# Intrinsic Ergodicity, Generators and Symbolic Representationsof Algebraic
Group Actions

Hanfeng Li  Hanfeng Li: Department of Mathematics, SUNY at Buffalo, NY
14260-2900, USA<EMAIL_ADDRESS>and  Klaus Schmidt  Klaus Schmidt:
Mathematics Institute, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090
Vienna, Austria<EMAIL_ADDRESS>Dedicated to Anatole M. Vershik on
the occasion of his 90th birthday

###### Abstract.

We construct natural symbolic representations of intrinsically ergodic, but
not necessarily expansive, principal algebraic actions of countably infinite
amenable groups and use these representations to find explicit generating
partitions (up to null-sets) for such actions.

The first author gratefully acknowledges partial support by the NSF grant
DMS-1900746

##  1\. Introduction

Let $\Gamma$ be a countably infinite discrete group with integral and real
group rings $\mathbb{Z}\Gamma\subset\mathbb{R}\Gamma$. Every
$g\in\mathbb{R}\Gamma$ is written as a finite formal sum
$g=\sum_{\gamma\in\Gamma}g_{\gamma}\cdot\gamma$ with $g_{\gamma}$ in
$\mathbb{Z}$ or $\mathbb{R}$, respectively, for every $\gamma$. We write
$\textup{supp}(g)=\\{\gamma\in\Gamma\mid g_{\gamma}\neq 0\\}$ for the
_support_ of $g$ and set
$g^{+}=\sum_{\gamma\in\Gamma}\max\\{g_{\gamma},0\\}\cdot\gamma$ and
$g^{-}=\sum_{\gamma\in\Gamma}\min\\{g_{\gamma},0\\}\cdot\gamma$.

For
$g=\sum_{\gamma\in\Gamma}g_{\gamma}\cdot\gamma,\,h=\sum_{\gamma\in\Gamma}h_{\gamma}\cdot\gamma$
in $\mathbb{R}\Gamma$ we denote by
$g+h=\sum_{\gamma\in\Gamma}(g_{\gamma}+h_{\gamma})\cdot\gamma$ their _sum_ ,
by $gh=\sum_{\gamma,\delta\in\Gamma}g_{\gamma}h_{\delta}\cdot\gamma\delta$
their _product_ , and by
$g^{*}=\sum_{\gamma\in\Gamma}g_{\gamma}\cdot\gamma^{-1}$ and
$h^{*}=\sum_{\gamma\in\Gamma}h_{\gamma}\cdot\gamma^{-1}$ their _adjoints_. The
adjoint map $g\mapsto g^{*}$ is an _involution_ on $\mathbb{R}\Gamma$, i.e.,
$(gh)^{*}=h^{*}g^{*}$.

An _algebraic $\Gamma$-action_ is a homomorphism
$\tau\colon\Gamma\rightarrow\textup{Aut}(X)$ from $\Gamma$ to the group of
continuous automorphisms of a compact metrizable abelian group $X$. If $\tau$
is such an algebraic $\Gamma$-action, then $\tau^{\gamma}\in\textup{Aut}(X)$
denotes the image of $\gamma\in\Gamma$, and
$\tau^{\gamma\delta}=\tau^{\gamma}\tau^{\delta}$ for every
$\gamma,\delta\in\Gamma$. The action $\tau$ induces an action of
$\mathbb{Z}\Gamma$ by group homomorphisms $\tau^{f}\colon X\rightarrow X$,
where $\tau^{f}=\sum_{\gamma\in\Gamma}f_{\gamma}\tau^{\gamma}$ for every
$f=\sum_{\gamma\in\Gamma}f_{\gamma}\cdot\gamma\in\mathbb{Z}\Gamma$. Clearly,
if $f,g\in\mathbb{Z}\Gamma$, then $\tau^{fg}=\tau^{f}\tau^{g}$.

Let $\hat{X}$ be the dual group of $X$. If $\hat{\tau}^{\gamma}$ is the
automorphism of $\hat{X}$ dual to $\tau^{\gamma}$, then the map
$\hat{\tau}\colon\Gamma\rightarrow\textup{Aut}(\hat{X})$ satisfies that
$\hat{\tau}^{\gamma\delta}=\hat{\tau}^{\delta}\hat{\tau}^{\gamma}$ for all
$\gamma,\delta\in\Gamma$. We denote by
$\hat{\tau}^{f}\colon\hat{X}\rightarrow\hat{X}$ the group homomorphism dual to
$\tau^{f}$ and set $f\cdot a=\hat{\tau}^{f^{*}}a$ for every
$f\in\mathbb{Z}\Gamma$ and $a\in\hat{X}$. The resulting map $(f,a)\mapsto
f\cdot a$ from $\mathbb{Z}\Gamma\times\hat{X}$ to $\hat{X}$ satisfies that
$(fg)\cdot a=f\cdot(g\cdot a)$ for all $f,g\in\mathbb{Z}\Gamma$ and turns
$\hat{X}$ into a left module over the group ring $\mathbb{Z}\Gamma$.
Conversely, if $M$ is a countable left module over $\mathbb{Z}\Gamma$, we set
$X=\widehat{M}$ and put $\hat{\tau}^{f}a=f^{*}\cdot a$ for
$f\in\mathbb{Z}\Gamma$ and $a\in M$. The maps
$\tau^{f}\colon\widehat{M}\rightarrow\widehat{M}$ dual to $\hat{\tau}^{f}$
define an action of $\mathbb{Z}\Gamma$ by homomorphisms of $\widehat{M}$,
which in turn induces an algebraic action $\tau$ of $\Gamma$ on
$X=\widehat{M}$.

The simplest examples of algebraic $\Gamma$-actions arise from
$\mathbb{Z}\Gamma$-modules of the form $M=\mathbb{Z}\Gamma/(f)$, where
$(f)=\mathbb{Z}\Gamma f$ is the principal left ideal generated by $f$: such
actions are called _principal_. For an explicit description of such actions we
put $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ and define the left and right shift-
actions $\lambda$ and $\rho$ of $\Gamma$ on $\mathbb{T}^{\Gamma}$ by

$(\lambda^{\gamma}x)_{\delta}=x_{\gamma^{-1}\delta}\quad\textup{and}\quad(\rho^{\gamma}x)_{\delta}=x_{\delta\gamma}$
(1.1)

for every $\gamma\in\Gamma$ and
$x=(x_{\delta})_{\delta\in\Gamma}\in\mathbb{T}^{\Gamma}$. The actions
$\lambda$ and $\rho$ extend to $\mathbb{Z}\Gamma$-actions on
$\mathbb{T}^{\Gamma}$ given by

$\lambda^{f}=\textstyle\sum_{\gamma\in\Gamma}f_{\gamma}\lambda^{\gamma},\qquad\rho^{f}=\textstyle\sum_{\gamma\in\Gamma}f_{\gamma}\rho^{\gamma},$

for every $f=\sum_{\gamma\in\Gamma}f_{\gamma}\cdot\gamma\in\mathbb{Z}\Gamma$.
These $\mathbb{Z}\Gamma$-actions obviously commute: for every
$f,g\in\mathbb{Z}\Gamma$ and $x\in\mathbb{T}^{\Gamma}$,

$(\lambda^{f}\circ\rho^{g})\,x=(\rho^{g}\circ\lambda^{f})\,x.$

The pairing $\langle
f,x\rangle=\sum_{\gamma\in\Gamma}f_{\gamma}x_{\gamma}=(\rho^{f}x)_{1_{\Gamma}}$
with $f=\sum_{\gamma\in\Gamma}f_{\gamma}\cdot\gamma\in\mathbb{Z}\Gamma$ and
$x=(x_{\gamma})\in\mathbb{T}^{\Gamma}$, identifies $\mathbb{Z}\Gamma$ with the
dual group $\widehat{\mathbb{T}^{\Gamma}}$ of $\mathbb{T}^{\Gamma}$ and has
the property that

$\displaystyle\langle h,\rho^{f}x\rangle$
$\displaystyle=\smash[t]{\Bigl{\langle}h,\sum\nolimits_{\delta\in\Gamma}f_{\delta}\rho^{\delta}x\Bigr{\rangle}=\sum\nolimits_{\gamma\in\Gamma}h_{\gamma}\sum\nolimits_{\delta\in\Gamma}f_{\delta}x_{\gamma\delta}}$
$\displaystyle=\sum\nolimits_{\gamma\in\Gamma}\sum\nolimits_{\delta\in\Gamma}h_{\gamma\delta^{-1}}f_{\delta}x_{\gamma}=\sum\nolimits_{\gamma\in\Gamma}(hf)_{\gamma}x_{\gamma}=\langle
hf,x\rangle$

for every $f,h\in\mathbb{Z}\Gamma$ and $x\in\mathbb{T}^{\Gamma}$. Every
$f\in\mathbb{Z}\Gamma$ defines a $\lambda$-invariant closed subgroup

$\displaystyle X_{f}$
$\displaystyle=\ker\rho^{f}=\\{x\in\mathbb{T}^{\Gamma}\mid\rho^{f}x=0\\}$
(1.2) $\displaystyle=\\{x\in\mathbb{T}^{\Gamma}\mid\langle
h,\rho^{f}x\rangle=\langle hf,x\rangle=0\enspace\textup{for every}\enspace
h\in\mathbb{Z}\Gamma\\}=(f)^{\perp}\subset\widehat{\mathbb{Z}\Gamma}=\mathbb{T}^{\Gamma}.$

We denote by $\lambda_{f}=\lambda_{X_{f}}$ the restriction of $\lambda$ to
$X_{f}$ and note that the normalized Haar measure $\mu_{f}$ on $X_{f}$ is
invariant under $\lambda_{f}$.

######  Definition 1.1.

For every $f\in\mathbb{Z}\Gamma$ we call the left shift-action $\lambda_{f}$
on the probability space $(X_{f},\mu_{f})$ the _principal algebraic
$\Gamma$-action_ defined by $f$.

Dynamical properties of algebraic actions of countably infinite groups — and,
in particular, of principal actions — have been investigated at various levels
of generality (cf. e.g., [DSAO], [Chung-Li], [Li-Thom], or [Hayes]). In this
paper we focus on symbolic representations of principal algebraic actions of
countably infinite amenable groups and on generators of such actions arising
from these representations.

Symbolic representations of algebraic actions have a long history. The first
such representations arose from geometrically constructed Markov partitions
around 1967–1970 (cf. [Sinai], [Bowen]) and helped to provide a crucial link
between smooth and symbolic dynamics. A different approach to symbolic
representations of toral automorphisms had its origins in the paper [Vershik]
by Vershik from 1992, where he represented the hyperbolic toral automorphism
$\bigl{(}\begin{smallmatrix}1&1\\\ 1&0\end{smallmatrix}\bigr{)}$ by the
_Golden Mean_ shift by using homoclinic points rather than Markov partitions.
Vershik’s original construction was subsequently extended to arbitrary
hyperbolic toral and solenoidal automorphisms in [K-V] and [L-B], and to the
‘homoclinic’ construction of symbolic covers of expansive111 A continuous
$\Gamma$-action $\tau$ on a compact metrizable space $Y$ with compatible
metric $\mathsf{d}$ is _expansive_ if there exists a $\delta>0$ such that
$\sup_{\gamma\in\Gamma}\mathsf{d}(\tau^{\gamma}y,\tau^{\gamma}y^{\prime})\geq\delta$
whenever $y,y^{\prime}$ are distinct points in $Y$.  principal algebraic
$\mathbb{Z}^{d}$-actions (cf. [E-S]). What these constructions have in common
is that they use a _summable homoclinic point_ $w\in X_{f}$ of a principal
algebraic action $\lambda_{f}$ of a countably infinite discrete group $\Gamma$
to define a shift-equivariant surjective map
$\xi_{w}\colon\ell^{\infty}(\Gamma,\mathbb{Z})\to X_{f}$, and to restrict this
map to a suitable compact shift-invariant subset
$\mathcal{V}\subset\ell^{\infty}(\Gamma,\mathbb{Z})$ (cf. e.g. [E-S],
[Goell1], or [Lind-S-2]).

While expansive principal algebraic actions always have summable homoclinic
points permitting such a construction, this is generally not the case for
nonexpansive actions (cf. [Chung-Li], [Deninger-S], [homoclinic]). In this
paper we obviate the need for summable homoclinic points and show directly
that, for every countably infinite discrete amenable group $\Gamma$ and every
$f\in\mathbb{Z}\Gamma$ for which the principal algebraic action $\lambda_{f}$
on $(X_{f},\mu_{f})$ is _intrinsically ergodic_ there exists a natural
isomorphism $(\textup{mod}\;\mu_{f})$ of $\lambda_{f}$ with the left shift
action $\bar{\lambda}$ of $\Gamma$ on a closed, shift-invariant subset
$\bar{Z}_{f}$ of the symbolic space
$\\{-\|f^{-}\|_{1},\dots,\|f^{+}\|_{1}\\}^{\Gamma}$, furnished with a shift-
invariant Borel probability measure $\nu_{f}^{\scriptscriptstyle\\#}$ (Theorem
3.18). As an obvious consequence of this isomorphism one obtains that the
‘alphabet’ $\mathcal{B}_{f}\subsetneq\\{-\|f^{-}\|_{1},\dots,\|f^{+}\|_{1}\\}$
of $\bar{Z}_{f}$ determines a natural generator222 If $\tau$ is a measure-
preserving action of a countably infinite group $\Gamma$ on a standard
probability space $(Y,\mathcal{B}_{Y},\mu)$, a countable Borel partition
$\mathcal{C}$ of $Y$ is a _generator_ for $\tau$ if the smallest
$\tau$-invariant sigma-algebra $\mathcal{T}\subset\mathcal{B}_{Y}$ containing
$\mathcal{C}$ is equal to $\mathcal{B}_{Y}$ $(\textup{mod}\;\mu)$, i.e. up to
$\mu$-null sets. for $\lambda_{f}$ on $(X_{f},\mu_{f})$ (Corollary 4.1). As a
further corollary of this construction we see that the partition
$\mathcal{C}_{f}=\\{C_{j}\mid j=0,\dots,\|f\|_{1}-1\\}$ of $X_{f}$, defined by

$C_{j}=\bigl{\\{}x=(x_{\gamma})_{\gamma\in\Gamma}\in
X_{f}\mid{\scriptstyle\tfrac{j}{\|f\|_{1}}}\leq
x_{1_{\Gamma}}<{\scriptstyle\tfrac{j+1}{\|f\|_{1}}}\enspace(\textup{mod}\;1)\bigr{\\}}$

for $j=0,\dots,\|f\|_{1}-1$, is a generator $(\textup{mod}\;\mu_{f})$ for
$\lambda_{f}$ (Corollary 4.2). If $\lambda_{f}$ is _expansive_ ,
$\mathcal{C}_{f}$ is obviously a generator without any conditions on $\Gamma$
(cf. Subsection 5.2.1), but for nonexpansive actions this result is
nontrivial.

In Section 5 we present examples of intrinsically ergodic principal algebraic
actions $\lambda_{f}$, $f\in\mathbb{Z}\Gamma$, of countably infinite discrete
amenable groups $\Gamma$. If $\Gamma=\mathbb{Z}^{d}$, $d\geq 1$, or if
$\Gamma$ is arbitrary and $\lambda_{f}$ is expansive, the situation is well-
understood (Subsections 5.1 or 5.2.1). For nonexpansive principal algebraic
actions intrinsic ergodicity is a more elusive property. A sufficient
condition for intrinsic ergodicity is that the group $\Delta^{1}(X_{f})$ of
summable homoclinic points of $\lambda_{f}$ is dense in the group $X_{f}$
carrying the action (Proposition 5.3). In Theorem 6.1 we verify the latter
condition for _well-balanced_ polynomials $f\in\mathbb{Z}\Gamma$, provided
that $\Gamma$ is not virtually $\mathbb{Z}$ or $\mathbb{Z}^{2}$ and the center
of $\Gamma$ contains an element of infinite order.

##  2\. Linearization of principal algebraic actions

Let $\Gamma$ be a countably infinite discrete group, and let
$\ell^{\infty}(\Gamma,\mathbb{R})$ be the space of all bounded maps
$v\colon\Gamma\to\mathbb{R}$, furnished with the norm
$\|v\|_{\infty}=\sup_{\gamma\in\Gamma}|v_{\gamma}|$. We write
$\eta\colon\ell^{\infty}(\Gamma,\mathbb{R})\to\mathbb{T}^{\Gamma}$ for the
weak∗-continuous map defined by

$\eta(w)_{\gamma}=w_{\gamma}\;\text{(mod 1)},\,\gamma\in\Gamma,$ (2.1)

and define the shift-actions $\bar{\lambda}$ and $\bar{\rho}$ of $\Gamma$ on
$\ell^{\infty}(\Gamma,\mathbb{R})$ as in (1.1) by

$(\bar{\lambda}^{\gamma}v)_{\delta}=v_{\gamma^{-1}\delta},\enspace\enspace(\bar{\rho}^{\gamma}v)_{\delta}=v_{\delta\gamma},$
(2.2)

for every
$v=(v_{\delta})_{\delta\in\Gamma}\in\ell^{\infty}(\Gamma,\mathbb{R})$ and
$\gamma\in\Gamma$. Again we extend these $\Gamma$-actions to
$\mathbb{Z}\Gamma$-actions on $\ell^{\infty}(\Gamma,\mathbb{R})$ by setting

$\bar{\lambda}^{h}=\sum\nolimits_{\gamma\in\Gamma}h_{\gamma}\bar{\lambda}^{\gamma}\quad\textup{and}\quad\bar{\rho}^{h}=\sum\nolimits_{\gamma\in\Gamma}h_{\gamma}\bar{\rho}^{\gamma}$

for every $h=\sum_{\gamma\in\Gamma}h_{\gamma}\cdot\gamma\in\mathbb{Z}\Gamma$.
These actions correspond to the usual convolutions

$\bar{\lambda}^{h}v=h\cdot v,\qquad\bar{\rho}^{h}v=v\cdot h^{*},$ (2.3)

for $h\in\mathbb{Z}\Gamma$ and $v\in\mathbb{R}\Gamma$, and extend further to
$h\in\ell^{1}(\Gamma,\mathbb{R})$ and $v\in\ell^{\infty}(\Gamma,\mathbb{R})$.

