# Coboundaries of commuting Borel automorphisms

Shrey Sanadhya  Department of Mathematics, University of Iowa, Iowa City,
52242 IA, USA<EMAIL_ADDRESS>
###### Abstract.

We show that if $S,T$ are two commuting automorphisms of standard Borel space
such that they generate a free Borel ${\mathbb{Z}}^{2}$-action then $S$ and
$T$ do not have same sets of real valued bounded coboundaries. We also prove a
weaker form of Rokhlin Lemma for Borel ${\mathbb{Z}}^{d}$-actions.

###### Key words and phrases:

coboundries, Rokhlin Lemma, Borel $\mathbb{Z}^{d}$-action

###### 2020 Mathematics Subject Classification:

37A40, 37A99, 37B99.

##  1\. Introduction

Let $T$ be an aperiodic Borel automorphism of a standard Borel space
$(X,\mathcal{B})$. A real valued Borel map $f$ on $X$ is called a (Borel)
coboundary for $T$ if there exists a Borel function $g$ such that

$f(x)=g(x)-g(Tx)$ (1.1)

for every $x\in X$. Two Borel functions $f$ and $h$ are called cohomologous if
$f-h$ is a coboundary. We denote by $Cob(T)$ the set of real valued bounded
(Borel) coboundaries of $T$.

Our motivation to study the cohomology theory of a Borel dynamical system
stems from the fact that a similar study in ergodic theory and topological
dynamics has proved to be very useful. Cohomology theory has been used in the
study of measure-theoretic dynamical systems up to orbit equivalence. It has
given rise to methods that have been widely applied in the classification of
dynamical systems. A complete reference to papers focused on the study of
cohomology in this context is too long to mention. We list few key reference
which include the seminal work of Ramsay [Ram71], Moore [Moo70], Schmidt
[Sch77, Sch90] and Zimmer [Zim84].

The goal of the current work is to extend the following theorem of I. Kornfeld
[Kor99] from ergodic theory to the domain of Borel dynamics.

######  Theorem 1.1 (Kornfeld [Kor99]).

Suppose $\sigma$ and $\tau$ are two commuting invertible ergodic measure
preserving transformations of a non-atomic probability space
$(X,{\mathcal{B}},\mu)$. They have the same coboundaries if and only if
$\sigma=\tau^{\pm 1}$.

The coboundries in Theorem 1.1 are measurable real valued functions that
satisfy $(\ref{eq_1})$ for $\mu$ a.e. $x\in X$. Following Kornfeld, the key
question we ask in this note is as follows: To what extent does the set
$Cob(T)$ determine $T$?

A similar result in the context of Cantor dynamics is due to N. Ormes (Theorem
1.2). We refer the reader to [Orm00] for details. Coboundaries considered in
Theorem 1.2 are continuous.

######  Theorem 1.2 (Ormes [Orm00]).

Let $(X,T)$ and $(Y,S)$ be two Cantor minimal systems. There is an orbit
equivalence $h:X\rightarrow Y$ which induces a bijection from the set of real
$S$-coboundaries to the set of real $T$-coboundaries if and only if $S$ and
$T$ are flip conjugate, i.e. $S$ is conjugate to $T$ or $S$ is conjugate to
$T^{-1}$.

There exist many similarities between Borel dynamics and ergodic theory but
the fact that in Borel dynamics we do not consider any prescribed measure on
the underlying space makes it significantly different from ergodic theory. For
example, in the study of Borel dynamics, we consider maps that are defined
everywhere (instead of a.e. as in ergodic theory). It is also significantly
different from topological dynamics since the underlying space is a non-
compact polish space and the maps (for example the coboundries) are Borel
(instead of continuous).

In this paper, we adopt I. Kornfeld’s ideas to the domain of Borel dynamics
and prove the following theorem :

######  Theorem 1.3.

Let $S,T$ be two commuting automorpisms of standard Borel space such that they
generate a free Borel ${\mathbb{Z}}^{2}$-action. Then $S$ and $T$ do not have
same sets of real valued bounded coboundaries.

######  Remark 1.4.

Note that $S=T^{-1}$ implies $Cob(T)=Cob(S)$. To see this, let $f\in Cob(T)$,
thus $f(x)=h(x)-h(Tx)$ for some Borel function $h$ and every $x\in X$. Then
$f$ can also be written as $f(x)=h_{1}(x)-h_{1}(Sx)$ where $h_{1}=-h\circ T$.
Theorem 1.3 gives us a criterion to determine when two commuting
transformations do not have same set of bounded coboundries.

A key component in the proof of Theorem 1.3 is a two-dimensional version of
the following theorem (also see [BDK06, Lemma 3.3]):

######  Theorem 1.5 (Weiss [Wei84]).

Let $S$ be an aperiodic Borel automorpisms on a standard Borel space
$(X,{\mathcal{B}})$. Then there exists a decreasing sequence $A_{n}$,
$n\in{\mathbb{N}}$, of Borel sets such that

$(i)$ for each $n$,
$\underset{i=-\infty}{\overset{\infty}{\bigcup}}S^{i}A_{n}=\underset{i=-\infty}{\overset{\infty}{\bigcup}}S^{i}(X-A_{n})$
= X,

$(ii)$ for each $n$, the sets $A_{n},S(A_{n}),S^{2}(A_{n}),...S^{n-1}A_{n}$
are pairwise disjoint,

$(iii)$ the intersection
$\underset{n=1}{\overset{\infty}{\bigcap}}A_{n}=\emptyset$.

The nested sequence of Borel sets $A_{n}$, satisfying conditions $(i)-(iii)$
of Theorem 1.5, are called a vanishing sequence of markers. We prove a finite-
dimensional version of this theorem (Theorem 1.6).

Let $\Gamma$ denote a Borel action of a countable group (see Definition 2.1),
then a Borel set $A$ is a complete section for $\Gamma$ if every
$\Gamma$-orbit intersects the set $A$. Following theorem is a finite-
dimensional version of Theorem 1.5.

######  Theorem 1.6.

Let $\Gamma$ be a free Borel ${\mathbb{Z}}^{d}$-action generated by commuting
Borel automorphisms, $\\{T_{i}\\}$, $i=1,2,..,d$, of a standard Borel space
$(X,{\mathcal{B}})$. Then for any $d$-tuple $(t_{1},...,t_{d})$ of positive
integers, there exists a Borel set $A$, such that

$(i)$ $A$ is a complete section for $\Gamma$,

$(ii)$ the sets $T_{1}^{k_{1}}...T_{d}^{k_{d}}A$, $0\leq k_{i}<t_{i}$,
$i=1,2,..,d$, are pairwise disjoint.

Theorem 1.6 can also be considered as a weaker version of Rokhlin Lemma for
Borel ${\mathbb{Z}}^{d}$-action. We should mention that Theorem 1.6 can be
derived from the work of S. Gao and S. Jackson [GJ15]. In this note, we
provide a new and purely dynamical proof.

The outline of the paper is as follows. In Section 2, we provide basic
definitions and preliminary results about groups of Borel automorphisms. In
Section 3, we provide proof of Theorem 1.6. In Section 4, we use Theorem 1.6
(for $d=2$), to prove Theorem 1.3.

Throughout the paper, we use the following notation:

  * •

$(X,\mathcal{B})$ is a standard Borel space with the $\sigma$-algebra of Borel
sets $\mathcal{B}=\mathcal{B}(X)$.

  * •

A one-to-one Borel map $T$ of space $(X,\mathcal{B})$ onto itself is called a
Borel automorphism of $X$. In this paper the term "automorphism" means a Borel
automorphism of $(X,\mathcal{B})$.

  * •

In this paper $Cob(T)$ denotes the set of real valued bounded coboundries (see
$(\ref{eq_1})$) of Borel automorphism $T$.

  * •

$Aut(X,\mathcal{B})$ is the group of all Borel automorphisms of $X$ with the
identity map ${\mathbb{I}}\in Aut(X,\mathcal{B})$.

##  2\. Preliminaries

In this section we provide the basic definitions from Borel dynamics and
descriptive set theory.

Borel dynamical system. Let $X$ be a separable completely metrizable
topological space (also called Polish space). Let ${\mathcal{B}}$ be the
$\sigma$-algebra generated by the open sets in $X$. Then we call the pair
$(X,{\mathcal{B}})$ a standard Borel space. Any two uncountable standard Borel
spaces are isomorphic. Any countable subgroup $\Gamma$ of $Aut(X,\mathcal{B})$
is called a Borel automorphism group, and the pair $(X,\Gamma)$ is referred to
as a Borel dynamical system.

######  Definition 2.1.

Let $G$ be any countable group and $e$ be the identity of $G$. For every $g\in
G$, define a map $\rho_{g}:X\rightarrow X$, such that $(i)$
$\rho_{gh}(x)=\rho_{g}(\rho_{h}(x))$ for every $g,h\in G$, $(ii)$
$\rho_{e}(x)=x$ for every $x\in X$ and $(iii)$ $x\mapsto\rho_{g}(x)$ is a one-
to-one, onto Borel map for each $g\in G$. Then $\rho(G)=\\{\rho_{g}:g\in G\\}$
is called a Borel action of the group $G$ on $(X,\mathcal{B})$.

Note that $\rho(G)$ is a countable subgroup of $Aut(X,\mathcal{B})$. We say
that $\rho$ is a free action (of group $G$) if $\rho_{g}(x)=x$ for some $x\in
X$, implies $g=e$. Thus a free action $\rho$ results in an injective group
homomorphism $\phi:G\rightarrow Aut(X,\mathcal{B}):g\mapsto\rho_{g}$. Any
Borel automorphism $T\in Aut(X,{\mathcal{B}})$ gives rise to a Borel action of
group $\mathbb{Z}$ (also referred as a Borel ${\mathbb{Z}}$-action) by
identifying $k\in\mathbb{Z}$ with $T^{k}\in Aut(X,{\mathcal{B}})$.

In this paper we work with free Borel ${\mathbb{Z}}^{d}$-actions generated by
$d$ commuting automorphisms. We make this notion precise in the following
definition. For simplicity we use $d=2$.

######  Definition 2.2.

Let $S$ and $T$ be two commuting Borel automorphism we say that they generate
a free Borel action $\rho({\mathbb{Z}}^{2})$ of group ${\mathbb{Z}}^{2}$ where
$\rho_{(n,m)}:=S^{n}T^{m}=T^{m}S^{n}\in Aut(X,{\mathcal{B}})$, if for every
$(n,m)\in{\mathbb{Z}}^{2}$ we have $\rho_{(n,m)}(x)\neq x$ for all $x\in X$
and any pair of integers $(m,n)\neq(0,0)$.

Note that the above definition immediately implies that $S$ is not a power of
$T$ for any $x\in X$. To see this assume there exist $x\in X$ such that
$Sx=T^{p}x$ for some $p\in{\mathbb{Z}}$. Hence $ST^{-p}x=x$, which contradicts
the assumption that $\rho$ is a free action.

