must contain the ID class,
$\bigl{(}x_{\mu},(x_{\eta}^{k})_{k\in{{\mathbbm{N}}}}\bigr{)}\mapsto\phi(x_{\mu})$
is measurable with respect to ${\cal A}_{{\cal F}}(X_{\mu}\times
X_{\eta}^{{{\mathbbm{N}}}},\lambda_{\infty}^{\eta},T\times
T^{{{\mathbbm{N}}}})$. Therefore, the same holds for
$\bigl{(}x_{\mu},(x_{\eta}^{k})_{k\in{{\mathbbm{N}}}}\bigr{)}\mapsto
f(x_{\mu})$, and this proves that $f$ is ${\cal A}_{{\cal
F}}(X,\mu,T)$-measurable.

Conversely, assume that $f$ is ${\cal A}_{{\cal F}}(X,\mu,T)$-measurable. We
will use the same joining $\lambda$ as before, but now we disintegrate it with