We fix a nonzero element $f\in\mathbb{Z}\Gamma$ and consider the left shift
action $\lambda_{f}$ on the compact group $X_{f}\subset\mathbb{T}^{\Gamma}$
defined in (1.1) – (1.2). The space

$\displaystyle W_{f}\coloneqq\eta^{-1}(X_{f})$
$\displaystyle=\\{w\in\ell^{\infty}(\Gamma,\mathbb{R})\mid\eta(w)\in X_{f}\\}$
(2.4)
$\displaystyle=\\{w\in\ell^{\infty}(\Gamma,\mathbb{R})\mid\bar{\rho}^{f}w=w\cdot
f^{*}\in\ell^{\infty}(\Gamma,\mathbb{Z})\\}$

is the _linearization_ of $X_{f}$, and the restriction of $\bar{\lambda}$ to
$W_{f}$ is the _linearization_ of $\lambda_{f}$. We set

$Y_{f}=W_{f}\cap[0,1)^{\Gamma}\subset\ell^{\infty}(\Gamma,\mathbb{R}),\qquad
Z_{f}=\bar{\rho}^{f}(Y_{f})\subset\ell^{\infty}(\Gamma,\mathbb{Z}),$ (2.5)

write $\bar{Y}_{f}$ and $\bar{Z}_{f}$ for the weak∗ closures of $Y_{f}$ and
$Z_{f}$ in $\ell^{\infty}(\Gamma,\mathbb{R})$. Since $0\leq y_{\gamma}<1$ for
every $y\in Y_{f}$ and $\gamma\in\Gamma$, it is clear that

$c_{f}^{-}\leq(\bar{\rho}^{f}y)_{\gamma}\leq c_{f}^{+}$

for every $y\in Y_{f}$ and $\gamma\in\Gamma$, where

$c_{f}^{-}=\min\,\\{0,1-\|f^{-}\|_{1}\\}\quad\textup{and}\quad
c_{f}^{+}=\max\,\\{0,\|f^{+}\|_{1}-1\\}.$

It follows that

$Z_{f}\subseteq\bar{Z}_{f}\subseteq\\{c_{f}^{-},\dots,c_{f}^{+}\\}^{\Gamma}.$
(2.6)

Finally we denote by

$K_{f}=\\{w\in\ell^{\infty}(\Gamma,\mathbb{R})\mid\bar{\rho}^{f}w=0\\}\subset
W_{f}$ (2.7)

the kernel of $\bar{\rho}^{f}$ in $\ell^{\infty}(\Gamma,\mathbb{R})$. By
[Deninger-S]*Theorem 3.2, the action $\lambda_{f}$ on $X_{f}$ is expansive if
and only if $K_{f}=\\{0\\}$.

If there is any danger of confusion we denote the restrictions of
$\bar{\lambda}$ to the $\bar{\lambda}$-invariant sets $\bar{Y}_{f}$,
$\bar{Z}_{f}$, and $K_{f}$ by $\bar{\lambda}_{\bar{Y}_{f}}$,
$\bar{\lambda}_{\bar{Z}_{f}}$ and $\bar{\lambda}_{K_{f}}$, respectively. The
map $\eta\colon\ell^{\infty}(\Gamma,\mathbb{R})\to\mathbb{T}^{\Gamma}$ in
(2.1) induces a left shift-equivariant, continuous, surjective map from
$\bar{Y}_{f}$ to $X_{f}$ whose restriction to $Y_{f}$ is bijective, and
$\bar{\rho}^{f}$ intertwines the $\Gamma$-actions
$\bar{\lambda}_{\bar{Y}_{f}}$ and $\bar{\lambda}_{\bar{Z}_{f}}$.

######  Proposition 2.1.

Let $\Gamma$ be a countably infinite discrete group, $0\neq
f\in\mathbb{Z}\Gamma$, and let $W_{f},\bar{Y}_{f},\bar{Z}_{f},K_{f}$ be the
closed, $\bar{\lambda}$-invariant subsets of
$\ell^{\infty}(\Gamma,\mathbb{R})$ defined in (2.4) – (2.7). Put
$\tilde{Z}_{f}=\bar{Z}_{f}\times K_{f}$, and denote by
$\tilde{\lambda}=\bar{\lambda}_{\bar{Z}_{f}}\times\bar{\lambda}_{K_{f}}$ the
product $\Gamma$-action on $\tilde{Z}_{f}$. Then there exists, for every
$\bar{\lambda}$-invariant Borel probability measure $\nu$ on $\bar{Y}_{f}$, a
$\tilde{\lambda}$-invariant Borel probability measure $\tilde{\nu}$ on
$\tilde{Z}_{f}$ with the following properties:

  1. (1)

$\pi^{(1)}_{*}\tilde{\nu}=\bar{\rho}^{f}_{*}\nu\eqqcolon\nu^{\scriptscriptstyle\\#}$,
where $\pi^{(1)}\colon\tilde{Z}_{f}\to\bar{Z}_{f}$ is the first coordinate
projection;

  2. (2)

$\tilde{\nu}(\bar{Z}_{f}\times B_{2}(K_{f}))=1$, where $B_{r}(K_{f})=\\{w\in
K_{f}\mid\|w\|_{\infty}\leq r\\}$ for every $r\geq 0$;

  3. (3)

The $\Gamma$-actions $\bar{\lambda}$ on $(\bar{Y}_{f},\nu)$ and
$\tilde{\lambda}$ on $(\tilde{Z}_{f},\tilde{\nu})$ are measurably conjugate.

###### Proof.

Since $\bar{\rho}^{f}$ is continuous, [Partha]*Theorem I.4.2 shows that there
exists a Borel map $\zeta\colon\bar{Z}_{f}\rightarrow\bar{Y}_{f}$ with
$\bar{\rho}^{f}\circ\zeta(z)=z$ for every $z\in\bar{Z}_{f}$. For every
$z\in\bar{Z}_{f}$ and $\gamma\in\Gamma$ we set

$c(\gamma,z)=\zeta\circ\bar{\lambda}^{\gamma}(z)-\bar{\lambda}^{\gamma}\circ\zeta(z)\in
B_{1}(K_{f}).$ (2.8)

Then

$c(\gamma\delta,z)=c(\gamma,\bar{\lambda}^{\delta}z)+\bar{\lambda}^{\gamma}c(\delta,z)$

for every $\gamma,\delta\in\Gamma$ and $z\in\bar{Z}_{f}$, i.e., the Borel map
$c\colon\Gamma\times\bar{Z}_{f}\rightarrow K_{f}$ is a cocycle taking values
in $B_{1}(K_{f})$. We define a Borel action $\tilde{\lambda}_{1}$ of $\Gamma$
on $\tilde{Z}_{f}$ by setting

$\tilde{\lambda}_{1}^{\gamma}(z,v)=(\bar{\lambda}^{\gamma}z,\bar{\lambda}^{\gamma}v-c(\gamma,z))$

for every $(z,v)\in\tilde{Z}_{f}$, and consider the injective Borel map
$\theta_{1}\colon\bar{Y}_{f}\rightarrow\bar{Z}_{f}\times
B_{1}(K_{f})\subset\tilde{Z}_{f}$ given by

$\theta_{1}(w)=(\bar{\rho}^{f}w,w-\zeta\circ\bar{\rho}^{f}(w))$ (2.9)

for every $w\in\bar{Y}_{f}$. Then

$\theta_{1}\circ\bar{\lambda}^{\gamma}=\tilde{\lambda}_{1}^{\gamma}\circ\theta_{1}$
(2.10)

for every $\gamma\in\Gamma$.

Let $\nu$ be a $\bar{\lambda}$-invariant Borel probability measure on
$\bar{Y}_{f}$ and let $\nu^{\scriptscriptstyle\\#}=\bar{\rho}^{f}_{*}\nu$. The
probability measure $\tilde{\nu}^{(1)}=(\theta_{1})_{*}\nu$ is
$\tilde{\lambda}_{1}$-invariant by (2.10), and is supported in the
weak∗-compact and metrizable set $\bar{Z}_{f}\times
B_{1}(K_{f})\subset\tilde{Z}_{f}$. Furthermore,
$\pi^{(1)}_{*}\tilde{\nu}^{(1)}=\nu^{\scriptscriptstyle\\#}$, where
$\pi^{(1)}\colon\tilde{Z}_{f}\to\bar{Z}_{f}$ is the first coordinate
projection. We decompose $\tilde{\nu}^{(1)}$ over $\bar{Z}_{f}$ by choosing a
Borel measurable family $\tilde{\nu}^{(1)}_{z},\,z\in\bar{Z}_{f}$, of Borel
probability measures on $K_{f}$ with $\tilde{\nu}^{(1)}_{z}(B_{1}(K_{f}))=1$
for every $z\in\bar{Z}_{f}$, and with

$\int
g(z,v)\,d\tilde{\nu}^{(1)}(z,v)=\int_{\bar{Z}_{f}}\int_{K_{f}}g(z,v)\,d\tilde{\nu}^{(1)}_{z}(v)\,d\nu^{\scriptscriptstyle\\#}(z)$

for every bounded Borel map $g\colon\tilde{Z}_{f}\rightarrow\mathbb{R}$. Since
$\tilde{\nu}^{(1)}$ is $\tilde{\lambda}_{1}$-invariant,

$\int h(v)\,d\tilde{\nu}^{(1)}_{z}(v)=\int
h(\bar{\lambda}^{\gamma}v-c(\gamma,\bar{\lambda}^{\gamma^{-1}}z))\,d\tilde{\nu}^{(1)}_{\bar{\lambda}^{\gamma^{-1}}z}(v)$
(2.11)

for every bounded Borel map $h\colon K_{f}\rightarrow\mathbb{R}$, every
$\gamma\in\Gamma$, and
$\nu^{\scriptscriptstyle\\#}\textit{-a.e.}\;z\in\bar{Z}_{f}$.

Define a Borel map $b\colon\bar{Z}_{f}\to K_{f}$ by setting
$b(z)=\int_{K_{f}}v\,d\tilde{\nu}^{(1)}_{z}(v)\in B_{1}(K_{f})$ for every
$z\in\bar{Z}_{f}$, where the integral is taken coordinate-wise (or,
equivalently, in the weak∗-topology) on $\ell^{\infty}(\Gamma,\mathbb{R})$.
Equation (2.11) shows that

$\displaystyle b(z)$
$\displaystyle=\int_{K_{f}}v\,d\tilde{\nu}^{(1)}_{z}(v)=\int_{K_{f}}(\bar{\lambda}^{\gamma}v-c(\gamma,\bar{\lambda}^{\gamma^{-1}}z))\,d\tilde{\nu}^{(1)}_{\bar{\lambda}^{\gamma^{-1}}z}(v)$
$\displaystyle=\int_{K_{f}}\bar{\lambda}^{\gamma}v\,d\tilde{\nu}^{(1)}_{\bar{\lambda}^{\gamma^{-1}}z}(v)-c(\gamma,\bar{\lambda}^{\gamma^{-1}}z)=\bar{\lambda}^{\gamma}b(\bar{\lambda}^{\gamma^{-1}}z)-c(\gamma,\bar{\lambda}^{\gamma^{-1}}z)$

for $\nu^{\scriptscriptstyle\\#}\textit{-a.e.}\;z\in\bar{Z}_{f}$. If we
replace $z$ by $\bar{\lambda}^{\gamma}z$ in the last equation we see that

$c(\gamma,\cdot)=\bar{\lambda}^{\gamma}\circ
b-b\circ\bar{\lambda}^{\gamma}\quad\nu^{\scriptscriptstyle\\#}\textit{-a.e.},\enspace\textup{for
every}\enspace\gamma\in\Gamma.$ (2.12)

In other words, the cocycle $c\colon\Gamma\times\bar{Z}_{f}\to K_{f}$ is a
coboundary $(\textup{mod}\;\nu^{\scriptscriptstyle\\#})$ with Borel cobounding
function $b\colon\bar{Z}_{f}\rightarrow B_{1}(K_{f})$.

Let $\theta_{2}\colon\tilde{Z}_{f}\rightarrow\tilde{Z}_{f}$ be the bijection
given by

$\theta_{2}(z,v)=(z,v-b(z))$

for every $(z,v)\in\tilde{Z}_{f}$, and put
$\tilde{\nu}=(\theta_{2})_{*}\tilde{\nu}^{(1)}=(\theta_{2}\circ\theta_{1})_{*}\nu$.
Then
$\pi^{(1)}_{*}\tilde{\nu}=\pi^{(1)}_{*}\tilde{\nu}^{(1)}=\nu^{\scriptscriptstyle\\#}$.
We set

$\theta=\theta_{2}\circ\theta_{1}\colon\bar{Y}_{f}\to\tilde{Z}_{f}$ (2.13)

and obtain that

$\theta(w)=\bigl{(}\bar{\rho}^{f}w,w-\zeta\circ\bar{\rho}^{f}(w)-b\circ\bar{\rho}^{f}(w)\bigr{)}\enspace\textup{for
every}\enspace w\in\bar{Y}_{f}.$

For $\nu\textit{-a.e.}\;w\in\bar{Y}_{f}$ we have that

$\displaystyle\tilde{\lambda}^{\gamma}\circ\theta(w)$
$\displaystyle=\bigl{(}\bar{\lambda}^{\gamma}\circ\bar{\rho}^{f}(w),\bar{\lambda}^{\gamma}w-\bar{\lambda}^{\gamma}\circ\zeta\circ\bar{\rho}^{f}(w)-\bar{\lambda}^{\gamma}\circ
b\circ\bar{\rho}^{f}(w)\bigr{)}$ (2.14)
$\displaystyle=\bigl{(}\bar{\rho}^{f}(\bar{\lambda}^{\gamma}w),\bar{\lambda}^{\gamma}w+c(\gamma,\bar{\rho}^{f}w)-\zeta\circ\bar{\rho}^{f}\circ\bar{\lambda}^{\gamma}(w)$
$\displaystyle\hskip
56.9055pt-c(\gamma,\bar{\rho}^{f}w)-b\circ\bar{\rho}^{f}\circ\bar{\lambda}^{\gamma}(w)\bigr{)}\qquad\textup{(by
\eqref{eq:cocycle} and \eqref{eq:coboundary})}$
$\displaystyle=\bigl{(}\bar{\rho}^{f}(\bar{\lambda}^{\gamma}w),\bar{\lambda}^{\gamma}w-\zeta\circ\bar{\rho}^{f}(\bar{\lambda}^{\gamma}w)-b\circ\bar{\rho}^{f}(\bar{\lambda}^{\gamma}w)\bigr{)}=\theta\circ\bar{\lambda}^{\gamma}(w),$

which proves that the $\Gamma$-actions $\bar{\lambda}$ on $(\bar{Y}_{f},\nu)$
and $\tilde{\lambda}$ on $(\tilde{Z}_{f},\tilde{\nu})$ are measurably
conjugate, as claimed in (3). ∎

In the next section we show that, if $\Gamma$ is amenable and $\lambda_{f}$
has finite and completely positive entropy, there exist unique
$\bar{\lambda}$-invariant Borel probability measures $\nu_{f}$ on
$\bar{Y}_{f}$ and $\nu_{f}^{\scriptscriptstyle\\#}$ on $\bar{Z}_{f}$ such that
the principal algebraic $\Gamma$-action $\lambda_{f}$ on $(X_{f},\mu_{f})$ is
measurably conjugate to the $\Gamma$-actions $\bar{\lambda}_{\bar{Y}_{f}}$ and
$\bar{\lambda}_{\bar{Z}_{f}}$ on $(\bar{Y}_{f},\nu_{f})$ and
$(\bar{Z}_{f},\nu_{f}^{\scriptscriptstyle\\#})$, respectively (cf. Theorem
3.18).

##  3\. Symbolic representation of intrinsically ergodic principal algebraic
actions

Throughout this section we assume that $\Gamma$ is a countably infinite
discrete amenable group and that $f\in\mathbb{Z}\Gamma$ is nonzero. We denote
by $\mu_{f}$ the normalized Haar measure on $X_{f}$ and define
$\bar{Y}_{f}\subset W_{f}\subset\ell^{\infty}(\Gamma,\mathbb{R})$ and
$\bar{Z}_{f}=\bar{\rho}^{f}(\bar{Y}_{f})\subset\ell^{\infty}(\Gamma,\mathbb{Z})$
as in (2.4) – (2.5).

######  Lemma 3.1.

The principal algebraic $\Gamma$-action $\lambda_{f}$ on $X_{f}$ has infinite
topological entropy if and only if $f$ is a left zero divisor in
$\mathbb{R}\Gamma$, i.e., if and only if there exists a nonzero
$g\in\mathbb{R}\Gamma$ with $fg=0$.

###### Proof.

This is a special case of [Chung-Li]*Theorem 4.11. For later reference we
include here an explicit proof of the fact that
$\textup{h}_{\textup{top}}(\lambda_{f})=\infty$ if $f$ is a left zero divisor
in $\mathbb{R}\Gamma$.

We embed $\mathbb{R}\Gamma$ in $\ell^{\infty}(\Gamma,\mathbb{R})$ in the
obvious manner by identifying each
$h=\sum_{\gamma\in\Gamma}h_{\gamma}\cdot\gamma\in\mathbb{R}\Gamma$ with
$(h_{\gamma})_{\gamma\in\Gamma}\in\ell^{\infty}(\Gamma,\mathbb{R})$.

If $f\in\mathbb{Z}\Gamma$ is a left zero divisor in $\mathbb{R}\Gamma$ we
choose $g=\sum_{\gamma\in\Gamma}g_{\gamma}\cdot\gamma\in\mathbb{R}\Gamma$ with
$1_{\Gamma}\in\textup{supp}(g)$ and $fg=0$. Then
$\bar{\rho}^{f}g^{*}=g^{*}f^{*}=0$ (cf. (2.3)), and hence
$\bar{\rho}^{f}(cg^{*})=cg^{*}f^{*}=0$ for every $c\in\mathbb{R}$. This shows
that $cg^{*}\in W_{f}$ and $\eta(cg^{*})\in X_{f}$ for every $c\in\mathbb{R}$
(cf. (2.4)).

If $I\subset\mathbb{R}$ is the open interval
$(-\frac{1}{2\|g\|_{\infty}},\frac{1}{2\|g\|_{\infty}})$, then the elements
$\eta(cg^{*})\in X_{f}$, $0\neq c\in I$, are all distinct with identical
supports $E=\textup{supp}(g^{*})=\textup{supp}(g)^{-1}$.

Choose a maximal set $D\subset\Gamma$ such that the translates $\\{\delta
E\mid\delta\in D\\}$ are disjoint. We claim that $DEE^{-1}=\bigcup_{\delta\in
D}\delta EE^{-1}=\Gamma$. Indeed, if $DEE^{-1}\neq\Gamma$, then there exists a
$\gamma\in\Gamma$ which is not equal to
$\delta\gamma^{\prime}\gamma^{\prime\prime-1}$ for any $\delta\in D$ and
$\gamma^{\prime},\gamma^{\prime\prime}\in E$. Then $\gamma E\cap\delta
E=\varnothing$ for every $\delta\in D$, which contradicts the maximality of
$D$. This proves the last claim.