In the study of Borel dynamical systems the theory of Countable Borel
equivalence relation (CBER) on $(X,\mathcal{B})$ plays an important role as it
provides a link between descriptive set theory and Borel actions (see Theorem
2.3). We call an equivalence relation $E$ on $(X,\mathcal{B})$ Borel if it is
a Borel subset of the product space $E\subset X\times X$. We say that an
equivalence relation $E$ on $(X,\mathcal{B})$ is countable if every
equivalence class $[x]_{E}:=\\{y\in X:(x,y)\in E\\}$ is a countable set for
all $x\in X$. Let $B\in{\mathcal{B}}$ be a Borel set, then the saturation of
$B$ with respect to a CBER $E$ on $(X,{\mathcal{B}})$ is the set containing
entire equivalence classes $[x]_{E}$ for every $x\in B$.

Let $\Gamma$ be a Borel automorphism group of $(X,{\mathcal{B}})$, then the
orbit equivalence relation generated by the action of $\Gamma$ on $X$ is given

$E_{X}(\Gamma)=\\{(x,y)\in X\times X:x=\gamma y\ \mbox{for\ some}\
\gamma\in\Gamma\\}.$

Note that $E_{X}(\Gamma)$ is a CBER. We will call an equivalence relation $E$
periodic at a point $x\in X$ if the equivalence class $[x]_{E}$ is finite.
Similarly, an equivalence relation $E$ is aperiodic at $x\in X$ if the
equivalence class $[x]_{E}$ is countably infinite. If every $E$-class is
countably infinite we will say that the equivalence relation $E$ is aperiodic.
In this paper we work with aperiodic CBERs. The study of aperiodic CBERs in
itself is an area of immense importance in descriptive set theory. We refer
our readers to [Kec19] for an up-to-date survey of the theory of countable
Borel equivalence relations.

The following theorem shows that all CBERs come from Borel actions of
countable groups.

######  Theorem 2.3 (Feldman–Moore [FM77]).

Let $E$ be a countable Borel equivalence relation on a standard Borel space
$(X,\mathcal{B})$. Then there is a countable group $\Gamma$ of Borel
automorphisms of $(X,\mathcal{B})$ such that $E=E_{X}(\Gamma)$.

A Borel set $C$ is called a complete section for an equivalence relation $E$
on $(X,{\mathcal{B}})$ if every $E$-class intersects $C$, in other words
$[C]_{E}=X$. Let $\Gamma\in Aut(X,{\mathcal{B}})$ be a countable Borel
automorphism group. Then we will denote by $\mathcal{C}_{\Gamma}$ the
collection of Borel subsets $C$ such that $C$ and $X\setminus C$ both are
complete section for $E_{X}(\Gamma)$.

A CBER, $E$ on $(X,{\mathcal{B}})$ is hyperfinite if
$E=\bigcup_{i=1}^{\infty}E_{i}$, where each $E_{i}$ is a finite Borel
equivalence relation and $E_{i}\subset E_{i+1}$ for all $i\in{\mathbb{N}}$. If
$G$ is a countable group such that $E_{X}(G)=E$ and it acts freely, then we
call $G$ to be a hyperfinite group of Borel automorphisms of
$(X,{\mathcal{B}})$.

Let $\Gamma_{1},\Gamma_{2}$ be two countable Borel automorphism groups of
$(X,{\mathcal{B}})$. We say that $\Gamma_{1}$ and $\Gamma_{2}$ are orbit
equivalent (also o.e.) if there exists a Borel isomorphism $\phi:X\rightarrow
X$ such that $\phi(\Gamma_{1}x)=\Gamma_{2}(\phi(x))$, $\forall x\in X$. In
other words $\Gamma_{1}$ orbit of $x$ is same as the $\Gamma_{2}$ orbit of
$\phi(x)$ for every $x\in X$. The following theorem gives an important
characterisation of a hyperfinite CBER.

######  Theorem 2.4 (Slaman-Steel [SS88], Weiss [Wei84]).

Suppose $E$ is a CBER. The following are equivalent:

$1.$ $E$ is hyperfinite.

$2.$ $E$ is generated by a Borel $\mathbb{Z}$-action.

##  3\. Weak Rokhlin Lemma for Borel ${\mathbb{Z}}^{d}$-actions

In this section, we prove Theorem 1.6, which can be considered as a weak
version of Rokhlin Lemma for free Borel ${\mathbb{Z}}^{d}$-action. Before we
prove Theorem 1.6 it is useful to discuss its one-dimensional version (Theorem
3.6) due to B. Weiss ([Wei84]). We have included the proof of Theorem 3.6 for
completion. The proof of Theorem 3.6 is taken from [Nad13, chapter 7].

######  Definition 3.1.

Let $S$ be an aperiodic Borel automorphism of a standard Borel space
$(X,{\mathcal{B}})$. A Borel set $W\in{\mathcal{B}}$ is said to be wandering
with respect to $S$ if the sets $S^{i}W$, $i\in{\mathbb{Z}}$, are pairwise
disjoint. We will denote by $\mathcal{W}_{S}$ (or $\mathcal{W}$ when $S$ is
obvious) the sigma ideal generated by all the wandering sets in
${\mathcal{B}}$.

By Poincare Recurrence Lemma, we can say that given $S\in
Aut(X,{\mathcal{B}})$ and $A\in{\mathcal{B}}$, there exists
$N\in\mathcal{W}_{S}$ such that for each $x\in A\setminus N$, the points
$S^{n}x$ return to $A$ for infinitely may positive $n$ and also for infinitely
many negative $n$. The elements in $A\setminus N$ are called non-wandering or
recurrent elements of $A$.

######  Definition 3.2.

Let $A\in{\mathcal{B}}$ be a Borel set such that every point of $A$ is non-
wandering with respect to $S\in Aut(X,{\mathcal{B}})$. The induced
automorphism on $A$, denoted by $S_{A}$ is defined by $S_{A}(x)=S^{n}(x)$,
$x\in A$, where $n=n(x)$ is the smallest positive integer such that
$S^{n}(x)\in A$.

Lemma 3.3 shows that given a Borel automorphism $S$ of a standard Borel space
$(X,{\mathcal{B}})$, we can find a family of generating sets for
${\mathcal{B}}$ such that none of the sets in the family contains a full
$S$-orbit.

######  Lemma 3.3 (Nadkarni [Nad13]).

Let $S$ be a free Borel automorphism on a standard Borel space
$(X,{\mathcal{B}})$. Then there exists a countable collection of Borel sets
$\\{A\\}_{i\in{\mathbb{N}}}$ such that it generate ${\mathcal{B}}$ and none of
the set in the collection contains a full $S$-orbit.

Proof. Since $(X,{\mathcal{B}})$ is standard Borel space, it is countably
generated and countably separated. Let $\\{A_{i}\\}_{i\in\mathbb{N}}$, be a
separating system of sets which are closed under complement and generate the
sigma algebra ${\mathcal{B}}$. Note that for each $i\in{\mathbb{N}}$, the set
$B_{i}=\cap_{k=-\infty}^{\infty}S^{k}A_{i}$ is the largest subset of $A_{i}$
that is invariant under $S$. Since the collection $A_{i}$ separates points,
given any $x\in X$ there is at least one $i\in{\mathbb{N}}$ such that $x\in
A_{i}\setminus B_{i}:=C_{i}$ (say). Otherwise, there exists some $x_{0}\in X$
such that $x_{0}\in B_{i}$ whenever $x\in A_{i}$. So, whenever $x_{0}\in
A_{i}$, orb$(x_{0},S)\subseteq B_{i}\subseteq A_{i}$ (here orb$(x,S)$ denotes
the orbit of $x$ under $S$). This is a contradiction since $S$ is free,
orb$(x_{0},S)$ has more than one point, while the collection
$\\{A_{i}\\}_{i\in{\mathbb{N}}}$ separates points. Thus the countable
collection of sets $C_{1},C_{2},C_{3},...$ together with the collection
$B_{i}\cap C_{j}$, $i,j\in{\mathbb{N}}$, generates ${\mathcal{B}}$ and none of
the sets in this collection contains a full $S$-orbit. We rename this
collection $\\{A_{i}\\}_{i\in{\mathbb{N}}}$ to obtain the statement of the
lemma. $\square$

######  Definition 3.4.

A set $A\in{\mathcal{B}}$ is said to be decomposable (mod $\mathcal{W}_{S}$)
if we can write $A$ as a disjoint union of two Borel sets $C$ and $D$ such
that saturation (mod $\mathcal{W}_{S}$) of $C$, $D$ and $A$ with respect to
$S$ are same. In other words,

$\underset{i=-\infty}{\overset{\infty}{\bigcup}}S^{i}C=\underset{i=-\infty}{\overset{\infty}{\bigcup}}S^{i}D=\underset{i=-\infty}{\overset{\infty}{\bigcup}}S^{i}A\,\,\,\,(\mathrm{mod}\,\,\mathcal{W}_{S}).$

######  Lemma 3.5 (Nadkarni [Nad13]).

Every set in ${\mathcal{B}}$ is decomposable (mod $\mathcal{W}_{S}$) where $S$
is a free Borel automorphism on a standard Borel space $(X,{\mathcal{B}})$.

Proof. We first show that $X$ is decomposable (mod $\mathcal{W}_{S}$). Let
$\\{A_{i}\\}_{i\in{\mathbb{N}}}$ be a family of Borel sets closed under
complement that generate ${\mathcal{B}}$. In light of above discussion we can
assume that no $A_{i}$ contains a full $S$-orbit. Let

$B_{1}=A_{1}\hskip
216.24094ptC_{1}=\bigg{(}\underset{i=-\infty}{\overset{\infty}{\bigcup}}S^{i}A_{1}\bigg{)}\setminus
A_{1}$
$B_{2}=A_{2}\setminus\bigg{(}\underset{i=-\infty}{\overset{\infty}{\bigcup}}S^{i}A_{1}\bigg{)}\hskip
142.26378ptC_{2}=\bigg{(}\underset{i=-\infty}{\overset{\infty}{\bigcup}}S^{i}B_{2}\bigg{)}\setminus
B_{2}$
. .
. .
. .
$B_{n}=A_{n}\setminus\Bigg{(}\underset{i=-\infty}{\overset{\infty}{\bigcup}}S^{i}\bigg{(}\underset{i=1}{\overset{n-1}{\cup}}B_{i}\bigg{)}\Bigg{)}\hskip
102.42992ptC_{n}=\bigg{(}\underset{i=-\infty}{\overset{\infty}{\bigcup}}S^{i}B_{n}\bigg{)}\setminus
B_{n}$
… …
… …

Now put $B=\underset{i=1}{\overset{\infty}{\cup}}B_{i}$ and
$C=\underset{i=1}{\overset{\infty}{\cup}}C_{i}$. Note that $B$ and $C$ are
disjoint and their union is $X$ (this is true since we chose the collection
$\\{A_{i}\\}_{i{\mathbb{N}}}$ to be closed under complement). Also note that

$\underset{i=-\infty}{\overset{\infty}{\bigcup}}S^{i}B=\underset{i=-\infty}{\overset{\infty}{\bigcup}}S^{i}C=X.$

Thus $X$ is decomposable (mod $\mathcal{W}_{S}$). To show that any Borel set
$A\in{\mathcal{B}}$ is decomposable (mod $\mathcal{W}_{S}$) we apply this
result with induced automorphism $S_{A}$. $\square$

######  Theorem 3.6 (Weiss [Wei84]).

Let $S$ be an aperiodic, Borel automorpisms on a standard Borel space
$(X,{\mathcal{B}})$. Then there exists a decreasing sequence $A_{n}$,
$n\in{\mathbb{N}}$, of Borel sets such that

$(i)$ for each $n$,
$\underset{i=-\infty}{\overset{\infty}{\bigcup}}S^{i}A_{n}=\underset{i=-\infty}{\overset{\infty}{\bigcup}}S^{i}(X-A_{n})=X$,
in other words, for each $n$, $A_{n}\in\mathcal{C}_{S}$,

$(ii)$ for each $n$, the sets $A_{n},S(A_{n}),S^{2}(A_{n}),...S^{n-1}A_{n}$
are pairwise disjoint,

$(iii)$ the intersection
$\underset{n=1}{\overset{\infty}{\bigcap}}A_{n}=:A_{\infty}$ is a wandering
set.