Since the sets $\delta E,\,\delta\in D$, are disjoint, we obtain for every
$z=(z_{\delta})_{\delta\in D}\in I^{D}$, a point $\tilde{z}\in
W_{f}\cap(-\frac{1}{2},\frac{1}{2})^{\Gamma}$ which coincides on each $\delta
E$ with $z_{\delta}\bar{\lambda}^{\delta}g^{*}$. This shows that the
restriction of $X_{f}$ to its coordinates in $DE$ contains — in essence — a
Cartesian product of the form $I^{D}$. Since $(DE)E^{-1}=\Gamma$ and $E^{-1}$
is finite, $\lambda_{f}$ must have infinite topological entropy on $X_{f}$. ∎

######  Proposition 3.2.

If the principal algebraic $\Gamma$-action $\lambda_{f}$ on $X_{f}$ has finite
topological entropy, then the restriction $\bar{\lambda}_{C}$ of
$\bar{\lambda}$ to every weak∗-closed, bounded, $\bar{\lambda}$-invariant
subset $C\subset K_{f}$ has topological entropy zero (cf. (2.7)).

For the proof of Proposition 3.2 we need a lemma.

######  Lemma 3.3.

Let $\tau\colon\Gamma\to\textup{Aut}(X)$ be an action of a countably infinite
discrete amenable group $\Gamma$ by continuous automorphisms of a compact
metrizable group $X$ such that $\textup{h}_{\textup{top}}(\tau)<\infty$. Then
there exists, for every $\varepsilon>0$, a neighbourhood $U$ of $1_{X}$ in $X$
such that the topological entropy $\textup{h}_{\textup{top}}(\tau_{C})$ of the
restriction of $\tau$ to any closed $\tau$-invariant subset $C\subset U$ is
less than $\varepsilon$.

###### Proof.

Choose a compatible left translation invariant metric $\mathsf{d}$ on $X$
(i.e., $\mathsf{d}(x,y)=\mathsf{d}(zx,zy)$ for all $x,y,z\in X$). For every
nonempty finite subset $F\Subset\Gamma$, put

$\mathsf{d}_{F}(x,y)=\max\nolimits_{\gamma\in
F}\mathsf{d}(\tau^{\gamma}x,\tau^{\gamma}y),\enspace x,y\in X.$

For each $\zeta>0$ we denote by $\textup{sep}(X,\mathsf{d},\zeta)$ the maximal
cardinality of subsets $Z\subset X$ which are $(\mathsf{d},\zeta)$-separated
in the sense that $\mathsf{d}(y,z)\geq\zeta$ for all distinct $y,z\in Z$.

Take a left Følner sequence $(F_{n})_{n\geq 1}$ for $\Gamma$, i.e., a sequence
of nonempty finite sets $F_{n}\subset\Gamma$ with
$\lim_{n\to\infty}\frac{|\gamma F_{n}\cap F_{n}|}{|F_{n}|}=0$ for every
$\gamma\in\Gamma$. Then there exists, for every $\varepsilon>0$, some
$\zeta>0$ such that

$\liminf_{n\to\infty}\frac{1}{|F_{n}|}\log\textup{sep}(X,\mathsf{d}_{F_{n}},\zeta)\geq\textup{h}_{\textup{top}}(\tau)-\varepsilon/2.$

For large enough $n$, take a $(\mathsf{d}_{F_{n}},\zeta)$-separated set
$X_{n}\subset X$ such that
$\frac{1}{|F_{n}|}\log|X_{n}|\geq\textup{h}_{\textup{top}}(\tau)-\varepsilon$.

Put $U=\\{x\in X\mid\mathsf{d}(x,1_{X})<\zeta/10\\}$. Let $Y\subset U$ be
closed and $\Gamma$-invariant, and let $\tau_{Y}$ be the restriction of $\tau$
to $Y$. In order to show that
$\textup{h}_{\textup{top}}(\tau_{Y})\leq\varepsilon$ it suffices to show that

$\displaystyle\limsup_{n\to\infty}\frac{1}{|F_{n}|}\log\textup{sep}(Y,\mathsf{d}_{F_{n}},\delta)\leq\varepsilon$
(3.1)

whenever $0<\delta<\zeta/10$. In order to verify (3.1) we choose, for each
$n$, a $(\mathsf{d}_{F_{n}},\delta)$-separated set $Y_{n}\subset Y$ of
cardinality $|Y_{n}|=\textup{sep}(Y,\mathsf{d}_{F_{n}},\delta)$. When $n$ is
large enough, then $|X_{n}Y_{n}|=|X_{n}|\cdot|Y_{n}|$ and $X_{n}Y_{n}$ is
$(\mathsf{d}_{F_{n}},\delta)$-separated: indeed,

$\mathsf{d}_{F_{n}}(xy,xz)=\mathsf{d}_{F_{n}}(y,z)\geq\delta,$

for $x\in X_{n}$ and distinct $y,z\in Y_{n}$, whereas

$\displaystyle\mathsf{d}_{F_{n}}(x_{1}y,x_{2}z)$
$\displaystyle\geq\mathsf{d}_{F_{n}}(x_{1},x_{2})-\mathsf{d}_{F_{n}}(x_{1}y,x_{1})-\mathsf{d}_{F_{n}}(x_{2}z,x_{2})$
$\displaystyle=\mathsf{d}_{F_{n}}(x_{1},x_{2})-\mathsf{d}_{F_{n}}(y,1_{X})-\mathsf{d}_{F_{n}}(z,1_{X})\geq\zeta-\zeta/10-\zeta/10\geq\delta$

for $y,z\in Y_{n}$ and distinct $x_{1},x_{2}\in X_{n}$.

Then

$\displaystyle\textup{h}_{\textup{top}}(\tau)$
$\displaystyle\geq\limsup_{n\to\infty}\tfrac{1}{|F_{n}|}\log\textup{sep}(X,\mathsf{d}_{F_{n}},\delta)\geq\limsup_{n\to\infty}\tfrac{1}{|F_{n}|}\log(|X_{n}|\cdot|Y_{n}|)$
$\displaystyle\geq\textup{h}_{\textup{top}}(\tau)-\varepsilon+\limsup_{n\to\infty}\tfrac{1}{|F_{n}|}\log\textup{sep}(Y,\mathsf{d}_{F_{n}},\delta),$

which implies (3.1). ∎

###### Proof of Proposition 3.2.

Since the $\Gamma$-actions $\bar{\lambda}_{B_{r}(K_{f})},\,r>0$, are all
conjugate to each other,
$\textup{h}_{\textup{top}}(\bar{\lambda}_{B_{r}(K_{f})})$ is the same for all
$r>0$.

For $0<r<1/2$, the map
$\eta\colon\ell^{\infty}(\Gamma,\mathbb{R})\rightarrow\mathbb{T}^{\Gamma}$ in
(2.1) embeds $B_{r}(K_{f})$ injectively as a closed $\Gamma$-invariant subset
of $X_{f}$. If $U\subset X_{f}$ is any open neighbourhood of $1_{X_{f}}$, then
$\eta(B_{r}(K_{f}))\subset U$ for all sufficiently small $r>0$.

Let $C\subset K_{f}$ be a weak∗-closed, bounded, $\bar{\lambda}$-invariant
subset. Then $C\subset B_{r}(K_{f})$ for some $r>0$, and
$\textup{h}_{\textup{top}}(\bar{\lambda}_{C})\leq\textup{h}_{\textup{top}}(\bar{\lambda}_{B_{r}(K_{f})})=\textup{h}_{\textup{top}}(\bar{\lambda}_{B_{r^{\prime}}(K_{f})})$
for every $r^{\prime}>0$.

Let $\varepsilon>0$. By Lemma 3.3 there is some neighbourhood $U$ of
$1_{X_{f}}$ in $X_{f}$ such that for any closed $\Gamma$-invariant subset $Y$
of $X_{f}$ contained in $U$ the restriction $(\lambda_{f})_{Y}$ of
$\lambda_{f}$ to $Y$ has entropy $\leq\varepsilon$. If $r^{\prime}>0$ is small
enough, $\eta(B_{r^{\prime}}(K_{f}))\subset U$, so that

$\textup{h}_{\textup{top}}(\bar{\lambda}_{C})\leq\textup{h}_{\textup{top}}(\bar{\lambda}_{B_{r}(K_{f})})=\textup{h}_{\textup{top}}(\bar{\lambda}_{B_{r^{\prime}}(K_{f})})=\textup{h}_{\textup{top}}((\lambda_{f})_{\eta(B_{r^{\prime}}(K_{f}))})\leq\varepsilon.$

As $\varepsilon>0$ is arbitrary, we conclude that
$\textup{h}_{\textup{top}}(\bar{\lambda}_{C})=0$. ∎

######  Proposition 3.4 (cf. [Leiden]*Proposition 8.7).

Let $Y_{1},Y_{2}$ be compact metrizable spaces, and let $\tau_{1},\tau_{2}$ be
continuous actions of a countably infinite discrete amenable group $\Gamma$ on
$Y_{1}$ and $Y_{2}$ such that the topological entropy
$\textup{h}_{\textup{top}}(\tau_{2})$ of $\tau_{2}$ is equal to zero. We write
$\pi^{(i)}\colon Y_{1}\times Y_{2}\rightarrow Y_{i}$ for the two coordinate
projections. If $\mu$ is a
$(\tau_{1}\negthinspace\times\negthinspace\tau_{2})$-invariant Borel
probability measure on $Y_{1}\times Y_{2}$ we set $\mu_{i}=\pi^{(i)}_{*}\mu$.
Then
$h_{\mu}(\tau_{1}\negthinspace\times\negthinspace\tau_{2})=h_{\mu_{1}}(\tau_{1})$.

###### Proof.

Let $\mathcal{P}$ and $\mathcal{Q}$ be finite Borel partitions of $Y_{1}$ and
$Y_{2}$, respectively, and set $\tilde{\mathcal{P}}=\\{P\times Y_{2}\mid
P\in\mathcal{P}\\}$ and $\tilde{\mathcal{Q}}=\\{Y_{1}\times Q\mid
Q\in\mathcal{Q}\\}$. If $(F_{n})$ is a left Følner sequence in $\Gamma$, then

$\displaystyle
h_{\mu}(\tau_{1}\negthinspace\times\negthinspace\tau_{2},\tilde{\mathcal{P}}\vee\tilde{\mathcal{Q}})$
$\displaystyle=\lim_{n\to\infty}\frac{1}{|F_{n}|}H_{\mu}\bigl{(}\textstyle\bigvee_{\gamma\in
F_{n}}(\tau_{1}\negthinspace\times\negthinspace\tau_{2})^{\gamma^{-1}}(\tilde{\mathcal{P}}\vee\tilde{\mathcal{Q}})\bigr{)}$
$\displaystyle=\lim_{n\to\infty}\frac{1}{|F_{n}|}H_{\mu}\bigl{(}\textstyle\bigvee_{\gamma\in
F_{n}}\tau_{1}^{\gamma^{-1}}(\tilde{\mathcal{P}})\vee\textstyle\bigvee_{\gamma\in
F_{n}}\tau_{2}^{\gamma^{-1}}(\tilde{\mathcal{Q}})\bigr{)}$
$\displaystyle=\lim_{n\to\infty}\frac{1}{|F_{n}|}\Bigl{(}H_{\mu}\bigl{(}\textstyle\bigvee_{\gamma\in
F_{n}}\tau_{1}^{\gamma^{-1}}(\tilde{\mathcal{P}})\bigr{)}$
$\displaystyle\qquad\qquad\qquad+H_{\mu}\bigl{(}\textstyle\bigvee_{\gamma\in
F_{n}}\tau_{2}^{\gamma^{-1}}(\tilde{\mathcal{Q}})\bigm{|}\textstyle\bigvee_{\gamma\in
F_{n}}\tau_{1}^{\gamma^{-1}}(\tilde{\mathcal{P}})\bigr{)}\Bigr{)}$
$\displaystyle\leq\lim_{n\to\infty}\frac{1}{|F_{n}|}\Bigl{(}H_{\mu}\bigl{(}\textstyle\bigvee_{\gamma\in
F_{n}}\tau_{1}^{\gamma^{-1}}(\tilde{\mathcal{P}})\bigr{)}+H_{\mu}\bigl{(}\textstyle\bigvee_{\gamma\in
F_{n}}\tau_{2}^{\gamma^{-1}}(\tilde{\mathcal{Q}})\bigr{)}\Bigr{)}$
$\displaystyle=\lim_{n\to\infty}\frac{1}{|F_{n}|}\Bigl{(}H_{\mu_{1}}\bigl{(}\textstyle\bigvee_{\gamma\in
F_{n}}\tau_{1}^{\gamma^{-1}}(\mathcal{P})\bigr{)}+H_{\mu_{2}}\bigl{(}\textstyle\bigvee_{\gamma\in
F_{n}}\tau_{2}^{\gamma^{-1}}(\mathcal{Q})\bigr{)}\Bigr{)}$ $\displaystyle\leq
h_{\mu_{1}}(\tau_{1},\mathcal{P})+\textup{h}_{\textup{top}}(\tau_{2})=h_{\mu_{1}}(\tau_{1},\mathcal{P})$

by the variational principle [KL16]*Theorem 9.48. By taking the supremum over
all finite partitions $\mathcal{P}$ and $\mathcal{Q}$ we obtain that
$h_{\mu}(\tau_{1}\negthinspace\times\negthinspace\tau_{2})\linebreak[0]\leq
h_{\mu_{1}}(\tau_{1})$. The reverse inequality $h_{\mu_{1}}(\tau_{1})\leq
h_{\mu}(\tau_{1}\negthinspace\times\negthinspace\tau_{2})$ is obvious. ∎

######  Proposition 3.5 (cf. [Leiden]*Corollary 8.9).

Suppose that the principal algebraic $\Gamma$-action $\lambda_{f}$ on $X_{f}$
has finite topological entropy. Then the following is true.

  1. (1)

For every $\bar{\lambda}$-invariant Borel probability measure $\nu$ on
$\bar{Y}_{f}$, the probability measure
$\nu^{\scriptscriptstyle\\#}=\bar{\rho}^{f}_{*}\nu$ on $\bar{Z}_{f}$ is
$\bar{\lambda}$-invariant, and
$h_{\nu}(\bar{\lambda}_{\bar{Y}_{f}})=h_{\nu^{\scriptscriptstyle\\#}}(\bar{\lambda}_{\bar{Z}_{f}})$.

  2. (2)

$\textup{h}_{\textup{top}}(\bar{\lambda}_{\bar{Y}_{f}})=\textup{h}_{\textup{top}}(\bar{\lambda}_{\bar{Z}_{f}})$.

###### Proof.

Since $\bar{\rho}^{f}$ induces a continuous surjective
$\bar{\lambda}$-equivariant map from $\bar{Y}_{f}$ to $\bar{Z}_{f}$,
$\textup{h}_{\textup{top}}(\bar{\lambda}_{\bar{Z}_{f}})\leq\textup{h}_{\textup{top}}(\bar{\lambda}_{\bar{Y}_{f}})$
and $h_{\nu^{\scriptscriptstyle\\#}}(\bar{\lambda}_{\bar{Z}_{f}})\leq
h_{\nu}(\bar{\lambda}_{\bar{Y}_{f}})$ for every $\bar{\lambda}$-invariant
Borel probability measure $\nu$ on $\bar{Y}_{f}$.

By applying the Propositions 3.2 and 3.4 with $Y_{1}=\bar{Z}_{f}$,
$Y_{2}=B_{2}(K_{f})$, $\tau_{1}=\bar{\lambda}_{\bar{Z}_{f}}$,
$\tau_{2}=\bar{\lambda}_{B_{2}(K_{f})}$, $\mu=\tilde{\nu}$, and
$\mu_{1}=\nu^{\scriptscriptstyle\\#}=\bar{\rho}^{f}_{*}\nu$, we obtain that
$h_{\nu}(\bar{\lambda}_{\bar{Y}_{f}})=h_{\tilde{\nu}}(\tilde{\lambda})=h_{\nu^{\scriptscriptstyle\\#}}(\bar{\lambda}_{\bar{Z}_{f}})$.
This proves (1).

(2): The variational principle [KL16]*Theorem 9.48 implies that

$\textup{h}_{\textup{top}}(\bar{\lambda}_{\bar{Y}_{f}})=\sup_{\mu}h_{\mu}(\bar{\lambda}_{\bar{Y}_{f}})=\sup_{\mu}h_{\bar{\rho}^{f}_{*}\mu}(\bar{\lambda}_{\bar{Z}_{f}})\leq\textup{h}_{\textup{top}}(\bar{\lambda}_{\bar{Z}_{f}}),$

where the supremum is taken over the set of $\bar{\lambda}$-invariant Borel
probability measures $\mu$ on $\bar{Y}_{f}$. Since the opposite inequality
$\textup{h}_{\textup{top}}(\bar{\lambda}_{\bar{Z}_{f}})\leq\textup{h}_{\textup{top}}(\bar{\lambda}_{\bar{Y}_{f}})$
is trivially satisfied, this completes the proof of the proposition. ∎

Proposition 3.2 yields a strengthening of Proposition 3.5 for
$\bar{\lambda}$-invariant probability measures $\nu$ on $\bar{Y}_{f}$ with
completely positive entropy. We use the same notation as in the Propositions
2.1 and 3.5.

######  Corollary 3.6.

Suppose that the principal algebraic $\Gamma$-action $\lambda_{f}$ on $X_{f}$
has finite topological entropy. Then the $\Gamma$-actions
$\bar{\lambda}_{\bar{Y}_{f}}$ on $(\bar{Y}_{f},\nu)$ and
$\bar{\lambda}_{\bar{Z}_{f}}$ on $(\bar{Z}_{f},\nu^{\scriptscriptstyle\\#})$
are measurably conjugate for every $\bar{\lambda}$-invariant Borel probability
measure $\nu$ on $\bar{Y}_{f}$ with completely positive entropy.

###### Proof.

As in the proof of Proposition 2.1 we define
$\theta_{1}\colon\bar{Y}_{f}\to\tilde{Z}_{f}$ by (2.9) and set
$\tilde{\nu}^{(1)}=(\theta_{1})_{*}\nu$ and
$\tilde{\nu}=(\theta_{2})_{*}\tilde{\nu}^{(1)}=\theta_{*}\nu$. Then
$\tilde{\nu}$ is $\tilde{\lambda}$-invariant by (2.14), and
$\pi^{(1)}_{*}\tilde{\nu}=\nu^{\scriptscriptstyle\\#}$. We write
$\pi^{(2)}\colon\tilde{Z}_{f}\to K_{f}$ for the second coordinate projection,
denote by $\xi_{f}=\pi^{(2)}_{*}\tilde{\nu}$ the projection of $\tilde{\nu}$
onto $K_{f}$, and note that the $\Gamma$-action $\tilde{\lambda}$ on
$(\tilde{Z}_{f},\tilde{\nu})$ has the zero-entropy $\Gamma$-action
$\bar{\lambda}$ on $(K_{f},\xi_{f})$ as a factor (cf. Proposition 3.2). Since
$\tilde{\lambda}$ on $(\tilde{Z}_{f},\tilde{\nu})$ is measurably conjugate to
$\bar{\lambda}$ on $(\bar{Y}_{f},\nu)$ and thus has completely positive
entropy, we obtain a contradiction unless $\xi_{f}$ is concentrated in a
single point.