Proof. By the lemma above choose $A$ such that $A$, $(X-A)$ decompose $X$ (in
other words $A\in\mathcal{C}_{S}$) and set $A_{1}=A$. For each $x\in A_{1}$,
let $n(x)$ be the first positive integer $n$ such that $S^{n}(X)\notin A_{1}$.
Since $A_{1}$ does not contain any full $S$-orbit, the existence of such a
positive integer $n(x)$ is guaranteed for each $x\in A_{1}$. Write

$E_{k}=\\{x\in A_{1}:n(x)=k\\},\,\,\,\,k=1,2,3..$

and put

$A_{2}=\bigcup_{k=1}^{\infty}S^{k-1}E_{k}\,\subseteq A_{1}.$

We note that $\underset{i=-\infty}{\overset{\infty}{\bigcup}}S^{i}A_{2}=X$ and
$A_{2}\cup SA_{2}=\emptyset$. Now we apply the same argument to $S_{A_{2}}$
and obtain a set $A_{3}\subseteq A_{2}$ such that $A_{3}\cap
S_{A_{2}}A_{3}=\emptyset$ and the saturation of $A_{3}$ under $S_{A_{2}}$ is
$A_{2}$, hence the saturation of $A_{3}$ under $S$ is entire $X$. Continuing
similarly we obtain $A_{4},A_{5},...$ a decreasing sequence $A_{n}$,
$n\in{\mathbb{N}}$, of Borel sets such that for each $n$

$(i)$
$\underset{i=-\infty}{\overset{\infty}{\bigcup}}S^{i}A_{n}=\underset{i=-\infty}{\overset{\infty}{\bigcup}}S^{i}(X-A_{n})=X$,

$(ii)$ $A_{n},S(A_{n}),S^{2}(A_{n}),...S^{n-1}A_{n}$ are pairwise disjoint.

This proves part $(i)$ and $(ii)$. Note that the set
$A_{\infty}=\underset{n=1}{\overset{\infty}{\bigcap}}A_{n}$ has the property
that $A_{\infty},SA_{\infty},...,S^{n-1}A_{\infty}$ are pairwise disjoint for
each $n$, thus $A_{\infty}$ is a wandering set. This shows $(iii)$. $\square$

######  Remark 3.7.

If we replace $A_{n}$ with $B_{n}$ such that $B_{n}=(A_{n}\setminus
A_{\infty})\cup_{|i|>n}S^{i}A_{\infty}$, then each point in $B_{n}$ is
recurrent and $B_{n+1}\subset B_{n}$, for every $n\in{\mathbb{N}}$. Moreover
we get that $\underset{n=1}{\overset{\infty}{\bigcap}}B_{n}=\emptyset$.

For ease of notation, we will prove Theorem 1.6 for $d=2$ (see Theorem 3.9
below).

######  Remark 3.8.

For $n,m\in{\mathbb{N}}$, let $I_{(n,m)}$ denote the index set containing $nm$
pairs of integers i.e.

$I_{(n,m)}=\\{(0,0),(0,1),...,(n-1,m-1)\\}.$

Each element of $I_{(n,m)}$ represents the powers to which $S$ and $T$ are
raised. Thus $(i,j)\in I_{(n,m)}$ corresponds to $S^{i}T^{j}$. For the rest of
the section, we work with the lexicographic order on ${\mathbb{Z}}^{2}$
(denoted by "$\prec$"). For $(i_{1},j_{1}),(i_{2},j_{2})\in{\mathbb{Z}}^{2}$,
we say $(i_{1},j_{1})\prec(i_{2},j_{2})$ if,

$(i)$ $i_{1}<i_{2}$ or

$(ii)$ $i_{1}=i_{2}$ and $j_{1}<j_{2}.$

######  Theorem 3.9.

Let $\Gamma$ be a free Borel ${\mathbb{Z}}^{2}$-action generated by two
commuting aperiodic Borel automorphisms $S,T$. Then for each pair
$(n,m)\in{\mathbb{N}}^{2}$ there exists a Borel set $A_{(n,m)}$, such that,

$(i)$ for each pair $(n,m)$, $A_{(n,m)}\in\mathcal{C}_{\Gamma}$,

$(ii)$ for each $(n,m)$, the sets $S^{i}T^{j}A_{(n,m)}$, $(i,j)\in I_{(n,m)}$,
are pairwise disjoint.

Proof. A Borel ${\mathbb{Z}}^{d}$-action defines a hyperfinite equivalence
relation (see [GJ15], [Wei84]). Since $\Gamma$ is a $\mathbb{Z}^{2}$-action,
$E_{X}(\Gamma)$ is hyperfinite, hence generated by a Borel $\mathbb{Z}$-action
(see Theorem 2.4). Thus there exists $R\in Aut(X,{\mathcal{B}})$, such that
for every $x\in X$, $\Gamma x=\\{R^{i}x:i\in{\mathbb{Z}}\\}$.

Using lemma 3.5, choose $A\in\mathcal{C}_{R}$ such that $A$ does not contain a
full $R$-orbit. Set $A_{(0,0)}=A$. Let $(n,m)$ be a pair of integers and let
$I_{(n,m)}$ be the corresponding index set (see Remark 3.8). As mentioned in
Remark 3.8 above, there are $nm$ elements in the set $I_{(n,m)}$, each
corresponding to a power of $S$ and $T$. We consider lexicographic order on
the index set. Hence the smallest element of $I_{(n,m)}$ is the pair $(0,0)$
(which corresponds to $S^{0}T^{0}$) the next element in the order is the pair
$(0,1)$ (which corresponds to $S^{0}T^{1}$) and so forth. We want to find
$A_{(0,1)}\subset A$ such that, $A_{(0,1)}\in\mathcal{C}_{R}$ and
$A_{(0,1)}\cap T(A_{(0,1)})=\emptyset$.

Define $\\{B_{j}\\}_{j\in{\mathbb{Z}}\setminus\\{0\\}}$ a partition of
$A=A_{(0,0)}$ as follows,

$B_{j}=\\{x\in A_{(0,0)}:Tx=R^{j}x\\},\,\,\,j\in{\mathbb{Z}}\setminus\\{0\\}.$

Since $A_{(0,0)}$ does not contain a full $R$-orbit, none of the $B_{j}$,
$j\in{\mathbb{Z}}\setminus\\{0\\}$ contains full $R$-orbit. We start with
$B_{-1}$. For $x\in B_{-1}$, let $-n(x)$ be the first negative integer $-n$
such that $R^{-n}(x)\notin B_{-1}$. Write

$E_{-k}^{-1}=\\{x\in B_{-1}:-n(x)=-k\\},\,\,\,k=1,2,3..$

Since $R^{-n(x)}(x)\notin B_{-1}$, there are two possibilities:

  1. (i)

$R^{-n(x)}(x)\in X\setminus A_{(0,0)}$ (we call such elements of $B_{-1}$ type
$a$ elements) or

  2. (ii)

$R^{-n(x)}(x)\in B_{i}$, for $i\neq-1$ (we call such elements of $B_{-1}$ type
$b$ elements).

We remove type $b$ elements from $B_{-1}$. Thus,

$B_{-1}=B_{-1}\setminus\\{x\in B_{-1}:\,x\,\textrm{is type}\,b\\},$

and

$A_{(0,0)}=A_{(0,0)}\setminus\\{x\in B_{-1}:\,x\,\textrm{is type}\,b\\}.$

Hence the set $E_{-k}^{-1}$ becomes

$E_{-k}^{-1}=\\{x\in B_{-1}:-n(x)=-k\,;\,\,R^{-n(x)}(x)\in X\setminus
A_{(0,0)}\\},\,\,\,k=1,2,3..$

Put

$A_{-1}=\bigcup_{-k=-1}^{-\infty}R^{(-k+1)}(E_{-k}^{-1})\subseteq B_{-1}$
(3.1)

Note that $A_{-1}\cap R^{-1}(A_{-1})=\emptyset$, and $R^{-1}(A_{-1})\cap
B_{i}=\emptyset$ for $i\in{\mathbb{Z}}\setminus\\{0\\}$, $i\neq-1$.

Now we repeat the same process for $B_{1}$. For $x\in B_{1}$, let $n(x)$ be
the first positive integer $n$ such that $R^{n}(x)\notin B_{1}$. Again there
are two possibilities:

  1. (i)

$R^{n(x)}(x)\in X\setminus A_{(0,0)}$ (again we denote such elements as type
$a$ elements of $B_{1}$) or

  2. (ii)

$R^{n(x)}(x)\in B_{i}$, for $i\neq-1,1$ (type $b$ elements of $B_{1}$).

We remove type $b$ elements from $B_{1}$. Thus

$B_{1}=B_{1}\setminus\\{x\in B_{1}:\,x\,\textrm{is type}\,b\\},$

and

$A_{(0,0)}=A_{(0,0)}\setminus\\{x\in B_{1}:\,x\,\textrm{is type}\,b\\}.$

The set $E_{k}^{1}$ is now defined as

$E_{k}^{1}=\\{x\in B_{1}:n(x)=k\,;\,\,R^{n(x)}(x)\in X\setminus
A_{(0,0)}\\},\,\,\,k=1,2,3..$

Put

$A_{1}=\bigcup_{k=1}^{\infty}R^{(k-1)}(E_{k}^{1})\subseteq B_{1}.$ (3.2)

Note that $A_{1}\cap R(A_{1})=\emptyset$, and $R(A_{1})\cap B_{i}=\emptyset$
for $i\in{\mathbb{Z}}\setminus\\{0\\}$, $i\neq 1$.