Since $\xi_{f}$ is a point mass, the first coordinate projection
$\pi^{(1)}\colon(\tilde{Z}_{f},\tilde{\nu})\to(\bar{Z}_{f},\nu^{\scriptscriptstyle\\#})$
is injective $(\textup{mod}\;\tilde{\nu})$, and the $\Gamma$-actions
$\tilde{\lambda}$ on $(\tilde{Z}_{f},\tilde{\nu})$ and $\bar{\lambda}$ on
$(\bar{Z}_{f},\nu^{\scriptscriptstyle\\#})$ are conjugate. This proves that
the $\Gamma$-actions $\bar{\lambda}$ on $(\bar{Y}_{f},\nu)$ and on
$(\bar{Z}_{f},\nu^{\scriptscriptstyle\\#})$ are measurably conjugate. ∎

Having discussed the relation between $\bar{\lambda}$-invariant probability
measures on $\bar{Y}_{f}$ and $\bar{Z}_{f}$ we turn to the corresponding
question for measures on $\bar{Y}_{f}$ and their images under $\eta$.

######  Lemma 3.7.

There exists a unique $\bar{\lambda}$-invariant Borel probability measure
$\nu_{f}$ on $\bar{Y}_{f}$ with $\eta_{*}\nu_{f}=\mu_{f}$, and the map
$\eta\colon\bar{Y}_{f}\to X_{f}$ induces a conjugacy of the $\Gamma$-actions
$\bar{\lambda}$ on $(\bar{Y}_{f},\nu_{f})$ and $\lambda_{f}$ on
$(X_{f},\mu_{f})$.

###### Proof.

Let $\nu$ be a $\bar{\lambda}$-invariant Borel probability measure on
$\bar{Y}_{f}$ such that $\eta_{*}\nu=\mu_{f}$. If
$\nu(\bar{Y}_{f}\smallsetminus Y_{f})>0$, the set

$V=\\{y\in\bar{Y}_{f}\mid y_{1_{\Gamma}}=1\\}$

must have positive $\nu$-measure, which implies that the closed subgroup

$H=\\{x\in X_{f}\mid x_{1_{\Gamma}}=0\\}\supset\eta(V)$

has positive $\mu_{f}$-measure. We set

$K=\pi_{1_{\Gamma}}(X_{f})\subset\mathbb{T},$

observe that $K$ is a closed subgroup of $\mathbb{T}$, and denote by $\mu_{K}$
the normalized Haar measure of $K$. Since
$\mu_{K}(\\{t\\})=\mu_{K}(\\{0\\})=\mu_{f}(H)>0$ for every $t\in K$, the group
$K$ must be finite, which implies that $X_{f}\subset K^{\Gamma}$ and hence
$Y_{f}=\bar{Y}_{f}$, contrary to our assumption that
$\nu(\bar{Y}_{f}\smallsetminus Y_{f})>0$. It follows that $\nu(B)=\nu(B\cap
Y_{f})=\mu_{f}(\eta(B))$ for every Borel set $B\subset\bar{Y}_{f}$, as
claimed. Hence the map $\eta\colon\bar{Y}_{f}\to X_{f}$ induces a measure
space isomorphism from $(\bar{Y}_{f},\nu_{f})$ to $(X_{f},\mu_{f})$ which
carries the $\Gamma$-action $\bar{\lambda}_{f}$ on $(\bar{Y}_{f},\nu_{f})$ to
$\lambda_{f}$ on $(X_{f},\mu_{f})$. ∎

######  Theorem 3.8.

Let $\Gamma$ be a countably infinite discrete amenable group,
$f\in\mathbb{Z}\Gamma$, and assume that the principal algebraic
$\Gamma$-action $\lambda_{f}$ on $X_{f}$ has finite topological entropy. Then
$\textup{h}_{\textup{top}}(\bar{\lambda}_{\bar{Z}_{f}})=\textup{h}_{\textup{top}}(\bar{\lambda}_{\bar{Y}_{f}})=\textup{h}_{\textup{top}}(\lambda_{f})$.

We start the proof of Theorem 3.8 with four lemmas. For any finite subset
$F\subset\Gamma$ containing $1_{\Gamma}$ and any $Q\subset\Gamma$ we put

$\textup{Int}_{F}Q=\\{\gamma\in\Gamma\mid\gamma F\subset Q\\}.$ (3.2)

######  Lemma 3.9 ([Leiden]*Lemma 6.4).

Let $V$ be a finite dimensional vector space over $\mathbb{R}$, and let
$k>\dim V$. Let $\phi_{1},\dots,\phi_{k}$ be affine functions on $V$ and
$b_{1},\dots,b_{k}\in\mathbb{R}$. Then there exist
$a_{1},\dots,a_{k}\in\\{0,1\\}$ such that
$\bigcap_{j=1}^{k}W_{j}(a_{j})=\varnothing$, where

$W_{j}(a_{j})=\begin{cases}\\{x\in
V\mid\phi_{j}(x)<b_{j}\\}&\textup{if}\enspace a_{j}=0,\\\ \\{x\in
V\mid\phi_{j}(x)\geq b_{j}\\}&\textup{if}\enspace a_{j}=1.\end{cases}$

######  Lemma 3.10.

Suppose that $f\in\mathbb{R}\Gamma$ satisfies that
$\textup{h}_{\textup{top}}(\lambda_{f})<\infty$, and that $1_{\Gamma}\in
E=\textup{supp}(f)$. Let $Q\Subset\Gamma$. For every nonzero
$v\in\mathbb{R}\Gamma$, the product $v\cdot f^{*}$ is nonzero (since $f$ is
not a left zero divisor), and the restriction of $v\cdot f^{*}$ to
$\textup{Int}_{E}Q$ depends only on the restriction $\pi_{Q}(v)$ of $v$ to
$Q$: for every $v,v^{\prime}\in\mathbb{R}\Gamma$ with
$\pi_{Q}(v)=\pi_{Q}(v^{\prime})$, $\pi_{\textup{Int}_{E}Q}(v\cdot
f^{*})=\pi_{\textup{Int}_{E}Q}(v^{\prime}\cdot f^{*})$. Since the map
$v\mapsto\bar{\rho}^{f}v=v\cdot f^{*}$ from $\mathbb{R}\Gamma$ to
$\mathbb{R}\Gamma$ in (2.3) induces an injective map from $\mathbb{R}^{Q}$ to
$\mathbb{R}^{QE^{-1}}$, the linear space

$V_{Q}=\bigl{\\{}v=(v_{\gamma})_{\gamma\in
Q}\in\mathbb{R}^{Q}\mid\pi_{\textup{Int}_{E}Q}(v\cdot f^{*})=0\bigr{\\}},$
(3.3)

has dimension $\dim V_{Q}\leq|QE^{-1}\smallsetminus\textup{Int}_{E}Q|$ (cf.
(3.2)).

###### Proof.

Since
$\dim\,(\\{w\in\mathbb{R}^{QE^{-1}}\mid\pi_{\textup{Int}_{E}Q}(w)=0\\})=|QE^{-1}\smallsetminus\textup{Int}_{E}Q|$
and the map $\mathbb{R}^{Q}\to\mathbb{R}^{QE^{-1}}$ induced by
$\bar{\rho}^{f}$ is injective, $\dim
V_{Q}\leq|QE^{-1}\smallsetminus\textup{Int}_{E}Q|$, as claimed. ∎

For the next lemma we recall that a family of subsets $\mathcal{Z}$ of a
finite set $Z$ is said to _scatter_ a set $J\subset Z$ if $\mathcal{Z}\cap
J=\\{C\cap J\mid C\in\mathcal{Z}\\}=\mathcal{P}(J)$, the set of all subsets of
$J$.

######  Lemma 3.11 (Sauer-Perles-Shelah [Pajor], [Sauer]*Theorem 1, [Shelah]).

Let $Z$ be a finite set with cardinality $n\geq 1$ and let $\mathcal{Z}$ be a
collection of subsets of $Z$. If
$|\mathcal{Z}|>\sum_{i=0}^{k-1}\binom{|Z|}{i}$ for some
$k\in\\{1,\dots,|Z|\\}$, then $\mathcal{Z}$ scatters a subset $J\subset Z$ of
size $k$.

###### Proof.

For a proof see, e.g., [Wiki]. ∎

######  Lemma 3.12 ([3]*Lemma A.1).

Let $0<\beta<1/2$. Then there exist $\kappa=\kappa(\beta)>0$ and
$m_{0}=m_{0}(\beta)\in\mathbb{N}=\\{1,2,\dots\\}$ with

$\lim\nolimits_{\beta\to 0}\kappa(\beta)=0,$

such that

$\sum\nolimits_{i=0}^{\lfloor\beta m\rfloor}\binom{m}{i}\leq e^{\kappa m}$

for all $m\in\mathbb{N}$ with $m\geq m_{0}$.

###### Proof of Theorem 3.8.

For $\delta\in\Gamma$, the spaces $X_{f},Y_{f}$ do not change if we replace
$f=\sum_{\gamma\in\Gamma}f_{\gamma}\cdot\gamma$ by $\delta
f=\sum_{\gamma\in\Gamma}f_{\gamma}\cdot\delta\gamma$, and $Z_{\delta
f}=\bar{\rho}^{\delta}(Z_{f})$. For the proof of this theorem we may therefore
assume without loss of generality that $1_{\Gamma}\in\textup{supp}(f)$.

Since the continuous shift-equivariant map
$\eta\colon\ell^{\infty}(\Gamma,\mathbb{R})\to\mathbb{T}^{\Gamma}$ in (2.1)
sends $\bar{Y}_{f}$ onto $X_{f}$, we know that
$\textup{h}_{\textup{top}}(\bar{\lambda}_{\bar{Y}_{f}})\geq\textup{h}_{\textup{top}}(\lambda_{f})$.
It will suffice to show that
$\textup{h}_{\textup{top}}(\bar{\lambda}_{\bar{Y}_{f}})\leq\textup{h}_{\textup{top}}(\lambda_{f})$.

Denote by $\mathsf{d}_{\mathbb{I}}(s,t)=|s-t|$ the Euclidean metric on the
closed unit interval $\mathbb{I}=[0,1]$ and by $\mathsf{d}_{\mathbb{T}}$ the
metric on $\mathbb{T}$ given by

$\mathsf{d}_{\mathbb{T}}(s+\mathbb{Z},t+\mathbb{Z})=\min_{k\in\mathbb{Z}}|s-t-k|.$

For any $F\Subset\Gamma$ we define continuous pseudometrics
$\mathsf{d}_{\mathbb{I}}^{(F)}$ and $\mathsf{d}_{\mathbb{T}}^{(F)}$ on
$\bar{Y}_{f}$ and $X_{f}$, respectively, by

$\displaystyle\mathsf{d}_{\mathbb{I}}^{(F)}(y,y^{\prime})\coloneqq\max_{\gamma\in
F^{-1}}\mathsf{d}_{\mathbb{I}}(y_{\gamma},y_{\gamma}^{\prime}),\enspace
y,y^{\prime}\in\bar{Y}_{f},$
$\displaystyle\mathsf{d}_{\mathbb{T}}^{(F)}(x,x^{\prime})\coloneqq\max_{\gamma\in
F^{-1}}\mathsf{d}_{\mathbb{T}}(x_{\gamma},x_{\gamma}^{\prime}),\enspace
x,x^{\prime}\in X_{f}.$

For every $\varepsilon>0$ we denote by
$\textup{sep}(\bar{Y}_{f},\mathsf{d}_{\mathbb{I}}^{(F)},\varepsilon)$ and
$\textup{sep}(X_{f},\mathsf{d}_{\mathbb{T}}^{(F)},\varepsilon)$ the maximal
cardinalities of $(\mathsf{d}_{\mathbb{I}}^{(F)},\varepsilon)$-separated
subsets of $\bar{Y}_{f}$ and
$(\mathsf{d}_{\mathbb{T}}^{(F)},\varepsilon)$-separated subsets of $X_{f}$,
respectively.

Let $(F_{n})$ be a left Følner sequence of $\Gamma$. By [D06]*Proposition 2.3
we have

$\displaystyle\textup{h}_{\textup{top}}(\bar{\lambda}_{\bar{Y}_{f}})=\sup_{\varepsilon>0}\limsup_{n\to\infty}\frac{\log\textup{sep}(\bar{Y}_{f},\mathsf{d}_{\mathbb{I}}^{(F_{n})},\varepsilon)}{|F_{n}|},$
$\displaystyle\textup{h}_{\textup{top}}(\lambda_{f})=\sup_{\varepsilon>0}\limsup_{n\to\infty}\frac{\log\textup{sep}(X_{f},\mathsf{d}_{\mathbb{T}}^{(F_{n})},\varepsilon)}{|F_{n}|}.$

Assume that
$\textup{h}_{\textup{top}}(\bar{\lambda}_{\bar{Y}_{f}})>\textup{h}_{\textup{top}}(\lambda_{f})$.
Then we can find some $0<\varepsilon<\frac{1}{\max(10,2\|f\|_{1})}$ and $c>0$
such that, passing to a subsequence of $(F_{n})$ if necessary, one has

$\textup{sep}(\bar{Y}_{f},\mathsf{d}_{\mathbb{I}}^{(F_{n})},\varepsilon)\geq\textup{sep}(X_{f},\mathsf{d}_{\mathbb{T}}^{(F_{n})},\varepsilon/3)\exp(c|F_{n}|)$
(3.4)

for all $n\geq 1$.

We fix $n\geq 1$ for the moment and choose a
$(\mathsf{d}_{\mathbb{I}}^{(F_{n})},\varepsilon)$-separated subset
$\mathcal{W}_{n}\subset\bar{Y}_{f}$ with
$|\mathcal{W}_{n}|=\textup{sep}(\bar{Y}_{f},\mathsf{d}_{\mathbb{I}}^{(F_{n})},\varepsilon)$.
Then $\mathcal{W}_{n}$ is
$(\mathsf{d}_{\mathbb{I}}^{(F_{n})},\varepsilon)$-spanning in $\bar{Y}_{f}$.
Since $\eta^{-1}(X_{f})\cap[0,1)^{\Gamma}$ is dense in $\bar{Y}_{f}$, we may
move some of the points in $\mathcal{W}_{n}$ by less than $\varepsilon/10$ in
the pseudometric $\mathsf{d}_{\mathbb{I}}^{(F_{n})}$, if necessary, and
without changing notation, so that
$\mathcal{W}_{n}\subset\bar{Y}_{f}\cap[0,1)^{\Gamma}$, while remaining
$(\mathsf{d}_{\mathbb{I}}^{(F_{n})},\varepsilon)$-spanning and
$(\mathsf{d}_{\mathbb{I}}^{(F_{n})},4\varepsilon/5)$-separated in
$\bar{Y}_{f}$. Similarly, if $\mathcal{V}_{n}\subset X_{f}$ is a maximal
$(\mathsf{d}_{\mathbb{T}}^{(F_{n})},\varepsilon/3)$-separated subset in
$X_{f}$, then

$X_{f}\subset\bigcup\nolimits_{x\in\mathcal{V}_{n}}B_{\mathbb{T}}^{(F_{n})}(x,\varepsilon/3),$

where $B_{\mathbb{T}}^{(F_{n})}(x,\varepsilon/3)$ is the open
$\mathsf{d}_{\mathbb{T}}^{(F_{n})}$-ball in $X_{f}$ with centre $x$ and radius
$\varepsilon/3$, and we can find, for every $n\geq 1$, a point
$z^{(n)}\in\mathcal{V}_{n}$ such that $|\eta(\mathcal{W}_{n})\cap
B_{\mathbb{T}}^{(F_{n})}(z^{(n)},\varepsilon/3)|\geq\exp(c|F_{n}|)$ (cf.
(3.4)). For every $n\geq 1$ we set
$\mathcal{W}_{n}^{\prime}=\\{y\in\mathcal{W}_{n}\mid\eta(y)\in
B_{\mathbb{T}}^{(F_{n})}(z^{(n)},\varepsilon/3)\\}$ and denote by
$\tilde{z}^{(n)}\in[0,1)^{F_{n}^{-1}}$ the unique point with
$z_{\gamma}^{(n)}=\tilde{z}_{\gamma}^{(n)}\;(\textup{mod\,1})$ for every
$\gamma\in F_{n}^{-1}$.

For every $y\in\mathcal{W}_{n}^{\prime}$ there is a unique
$\tilde{y}\in\\{-1,0,1\\}^{F_{n}^{-1}}$ such that
$|y_{\gamma}-\tilde{y}_{\gamma}-\tilde{z}_{\gamma}^{(n)}|<\varepsilon/3$ for
every $\gamma\in F_{n}^{-1}$. We set

$G_{y}^{+}=\\{\gamma\in F_{n}^{-1}\mid\tilde{y}_{\gamma}=1\\},\qquad
G_{y}^{\circ}=\\{\gamma\in F_{n}^{-1}\mid\tilde{y}_{\gamma}=0\\},\qquad
G_{y}^{-}=\\{\gamma\in F_{n}^{-1}\mid\tilde{y}_{\gamma}=-1\\}.$

Since $\mathcal{W}_{n}^{\prime}$ is
$(\mathsf{d}_{\mathbb{I}}^{(F_{n})},4\varepsilon/5)$-separated and
$G_{y}^{+}\cup G_{y}^{\circ}\cup G_{y}^{-}=F_{n}^{-1}$, it is clear that
$\tilde{y}\neq\tilde{y}^{\prime}$ and hence
$(G_{y}^{+},G_{y}^{-})\neq(G_{y^{\prime}}^{+},G_{y^{\prime}}^{-})$ for any
$y\neq y^{\prime}$ in $\mathcal{W}_{n}^{\prime}$.

We recall that $1_{\Gamma}\in E=\textup{supp}(f)$ and define
$\textup{Int}_{E}F_{n}^{-1}$ as in (3.2). For any $y\in\mathcal{W}_{n}$, the
restrictions of $y\cdot f^{*}$ and $y|_{F_{n}^{-1}}\cdot f^{*}$ to
$\textup{Int}_{E}F_{n}^{-1}$ coincide; since $y\cdot
f^{*}\in\ell^{\infty}(\Gamma,\mathbb{Z})$, this implies that
$y|_{F_{n}^{-1}}\cdot f^{*}$ and $(y|_{F_{n}^{-1}}-\tilde{z}^{(n)})\cdot
f^{*}$ have integral coordinates on $\textup{Int}_{E}F_{n}^{-1}$. Furthermore,
since $|y_{\gamma}-\tilde{y}_{\gamma}-\tilde{z}^{(n)}_{\gamma}|<\varepsilon/3$
for every $\gamma\in F_{n}^{-1}$, we obtain that

$\bigl{\|}\bigl{(}(y|_{F_{n}^{-1}}-\tilde{y}-\tilde{z}^{(n)})\cdot
f^{*}\bigr{)}\big{|}_{\textup{Int}_{E}F_{n}^{-1}}\bigr{\|}_{\infty}<\frac{\varepsilon}{3}\cdot\|f\|_{1}<1,$

so that

$v(y)\coloneqq y|_{F_{n}^{-1}}-\tilde{y}-\tilde{z}^{(n)}\in V_{F_{n}^{-1}}$

for every $y\in\mathcal{W}_{n}^{\prime}$ (cf. (3.3)).