We now repeat this process for $B_{-2}$ and then $B_{2}$ and so on. Finally,
for every $j\in{\mathbb{Z}}\setminus\\{0\\}$ we obtain $A_{j}\subseteq B_{j}$
such that, $R^{j}(A_{j})\cap A_{j}=\emptyset$ and $R^{j}(A_{j})\cap
B_{i}=\emptyset$ where $i\neq j$, $i\in{\mathbb{Z}}\setminus\\{0\\}$. Put

$A_{(0,1)}=\underset{i\in{\mathbb{Z}}\setminus\\{0\\}}{\bigcup}A_{i}\subseteq
A_{(0,0)},\,\,\textrm{then}\,\,A_{(0,1)}\in\mathcal{C}_{R}\,\,\textrm{and}\,A_{(0,1)}\cap
T(A_{(0,1)})=\emptyset.$

Thus, we found a set $A_{(0,1)}\in\mathcal{C}_{R}$ such that $A_{(0,1)}\cap
T(A_{(0,1)})=\emptyset$. Now we move on to next power in the lexicographic
order, i.e. $(0,2)$. Instead of working with $A$ we now work with
$A_{(0,1)}\subset A$ and repeat the above procedure to find $A_{(0,2)}\subset
A_{(0,1)}$, such that $A_{(0,2)}\in\mathcal{C}_{R}$ and $A_{(0,2)}\cap
T(A_{(0,2)})=\emptyset$. Thus, we have obtained set $A_{(0,2)}\subset A$ such
that $A_{(0,2)}\in\mathcal{C}_{R}$ and $A_{(0,2)}\cap T(A_{(0,2)})\cap
T^{2}(A_{(0,2)})=\emptyset$. We continue this process for all $nm$ pairs in
the index set $I_{(n,m)}$ (which corresponds to powers of $S$ and $T$) and
obtain set $A_{(n-1,m-1)}$. Rename this set as $A_{(n,m)}$ to be consistent
with statement of Theorem 3.9. The set $A_{(n,m)}$ satisfies $(i)$ and $(ii)$.
This completes the proof. $\square$

With help of an example, we illustrate the proof of Theorem 3.9.

######  Example 3.10.

Assume $(n,m)=(2,3)$. We want to describe sets $A_{(i,j)}$’s ($0\leq i<2$,
$0\leq j<3$), that we obtain at each step. The elements of $I_{(2,3)}$ in
lexicographic order are $\\{(0,0),(0,1),(0,2),(1,0),(1,1),(1,2)\\}$ (they
correspond to $S^{0}T^{0}=e$, $T$, $T^{2}$, $S$, $ST$, $ST^{2}$ respectively).
In the first step we obtain set $A_{(0,1)}$ such that $A_{(0,1)}\cap
TA_{(0,1)}=\emptyset$. In the second step we obtain set $A_{(0,2)}$ such that
the sets $A_{(0,2)},TA_{(0,2)},T^{2}A_{(0,2)}$ are mutually disjoint.

The next element in the index set is $(1,0)$. So we have to go from $T^{2}$ to
$S$, hence we would have to work with powers of $R$ that corresponds to
$ST^{-2}$. In other words the partition $B_{j}$ will be $B_{j}=\\{x\in
A_{(0,2)}:ST^{-1}x=R^{j}x\\}$. This step will yield set $A_{(1,0)}$ such that
sets

$\\{A_{(1,0)},ST^{-2}A_{(1,0)},ST^{-1}A_{(1,0)},SA_{(1,0)}\\}$

are mutually disjoint. Similarly, the next step will yield set $A_{(1,1)}$
such that sets

$\\{A_{(1,1)},TA_{(1,1)},ST^{-1}A_{(1,1)},SA_{(1,1)},STA_{(1,1)}\\}$

are mutually disjoint. Finally we will obtain set $A_{(1,2)}$ such that sets

$\\{A_{(1,2)},TA_{(1,2)},T^{2}A_{(1,2)},SA_{(1,2)},STA_{(1,2)},ST^{2}A_{(1,2)}\\}$

are mutually disjoint. Denote $A_{(2,3)}=A_{(1,2)}$, thus sets
$S^{i}T^{j}A_{(2,3)}$ are disjoint for $0\leq i<2$, $0\leq j<3$ as needed.

Proof of Theorem 1.6: The proof of Theorem 1.6 is identical to the proof of
Theorem 3.9. Instead of working with $2$-dimensional index set $I_{n,m}$, we
will work with $d$-dimensional index set (with lexicographic ordering).
Everything else remains the same. $\square$

##  4\. Commuting automorphims and their coboundaries

In this section, we provide proof for Theorem 1.3. As mentioned in the
introduction our work is an extension of Kornfeld’s ideas (see [Kor99, Theorem
1]) to the domain of Borel dynamics. Although similar in nature our work
differs from [Kor99] in the following manner. In [Kor99] the author used a
version of ${\mathbb{Z}}^{2}$-Rokhlin Lemma (see [Kor99, Lemma 1]) which he
called a weak form of Rokhlin Lemma for a measure preserving
${\mathbb{Z}}^{2}$-action (attributed to Conze [Con73]). We use Theorem 3.9,
which can be considered as a weak form of Rokhlin Lemma for free Borel
${\mathbb{Z}}^{2}$-actions.

The other principal difference is that in [Kor99], the author worked with an
invariant ergodic probability measure on the ambient space. We do not work
with any prescribed measure on the standard Borel space $(X,{\mathcal{B}})$.
However, in the proof of Theorem 1.3 we use the existence of $\Gamma$-quasi-
invariant probability measure on $(X,{\mathcal{B}})$ (where $\Gamma$ is a free
Borel ${\mathbb{Z}}^{2}$-action).

If $\Gamma$ is a Borel-action of $(X,{\mathcal{B}})$ (and $E_{X}(\Gamma)$
denotes the corresponding orbit equivalence relation on $X$), we say that
$\mu$ is $E_{X}(\Gamma)$-quasi invariant (or $\Gamma$-quasi invariant for
short) if $\mu(B)>0\,\Longleftrightarrow\mu(T(B))>0$ for all Borel sets
$B\in{\mathcal{B}}$ and Borel maps $T:X\rightarrow X$ whose graphs are
contained in $E_{X}(\Gamma)$.

######  Remark 4.1.

The following statement is a consequence of the fact that $\mu$ is
$\Gamma$-quasi-invariant: Fix $\gamma\in\Gamma$ then for $\epsilon>0$ there
exists $\delta_{\gamma}$ such that for every $B\in{\mathcal{B}}$ with
$\mu(B)<\delta_{\gamma}$ we have $\mu(\gamma B)<\epsilon$.

Proof of Theorem 1.3. By assumption $S$ and $T$ generate a free Borel
${\mathbb{Z}}^{2}$-action i.e. $T^{m}S^{n}x\neq x$, for all $x\in X$ and any
pair of integers $(m,n)\neq(0,0)$. We will denote this Borel
${\mathbb{Z}}^{2}$-action by $\Gamma$ in rest of the proof. Also note that
since $\Gamma$ is assumed to be free we can use Theorem 3.9.

Without loss of generality we will construct a Borel function
$f:X\rightarrow\mathbb{R}$, such that $f\in Cob(S)$ but $f\notin Cob(T)$. In
particular, we will construct $f$ with following properties :

$(a)$ There exists a constant $M\in\mathbb{R}$, such that

$\left|\underset{k=0}{\overset{n-1}{\sum}}f\circ S^{k}(x)\right|\leq M,$ (4.1)

for every $x\in X$ and $n\in\mathbb{N}$.

$(b)$ Let $\mu$ be a $\Gamma$-quasi-invariant probability measure on
$(X,{\mathcal{B}})$. For every $r\in{\mathbb{N}}$, there exists
$m_{r}\in\mathbb{N}$ and a set $A_{r}\in{\mathcal{B}}$, $\mu(A_{r})>\beta$
(for some $\beta>0$) such that

$\left|\underset{k=0}{\overset{m_{r}-1}{\sum}}f\circ T^{k}(x)\right|\geq r$
(4.2)

for all $x\in A_{r}$.

Since $\Gamma$ is a free Borel ${\mathbb{Z}}^{2}$-action, in particular this
implies that $T$ is not a power of $S$ hence $(\ref{eq 5.1})$ and $(\ref{eq
5.2})$ can hold simultaneously. Observe that $(\ref{eq 5.1})$ implies that
$f\in Cob(S)$. To see this, set

$g(x)=\underset{n\geq
1}{\rm{sup}}\,\Bigg{(}\underset{k=0}{\overset{n-1}{\sum}}f\circ
S^{k}(x)\Bigg{)}.$

Thus $f(x)=g(x)-g(Sx)$ for all $x\in X$.

We claim that property $(b)$ implies that $f\notin Cob(T)$. To see this assume
by contradiction that $f\in Cob(T)$. Thus, there exists a transfer function
$g(x)$ such that $f(x)=g(x)-g(Tx)$. For $n\in{\mathbb{N}}$, we can write
$\underset{k=0}{\overset{n-1}{\sum}}f\circ T^{k}(x)=g(x)-g\circ T^{n}(x)$. We
will use the statement in Remark 4.1 as follows : Fix $\gamma\in\Gamma$ then
for $\beta>0$ (where $\beta$ is as in $(b)$ above) and every
$0<\epsilon<\beta$, there exists a $\delta_{\gamma}>0$, such that
$\epsilon+\delta_{\gamma}<\beta$ and for every $B\in{\mathcal{B}}$ with
$\mu(B)<\delta_{\gamma}$, we have $\mu(\gamma B)<\epsilon$.

For $\gamma=T^{-1}$ and $\epsilon>0$ such that $0<\epsilon<\beta$ (where
$\beta$ is as in $(b)$ above) and consider $\delta_{T^{-1}}$ (write
$\delta_{1}$ for simplicity) as described above. For $K_{1}\in{\mathbb{R}}$
such that

$\mu(\\{x:|g(x)|\geq K_{1}\\})<\delta_{1},$ (4.3)

using the fact that $\mu$ is $\Gamma$-quasi-invariant we get
$\mu(\\{x:|g(Tx)|\geq K_{1}\\})<\epsilon$. We choose $\delta_{1}$ to be such
that $0<\epsilon+\delta_{1}<\beta$ and $K_{1}$ to be the infimum possible
corresponding to $\delta_{1}$. Thus we get

$\mu\Big{(}\Big{\\{}x:\left|g(x)-g(Tx)\right|\geq
2K_{1}\Big{\\}}\Big{)}<\epsilon+\delta_{1}<\beta.$

i.e. $\mu(\\{x:\left|f(x)\right|\geq 2K_{1}\\})<\beta$. Now we repeat the
process for every $n>1$ (put $\gamma=T^{-n}$). For same $\epsilon>0$ as above
we obtain a sequence of $\delta_{n}>0$. For every $n>1$, we choose each
$\delta_{n}$ such that $0<\epsilon+\delta_{n}<\beta$ and
$\delta_{n}<\delta_{n-1}$. Thus for $K_{n}\in{\mathbb{R}}$ such that

$\mu(\\{x:|g(x)|\geq K_{n}\\})<\delta_{n},$

we get,

$\mu(\\{x:|g(T^{n}x)|\geq K_{n}\\})<\epsilon$

for every $n>1$. As before we work with infimum possible $K_{n}$ corresponding
to $\delta_{n}$. Since $\\{\delta_{n}\\}_{n=1}^{\infty}$ is a decreasing
sequence we get that the corresponding sequence $\\{K_{n}\\}_{n=1}^{\infty}$
is increasing. Hence for every $n>1$ we obtain