Put $\mathscr{W}_{n}^{\prime\prime}=\\{(G_{y}^{+},G_{y}^{-})\mid
y\in\mathscr{W}_{n}^{\prime}\\}$, $\mathscr{W}_{n}^{+}=\\{G_{y}^{+}\mid
y\in\mathscr{W}_{n}^{\prime}\\}$ and $\mathscr{W}_{n}^{-}=\\{G_{y}^{-}\mid
y\in\mathscr{W}_{n}^{\prime}\\}$. Since
$|\mathcal{W}_{n}^{\prime\prime}|=|\mathcal{W}_{n}^{\prime}|\geq\exp(c|F_{n}|)$
it is clear that
$\max(|\mathcal{W}_{n}^{+}|,|\mathcal{W}_{n}^{-}|)\geq\exp(c|F_{n}|/2)$.

Suppose that $|\mathcal{W}_{n}^{+}|\geq\exp(c|F_{n}|/2)$ for infinitely many
$n\geq 1$ (if $|\mathcal{W}_{n}^{-}|\geq\exp(c|F_{n}|/2)$ for infinitely many
$n$ the proof is completely analogous). By passing to a subsequence we may
assume that $|\mathcal{W}_{n}^{+}|\geq\exp(c|F_{n}|/2)$ for every $n\geq 1$.
By Lemma 3.12 there exists $\beta>0$ such that $\kappa=\kappa(\beta)<c/2$ and
$\sum\nolimits_{i=0}^{\lfloor\beta|F_{n}|\rfloor}{\tbinom{|F_{n}|}{i}}<\exp(\kappa|F_{n}|)<\exp(c|F_{n}|/2)\leq|\mathcal{W}_{n}^{+}|$
for every sufficiently large $n\geq 1$. Lemma 3.11 implies that
$\mathcal{W}_{n}^{+}$ scatters a subset $J_{n}^{+}\subset F_{n}^{-1}$ of size
$\geq\beta|F_{n}^{-1}|$.

We are going to show that $\dim V_{F_{n}^{-1}}\geq|J_{n}^{+}|$ for infinitely
many $n\geq 1$, thereby contradicting Lemma 3.10. For this we define, for
every $\gamma\in F_{n}^{-1}$, a linear functional $\phi_{\gamma}\colon
V_{F_{n}^{-1}}\to\mathbb{R}$ by setting $\phi_{\gamma}(v)=v_{\gamma}$ for
every $v\in V_{F_{n}^{-1}}$. For every $y\in\mathcal{W}_{n}^{\prime}$ and
$\gamma\in F_{n}^{-1}$, we have the following possibilities:

$\displaystyle\gamma\in
G_{y}^{+}\enspace\textup{and}\enspace\phi_{\gamma}(v(y))+\tilde{z}_{\gamma}^{(n)}=y_{\gamma}-1<0,$
$\displaystyle\gamma\in G_{y}^{\circ}\enspace\textup{and}\enspace
1>\phi_{\gamma}(v(y))+\tilde{z}_{\gamma}^{(n)}=y_{\gamma}\geq 0,$
$\displaystyle\gamma\in
G_{y}^{-}\enspace\textup{and}\enspace\phi_{\gamma}(v(y))+\tilde{z}_{\gamma}^{(n)}=y_{\gamma}+1\geq
1.$

In particular, $\phi_{\gamma}(v(y))<-\tilde{z}_{\gamma}^{(n)}$ if $\gamma\in
G_{y}^{+}$, and $\phi_{\gamma}(v(y))\geq-\tilde{z}_{\gamma}^{(n)}$ if
$\gamma\in F_{n}^{-1}\smallsetminus G_{y}^{+}$.

We can thus find, for every subset $H\subset J_{n}^{+}$, a
$y\in\mathcal{W}_{n}^{\prime}$ for which

$\phi_{\gamma}(v(y))<-\tilde{z}_{\gamma}^{(n)}\quad\textup{if}\quad\gamma\in
H,$

and

$\phi_{\gamma}(v(y))\geq-\tilde{z}_{\gamma}^{(n)}\quad\textup{if}\quad\gamma\in
J_{n}^{+}\smallsetminus H.$

According to Lemma 3.9 this means that $\dim
V_{F_{n}^{-1}}\geq|J_{n}^{+}|\geq\beta|F_{n}|$. If we set $Q=F_{n}^{-1}$,
where $n$ is sufficiently large, we obtain a contradiction to Lemma 3.10. This
contradiction shows that
$\textup{h}_{\textup{top}}(\bar{\lambda}_{\bar{Y}_{f}})\leq\textup{h}_{\textup{top}}(\lambda_{f})$
and completes the proof of Theorem 3.8. ∎

######  Lemma 3.13.

Any $\bar{\lambda}$-invariant Borel probability measure $\nu$ on $\bar{Y}_{f}$
satisfies that
$h_{\nu}(\bar{\lambda}_{\bar{Y}_{f}})=\linebreak[4]h_{\eta_{*}\nu}(\lambda_{f})$.

###### Proof.

For every $x\in X_{f}$ we denote by
$\textup{h}_{\textup{top}}\bigl{(}\bar{\lambda}_{\bar{Y}_{f}}|\eta^{-1}(x)\bigr{)}$
the fibre entropy of $\bar{\lambda}_{\bar{Y}_{f}}$, given $x$, defined in
[Li]*Definition 6.7. The proof of Theorem 3.8 shows that
$\textup{h}_{\textup{top}}\bigl{(}\bar{\lambda}_{\bar{Y}_{f}}|\eta^{-1}(x)\bigr{)}=0$
for every $x\in X_{f}$. By [Li]*Lemmas 6.8 and 6.9,
$h_{\nu}(\bar{\lambda}_{\bar{Y}_{f}}|\eta^{-1}(\mathcal{B}_{X_{f}}))=0$ for
every $\bar{\lambda}$-invariant Borel probability measure $\nu$ on
$\bar{Y}_{f}$, where $\mathcal{B}_{X_{f}}$ is the Borel $\sigma$-algebra of
$X_{f}$. By [Danilenko]*Theorem 0.2 or [KL16]*Theorem 9.16,
$h_{\nu}(\bar{\lambda}_{\bar{Y}_{f}})=h_{\nu}(\bar{\lambda}_{\bar{Y}_{f}}|\mathcal{B}_{X_{f}})+h_{\eta_{*}\nu}(\lambda_{f})=h_{\eta_{*}\nu}(\lambda_{f})$.
∎

The coincidence of topological entropies of the $\Gamma$-actions $\lambda_{f}$
and $\bar{\lambda}_{\bar{Y}_{f}}$ in Theorem 3.8 is not quite as obvious as
one might think. As noted in the proof of Lemma 3.13, the conditional fibre
entropy
$\textup{h}_{\textup{top}}(\bar{\lambda}_{\bar{Y}_{f}}|\linebreak[0]\eta^{-1}(x))$
is equal to zero for every $x\in X_{f}$ whenever $\Gamma$ is amenable and
$f\in\mathbb{Z}\Gamma$ is not a left zero divisor. This is no longer true if
$f$ is a left zero divisor (in which case the topological entropies
$\textup{h}_{\textup{top}}(\lambda_{f})$ and
$\textup{h}_{\textup{top}}(\bar{\lambda}_{\bar{Y}_{f}})$ are infinite by Lemma
3.1). A slight modification of the proof of Lemma 3.1 yields the following
result.

######  Proposition 3.14.

Let $\Gamma$ be a countably infinite amenable group, and let
$f\in\mathbb{Z}\Gamma$ be a left zero divisor in $\mathbb{R}\Gamma$. Then the
fibre entropy
$\textup{h}_{\textup{top}}(\bar{\lambda}_{\bar{Y}_{f}}|\eta^{-1}(0_{X_{f}}))$
is positive.

###### Proof.

Take a compatible metric $\mathsf{d}$ on $\bar{Y}_{f}$ such that
$\mathsf{d}(y,z)\geq|y_{1_{\Gamma}}-z_{1_{\Gamma}}|$ for all
$y,z\in\bar{Y}_{f}$.

If $f\in\mathbb{Z}\Gamma$ is a left zero divisor we choose
$g=\sum_{\gamma\in\Gamma}g_{\gamma}\cdot\gamma\in\mathbb{R}\Gamma$ with
$g_{1_{\Gamma}}=\|g\|_{\infty}>0$ and $fg=0$. Following the proof of Lemma 3.1
we note that $cg^{*}\in W_{f}$ and $\eta(cg^{*})\in X_{f}$ for every
$c\in\mathbb{R}$. Put $E=\textup{supp}(g^{*})$ and choose a maximal set
$D\subset\Gamma$ such that the translates $\\{\delta E\mid\delta\in D\\}$ are
disjoint. Then $DEE^{-1}=\Gamma$ (cf. the proof of Lemma 3.1). Since the sets
$\delta E,\delta\in D$, are disjoint, we obtain, for every
$z=(z_{\delta})_{\delta\in D}\in\\{-1,1\\}^{D}$ and every $c\in\mathbb{R}$
with $0<c<\frac{1}{2\|g\|_{\infty}}$, a point $w^{(c,z)}=c\cdot\sum_{\delta\in
D}z_{\delta}\bar{\lambda}^{\delta}g^{*}\in W_{f}$ with
$\|w^{(c,z)}\|_{\infty}=c\|g\|_{\infty}$ and
$w^{(c,z)}_{\delta}=cz_{\delta}\|g\|_{\infty}$ for every $\delta\in D$.

We set $x^{(c,z)}=\eta(w^{(c,z)})\in X_{f}$ and denote by $y^{(c,z)}$ the
unique point in $Y_{f}$ satisfying
$\eta(y^{(c,z)})\linebreak[0]=x^{(c,z)}=\eta(w^{(c,z)})$. For every $\delta\in
D$,

$y^{(c,z)}_{\delta}=\begin{cases}c\|g\|_{\infty}&\textup{if}\enspace
z_{\delta}=1,\\\ 1-c\|g\|_{\infty}&\textup{if}\enspace
z_{\delta}=-1.\end{cases}$

As $c\searrow 0$, $y^{(c,z)}$ converges coordinate-wise to a point
$y^{(z)}\in\bar{Y}_{f}$ with

$y^{(z)}_{\delta}=\begin{cases}0&\textup{if}\enspace z_{\delta}=1,\\\
1&\textup{if}\enspace z_{\delta}=-1,\end{cases}$

for $\delta\in D$. With the exception of the single point
$z^{\prime}=(z_{\delta}^{\prime})_{\delta\in D}$ with $z_{\delta}^{\prime}=1$
for every $\delta\in D$, all the points $y^{(z)},z\in\\{-1,1\\}^{D}$ lie in
$\bar{Y}_{f}\smallsetminus Y_{f}$ and satisfy that $\eta(y^{(z)})=0_{X_{f}}$.
As in the proof of Lemma 3.1 we conclude that the fibre entropy
$\textup{h}_{\textup{top}}(\bar{\lambda}_{\bar{Y}_{f}}|\eta^{-1}(0_{X_{f}}))$
is positive. ∎

######  Definition 3.15 ([Weiss]).

A continuous action $\tau$ of a countably infinite amenable group $\Gamma$ on
a compact metrizable space $X$ is _intrinsically ergodic_ if it has finite
topological entropy and there exists a unique $\tau$-invariant Borel
probability measure $\mu$ on $X$ with
$h_{\mu}(\tau)=\textup{h}_{\textup{top}}(\tau)$.

If $\Gamma$ is a countably infinite amenable group, and if
$f\in\mathbb{Z}\Gamma$ satisfies that
$\textup{h}_{\textup{top}}(\lambda_{f})<\infty$, then the principal algebraic
action $\lambda_{f}$ on $X_{f}$ is intrinsically ergodic (with unique maximal
measure $\mu_{f}$) if and only if $\lambda_{f}$ has completely positive
entropy w.r.t. $\mu_{f}$ ([Chung-Li]*Theorem 8.6). If $\lambda_{f}$ is
intrinsically ergodic on $X_{f}$, the next result extends this property to the
$\Gamma$-actions $\bar{\lambda}_{\bar{Y}_{f}}$ and
$\bar{\lambda}_{\bar{Z}_{f}}$.

######  Proposition 3.16.

Suppose that $\Gamma$ is a countably infinite discrete amenable group,
$f\in\mathbb{Z}\Gamma$, and $\lambda_{f}$ is intrinsically ergodic on $X_{f}$.
Then the following are true.

  1. (1)

The $\Gamma$-actions $\bar{\lambda}_{\bar{Y}_{f}}$ and
$\bar{\lambda}_{\bar{Z}_{f}}$ are intrinsically ergodic;

  2. (2)

The maximal entropy measures of the $\Gamma$-actions
$\bar{\lambda}_{\bar{Y}_{f}}$ and $\bar{\lambda}_{\bar{Z}_{f}}$ have
completely positive entropy.

The proof of Proposition 3.16 consists of three lemmas.

######  Lemma 3.17.

If $\lambda_{f}$ is intrinsically ergodic on $X_{f}$, then the
$\Gamma$-actions $\bar{\lambda}_{\bar{Y}_{f}}$ on $(\bar{Y}_{f},\nu_{f})$ and
$\bar{\lambda}_{\bar{Z}_{f}}$ on
$(\bar{Z}_{f},\nu_{f}^{\scriptscriptstyle\\#})$ (with
$\nu_{f}^{\scriptscriptstyle\\#}\coloneqq\bar{\rho}^{f}_{*}\nu_{f}$) have
completely positive entropy.

###### Proof.

Since $\lambda_{f}$ on $(X_{f},\mu_{f})$ is measurably conjugate to
$\bar{\lambda}_{\bar{Y}_{f}}$ on $(\bar{Y}_{f},\nu_{f})$ by Lemma 3.7, and
$\bar{\lambda}_{\bar{Z}_{f}}$ on
$(\bar{Z}_{f},\nu_{f}^{\scriptscriptstyle\\#})$ is a factor of
$\bar{\lambda}_{\bar{Y}_{f}}$ on $(\bar{Y}_{f},\nu_{f})$, all these actions
have completely positive entropy. ∎

###### Proof of Proposition 3.16.

If $\nu$ is a $\bar{\lambda}$-invariant Borel probability measure on
$\bar{Y}_{f}$ with entropy
$h_{\nu}(\bar{\lambda}_{\bar{Y}_{f}})=\textup{h}_{\textup{top}}(\bar{\lambda}_{\bar{Y}_{f}})=\textup{h}_{\textup{top}}(\lambda_{f})$
(cf. Theorem 3.8), then Lemma 3.13 implies that $\eta_{*}\nu=\mu_{f}$, the
unique $\lambda_{f}$-invariant Borel probability measure on $X_{f}$ with
maximal entropy. By Lemma 3.7, $\nu=\nu_{f}$, and the $\Gamma$-actions
$\lambda_{f}$ and $\bar{\lambda}_{\bar{Y}_{f}}$ on $(X_{f},\mu_{f})$ and
$(\bar{Y}_{f},\nu_{f})$ are conjugate. Lemma 3.17 completes the proof of
Proposition 3.16. ∎

######  Theorem 3.18.

Suppose that $\Gamma$ is a countably infinite amenable group,
$f\in\mathbb{Z}\Gamma$, and the principal algebraic $\Gamma$-action
$\lambda_{f}$ on $X_{f}$ is intrinsically ergodic. Then the principal
algebraic $\Gamma$-action $\lambda_{f}$ on $(X_{f},\mu_{f})$ is measurably
conjugate to the $\Gamma$-actions $\bar{\lambda}_{\bar{Y}_{f}}$ and
$\bar{\lambda}_{\bar{Z}_{f}}$ on $(\bar{Y}_{f},\nu_{f})$ and
$(\bar{Z}_{f},\nu_{f}^{\scriptscriptstyle\\#})$, respectively.

###### Proof.

If $\lambda_{f}$ is intrinsically ergodic on $X_{f}$, then
$\textup{h}_{\textup{top}}(\lambda_{f})<\infty$ and $\mu_{f}$ has c.p.e. (cf.
Definition 3.15). Lemma 3.7 shows that the $\Gamma$-actions $\lambda_{f}$ on
$(X_{f},\mu_{f})$ and $\bar{\lambda}$ on $(\bar{Y}_{f},\nu_{f})$ are
measurably conjugate, and the $\Gamma$-actions $\bar{\lambda}_{\bar{Y}_{f}}$
and $\bar{\lambda}_{\bar{Z}_{f}}$ on $(\bar{Y}_{f},\nu_{f})$ and
$(\bar{Z}_{f},\nu_{f}^{\scriptscriptstyle\\#})$ are measurably conjugate by
Corollary 3.6. ∎

##  4\. Generators of intrinsically ergodic principal algebraic actions

In this section we apply Theorem 3.18 to find generators of intrinsically
ergodic principal algebraic actions of a countably infinite amenable group
$\Gamma$.

Let $f\in\mathbb{Z}\Gamma$ be a nonzero element such that the principal
algebraic $\Gamma$-action $\lambda_{f}$ on $X_{f}$ is intrinsically ergodic.
We view $X_{f}\subset\mathbb{T}^{\Gamma}$ as a subset of $[0,1)^{\Gamma}$ as
in (2.1) – (2.4) by identifying $Y_{f}$ and $X_{f}$ through $\eta$ and set,
for every $j\in\mathbb{Z}$,

$B[j]=\bigl{\\{}x\in
X_{f}\,\mid\,\textstyle{\sum_{\gamma\in\textup{supp}(f)}}f_{\gamma}x_{\gamma}=j\bigr{\\}}=\bigl{\\{}x\in
X_{f}\mid(\bar{\rho}^{f}x)_{1_{\Gamma}}=j\bigr{\\}}.$ (4.1)

The following corollaries are immediate consequences of Theorem 3.18.

######  Corollary 4.1.

Put

$\mathcal{B}_{f}=\begin{cases}\\{B[j]\mid
j=-\|f^{-}\|_{1}+1,\dots,\|f^{+}\|_{1}-1\\}&\textup{if both $f^{+}$ and
$f^{-}$ are nonzero},\\\ \\{B[j]\mid j=0,\dots,\|f^{+}\|_{1}-1\\}&\textup{if
$f^{+}\neq 0$ and $f^{-}=0$},\\\ \\{B[j]\mid
j=-\|f^{-}\|_{1}+1,\dots,0\\}&\textup{if $f^{+}=0$ and $f^{-}\neq
0$}.\end{cases}$ (4.2)

Then $\mathcal{B}_{f}$ is a Borel partition of $X_{f}$ which is a generator
$(\textup{mod}\;\mu_{f})$ for $\lambda_{f}$.