$\mu\Big{(}\Big{\\{}x:\left|g(x)-g(T^{n}x)\right|\geq
2K_{n}\Big{\\}}\Big{)}<\epsilon+\delta_{n}<\beta.$

Thus for every $n\in{\mathbb{N}}$ we obtain $\delta_{n}$ and $K_{n}$ such
that,

$\mu\Big{(}\Big{\\{}x:\left|\underset{k=0}{\overset{n-1}{\sum}}f\circ
T^{k}(x)\right|\geq 2K_{n}\Big{\\}}\Big{)}<\epsilon+\delta_{n}<\beta.$ (4.4)

Assume that $\underset{n}{\mathrm{sup}}\,\,K_{n}=:K$ exists (we will prove
this later). Thus for every $n\in{\mathbb{N}}$,

$\mu(\\{x:|g(x)|\geq K\\})<\delta_{n},$

we get,

$\mu(\\{x:|g(T^{n}x)|\geq K\\})<\epsilon.$

Thus for every $n\in{\mathbb{N}}$

$\mu\Big{(}\Big{\\{}x:\left|\underset{k=0}{\overset{n-1}{\sum}}f\circ
T^{k}(x)\right|\geq 2K\Big{\\}}\Big{)}<\epsilon+\delta_{n}<\beta.$ (4.5)

which contradicts $(\ref{eq 5.2})$. We need to show that
$K:=\underset{n}{\mathrm{sup}}\,\,K_{n}$ exists. To see this assume by
contradiction that the supremum does not exists. Hence for every
$M\in{\mathbb{R}}_{+}$ there exists $n\in{\mathbb{N}}$ such that,

$\mu(\\{x:|g(x)|\geq M/2\\})>\delta_{n}.$

Since $\mu$ is $\Gamma$-quasi-invariant we get

$\mu(\\{x:|g(Tx)|\geq M/2\\})>0.$

Hence $\mu(\\{x:|g(x)-g(Tx)|\geq M\\})>0$, i.e. $\mu(\\{x:|f(x)|\geq M\\})>0$.
Since we can do this for every $M\in{\mathbb{R}}_{+}$, $f$ is essentially
unbounded, which is a contradiction.

Now we construct a Borel function $f$ with properties $(a)$, $(b)$. The
function $f$ will be constructed as the sum of infinite series
$f:=\underset{r=1}{\overset{\infty}{\sum}}f_{r}$, in which every term $f_{r}$
is associated with a certain Rokhlin tower $\xi_{r}$ of the
$\mathbb{Z}^{2}$-action generated by commuting Borel automorphisms $S$ and
$T$. Below we describe the construction of tower $\xi_{r}$ and the associated
function $f_{r}$.

The size of tower $\xi_{r}$ (given by $(n_{r}m_{r})$) is determined by two
increasing sequence $\\{n_{r}\\}$ and $\\{m_{r}\\}$ of natural numbers. We
also associate a decreasing sequence of positive real numbers
$\\{\alpha_{r}\\}$ with towers $\xi_{r}$ (we assume $\alpha_{r}\rightarrow
0$). The only restriction on sequence $\\{n_{r}\\}$ is that it is an
increasing sequence of natural numbers. On the other hand, we have the
following assumptions for sequence $\\{m_{r}\\}$ and $\\{\alpha_{r}\\}$:

$m_{r}\Big{(}\underset{s=r+1}{\overset{\infty}{\sum}}\alpha_{s}\Big{)}\leq
1,\hskip 42.67912ptr=\\{1,2,...\\}$ (4.6)

$m_{r}\alpha_{r}\geq
2r+\Big{(}\underset{t=1}{\overset{r-1}{\sum}}\alpha_{t}m_{t}\Big{)},\,\,\,\,r=\\{2,3,...\\}$
(4.7)

The sequences $\\{m_{r}\\}$ and $\\{\alpha_{r}\\}$ with above properties can
be constructed inductively. We use Theorem 3.9 to define tower $\xi_{r}$ as
follows. Set $n=n_{r}$ and $m=m_{r}$ in Theorem 3.9 to obtain a set
$A_{r}=A_{(n,m)}$, which is the base of tower $\xi_{r}$ (the bottom left block
in Figure 1). This tower is a rectangle made up of $n_{r}m_{r}$ disjoint
blocks each representing a set of the form $S^{i}T^{j}A_{r}$, where $0\leq
i<n_{r}$ and $0\leq j<m_{r}$ (see Figure 1). The horizontal direction in the
tower corresponds to the transformation $S$ and the vertical corresponds to
the transformation $T$.

$\vdots$ $\vdots$ $\vdots$ $\ddots$ $\alpha_{r}$ \- $\alpha_{r}$ $\alpha_{r}$
$\cdots$ $\alpha_{r}$ \- $\alpha_{r}$ $\alpha_{r}$ $\cdots$ $\alpha_{r}$ \-
$\alpha_{r}$ $\alpha_{r}$ $\cdots$ Figure 1. The ${\mathbb{Z}}^{2}$ tower
$\xi_{r}$.

Function $f_{r}$ is zero outside the tower $\xi_{r}$. On the tower $\xi_{r}$
it is defined to be constant on each square (in other words $f_{r}$ is
constant on each set $S^{i}T^{j}A_{r}\subset\xi_{r}$). The value of $f_{r}$ on
each square is defined as follows : In each row the value alternates
$\alpha_{r}$, $-\alpha_{r}$ starting with plus sign on the leftmost set. (see
Figure 1).

We now estimate the sums $\left|\underset{k=0}{\overset{n-1}{\sum}}f_{r}\circ
S^{k}(x)\right|$ and $\left|\underset{k=0}{\overset{n-1}{\sum}}f_{r}\circ
T^{k}(x)\right|$ for fixed $r$. The first sum will be estimated from below to
prove that $f$ satisfies property $(a)$ and the second sum will be estimated
from above (for $x\in A_{r}$) to show that $f$ satisfies property $(b)$. Note
for any $x\in\xi_{r}$ we have
$\left|\underset{k=0}{\overset{n-1}{\sum}}f_{r}\circ
S^{k}(x)\right|\leq\alpha_{r}$. By definition $f_{r}=0$ outside the tower
$\xi_{r}$, thus we get

$\left|\underset{k=0}{\overset{n-1}{\sum}}f_{r}\circ
S^{k}(x)\right|\leq\alpha_{r}\,\,\,\textrm{for}\,\,\,x\in X.$

Hence for $x\in X$,

$\left|\underset{k=0}{\overset{n-1}{\sum}}f\circ
S^{k}(x)\right|=\left|\underset{k=0}{\overset{n-1}{\sum}}\Bigg{(}\underset{r=1}{\overset{\infty}{\sum}}f_{r}\Bigg{)}\circ
S^{k}(x)\right|\leq\underset{r=1}{\overset{\infty}{\sum}}\left|\underset{k=0}{\overset{n-1}{\sum}}f_{r}\circ
S^{k}(x)\right|\leq\underset{r=1}{\overset{\infty}{\sum}}\alpha_{r}=:M$

Therefore $f$ satisfies property $(a)$.

To see that $f$ satisfies $(b)$, note that to every $r\in\mathbb{N}$ we have
associated a pair of natural number $(n_{r},m_{r})$ and a tower $\xi_{r}$ with
base $A_{r}$. Since $A_{r}$ is a complete section with respect to $\Gamma$ and
$\mu$ is a $\Gamma$-quasi-invariant probability measure, $\mu(A_{r})>0$. We
work with $A_{r}$ and the corresponding tower $\xi_{r}$. Note that
$\left|\underset{k=0}{\overset{m_{r}-1}{\sum}}f_{r}\circ
T^{k}(x)\right|=m_{r}\alpha_{r}$, for all $x\in A_{r}$. Also we can write

$\left|\underset{k=0}{\overset{m_{r}-1}{\sum}}f\circ
T^{k}(x)\right|\geq\left|\underset{k=0}{\overset{m_{r}-1}{\sum}}f_{r}\circ
T^{k}(x)\right|-\underset{t=1}{\overset{r-1}{\sum}}\left|\underset{k=0}{\overset{m_{r}-1}{\sum}}f_{t}\circ
T^{k}(x)\right|-\underset{s=r+1}{\overset{\infty}{\sum}}\left|\underset{k=0}{\overset{m_{r}-1}{\sum}}f_{s}\circ
T^{k}(x)\right|.$ (4.8)

Since $f_{t}$ is zero outside the tower $\xi_{t}$ and $t<r$ we get

$\left|\underset{k=0}{\overset{m_{r}-1}{\sum}}f_{t}\circ
T^{k}(x)\right|=\left|\underset{k=0}{\overset{m_{t}-1}{\sum}}f_{t}\circ
T^{k}(x)\right|\leq\alpha_{t}m_{t}.$ (4.9)

for $x\in A_{r}$. In $(\ref{t})$ if towers $\xi_{r}$ and $\xi_{t}$ are
disjoint then the sum is zero. If towers $\xi_{r}$ and $\xi_{t}$ are not
disjoint than the sum can be at most $\alpha_{t}m_{t}$. This implies for $x\in
A_{r}$,

$-\underset{t=1}{\overset{r-1}{\sum}}\left|\underset{k=0}{\overset{m_{r}-1}{\sum}}f_{t}\circ
T^{k}(x)\right|\geq-\underset{t=1}{\overset{r-1}{\sum}}\alpha_{t}m_{t}.$

Since $f_{s}$ is zero outside the tower $\xi_{s}$ and $s>r$ similarly we get

$\left|\underset{k=0}{\overset{m_{r}-1}{\sum}}f_{s}\circ
T^{k}(x)\right|\leq\alpha_{s}m_{r}$

for $x\in A_{r}$. This implies

$-\underset{s=r+1}{\overset{\infty}{\sum}}\left|\underset{k=0}{\overset{m_{r}-1}{\sum}}f_{s}\circ
T^{k}(x)\right|\geq-\underset{s=r+1}{\overset{\infty}{\sum}}\alpha_{s}m_{r}$

for $x\in A_{r}$. Thus by $(\ref{eq 5.5})$

$\left|\underset{k=0}{\overset{m_{r}-1}{\sum}}f\circ T^{k}(x)\right|\geq
m_{r}\alpha_{r}-\underset{t=1}{\overset{r-1}{\sum}}\alpha_{t}m_{t}-\underset{s=r+1}{\overset{\infty}{\sum}}\alpha_{s}m_{r}$
(4.10)

for $x\in A_{r}$. Hence by $(\ref{eq 5.4})$ and $(\ref{eq 5.3})$

$\left|\underset{k=0}{\overset{m_{r}-1}{\sum}}f\circ T^{k}(x)\right|\geq
2r-\underset{s=r+1}{\overset{\infty}{\sum}}\alpha_{s}m_{r}\geq 2r-1\geq r$
(4.11)

for $x\in A_{r}$. This shows that $f$ satisfies property $(b)$, which
completes the proof.

$\square$

Acknowledgments. I would like to thank my advisor, Sergii Bezuglyi for
suggesting this problem and many helpful discussions.

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NL}(\bm{p},-\bm{p})}{M^{2}(p)}\left[P_{\rm
shell}(p)\right]^{2}+\mathcal{O}[\delta^{(1)}_{\Lambda}]\,,$ (C.18)

while for the Gaussian case this term is zero [45]. For local non-Gaussianity,
with $K_{\rm NL}=1$, this becomes

$\displaystyle\mathcal{S}_{\delta\delta^{2}}^{22}(\bm{k},\bm{k}^{\prime})=2M(k)\hat{\delta}_{\rm
D}(\bm{k}+\bm{k}^{\prime})P_{\varphi}(\Lambda)\frac{d\sigma^{2}_{\Lambda}}{d\Lambda}\lambda\,,$
(C.19)

leading to a non-zero contribution to the running of stochastic terms. We
leave a more detailed analysis of stochasticity in the presence of non-
Gaussianities for a future work.

##  Appendix D Evaluation of $\mathcal{S}_{O}$

In this appendix we evaluate the $\mathcal{S}_{O}$ in the presence of PNG
interactions. For a complete calculation without interactions, see [44]. We
expand the shell integrals

$\displaystyle\mathcal{S}_{O}[\delta^{(1)}_{\Lambda}](\bm{k})$
$\displaystyle=\sum_{\ell\geq
0}\left(\mathcal{S}_{O}[\delta^{(1)}_{\Lambda}](\bm{k})\right)_{(\ell)_{\rm
legs}}\,,$ (D.1)

in term of the number of external legs $\ell$ (and therefore contributions to
operators starting from $\ell$-th order) with

$\displaystyle\left(\mathcal{S}_{O}[\delta^{(1)}_{\Lambda}](\bm{k})\right)_{(\ell)_{\rm
legs}}=\raisebox{0.0pt}{\parbox[c]{0.0pt}{\includegraphics[scale={.9}]{figs/diags/diag_S2_ell_legs.pdf}}}+\raisebox{0.0pt}{\parbox[c]{0.0pt}{\includegraphics[scale={.9}]{figs/diags/NG_inter.pdf}}}\,.$
(D.2)

Note that the first type of diagrams correspond to the usual corrections from
Gaussian initial conditions considered in [44]. The second type of diagrams
arise due to the interacting non-Gaussian vertex presented in this work. In
App. D.2 and App. D.3 we evaluate these respectively for spin-0 and spin-2
non-Gaussianities.

###  D.1 Displacements of $\Psi$

Before evaluating the shell integrals, we discuss one important point when
evaluating expectation values including $\Psi$. It is important to evaluate
the operator $\Psi$ at the (Lagrangian) position $\bm{q}$ in the initial
conditions that corresponds to a given final-time (Eulerian) position
$\bm{x}$. Physically, this is due to the fact that the PNG interactions are
imprinted at initial time. Formally, it is required to cancel the non-
Galilean-invariant displacement terms involving the gradient of the potential
$\partial_{i}\Phi_{g}$ [22, 35]. So at the Lagrangian position
$\bm{q}=\mathbf{x}-\mathbf{s}$, with $\mathbf{s}$ the displacements, we can
write

$\Psi(\bm{q})=\Psi(\bm{x}-\mathbf{s}(\bm{q}))\,.$ (D.3)

Note that the displacement field admits a perturbative expansion

$\bm{s}(\bm{q})=\bm{s}_{(1)}(\bm{q})+\bm{s}_{(2)}(\bm{q})+\bm{s}_{(3)}(\bm{q})+\dots\,,$
(D.4)

and is also evaluated at the Lagrangian position $\bm{q}$, meaning that it
should also be expanded by

$\bm{s}(\bm{q})=\bm{s}(\bm{x}-\bm{s})=\bm{s}(\bm{x})-\bm{s}(\bm{q})\nabla\bm{s}(\bm{x})+\dots\,.$
(D.5)

With the above, we have at second-order in perturbation theory [29]

$\displaystyle\Psi^{[2]}(\bm{q})=\Psi(\bm{x})-s_{(1)}^{i}\partial_{i}\Psi(\bm{x})\;,$
(D.6)

with

$\mathbf{s}_{(1)}=-\frac{\nabla}{\nabla^{2}}\delta^{(1)}\;.$ (D.7)

At third order, we have

$\displaystyle\Psi^{[3]}(\mathbf{\bm{q}})$
$\displaystyle=\Psi(\mathbf{x}-\mathbf{s})$
$\displaystyle=\Psi(\mathbf{x})-s_{(1)}^{i}\partial_{i}\Psi(\mathbf{x})-s_{(2)}^{i}\partial_{i}\Psi(\mathbf{x})+s^{i}_{(1)}\partial_{i}(s^{j}_{(1)})\partial_{j}\Psi(\mathbf{x})+\frac{1}{2}s_{(1)}^{i}s_{(1)}^{j}\partial_{i}\partial_{j}\Psi(\mathbf{x})\;,$
(D.8)

with

$\mathbf{s}_{(2)}=\frac{3}{14}\frac{\nabla}{\nabla^{2}}\mathcal{G}_{2}\;.$
(D.9)

For the corrections of $\Psi$ we have to consider its expansion at fourth
order, including terms up to third order in the displacements

$\displaystyle\Psi^{[4]}(\bm{q})=\Psi^{[3]}(\bm{x})-s_{(3)}^{i}\partial_{i}\Psi+s_{(1)}^{j}\partial_{j}s_{(2)}^{i}\partial_{i}\Psi+s_{(2)}^{j}\partial_{j}s_{(1)}^{i}\partial_{i}\Psi-\frac{1}{2}s^{j}_{(1)}s^{k}_{(1)}\left(\partial_{j}\partial_{k}s^{i}_{(1)}\right)\partial_{i}\Psi$
$\displaystyle+s^{i}_{(2)}s^{j}_{(1)}\partial_{i}\partial_{j}\Psi-s^{k}_{(1)}\partial_{k}s^{j}_{(1)}\partial_{j}s^{i}_{(1)}\partial_{i}\Psi-s^{k}_{(1)}\partial_{k}s^{i}_{(1)}s^{j}_{(1)}\partial_{i}\partial_{j}\Psi-\frac{1}{6}s^{i}_{(1)}s^{j}_{(1)}s^{k}_{(1)}\partial_{i}\partial_{j}\partial_{k}\Psi\,,$
(D.10)

with [81]

$\displaystyle\bm{s}_{(3)}=\frac{i}{6}\frac{\bm{k}}{k^{2}}\int_{\bm{p}_{1},\bm{p},\bm{p}_{3}}\hat{\delta}_{\rm
D}(\bm{k}-\bm{p}_{123})\delta^{(1)}(\bm{p}_{1})\delta^{(1)}(\bm{p}_{2})\delta^{(1)}(\bm{p}_{3})$
(D.11) $\displaystyle\hskip
85.35826pt\times\left[\frac{5}{7}\sigma^{2}_{\bm{p}_{1},\bm{p}_{2}}\sigma^{2}_{\bm{p}_{1}+\bm{p}_{2},\bm{p}_{3}}-\frac{1}{3}\left(-2-3\sigma^{2}_{\bm{p}_{1},\bm{p}_{2}}+2\frac{(\bm{p}_{1}\cdot\bm{p}_{2})\;(\bm{p}_{2}\cdot\bm{p}_{3})\;(\bm{p}_{1}\cdot\bm{p}_{3})}{p_{1}^{2}p_{2}^{2}p_{3}^{2}}\right)\right]\,.$

Note that corresponding displacement contributions also exist for the Gaussian
operators such as $\delta$, $\mathcal{G}_{2}$, …, but these are included in
the perturbation theory kernels.

###  D.2 Spin-0 non-Gaussianity

#### Calculation of $\left(\mathcal{S}_{O}\right)_{(0)_{\rm legs}}$.

For the Gaussian operators, it is impossible to source a zero-leg leading-
order contribution by the structure of the vertex App. C, since those would
scale as $\lambda^{2}$.

For $\Psi$, despite this operator being linear, it admits an expansion with
respect to the displacements as in Eqs. (D.6)–(D.10). Then we see that at
second order in $\Psi$

$\displaystyle\left(\mathcal{S}_{\Psi}\right)_{(0)_{\rm
legs}}(\bm{k})=-\hat{\delta}_{\rm D}(\bm{k})a_{0}f_{\rm NL}\int_{\bm{p}}P_{\rm
shell}^{\Psi}(p)\,,$ (D.12)

which will vanish after subtracting the tadpole contribution
$\Psi\to\Psi+\hat{\delta}_{\rm D}(\bm{k})\int_{\bm{p}}P_{\rm 1\Psi}(p)$ via
Eq. (1.21).

Non-Gaussian contributions involving $O_{G}\Psi$ also vanish after using Eq.
(1.21), i.e. after subtracting the tadpole contribution

$\displaystyle\left(\mathcal{S}_{\delta\Psi}\right)_{(0)_{\rm
legs}}(\bm{k})=\hat{\delta}_{\rm D}(\bm{k})a_{0}f_{\rm NL}\int_{\bm{p}}P_{\rm
shell}^{\Psi}(p)\,,$ (D.13)

from $\delta\Psi$ via $\delta\Psi\to\delta\Psi-\hat{\delta}_{\rm
D}(\bm{k})\int_{\bm{p}}P_{\rm 1\Psi}(p)$.

#### Calculation of $\left(\mathcal{S}_{O}\right)_{(1)_{\rm legs}}$.

For operators that do not include $\Psi$, we have

$\displaystyle\left(\mathcal{S}_{\delta}\right)_{(1)_{\rm legs}}(\bm{k})$
$\displaystyle=4f_{\rm
NL}\varphi_{\Lambda}(\bm{k})\int_{\bm{p}}F_{2}(\bm{k}+\bm{p},-\bm{p})K_{\rm
NL}(\bm{k},\bm{p})\frac{M(|\bm{p}+\bm{k}|)}{M(p)}P_{\rm shell}(p)$
$\displaystyle=4f_{\rm
NL}a_{0}\Psi_{\Lambda}(\bm{k})\int_{\bm{p}}F_{2}(\bm{k}+\bm{p},-\bm{p})\left(\frac{k_{\rm
NG}}{p}\right)^{\Delta}P_{\rm shell}(p)\,,$ (D.14)
$\displaystyle\left(\mathcal{S}_{\delta^{2}}\right)_{(1)_{\rm legs}}(\bm{k})$
$\displaystyle=4f_{\rm
NL}a_{0}\Psi_{\Lambda}(\bm{k})\int_{\bm{p}}\left(\frac{k_{\rm
NG}}{p}\right)^{\Delta}P_{\rm shell}(p)\,,$ (D.15)
$\displaystyle\left(\mathcal{S}_{\mathcal{G}_{2}}\right)_{(1)_{\rm
legs}}(\bm{k})$ $\displaystyle=4f_{\rm
NL}a_{0}\Psi_{\Lambda}(\bm{k})\int_{\bm{p}}\sigma^{2}_{\bm{p}+\bm{k},-\bm{p}}\left(\frac{k_{\rm
NG}}{p}\right)^{\Delta}P_{\rm shell}(p)+\text{h.d.}\,.$ (D.16)

Eq. (D.15) is the leading source for the non-Gaussian bias coefficient
$b_{\Psi}$, and coincides with the counter-term derived in [28] and [35], with
a factor 4 difference with respect to the latter, presumably originating from
the possible permutations of the kernel and the number of equivalent
contractions. The terms with
$F_{2}(\bm{k}+\bm{p},-\bm{p})\propto\left(k/p\right)^{2}$ and
$\sigma_{\bm{k}+\bm{p},-\bm{p}}^{2}\propto\left(k/p\right)^{2}$ both source
the higher-derivative operators $\nabla^{2}\Psi_{\Lambda}$.