######  Corollary 4.2.

The Borel partition $\mathcal{C}_{f}=\\{C_{j}\mid j=0,\dots,\|f\|_{1}-1\\}$ of
$X_{f}$ with

$C_{j}=\bigl{\\{}x\in X_{f}\mid j/\|f\|_{1}\leq
x_{1_{\Gamma}}<(j+1)/\|f\|_{1}\;(\textup{mod}\;1)\\}\enspace\textup{for}\enspace
j=0,\dots,\|f\|_{1}-1,$

is a generator $(\textup{mod}\;\mu_{f})$ for $\lambda_{f}$.

By imposing further conditions on $\Gamma$ and $f$ we can sometimes find
slightly smaller generators $(\textup{mod}\;\mu_{f})$ for $\lambda_{f}$ in
Corollary 4.1.

######  Corollary 4.3.

Suppose that the group $\Gamma$ in Theorem 3.18 is left (or, equivalently,
right) orderable. If $f\in\mathbb{Z}\Gamma$ satisfies that
$|\textup{supp}(f)|\geq 2$, then the collection of sets

$\mathcal{B}_{f}^{\prime}=\bigl{\\{}B[j]\mid
j=-\|f^{-}\|_{1}+1,\dots,\|f^{+}\|_{1}-1\bigr{\\}},$

defined as in (4.1), is a generator $(\textup{mod}\;\mu_{f})$ for
$\lambda_{f}$.

For the proof of Corollary 4.3 we require an additional lemma. For notation we
refer to Lemma 3.7.

######  Lemma 4.4.

Suppose that the group $\Gamma$ in Theorem 3.18 is left (or, equivalently,
right) orderable. If $f\in\mathbb{Z}\Gamma$ satisfies that
$|\textup{supp}(f)|\geq 2$, then $K=\pi_{1_{\Gamma}}(X_{f})=\mathbb{T}$ and
hence $(\pi_{1_{\Gamma}})_{*}\mu_{f}=\mu_{\mathbb{T}}$, the Lebesgue measure
on $\mathbb{T}$.

###### Proof.

Since $K\subset\mathbb{T}$ is a closed subgroup, it is either finite or equal
to $\mathbb{T}$. If $K$ is finite we choose $L\geq 1$ such that $LK=\\{Lt\mid
t\in K\\}=\\{0\\}$ and conclude from (1.2) that $L$ lies in the left ideal
$(f)\subset\mathbb{Z}\Gamma$ generated by $f$. Then there exists
$h\in\mathbb{Z}\Gamma$ such that $L=hf$ or, equivalently,
$1=h\cdot\frac{1}{L}f$. In other words, the rational group ring
$\mathbb{Q}\Gamma$ contains the nontrivial unit $\frac{1}{L}f$, in violation
of [Passman]*Lemmas 13.1.7 and 13.1.10.

If $\pi_{1_{\Gamma}}(X_{f})=\mathbb{T}$, then obviously
$(\pi_{1_{\Gamma}})_{*}\mu_{f}=\mu_{\mathbb{T}}$. ∎

###### Proof of Corollary 4.3.

If both $f^{+}$ and $f^{-}$ are nonzero our assertion follows from Corollary
4.1. If $f^{-}=0$, then

$B[0]=\\{x\in
X_{f}\mid\textstyle\sum_{\gamma\in\textup{supp}(f)}f_{\gamma}x_{\gamma}=0\\}=\\{x\in
X_{f}\mid x_{\gamma}=0\enspace\textup{for
every}\enspace\gamma\in\textup{supp}(f)\\}.$

Hence $\mu_{f}(B[0])\leq\mu_{f}(\\{x\in X_{f}\mid x_{\gamma}=0\\})$ for every
$\gamma\in\Gamma$, so that $\mu_{f}(B[0])=0$ by Lemma 4.4. By Corollary 4.1,
$\\{B[j]\mid j=1,\dots,\|f^{+}\|_{1}-1\\}$ is a generator
$(\textup{mod}\;\mu_{f})$ for $\lambda_{f}$.

If $f^{+}=0$ the proof is completely analogous. ∎

##  5\. Examples

Let $\Gamma$ be a countably infinite discrete amenable group,
$f\in\mathbb{Z}\Gamma$, and let $\lambda_{f}$ be the principal algebraic
$\Gamma$-action on $X_{f}$ in Definition 1.1. In order to apply Theorem 3.18
and its corollaries to $\lambda_{f}$ we require the action $\lambda_{f}$ to be
intrinsically ergodic.

###  5.1. Intrinsically ergodic principal algebraic $\mathbb{Z}^{d}$-actions

If $\Gamma=\mathbb{Z}^{d}$ for some $d\geq 1$, the conditions for principal
algebraic $\Gamma$-actions to be intrinsically ergodic are well understood: if
$f$ is nonzero and not divisible by a generalized cyclotomic polynomial, then
$\lambda_{f}$ is intrinsically ergodic [DSAO]*Theorem 11.2, Propositions 19.4
and 20.5.

######  Example 5.1.

The matrix $M=\left[\begin{smallmatrix}\hphantom{-}0&1&0&0\\\
\hphantom{-}0&0&1&0\\\ \hphantom{-}0&0&0&1\\\
-1&1&1&1\end{smallmatrix}\right]\in\textup{SL}(4,\mathbb{Z})$ defines a
nonhyperbolic ergodic automorphism $\alpha_{M}$ of $\mathbb{T}^{4}$. The
question of finding ‘nice’ finite generating partitions for such automorphisms
was discussed in [Lind]*Theorem 1. By observing that $\alpha_{M}$ is
algebraically conjugate to the intrinsically ergodic algebraic
$\mathbb{Z}$-action $\lambda_{f}$ on $(X_{f},\mu_{f})$ for the characteristic
polynomial $f=u^{4}-u^{3}-u^{2}-u+1$ of $M$ and applying Corollary 4.2 we see
that the sets

$C_{j}=\bigl{\\{}x\in X_{f}\mid\tfrac{j}{5}\leq
x_{0}<\tfrac{j+1}{5}\enspace(\textup{mod}\;1)\\},\;0\leq j\leq 4,$

form a generating partition for $\lambda_{f}$ w.r.t. $\mu_{f}$ on $X_{f}$.
When translating this information back to $\alpha_{M}$ we obtain the generator

$\mathcal{D}=\bigl{\\{}D_{j}=\\{t=(t_{1},t_{2},t_{3},t_{4})\in\mathbb{T}^{4}\mid\tfrac{j}{5}\leq
t_{1}<\tfrac{j+1}{5}\enspace(\textup{mod}\;1)\\}\bigm{|}j=0,\dots,4\bigr{\\}}$

for $\alpha_{M}$ w.r.t. Lebesgue measure $\mu_{\mathbb{T}^{4}}$ on
$\mathbb{T}^{4}$.

Corollary 4.1 shows that $\alpha_{M}$ on
$(\mathbb{T}^{4},\mu_{\mathbb{T}^{4}})$ also has the $4$-element generator
corresponding to $\\{\negthinspace B[-2],B[-1],B[0],B[1]\\}$ in (4.2).

######  Example 5.2.

Let $\Gamma=\mathbb{Z}^{2}$, and let $f=1-u_{1}-u_{2}\in\mathbb{Z}[u_{1}^{\pm
1},u_{2}^{\pm 1}]\cong\mathbb{Z}[\mathbb{Z}^{2}]$. By [DSAO]*Proposition 19.7,
$\textup{h}_{\textup{top}}(\lambda_{f})=\frac{3\sqrt{3}}{4\pi}L(2,\chi_{3})>0$,
where $L(2,\chi_{3})$ is the Dirichlet $L$-function defined there. Since
$\lambda_{f}$ is intrinsically ergodic, Corollary 4.1 shows that $\lambda_{f}$
on $(X_{f},\mu_{f})$ has the 2-element generator $\\{\negthinspace
B[-1],B[0]\\}$ $(\textup{mod}\;\mu_{f})$ defined as in (4.1).

###  5.2. Intrinsically ergodic principal algebraic actions of amenable groups

If $\Gamma$ is a countably infinite discrete amenable group, establishing the
intrinsic ergodicity of a principal algebraic action $\lambda_{f}$,
$f\in\mathbb{Z}\Gamma$, is much more delicate than for
$\Gamma=\mathbb{Z}^{d}$. A sufficient condition for intrinsic ergodicity of
$\lambda_{f}$ can be expressed in terms of homoclinic points: a point
$x=(x_{s})\in X_{f}$ is _summable homoclinic_ if
$\sum_{s\in\Gamma}|\negthinspace|x_{s}|\negthinspace|<\infty$, where
$|\negthinspace|t|\negthinspace|$ denotes the distance from $0$ of a point
$t\in\mathbb{T}$ (cf. [LSV2]). Clearly, every summable homoclinic point $x\in
X_{f}$ is _homoclinic_ , i.e., $\lim_{s\to\infty}\lambda_{f}^{s}x=0$ (cf.
e.g., [homoclinic]*Definition 3.1 or [Chung-Li]).

######  Proposition 5.3.

Let $\Gamma$ be a countably infinite amenable discrete group,
$f\in\mathbb{Z}\Gamma$, and let $\Delta^{1}(X_{f})\linebreak[0]\subset X_{f}$
be the group of summable homoclinic points of the principal algebraic
$\Gamma$-action $\lambda_{f}$ on $X_{f}$. If $\Delta^{1}(X_{f})$ is dense in
$X_{f}$ and $\textup{h}_{\textup{top}}(\lambda_{f})<\infty$ then $\lambda_{f}$
is intrinsically ergodic.

The converse of Proposition 5.3 is clearly not true: the principal algebraic
$\mathbb{Z}$-action $\lambda_{f}$ on $X_{f}$ (or, equivalently, the
automorphism $\alpha_{M}$ of $\mathbb{T}^{4}$) in Example 5.1 is intrinsically
ergodic, but has no nonzero homoclinic points (cf. [homoclinic]*Example 3.4).

###### Proof of Proposition 5.3.

According to (1.2), the dual group $\widehat{X_{f}}$ is given by
$\widehat{X_{f}}=\mathbb{Z}\Gamma/(f)$ and is, in particular, a finitely
generated left $\mathbb{Z}\Gamma$-module. By [Chung-Li]*Theorem 7.8,
$\Delta^{1}(X_{f})\subset\textup{IE}(X_{f})$, the closed subgroup of $X_{f}$
defined in [Chung-Li]*Definition 7.2, and hence $\textup{IE}(X_{f})=X_{f}$ by
assumption. By [Chung-Li]*Corollary 8.4 and Theorem 8.6, $\lambda_{f}$ is
intrinsically ergodic. ∎

####  5.2.1. Expansive principal algebraic actions

For every countably infinite discrete group $\Gamma$ and every
$f\in\mathbb{Z}\Gamma$, the principal algebraic $\Gamma$-action $\lambda_{f}$
on $X_{f}$ in Definition 1.1 is expansive if and only if the map
$\bar{\rho}^{f}\colon\ell^{\infty}(\Gamma,\mathbb{R})\to\ell^{\infty}(\Gamma,\mathbb{R})$
in (2.3) is injective or, equivalently, if $f$ is invertible in
$\ell^{1}(\Gamma,\mathbb{R})$ ([Deninger-S]*Theorem 3.2). If this is the case,
$w^{\Delta}\coloneqq(f^{*})^{-1}\in W_{f}$ since
$\bar{\rho}^{f}(w^{\Delta})=1_{\Gamma}$. By [Deninger-S]*Proposition 4.2, the
map
$\bar{\rho}^{w^{\Delta}}\colon\ell^{\infty}(\Gamma,\mathbb{Z})\to\ell^{\infty}(\Gamma,\mathbb{R})$
is continuous in the weak∗-topology on closed, bounded subsets of
$\ell^{\infty}(\Gamma,\mathbb{Z})$, and the map
$\xi\coloneqq\eta\circ\bar{\rho}^{w^{\Delta}}\colon\ell^{\infty}(\Gamma,\mathbb{Z})\to\mathbb{T}^{\Gamma}$
satisfies that
$\xi(\\{v\in\ell^{\infty}(\Gamma,\mathbb{Z})\mid\|v\|_{\infty}\leq\|f\|_{1}/2\\})=X_{f}$
(cf. [Deninger-S]*Lemma 4.5). Since
$\xi(\mathbb{Z}\Gamma)\subset\Delta^{1}(X_{f})$, the continuity of $\xi$
implies that $\Delta^{1}(X_{f})$ is dense in $X_{f}$. If $\Gamma$ is amenable,
we conclude from Proposition 5.3 that every expansive principal algebraic
$\Gamma$-action is intrinsically ergodic.

If $\Gamma$ is amenable and $\lambda_{f}$ is expansive, the partitions
$\mathcal{B}_{f}$ and $\mathcal{C}_{f}$ in the Corollaries 4.1 and 4.2 are
obviously generators (and not only generators $(\textup{mod}\;\mu_{f})$) for
$\lambda_{f}$.

We mention two examples, taken from [E-R]. Let
$\mathbb{H}\subset\textup{SL}(3,\mathbb{Z})$ be the discrete Heisenberg group,
generated by

$\smash[b]{u_{1}=\left[\begin{smallmatrix}1&0&0\\\ 0&1&1\\\
0&0&1\end{smallmatrix}\right],\quad u_{2}=\left[\begin{smallmatrix}1&1&0\\\
0&1&0\\\ 0&0&1\end{smallmatrix}\right],\quad\text{and}\quad
u_{3}=\left[\begin{smallmatrix}1&0&1\\\ 0&1&0\\\
0&0&1\end{smallmatrix}\right].}$ (5.1)

######  Example 5.4 ([E-R]*Example 8.4).

Let $f=|a_{1}|+|a_{2}|+|a_{3}|+a_{1}\cdot u_{1}+a_{2}\cdot u_{2}+a_{3}\cdot
u_{3}\in\mathbb{Z}\mathbb{H}$. Then the principal algebraic
$\mathbb{H}$-action $\lambda_{f}$ is expansive if and only if $a_{1}a_{2}\neq
0$ and $a_{3}>0$. In these cases the partitions $\mathcal{B}_{f}$ and
$\mathcal{C}_{f}$ in the Corollaries 4.1 – 4.2 are generators for
$\lambda_{f}$.

######  Example 5.5.

Let $f=1-u_{1}-u_{2}\in\mathbb{Z}\mathbb{H}$. The principal algebraic
$\mathbb{H}$-action $\lambda_{f}$ on $(X_{f},\mu_{f})$ has zero entropy by
[Deninger]*Theorem 11 or [Lind-S]*Theorem 9.2, so that the hypotheses of
Theorem 3.18 are not satisfied. We do not know whether the
$\mathbb{H}$-actions $\lambda_{f}$ on $(X_{f},\mu_{f})$ and
$\bar{\lambda}_{\bar{Z}_{f}}$ on
$(\bar{Z}_{f},\nu^{\scriptscriptstyle\\#}_{f})$ are measurably conjugate and
whether $\mathcal{B}_{f}=\\{\negthinspace B[-1],B[0]\\}$ is a generator
$(\textup{mod}\;\mu_{f})$ for $\lambda_{f}$.

##  6\. Summable homoclinic points of nonexpansive principal algebraic actions

The existence of a nonzero summable homoclinic point for a nonexpansive
principal algebraic action in the second part of Example 5.5 is an interesting
fact. For $\Gamma=\mathbb{Z}^{d}$ this phenomenon is well understood: for
$f=\sum_{\mathbf{n}\in\mathbb{Z}^{d}}f_{\mathbf{n}}u^{\mathbf{n}}\in\mathbb{Z}[\mathbb{Z}^{d}]$
we denote by

$\mathsf{U}(f)=\\{\mathbf{z}=(z_{1},\dots,z_{d})\in\mathbb{S}^{d}\mid
f(\mathbf{z})=0\\}$ (6.1)

the _unitary variety_ of $f$, where
$\mathbb{S}=\\{z\in\mathbb{C}\mid|z|=1\\}$. According to [DSAO]*Theorem 6.5,
$\lambda_{f}$ is expansive if and only if $\mathsf{U}(f)=\varnothing$. If $f$
is nonzero and irreducible, then [LSV2]*Theorem 3.2 shows that
$\Delta^{1}(X_{f})\neq\\{0\\}$ if and only if the dimension of
$\mathsf{U}(f)\subset\mathbb{S}^{d}$ is $\leq d-2$. In this case
$\Delta^{1}(X_{f})$ is dense in $X_{f}$, $\lambda_{f}$ is intrinsically
ergodic, and the partitions $\mathcal{B}_{f}$ and $\mathcal{C}_{f}$ are
generators $(\textup{mod}\;\mu_{f})$ for $\lambda_{f}$.

For nonabelian groups $\Gamma$, examples of nonexpansive principal algebraic
actions with summable homoclinic points are much harder to come by. In order
to present a class of such actions we assume for the remainder of this section
that $\Gamma$ is a countably infinite discrete group with center $H$. We say
$f\in\mathbb{R}\Gamma$ is _well-balanced_ ([Bowen-Li]*Definition 1.2) if

  1. (1)

$\sum_{s\in\Gamma}f_{s}=0$,

  2. (2)

$f_{s}\leq 0$ for every $s\in\Gamma\smallsetminus\\{1_{\Gamma}\\}$,

  3. (3)

$f=f^{*}$,

  4. (4)

$\textup{supp}(f)$ generates $\Gamma$.

We shall prove the following theorem.

######  Theorem 6.1.

Assume that for any finite $F\subseteq\Gamma$ there is some $s$ in the center
$H$ of $\Gamma$ such that none of $s,s^{2},s^{3}$ is in $F$ (this happens for
example when $H^{6}=\\{s^{6}\mid s\in H\\}$ is infinite). Also assume that
$\Gamma$ is not virtually $\mathbb{Z}$ or $\mathbb{Z}^{2}$. Let
$f\in\mathbb{Z}\Gamma$ be well-balanced. Then $\Delta^{1}(X_{f})$ is dense in
$X_{f}$.

######  Corollary 6.2.

Assume that the group $\Gamma$ in Theorem 6.1 is amenable. If
$f\in\mathbb{Z}\Gamma$ is well-balanced, then the principal algebraic
$\Gamma$-action $\lambda_{f}$ on $X_{f}$ is intrinsically ergodic.

###### Proof.

The proof of Corollary 6.11 implies that $f$ is not a left zero divisor in
$\mathbb{Z}\Gamma$, so that $\textup{h}_{\textup{top}}(\lambda_{f})<\infty$ by
Lemma 3.1. Now apply Theorem 6.1 and Proposition 5.3. ∎

The proof of Theorem 6.1 requires a brief excursion into Banach algebras.

######  Lemma 6.3.

Let $\mathcal{A}$ be a unital Banach algebra such that
$\|ab\|\leq\|a\|\cdot\|b\|$ for all $a,b\in\mathcal{A}$. Let $0<c<1$ and
$q,r\in\mathcal{A}$ such that $\|r\|\leq 1-c$ and $\|q\|\leq c$. Then

$\sum_{k=0}^{\infty}\|r^{k}(c-q)^{3}(1-q)^{-(k+1)}\|<\infty.$

The proof of Lemma 6.3 requires some auxiliary results.

######  Lemma 6.4.

Let $0<c<1$. For any $x,y>0$ one has

$\frac{c^{x}(1-c)^{y}(x+y)^{x+y}}{x^{x}y^{y}}\leq 1.$

###### Proof.

Fix $y>0$ and put

$\phi(x)=\log\frac{c^{x}(1-c)^{y}(x+y)^{x+y}}{x^{x}y^{y}}=x\log
c+y\log(1-c)+(x+y)\log(x+y)-x\log x-y\log y$

for $x>0$. Then

$\phi^{\prime}(x)=\log c+\log(x+y)-\log x=\log\frac{c(x+y)}{x}.$

Clearly $\phi^{\prime}>0$ on the interval $(0,\frac{cy}{1-c})$,
$\phi^{\prime}=0$ at $\frac{cy}{1-c}$, and $\phi^{\prime}<0$ on
$(\frac{cy}{1-c},\infty)$. Thus $\phi$ attains its maximum value at
$\frac{cy}{1-c}$. Since $\phi(\frac{cy}{1-c})=0$, one concludes that
$\phi(x)\leq 0$ for all $x>0$. ∎

######  Lemma 6.5.

Let $0<c<1$ and $k\in\mathbb{N}$. Let $g_{k}$ be the cubic polynomial given by

$\displaystyle g_{k}(x)$ $\displaystyle=-(x+1)(x+2)(x+3)+3c(x+k+1)(x+2)(x+3)$
(6.2)
$\displaystyle\quad-3c^{2}(x+k+1)(x+k+2)(x+3)+c^{3}(x+k+1)(x+k+2)(x+k+3)$
$\displaystyle=(c-1)^{3}x^{3}+(3c(c-1)^{2}k+6(c-1)^{3})x^{2}$
$\displaystyle\quad+(3c^{2}(c-1)k^{2}+3c(c-1)(4c-5)k+11(c-1)^{3})x$
$\displaystyle\quad+c^{3}k^{3}+(6c^{3}-9c^{2})k^{2}+(11c^{3}-27c^{2}+18c)k+6(c-1)^{3}.$

For $\eta>0$, put

$\displaystyle y_{k,\eta,\pm}=\frac{ck}{1-c}-2\pm\frac{1}{1-c}\sqrt{\eta
k+\tfrac{1}{3}(1-c)^{2}}.$ (6.3)

Then

$\displaystyle
g_{k}(y_{k,\eta,\pm})=(c^{2}+c)k\pm\bigl{(}(3c-\eta)k+\tfrac{2}{3}(c-1)^{2}\bigr{)}\sqrt{\eta
k+\tfrac{1}{3}(1-c)^{2}}.$

###### Proof.

This is a direct computation:

$\displaystyle g_{k}(y_{k,\eta,\pm})$
$\displaystyle=(c-1)^{3}y_{k,\eta,\pm}^{3}+(3c(c-1)^{2}k+6(c-1)^{3})y_{k,\eta,\pm}^{2}$
$\displaystyle\quad+(3c^{2}(c-1)k^{2}+3c(c-1)(4c-5)k+11(c-1)^{3})y_{k,\eta,\pm}$
$\displaystyle\quad+c^{3}k^{3}+(6c^{3}-9c^{2})k^{2}+(11c^{3}-27c^{2}+18c)k+6(c-1)^{3}$
$\displaystyle=(c-1)^{3}(\tfrac{ck}{1-c}-2)((\tfrac{ck}{1-c}-2)^{2}+3\tfrac{1}{(1-c)^{2}}(\eta
k+\tfrac{1}{3}(1-c)^{2}))$
$\displaystyle\quad+(3c(c-1)^{2}k+6(c-1)^{3})((\tfrac{ck}{1-c}-2)^{2}+\tfrac{1}{(1-c)^{2}}(\eta
k+\tfrac{1}{3}(1-c)^{2}))$
$\displaystyle\quad+(3c^{2}(c-1)k^{2}+3c(c-1)(4c-5)k+11(c-1)^{3})(\tfrac{ck}{1-c}-2)$
$\displaystyle\quad+c^{3}k^{3}+(6c^{3}-9c^{2})k^{2}+(11c^{3}-27c^{2}+18c)k+6(c-1)^{3}$
$\displaystyle\quad\pm(c-1)^{3}(\tfrac{ck}{1-c}-2)^{2}\tfrac{3}{1-c}\sqrt{\eta
k+\tfrac{1}{3}(1-c)^{2}}$
$\displaystyle\quad\pm(c-1)^{3}\tfrac{1}{(1-c)^{3}}(\eta
k+\tfrac{1}{3}(1-c)^{2})\sqrt{\eta k+\tfrac{1}{3}(1-c)^{2}}$
$\displaystyle\quad\pm
2(3c(c-1)^{2}k+6(c-1)^{3})(\tfrac{ck}{1-c}-2)\tfrac{1}{1-c}\sqrt{\eta
k+\tfrac{1}{3}(1-c)^{2}}$
$\displaystyle\quad\pm(3c^{2}(c-1)k^{2}+3c(c-1)(4c-5)k+11(c-1)^{3})\tfrac{1}{1-c}\sqrt{\eta
k+\tfrac{1}{3}(1-c)^{2}}$ $\displaystyle=-(ck+2(c-1))((ck+2(c-1))^{2}+(3\eta
k+(1-c)^{2}))$ $\displaystyle\quad+(ck+2(c-1))(3(ck+2(c-1))^{2}+(3\eta
k+(1-c)^{2}))$
$\displaystyle\quad-(3c^{2}k^{2}+3c(4c-5)k+11(c-1)^{2})(ck+2(c-1))$
$\displaystyle\quad+c^{3}k^{3}+(6c^{3}-9c^{2})k^{2}+(11c^{3}-27c^{2}+18c)k+6(c-1)^{3}$
$\displaystyle\quad\mp 3(ck+2(c-1))^{2}\sqrt{\eta k+\tfrac{1}{3}(1-c)^{2}}$
$\displaystyle\quad\mp(\eta k+\tfrac{1}{3}(1-c)^{2})\sqrt{\eta
k+\tfrac{1}{3}(1-c)^{2}}$ $\displaystyle\quad\pm 6(ck+2(c-1))^{2}\sqrt{\eta
k+\tfrac{1}{3}(1-c)^{2}}$
$\displaystyle\quad\mp(3c^{2}k^{2}+3c(4c-5)k+11(c-1)^{2})\sqrt{\eta
k+\tfrac{1}{3}(1-c)^{2}}$
$\displaystyle=(c^{2}+c)k\pm((3c-\eta)k+\tfrac{2}{3}(c-1)^{2})\sqrt{\eta
k+\tfrac{1}{3}(1-c)^{2}}.$ ∎

∎

######  Lemma 6.6.

Fix $0<c<1$. Then there is some $k_{c}\in\mathbb{N}$ such that for each
$k\in\mathbb{N}$ with $k\geq k_{c}$ the following hold:

  1. (1)

the polynomial $g_{k}$ given by (6.2) has $3$ roots $t_{k,1}<t_{k,2}<t_{k,3}$
such that

$1<y_{k,4c,-}<t_{k,1}<y_{k,c,-}<t_{k,2}<y_{k,c,+}<t_{k,3}<y_{k,4c,+},$

where $y_{k,\eta,\pm}$ is given in (6.3);

  2. (2)

$g_{k}>0$ on $(-\infty,t_{k,1})\cup(t_{k,2},t_{k,3})$ and $g_{k}<0$ on
$(t_{k,1},t_{k,2})\cup(t_{k,3},\infty)$.

###### Proof.

Take $k_{c}\in\mathbb{N}$ such that for each $k\geq k_{c}$ one has

$\tfrac{ck}{1-c}-2-\tfrac{1}{1-c}\sqrt{4ck+\tfrac{1}{3}(1-c)^{2}}>1,$

and

$(c^{2}+c)k-(-ck+\tfrac{2}{3}(c-1)^{2})\sqrt{4ck+\tfrac{1}{3}(1-c)^{2}}>0,$

and

$(c^{2}+c)k-(2ck+\tfrac{2}{3}(c-1)^{2})\sqrt{ck+\tfrac{1}{3}(1-c)^{2}}<0,$

and

$(c^{2}+c)k+(-ck+\tfrac{2}{3}(c-1)^{2})\sqrt{4ck+\tfrac{1}{3}(1-c)^{2}}<0.$

Then for each $k\in\mathbb{N}$ with $k\geq k_{c}$, one has $y_{k,4c,-}>1$ and
by Lemma 6.5 one has $g_{k}(y_{k,4c,-})>0$, $g_{k}(y_{k,c,-})<0$,
$g_{k}(y_{k,c,+})>0$, and $g_{k}(y_{k,4c,+})<0$. Since
$y_{k,4c,-}<y_{k,c,-}<y_{k,c,+}<y_{k,4c,+}$, it follows that (1) holds. As
$g_{k}$ is a cubic polynomial, (2) must also hold. ∎

######  Lemma 6.7.

Fix $0<c<1$. For $k\in\mathbb{N}$ set

$\displaystyle
f_{k}(m)=c^{m}\tbinom{m+k}{k}-2c^{m+1}\tbinom{m+k+1}{k}+c^{m+2}\tbinom{m+k+2}{k}$
(6.4)

and

$\displaystyle h_{k}(m)=(1-c)^{k}f_{k}(m)$ (6.5)

for $m\in\mathbb{Z}_{\geq 0}$. Let $\xi:\mathbb{N}\rightarrow\mathbb{N}$ such
that $\xi(k)=\frac{ck}{1-c}+\mathcal{O}(k^{1/2})$ as $k\to\infty$. Then

$\displaystyle h_{k}(\xi(k))=\mathcal{O}(k^{-3/2})$

as $k\to\infty$.

###### Proof.

For $k\in\mathbb{N}$ define
$\varphi_{k},\psi_{k}:\mathbb{N}\rightarrow\mathbb{R}$ by

$\varphi_{k}(m)=(1-c)^{k}c^{m}\tbinom{m+k}{k}$

and

$\psi_{k}(m)=\tfrac{(m+1)(m+2)-2c(m+k+1)(m+2)+c^{2}(m+k+1)(m+k+2)}{(m+1)(m+2)}.$

For each $m\in\mathbb{N}$, one has

$f_{k}(m)=c^{m}\tbinom{m+k}{k}\bigl{(}1-2c\tfrac{m+k+1}{m+1}+c^{2}\tfrac{(m+k+1)(m+k+2)}{(m+1)(m+2)}\bigr{)}=c^{m}\tbinom{m+k}{k}\psi_{k}(m),$

whence

$h_{k}(m)=\varphi_{k}(m)\psi_{k}(m).$

Therefore it suffices to show that $\varphi_{k}(\xi(k))=\mathcal{O}(k^{-1/2})$
and $\psi_{k}(\xi(k))=\mathcal{O}(k^{-1})$.

By Stirling’s approximation formula there are constants $C_{1},C_{2}>0$ such
that

$C_{1}\sqrt{2\pi n}(\tfrac{n}{e})^{n}\leq n!\leq C_{2}\sqrt{2\pi
n}(\tfrac{n}{e})^{n}$

for all $n\in\mathbb{N}$. Then

$\varphi_{k}(m)=(1-c)^{k}c^{m}\tfrac{(m+k)!}{m!k!}\leq(1-c)^{k}c^{m}\tfrac{C_{2}\sqrt{2\pi(m+k)}(m+k)^{m+k}}{C_{1}^{2}2\pi\sqrt{mk}m^{m}k^{k}}\leq\tfrac{C_{2}\sqrt{2\pi(m+k)}}{C_{1}^{2}2\pi\sqrt{mk}},$

where the 2nd inequality comes from taking $x=m$ and $y=k$ in Lemma 6.4. Thus

$\varphi_{k}(\xi(k))\leq\tfrac{C_{2}\sqrt{2\pi(\xi(k)+k)}}{C_{1}^{2}2\pi\sqrt{\xi(k)k}}=\mathcal{O}(k^{-1/2}),$

whence $\varphi_{k}(\xi(k))=\mathcal{O}(k^{-1/2})$.

Write $\xi(k)$ as $\frac{ck}{1-c}+\lambda_{k}k^{1/2}$ with
$\lambda_{k}=\mathcal{O}(1)$ as $k\to\infty$. Then

$\displaystyle(\xi(k)+1)(\xi(k)+2)$
$\displaystyle=\tfrac{c^{2}k^{2}}{(1-c)^{2}}+\tfrac{2c\lambda_{k}}{1-c}k^{3/2}+\mathcal{O}(k),$
$\displaystyle 2c(\xi(k)+k+1)(\xi(k)+2)$
$\displaystyle=\tfrac{2c^{2}k^{2}}{(1-c)^{2}}+\tfrac{2c(1+c)\lambda_{k}}{1-c}k^{3/2}+\mathcal{O}(k),$
$\displaystyle c^{2}(\xi(k)+k+1)(\xi(k)+k+2)$
$\displaystyle=\tfrac{c^{2}k^{2}}{(1-c)^{2}}+\tfrac{2c^{2}\lambda_{k}}{1-c}k^{3/2}+\mathcal{O}(k),$

whence

$(\xi(k)+1)(\xi(k)+2)-2c(\xi(k)+k+1)(\xi(k)+2)+c^{2}(\xi(k)+k+1)(\xi(k)+k+2)=\mathcal{O}(k).$

It follows that $\psi_{k}(\xi(k))=\mathcal{O}(k^{-1})$. ∎

For a power series $\phi(x)=\sum_{m=0}^{\infty}a_{m}x^{m}\in\mathbb{C}[[x]]$,
we set $|\phi|$ to be the power series

$|\phi|(x)=\sum_{m=0}^{\infty}|a_{m}|x^{m}.$

######  Lemma 6.8.

Fix $0<c<1$. For each $k\in\mathbb{Z}_{\geq 0}$ let $\phi_{k}$ be the power
series given by

$\displaystyle\phi_{k}(x)=(c-x)^{3}\sum_{m=0}^{\infty}\tbinom{m+k}{k}x^{m}.$
(6.6)

Then $(1-c)^{k}|\phi_{k}|(c)=\mathcal{O}(k^{-3/2})$ as $k\to\infty$.

###### Proof.

Let $k\in\mathbb{Z}_{\geq 0}$. For each $m\in\mathbb{Z}_{\geq 0}$, put

$\displaystyle b_{k,m}$
$\displaystyle=-\tbinom{m+k}{k}+3c\tbinom{m+k+1}{k}-3c^{2}\tbinom{m+k+2}{k}+c^{3}\tbinom{m+k+3}{k}$
$\displaystyle=\tbinom{m+k}{k}\bigl{(}-1+\tfrac{3c(m+k+1)}{m+1}-\tfrac{3c^{2}(m+k+1)(m+k+2)}{(m+1)(m+2)}+\tfrac{c^{3}(m+k+1)(m+k+2)(m+k+3)}{(m+1)(m+2)(m+3)}\bigr{)}$
$\displaystyle=\tbinom{m+k}{k}\tfrac{g_{k}(m)}{(m+1)(m+2)(m+3)},$

where $g_{k}$ is given by (6.2). Then

$\phi_{k}(x)=c^{3}+x(c^{3}(k+1)-3c^{2})+x^{2}\bigl{(}c^{3}\tfrac{(k+1)(k+2)}{2}-3c^{2}(k+1)+3c\bigr{)}+\sum_{m=0}^{\infty}x^{m+3}b_{k,m}.$
(6.7)

Let $k_{c}\in\mathbb{N}$ be given by Lemma 6.6. Let $k\in\mathbb{N}$ with
$k\geq k_{c}$. Then $g_{k}$ has the roots $t_{k,i}$ for $i=1,2,3$ described in
Lemma 6.6. Put $m_{k,i}=\lfloor t_{k,i}\rfloor$ for $i=1,3$, and
$m_{k,2}=\lceil t_{k,2}\rceil$. Then $b_{k,m}\geq 0$ exactly when $0\leq m\leq
m_{k,1}$ or $m_{k,2}\leq m\leq m_{k,3}$. Increasing $k_{c}$ if necessary, we
may assume that
$c^{3}(k+1)-3c^{2},c^{3}\frac{(k+1)(k+2)}{2}-3c^{2}(k+1)+3c>0$. Then

$\displaystyle|\phi_{k}|(x)$
$\displaystyle=|c^{3}|+x|c^{3}(k+1)-3c^{2}|+x^{2}\big{|}c^{3}\tfrac{(k+1)(k+2)}{2}-3c^{2}(k+1)+3c\big{|}+\sum_{m=0}^{\infty}x^{m+3}|b_{k,m}|$
$\displaystyle=c^{3}+x(c^{3}(k+1)-3c^{2})+x^{2}\big{(}c^{3}\tfrac{(k+1)(k+2)}{2}-3c^{2}(k+1)+3c\big{)}$
$\displaystyle\quad+\sum_{0\leq m\leq m_{k,1}\textup{ \scriptsize or
}m_{k,2}\leq m\leq m_{k,3}}x^{m+3}b_{k,m}-\sum_{m_{k,1}<m<m_{k,2}\textup{
\scriptsize or }m>m_{k,3}}x^{m+3}b_{k,m}.$

Note that $\phi_{k}$ converges absolutely on the open interval $(-1,1)$.
Taking $x=c$ in (6.7) we get

$\displaystyle
0=c^{3}+c(c^{3}(k+1)-3c^{2})+c^{2}\big{(}c^{3}\tfrac{(k+1)(k+2)}{2}-3c^{2}(k+1)+3c\big{)}+\sum_{m=0}^{\infty}c^{m+3}b_{k,m}.$

Thus

$\displaystyle|\phi_{k}|(c)$
$\displaystyle=2\Big{(}c^{3}+c(c^{3}(k+1)-3c^{2})+c^{2}\big{(}c^{3}\tfrac{(k+1)(k+2)}{2}-3c^{2}(k+1)+3c\big{)}$
(6.8) $\displaystyle\quad+\sum_{0\leq m\leq m_{k,1}\mbox{ \scriptsize or
}m_{k,2}\leq m\leq m_{k,3}}c^{m+3}b_{k,m}\Big{)}.$