For operators including $\Psi$ we have to go up to third order in $\Psi$
including displacements Eq. (D.6).

Starting from the contribution of $\Psi$, we find

$\displaystyle\left(\mathcal{S}_{\Psi}\right)_{(1)_{\rm
legs}}(\bm{k})=-\frac{13}{21}a_{0}f_{\rm
NL}\delta^{(1)}_{\Lambda}(\bm{k})\int_{\bm{p}}P_{\rm shell}^{\Psi}(p)\,,$
(D.17)

noticing that it contributes to $\delta$. This is different than what is
observed for $\delta$, which only sources higher-derivative contributions as a
consequence of mass and momenta conservation via the equations of motion.
Therefore, for PNGs, we find also first-order operators acting as a source.
For the calculation of $\Psi\delta$, we start with the undisplaced

field $\Psi$ in Eq. (D.6), for which

$\displaystyle\left(\mathcal{S}_{\delta\Psi}\right)_{(1)_{\rm legs}}(\bm{k})$
$\displaystyle\supset\frac{34}{21}f_{\rm
NL}a_{0}\delta^{(1)}_{\Lambda}(\bm{k})\int_{\bm{p}}P_{\rm shell}^{\Psi}(p)\,.$
(D.18)

The second term in Eq. (D.6) gives

$\displaystyle
s_{(1)}^{i}\left[\partial_{i}\Psi(\bm{x})\right]\delta(\bm{x})=\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}_{3}}\hat{\delta}_{\rm
D}(\bm{k}-\bm{p}_{123})\frac{\bm{p}_{1}\cdot\bm{p}_{2}}{p_{1}^{2}}\delta^{(1)}_{\Lambda}(\bm{p}_{1})\Psi_{\Lambda}(\bm{p}_{2})\delta[\delta^{(1)}_{\Lambda}](\bm{p}_{3})\,,$
(D.19)

leading to [at linear order in
$\delta[\delta^{(1)}_{\Lambda}](\bm{p}_{3})=\delta^{(1)}_{\Lambda}(\bm{p}_{3})$]

$\displaystyle\left(\mathcal{S}_{\delta\Psi}^{2}\right)_{(1)_{\rm
legs}}(\bm{k})$ $\displaystyle\supset-f_{\rm
NL}a_{0}\delta^{(1)}_{\Lambda}(\bm{k})\int_{\bm{p}}P_{\rm shell}^{\Psi}(p)\,.$
(D.20)

Therefore, we find that the displaced field in that case changes the overall
prefactor of the $\left(\mathcal{S}_{\delta\Psi}^{2}\right)_{(1)_{\rm legs}}$
calculation. After taking the displacements of $\Psi$ in account we find

$\displaystyle\left(\mathcal{S}_{\delta\Psi}\right)_{(1)_{\rm legs}}(\bm{k})$
$\displaystyle=\frac{13}{21}f_{\rm
NL}a_{0}\delta^{(1)}_{\Lambda}(\bm{k})\int_{\bm{p}}P_{\rm shell}^{\Psi}(p)\,.$
(D.21)

For the other terms involving $\Psi$

$\displaystyle\left(\mathcal{S}_{\delta^{2}\Psi}\right)_{(1)_{\rm
legs}}(\bm{k})$ $\displaystyle=a_{0}f_{\rm
NL}\Psi_{\Lambda}(\bm{k})\int_{\bm{p}}P_{\rm shell}(p)+2a_{0}f_{\rm
NL}\delta^{(1)}_{\Lambda}(\bm{k})\int_{\bm{p}}P_{\rm shell}^{\Psi}(p)\,,$
(D.22) $\displaystyle\left(\mathcal{S}_{\mathcal{G}_{2}\Psi}\right)_{(1)_{\rm
legs}}(\bm{k})$ $\displaystyle=-a_{0}f_{\rm
NL}\frac{4}{3}\delta^{(1)}_{\Lambda}(\bm{k})\int_{\bm{p}}P_{\rm
shell}^{\Psi}(p)\,.$ (D.23)

#### Calculation of $\left(\mathcal{S}_{O}\right)_{(2)_{\rm legs}}$.

For terms that do not involve $\Psi$ and ignoring the terms sourcing higher
derivative operators we get

$\displaystyle\left(\mathcal{S}_{\delta^{2}}\right)_{(2)_{\rm legs}}(\bm{k})$
$\displaystyle=8f_{\rm NL}\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}}\hat{\delta}_{\rm
D}(\bm{k}-\bm{p}_{12})\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
$\displaystyle\quad\times K_{\rm
NL}(\bm{p}_{2},\bm{p})\frac{M(|\bm{p}_{2}+\bm{p}|)}{M(p_{2})M(p)}\left[F_{2}(\bm{p}+\bm{p}_{2},\bm{p}_{1})+F(-\bm{p},\bm{p}_{1})\right]P_{\rm
shell}(p)$ $\displaystyle=f_{\rm
NL}a_{0}\left(\frac{272}{21}[\Psi\delta](\bm{k})-4\left[s^{i}\partial_{i}\Psi\right](\bm{k})\right)\int_{\bm{p}}\left(\frac{k_{\rm
NG}}{p}\right)^{\Delta}P_{\rm shell}(p)+\text{h.d.}\;,$ (D.24)
$\displaystyle\left(\mathcal{S}_{\delta^{3}}\right)_{(2)_{\rm legs}}(\bm{k})$
$\displaystyle=12f_{\rm
NL}a_{0}[\Psi\delta](\bm{k})\int_{\bm{p}}\left(\frac{k_{\rm
NG}}{p}\right)^{\Delta}P_{\rm shell}(p)\;,$ (D.25)
$\displaystyle\left(\mathcal{S}_{\mathcal{G}_{2}\delta}\right)_{(2)_{\rm
legs}}(\bm{k})$ $\displaystyle=-\frac{8}{3}f_{\rm
NL}a_{0}[\Psi\delta](\bm{k})\int_{\bm{p}}\left(\frac{k_{\rm
NG}}{p}\right)^{\Delta}P_{\rm shell}(p)+\text{h.d.}\;,$ (D.26)

where we see that the prefactors of $\Psi$ in Eq. (D.15) and
$s_{(1)}^{i}\partial_{i}\Psi$ in Eq. (D.2) match as expected. Adding them
together, we get

$\displaystyle
4\left[\Psi(\bm{k})-\left[s^{i}\partial_{i}\Psi\right](\bm{k})\right]\int_{\bm{p}}\left(\frac{k_{\rm
NG}}{p}\right)^{\Delta}P_{\rm shell}(p)\,,$ (D.27)

corresponding then to $\Psi(\bm{q})$ and demonstrates again the importance of
calculating $\Psi$ at the Lagrangian point $\bm{q}$ in order to absorb all
terms appearing from the dynamics. Once more, Eq. (D.2) agrees (up to a factor
4) with the second-order counterterm for $\delta^{2}$ calculated in [35].

For operators that involve $\Psi$ it is again crucial to calculate the
displaced operator. If we calculate the operator $\Psi$ or $\Psi\delta$ at the
point $\bm{x}$ this will lead to terms violating the Galilean invariance. For
the operator $\Psi$, we have to consider the fourth-order expansion Eq. (D.10)
to find

$\displaystyle\left(\mathcal{S}_{\Psi}\right)_{(2)_{\rm legs}}(\bm{k})$
$\displaystyle=a_{0}f_{\rm
NL}\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}}\hat{\delta}_{\rm
D}(\bm{k}-\bm{p}_{12})\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
$\displaystyle\hskip
42.67912pt\times\left(\frac{1}{210}-\frac{13}{42}\frac{\bm{p}_{1}\cdot\bm{p}_{2}}{p_{1}^{2}}-\frac{13}{42}\frac{\bm{p}_{1}\cdot\bm{p}_{2}}{p_{2}^{2}}-\frac{577}{1890}\frac{\left(\bm{p}_{1}\cdot\bm{p}_{2}\right)^{2}}{p_{1}^{2}p_{2}^{2}}\right)P_{\rm
shell}^{\Psi}(p)$ $\displaystyle=a_{0}f_{\rm
NL}\left(-\frac{13}{21}\delta^{(2)}(\bm{k})+\frac{43}{135}\left(\delta^{2}\right)^{(2)}(\bm{k})-\frac{1699}{13230}\left(\mathcal{G}_{2}\right)^{(2)}(\bm{k})\right)\int_{\bm{p}}P_{\rm
shell}^{\Psi}(p)+\text{h.d.}\,.$ (D.28)

We calculate the corrections from $\Psi\delta$ more carefully to show the
cancellation of Galilean breaking terms. Evaluating it in the Eulerian
position, we find:

$\displaystyle\left(\mathcal{S}_{\delta\Psi|_{x}}\right)_{(2)\rm{legs}}=f_{\rm
NL}\int_{\bm{p}_{1},\bm{p}_{2}}\hat{\delta}_{\rm
D}(\bm{k}-\bm{p}_{12})\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\int\frac{p^{2}dp}{2\pi^{2}}P_{\rm
shell}^{\Psi}(p)$ $\displaystyle\hskip
56.9055pt\times\left[\frac{172}{105}+\frac{41}{42}\frac{\bm{p}_{1}\cdot\bm{p}_{2}}{p_{1}^{2}}+\frac{41}{42}\frac{\bm{p}_{1}\cdot\bm{p}_{2}}{p_{2}^{2}}+\frac{p^{2}}{6}\frac{\bm{p}_{1}\cdot\bm{p}_{2}}{p_{1}^{2}p_{2}^{2}}+\frac{157}{315}\frac{\left(\bm{p}_{1}\cdot\bm{p}_{2}\right)^{2}}{p_{1}^{2}p_{2}^{2}}\right]\,,$
(D.29)

where we see terms such as $\frac{\bm{p}_{1}\cdot\bm{p}_{2}}{p_{2}^{2}}$ and
$\frac{p^{2}}{6}\frac{\bm{p}_{1}\cdot\bm{p}_{2}}{p_{1}^{2}p_{2}^{2}}$ that
break Galilean symmetry, corresponding to operators such as
$(\partial_{i}\Phi_{g})^{2}$ and $\partial_{i}\Phi_{g}\partial^{i}\delta$.
Notice that after including displacements via the second term in Eq. (D.6), we
first have to expand Eq. (D.19) at third order, and this term leads to

$\displaystyle\left(\mathcal{S}_{\delta\Psi}\right)_{2\rm{legs}}(\bm{k})\supset-
f_{\rm NL}\int_{\bm{p}_{1},\bm{p}_{2}}\hat{\delta}_{\rm
D}(\bm{k}-\bm{p}_{12})\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})$
$\displaystyle\hskip 142.26378pt\times\int\frac{p^{2}dp}{2\pi^{2}}P_{\rm
shell}^{\Psi}(p)\left[\frac{\bm{p}_{1}\cdot\bm{p}_{2}}{6p_{1}^{2}}+\frac{\bm{p}_{1}\cdot\bm{p}_{2}}{6p_{2}^{2}}+\frac{p^{2}}{3}\frac{\bm{p}_{1}\cdot\bm{p}_{2}}{p_{1}^{2}p_{2}^{2}}\right]\,.$
(D.30)

Now we also have to consider the third-order displacement expansion Eq. (D.1)
that has terms such as

$\displaystyle\frac{1}{2}s_{(1)}^{i}s_{(1)}^{j}\left[\partial_{i}\partial_{j}\Psi(\mathbf{x})\right]\delta(\bm{x})$
$\displaystyle=\frac{1}{2}\int_{\bm{p}_{1},\bm{p}_{2},\bm{p}_{3},\bm{p}_{4}}\hat{\delta}_{\rm
D}(\bm{k}-\bm{p}_{1234})\frac{\bm{p}_{1}\cdot\bm{p}_{3}}{p_{1}^{2}}\frac{\bm{p}_{2}\cdot\bm{p}_{3}}{p_{2}^{2}}$
$\displaystyle\hskip
85.35826pt\times\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\Psi_{\Lambda}(\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{4})\,,$
(D.31)

that leads to

$\displaystyle\left(\mathcal{S}_{\delta\Psi}\right)_{2\rm{legs}}(\bm{k})\supset
f_{\rm NL}\int_{\bm{p}_{1},\bm{p}_{2}}\hat{\delta}_{\rm
D}(\bm{k}-\bm{p}_{12})\delta^{(1)}_{\Lambda}(\bm{p}_{1})\delta^{(1)}_{\Lambda}(\bm{p}_{2})\int\frac{p^{2}dp}{2\pi^{2}}P_{\rm
shell}^{\Psi}(p)\left[\frac{p^{2}}{6}\frac{\bm{p}_{1}\cdot\bm{p}_{2}}{p_{1}^{2}p_{2}^{2}}\right]\,.$
(D.32)

We see that by displacing the field $\Psi$ we exactly cancel the non-Galilean
term $\frac{p^{2}}{6}\frac{\bm{p}_{1}\cdot\bm{p}_{2}}{p_{1}^{2}p_{2}^{2}}$
that would violate the Galilean-invariance. By consistently carrying out the
rest of the calculations one can show that the other violating terms will be
generated with the correct pre-factor in order to be absorbed in the already
displaced operators $\delta,\;\delta^{2},\;\mathcal{G}_{2}$. In conclusion,
after including the expansion of $\Psi$ up to third order in displacements Eq.
(D.1) we have

$\displaystyle\left(\mathcal{S}_{\delta\Psi}\right)_{(2)_{\rm legs}}(\bm{k})$
$\displaystyle=f_{\rm
NL}a_{0}\left[\frac{13}{21}\delta^{(2)}(\bm{k})+\frac{478}{135}\left(\delta^{2}\right)^{(2)}(\bm{k})+\frac{79}{2205}\mathcal{G}_{2}^{(2)}(\bm{k})\right]\int_{\bm{p}}P_{\rm
shell}^{\Psi}(p)+\text{h.d.}\,,$ (D.33)
$\displaystyle\left(\mathcal{S}_{\delta^{2}\Psi}\right)_{(2)_{\rm
legs}}(\bm{k})$ $\displaystyle=f_{\rm
NL}a_{0}\left(2\delta^{(2)}(\bm{k})+\frac{47}{21}\left(\delta^{2}\right)^{(2)}(\bm{k})\right)\int_{\bm{p}}P_{\rm
shell}^{\Psi}(p)$ $\displaystyle\quad+a_{0}f_{\rm
NL}\left(\frac{68}{21}[\Psi\delta](\bm{k})-s^{i}\partial_{i}\Psi(\bm{k})\right)\int_{\bm{p}}P_{\rm
shell}(p)+\text{h.d.}\,,$ (D.34)
$\displaystyle\left(\mathcal{S}_{\mathcal{G}_{2}\Psi}\right)_{(2)_{\rm
legs}}(\bm{k})$ $\displaystyle=a_{0}f_{\rm
NL}\left[-\frac{4}{3}\delta^{(2)}(\bm{k})-\frac{31}{21}\left[\delta^{2}\right]^{(2)}(\bm{k})+\frac{-1}{21}\mathcal{G}_{2}^{(2)}(\bm{k})\right]\int_{\bm{p}}P_{\rm
shell}^{\Psi}(p)+\text{h.d.}\,.$ (D.35)

Notice here that the prefactors of $\delta^{(2)}$ exactly match with those of
$\delta^{(1)}$ calculated in Eqs. (D.21)–(D.23), as a consequence of the
Galilean invariance we mentioned before. Moreover, no $\Psi$ term is sourced
with two legs, as expected since this is a linear operator.

###  D.3 Spin-2 non-Gaussianity

#### Calculation of $\left(\mathcal{S}_{O}\right)_{(1)_{\rm legs}}$.

For these kind of terms, we again have to consider $\Psi_{ij}(\bm{q})$ at
second order in displacements. Collecting the contributions, we get

$\displaystyle\left(\mathcal{S}_{\rm Tr[\Psi\Pi^{(1)}]}\right)_{(1)_{\rm
legs}}(\bm{k})$ $\displaystyle=a_{2}f_{\rm
NL}\frac{34}{21}\delta^{(1)}(\bm{k})\int_{\bm{p}}P_{\rm shell}^{\Psi}(p)\,,$
(D.36) $\displaystyle\left(\mathcal{S}_{\rm\delta
Tr[\Psi\Pi^{(1)}]}\right)_{(1)_{\rm legs}}(\bm{k})$ $\displaystyle=a_{2}f_{\rm
NL}\delta^{(1)}(\bm{k})\int_{\bm{p}}P_{\rm shell}^{\Psi}(p)\,,$ (D.37)
$\displaystyle\left(\mathcal{S}_{\rm Tr[\Psi\Pi^{(2)}]}\right)_{(1)_{\rm
legs}}(\bm{k})$ $\displaystyle=a_{2}f_{\rm
NL}\frac{34}{21}\delta^{(1)}(\bm{k})\int_{\bm{p}}P_{\rm shell}^{\Psi}(p)\,.$
(D.38)

#### Calculation of $\left(\mathcal{S}_{O}\right)_{(2)_{\rm legs}}$.

We first consider operators that do not include $\rm Tr[\Psi\Pi^{(i)}]$. The
terms that source leading-in-derivatives operators are

$\displaystyle\left(\mathcal{S}_{\delta^{2}}\right)_{(2)_{\rm legs}}(\bm{k})$
$\displaystyle=16a_{2}f_{\rm NL}\int_{\bm{p}_{1},\bm{p}_{3}}\hat{\delta}_{\rm
D}(\bm{k}-\bm{p}_{1}-\bm{p}_{3})\delta^{(1)}_{\Lambda}(\bm{p}_{1})\Psi_{\Lambda}(\bm{p}_{3})\left(\frac{-2}{105}+\frac{2}{35}\frac{\left(\bm{p}_{3}\cdot\bm{p}_{1}\right)^{2}}{p_{3}^{2}p_{1}^{2}}\right)$
$\displaystyle\hskip 142.26378pt\times\int_{\bm{p}}\left(\frac{k_{\rm
NG}}{p}\right)^{\Delta^{\prime}}P_{\rm shell}(p)+\text{h.d.}$
$\displaystyle=a_{2}f_{\rm
NL}\frac{64}{105}\operatorname{Tr}\left[\Psi\Pi^{[1]}\right](\bm{k})\times\int_{\bm{p}}\left(\frac{k_{\rm
NG}}{p}\right)^{\Delta^{\prime}}P_{\rm shell}(p)+\text{h.d.}\;,$ (D.39)
$\displaystyle\left(\mathcal{S}_{\delta\mathcal{G}_{2}}\right)_{(2)_{\rm
legs}}(\bm{k})$ $\displaystyle=a_{2}f_{\rm
NL}\frac{16}{15}\operatorname{Tr}\left[\Psi\Pi^{[1]}\right](\bm{k})\times\int_{\bm{p}}\left(\frac{k_{\rm
NG}}{p}\right)^{\Delta^{\prime}}P_{\rm shell}(p)+\text{h.d.}\,.$ (D.40)

Calculating now for the terms that include $\rm Tr[\Psi\Pi^{(i)}]$, we find
again that we have to consider $\Psi_{ij}(\bm{q})$ at third order in
displacements (similarly to $\Psi\delta$ in the spin-0 case). Collecting the
contributions, we get

$\displaystyle\left(\mathcal{S}_{\rm Tr[\Psi\Pi^{(1)}]}\right)_{(2)_{\rm
legs}}(\bm{k})$ $\displaystyle=a_{2}f_{\rm
NL}\left[\frac{34}{21}\delta^{(2)}(\bm{k})+\frac{124}{315}\left[\delta^{2}\right]^{(2)}(\bm{k})-\frac{661}{4410}\mathcal{G}_{2}^{(2)}(\bm{k})\right]\int_{\bm{p}}P_{\rm
shell}^{\Psi}(p)$ $\displaystyle\hskip 256.0748pt+\text{h.d.}\,,$ (D.41)
$\displaystyle\left(\mathcal{S}_{\rm Tr[\delta\Psi\Pi^{(1)}]}\right)_{(2)_{\rm
legs}}(\bm{k})$ $\displaystyle=a_{2}f_{\rm
NL}\left[\delta^{(2)}(\bm{k})+\frac{178}{105}\left(\delta^{2}\right)^{(2)}(\bm{k})+\frac{4}{35}\mathcal{G}_{2}^{(2)}(\bm{k})\right]\int_{\bm{p}}P_{\rm
shell}^{\Psi}(p)$ $\displaystyle\quad+a_{2}f_{\rm
NL}\frac{8}{105}\operatorname{Tr}\left[\Psi\Pi^{[1]}\right](\bm{k})\int_{\bm{p}}P_{\rm
shell}(p)+\text{h.d.}\,,$ (D.42) $\displaystyle\left(\mathcal{S}_{\rm
Tr[\Psi\Pi^{(2)}]}\right)_{(2)_{\rm legs}}(\bm{k})$ $\displaystyle=a_{2}f_{\rm
NL}\left[\frac{58}{105}\operatorname{Tr}\left[\Psi\Pi^{[1]}\right](\bm{k})\right]\int_{\bm{p}}P_{\rm
shell}(p)$ $\displaystyle+a_{2}f_{\rm
NL}\left[\frac{34}{21}\delta^{(2)}(\bm{k})+\frac{14347}{6027}\left(\delta^{2}\right)^{(2)}(\bm{k})-\frac{241}{735}\mathcal{G}_{2}^{(2)}(\bm{k})\right]\int_{p}\frac{p^{2}dp}{2\pi}P_{\rm
shell}^{\Psi}(p)+\text{h.d.}\,.$ (D.43)

Notice again that the prefactors exactly match with those calculated in Eqs.
(D.36)–(D.38).

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