Similar to (6.7), we have

$\displaystyle(c-x)^{3}$ $\displaystyle\sum_{0\leq m\leq
m_{k,1}}x^{m}\tbinom{m+k}{k}=c^{3}+x(c^{3}(k+1)-3c^{2})+x^{2}\big{(}c^{3}\tfrac{(k+1)(k+2)}{2}-3c^{2}(k+1)+3c\big{)}$
$\displaystyle\qquad+\sum_{0\leq m\leq
m_{k,1}}x^{m+3}b_{k,m}-(c^{3}-3c^{2}x+3cx^{2})x^{m_{k,1}+1}\tbinom{m_{k,1}+k+1}{k}$
$\displaystyle\qquad-(c^{3}-3c^{2}x)x^{m_{k,1}+2}\tbinom{m_{k,1}+k+2}{k}-c^{3}x^{m_{k,1}+3}\tbinom{m_{k,1}+k+3}{k},$

and

$\displaystyle(c-x)^{3}$ $\displaystyle\sum_{m_{k,2}\leq m\leq
m_{k,3}}x^{m}\tbinom{m+k}{k}=(c^{3}-3c^{2}x+3cx^{2})x^{m_{k,2}}\tbinom{m_{k,2}+k}{k}$
$\displaystyle\qquad+(c^{3}-3c^{2}x)x^{m_{k,2}+1}\tbinom{m_{k,2}+k+1}{k}+c^{3}x^{m_{k,2}+2}\tbinom{m_{k,2}+k+2}{k}$
$\displaystyle\qquad+\sum_{m_{k,2}\leq m\leq
m_{k,3}}x^{m+3}b_{k,m}-(c^{3}-3c^{2}x+3cx^{2})x^{m_{k,3}+1}\tbinom{m_{k,3}+k+1}{k}$
$\displaystyle\qquad-(c^{3}-3c^{2}x)x^{m_{k,3}+2}\tbinom{m_{k,3}+k+2}{k}-c^{3}x^{m_{k,3}+3}\tbinom{m_{k,3}+k+3}{k}.$

Taking $x=c$ in the two identities above, we get

$\displaystyle 0$
$\displaystyle=c^{3}+c(c^{3}(k+1)-3c^{2})+c^{2}\big{(}c^{3}\tfrac{(k+1)(k+2)}{2}-3c^{2}(k+1)+3c\big{)}$
$\displaystyle\qquad+\sum_{0\leq m\leq
m_{k,1}}c^{m+3}b_{k,m}-c^{3}f_{k}(m_{k,1}+1),$

and

$\displaystyle 0=c^{3}f_{k}(m_{k,2})+\sum_{m_{k,2}\leq m\leq
m_{k,3}}c^{m+3}b_{k,m}-c^{3}f_{k}(m_{k,3}+1),$

where $f_{k}$ is defined in (6.4). Then (6.8) becomes

$\displaystyle|\phi_{k}|(c)=2c^{3}(f_{k}(m_{k,1}+1)-f_{k}(m_{k,2})+f_{k}(m_{k,3}+1)),$

whence

$(1-c)^{k}|\phi_{k}|(c)=2c^{3}(h_{k}(m_{k,1}+1)-h_{k}(m_{k,2})+h_{k}(m_{k,3}+1)),$

where $h_{k}$ is defined in (6.5). Note that the two sequences
$\\{y_{k,4c,-}\\}$ and $\\{y_{k,4c,+}\\}$ are both
$\frac{ck}{1-c}+\mathcal{O}(k^{1/2})$. It follows from Lemma 6.6.(1) that the
three sequences $\\{m_{k,1}+1\\}$, $\\{m_{k,2}\\}$ and $\\{m_{k,3}+1\\}$ are
all $\frac{ck}{1-c}+\mathcal{O}(k^{1/2})$. Thus from Lemma 6.7 we conclude
that the three sequences $\\{h_{k}(m_{k,1}+1)\\},\\{h_{k}(m_{k,2})\\}$ and
$\\{h_{k}(m_{k,3}+1)\\}$ are all $\mathcal{O}(k^{-3/2})$. Therefore
$(1-c)^{k}|\phi_{k}|(c)=\mathcal{O}(k^{-3/2})$. ∎

We are ready to prove Lemma 6.3.

###### Proof of Lemma 6.3.

Let $k\in\mathbb{N}$. One has

$\displaystyle(1-q)^{-(k+1)}=\Bigl{(}\sum_{j=0}^{\infty}q^{j}\Bigr{)}^{k+1}=\sum_{m=0}^{\infty}q^{m}\sum_{\begin{subarray}{c}j_{1}+\dots+j_{k+1}=m\\\
j_{1},\dots,j_{k+1}\geq
0\end{subarray}}1=\sum_{m=0}^{\infty}q^{m}\tbinom{m+k}{k},$

whence

$(c-q)^{3}(1-q)^{-(k+1)}=\phi_{k}(q),$

where $\phi_{k}$ is defined in (6.6). Write $\phi_{k}$ as
$\sum_{m=0}^{\infty}\lambda_{m}x^{m}$ with $\lambda_{m}\in\mathbb{C}$. Then

$\displaystyle\|r^{k}(c-q)^{3}(1-q)^{-(k+1)}\|$
$\displaystyle=\Big{\|}\sum_{m=0}^{\infty}\lambda_{m}r^{k}q^{m}\Big{\|}\leq\sum_{m=0}^{\infty}|\lambda_{m}|\cdot\|r\|^{k}\|q\|^{m}$
$\displaystyle\leq\sum_{m=0}^{\infty}|\lambda_{m}|(1-c)^{k}c^{m}=(1-c)^{k}|\phi_{k}|(c).$

Now the assertion follows from Lemma 6.8. ∎

Lemma 6.3 implies the following proposition.

######  Proposition 6.9.

Let $\mathcal{A}$ be a unital Banach algebra such that
$\|ab\|\leq\|a\|\cdot\|b\|$ for all $a,b\in\mathcal{A}$. Let $0<c<1$ and
$q,r\in\mathcal{A}$ such that $\|r\|\leq 1-c$, $\|q\|\leq c$ and $qr=rq$. Then
there is some $a\in\mathcal{A}$ such that

$a(1-(q+r))=(1-(q+r))a=(c-q)^{3}.$ (6.9)

###### Proof.

Formally, the element $a$ in (6.9) is given by

$\displaystyle a$
$\displaystyle=(c-q)^{3}(1-(q+r))^{-1}=(c-q)^{3}((1-q)-r)^{-1}$
$\displaystyle=\smash[b]{(c-q)^{3}(1-q)^{-1}(1-r(1-q)^{-1})^{-1}=(c-q)^{3}(1-q)^{-1}\sum_{k=0}^{\infty}r^{k}(1-q)^{-k}}$
$\displaystyle=\sum_{k=0}^{\infty}r^{k}(c-q)^{3}(1-q)^{-(k+1)}.$

By Lemma 6.3, the last series in this expression for $a$ converges in norm, so
that $a\in\mathcal{A}$ is well defined. Since $qr=rq$, one has $qa=aq$,
$ra=ar$, and $(1-q)^{-1}r=r(1-q)^{-1}$. Thus

$\displaystyle a(1-(q+r))$ $\displaystyle=a(1-q)-ar$
$\displaystyle=\sum_{k=0}^{\infty}r^{k}(c-q)^{3}(1-q)^{-k}-\sum_{k=0}^{\infty}r^{k+1}(c-q)^{3}(1-q)^{-(k+1)}=(c-q)^{3},$

and similarly $(1-(q+r))a=(c-q)^{3}$. ∎

######  Example 6.10.

Let $\mathbb{H}$ be the discrete Heisenberg group with canonical generators
$u_{1},u_{2},u_{3}$ defined in (5.1), and let
$f=4-u_{1}-u_{1}^{-1}-u_{2}-u_{2}^{-1}\in\mathbb{Z}\mathbb{H}$. Then
$f=4(1-p)$, where
$p=\frac{1}{4}(u_{1}+u_{2}+u_{1}^{-1}+u_{2}^{-1})\in\mathbb{Q}\mathbb{H}$ can
be viewed as a symmetric probability measure on $\mathbb{H}$. The polynomial
$p^{4}\in\mathbb{Q}\mathbb{H}$ can again be viewed as a probability measure on
$\mathbb{H}$, and the coefficient $c\coloneq p^{4}_{u_{3}}$ of $p^{4}$ at
$u_{3}$ is strictly positive. If we set $q=c\cdot u_{3}$ and $r=p^{4}-q$, then
$qr=rq$ and Proposition 6.9 yields an element
$a\in\ell^{1}(\mathbb{H},\mathbb{R})$ such that

$a(1-p^{4})=a(1-(q+r))=c^{3}(1-u_{3})^{3}.$

Then $b=a(1+p+p^{2}+p^{3})/4c^{3}\in\ell^{1}(\mathbb{H},\mathbb{R})$ satisfies
that

$fb=bf=bf^{*}=4(1-p)b=(1-u_{3})^{3}.$ (6.10)

It follows that $b\in W_{f}$ and $\eta(b)\in\Delta^{1}(X_{f})$ (cf. (2.4) and
Proposition 5.3).

We remark in passing that the existence of such a homoclinic point $b$ was
conjectured in [12]*page 130; in [12]*Theorem 4.1.2 it was shown that there is
some $b^{\prime}\in\ell^{1}(\mathbb{H},\mathbb{R})$ satisfying
$b^{\prime}f^{*}=(1-u_{3})^{9}$.

Having found a nonzero element of $\Delta^{1}(X_{f})$ we claim that
$\Delta^{1}(X_{f})$ is actually dense in $X_{f}$. To verify this we give an
ad-hoc proof based on [Goell1]*Theorem 5.1: for every $x\in X_{f}$ there
exists a $y\in Y_{f}$ with $\eta(y)=x$ (for notation we refer to (2.1)). Then
$v\coloneqq\bar{\rho}^{f}y\in\\{-3,\dots,3\\}^{\mathbb{H}}\subset\ell^{\infty}(\mathbb{H},\mathbb{Z})$
(cf. (2.4)) and
$(\bar{\rho}^{f}\circ\bar{\rho}^{b})(v)=\bar{\rho}^{fb}v=\bar{\rho}^{(1-u_{3})^{3}}v\in\ell^{\infty}(\mathbb{H},\mathbb{Z})$.
It follows that $\bar{\rho}^{b}v\in W_{f}$ and
$(\eta\circ\bar{\rho}^{b})(v)=(\eta\circ\bar{\rho}^{bf})(y)=\rho^{bf}(\eta(y))=\rho^{(1-u_{3})^{3}}x$
by (6.10). We set
$\mathcal{V}=\\{-3,\dots,3\\}^{\mathbb{H}}\subset\ell^{\infty}(\mathbb{H},\mathbb{Z})$
and conclude that
$(\eta\circ\bar{\rho}^{b})(\mathcal{V})\supseteq\rho^{(1-u_{3})^{3}}(X_{f})$.

We recall that
$X_{f}=\widehat{\mathbb{Z}\mathbb{H}/(f)}=(f)^{\perp}\subset\widehat{\mathbb{Z}\mathbb{H}}$
(cf. (1.2)). If $\rho^{1-u_{3}}(X_{f})\subsetneq X_{f}$ there exists an
element $h\in\mathbb{Z}\mathbb{H}$ such that $h\notin(f)$ and $\langle
h,\rho^{1-u_{3}}x\rangle=\langle h(1-u_{3}),x\rangle=1$ for every $x\in
X_{f}$. Hence $h(1-u_{3})=(1-u_{3})h\in(f)$, i.e. $(1-u_{3})h=gf$ for some
$g\in\mathbb{Z}\mathbb{H}$.

We denote by $\langle u_{3}\rangle$ the subgroup of $\mathbb{H}$ generated by
$u_{3}$, set $\mathbb{H}^{\prime}=\mathbb{H}/\langle
u_{3}\rangle\cong\mathbb{Z}^{2}$, and denote by
$\pi\colon\mathbb{Z}\mathbb{H}\to\mathbb{Z}\mathbb{H}^{\prime}\cong\mathbb{Z}\mathbb{H}/(z_{3}-1)\mathbb{Z}\mathbb{H}$
the group ring homomorphism corresponding to the quotient map
$\mathbb{H}\to\mathbb{H}^{\prime}$. As $f$ is not divisible by $1-u_{3}$,
$\pi(f)\neq 0$, but $\pi(gf)=\pi(g)\pi(f)=0$. Since
$\mathbb{Z}\mathbb{H}^{\prime}\cong\mathbb{Z}\mathbb{Z}^{2}$ is an integral
domain we obtain that $\pi(g)=0$, i.e. that $g=(1-u_{3})g^{\prime}$ for some
$g^{\prime}\in\mathbb{Z}\mathbb{H}$. Since $\mathbb{Z}\mathbb{H}$ has no
nontrivial zero divisors (cf. e.g. [Passman]*Theorem 13.1.11) we obtain that
$h=g^{\prime}f$, contrary to our hypothesis that
$h\notin\mathbb{Z}\mathbb{H}f$.

This contradiction implies that
$X_{f}=\rho^{1-u_{3}}(X_{f})=\rho^{(1-u_{3})^{3}}(X_{f})=(\eta\circ\bar{\rho}^{b})(\mathcal{V})$.
Since $\eta\circ\bar{\rho}^{b}\colon\mathcal{V}\to X_{f}$ is continuous and
$\mathbb{Z}\mathbb{H}\cap\mathcal{V}$ is dense in $\mathcal{V}$ (both in the
product topology on $\mathcal{V}$) we conclude that
$(\eta\circ\bar{\rho}^{b})(\mathbb{Z}\mathbb{H})$ is dense in $X_{f}$. Finally
we note that
$(\eta\circ\bar{\rho}^{b})(\mathbb{Z}\mathbb{H})\subset\Delta^{1}(X_{f})$, so
that $\Delta^{1}(X_{f})$ is dense in $X_{f}$, as promised.

According to Proposition 5.3 this shows that $\lambda_{f}$ is intrinsically
ergodic.

The following corollary of Proposition 6.9 will allow us to extend the
argument in Example 6.10 to the much more general setting of Theorem 6.1.

######  Corollary 6.11.

Assume that $\Gamma$ is infinite and not virtually $\mathbb{Z}$ or
$\mathbb{Z}^{2}$. Let $p$ be a finitely supported symmetric probability
measure on $\Gamma$ such that $\textup{supp}(p)$ generates $\Gamma$. By a
result of Varopoulos (cf. [Varopoulos], [Furman]*Theorem 2.1, [Woess]*Theorem
3.24), $\sum_{j=0}^{\infty}p^{j}$ converges in $\|\cdot\|_{\infty}$ to some
$\omega$ in $C_{0}(\Gamma,\mathbb{R})$. Then
$(1-s)^{3}\omega\in\ell^{1}(\Gamma,\mathbb{R})$ for every $s$ in the center of
$\Gamma$.

###### Proof.

It is easily checked that

$\omega(1-p)=1.$

Let $s\neq 1_{\Gamma}$ be a central element of $\Gamma$. Then
$s\in\textup{supp}(p^{k})\smallsetminus\\{1_{\Gamma}\\}$ for some
$k\in\mathbb{N}$. As in Example 6.10 there is some $a$ in
$\ell^{1}(\Gamma,\mathbb{R})$ such that

$a(1-p^{k})=(1-s)^{3}.$

Then $b=a\sum_{j=0}^{k-1}p^{j}$ is in $\ell^{1}(\Gamma,\mathbb{R})$ and

$b(1-p)=(1-s)^{3}.$

Note that $(1-s)^{3}\omega\in C_{0}(\Gamma,\mathbb{R})$ and

$(1-s)^{3}\omega(1-p)=(1-s)^{3}=b(1-p).$

It is well known that if $x\in C_{0}(\Gamma,\mathbb{R})$ satisfies $x(1-p)=0$,
then $x=0$. Thus

$(1-s)^{3}\omega=b\in\ell^{1}(\Gamma,\mathbb{R}).$

∎

###### Proof of Theorem 6.1.

Since $f$ is well-balanced, one has $f=f_{1_{\Gamma}}(1-p)$ for some symmetric
probability measure $p$ on $\Gamma$ such that $\textup{supp}(p)$ generates
$\Gamma$.

Since $\Gamma$ is not virtually $\mathbb{Z}$ or $\mathbb{Z}^{2}$,
$\omega=\sum_{j=0}^{\infty}p^{j}$ is in $C_{0}(\Gamma,\mathbb{R})$. Then

$(f_{1_{\Gamma}}^{-1}\omega)f=1.$

By [Bowen-Li]*Theorem 4.1 and Lemma 4.10, the group $\Delta(X_{f})$ of
homoclinic points of $\lambda_{f}$ is dense in $X_{f}$ and is the
$\Gamma$-invariant subgroup of $X_{f}$ generated by
$\eta(f_{1_{\Gamma}}^{-1}\omega)$. Thus it suffices to show that
$\eta(f_{1_{\Gamma}}^{-1}\omega)$ is in the closure of $\Delta^{1}(X_{f})$.

By assumption we can find a sequence $(s_{n})_{n\geq 1}$ in the center $H$ of
$\Gamma$ such that for any finite subset $F$ of $\Gamma$ one has
$s_{n},s_{n}^{2},s_{n}^{3}\not\in F$ for all large enough $n$. Then
$\eta((1-s_{n})^{3}f_{1_{\Gamma}}^{-1}\omega)$ converges to
$\eta(f_{1_{\Gamma}}^{-1}\omega)$ as $n\to\infty$. By Corollary 6.11 one has
$(1-s_{n})^{3}f_{1_{\Gamma}}^{-1}\omega\in\ell^{1}(\Gamma,\mathbb{R})$ and
hence $\eta((1-s_{n})^{3}f_{1_{\Gamma}}^{-1}\omega)\in\Delta^{1}(X_{f})$ for
each $n$. Therefore $\eta(f_{1_{\Gamma}}^{-1}\omega)$ lies in the closure of
$\Delta^{1}(X_{f})$, as desired. ∎

######  Remarks 6.12.

(1) For $\Gamma=\mathbb{Z}^{d}$ with $d\geq 1$, [LSV2]*Corollary 3.4 shows
that any atoral polynomial $f\in\mathbb{Z}[\mathbb{Z}^{d}]$ which is not a
unit in $\mathbb{Z}[\mathbb{Z}^{d}]$ satisfies that $\Delta^{1}(X_{f})$ is
dense in $X_{f}$ (for the definition of atorality we refer to
[LSV2]*Definition 2.1 and Proposition 2.2). In particular, Theorem 6.1 also
holds for $\Gamma=\mathbb{Z}^{2}$. Does Theorem 6.1 hold for virtually
$\mathbb{Z}^{2}$?

(2) For the polynomial $h=2-u_{1}-u_{2}\in\mathbb{Z}\mathbb{H}$ there exists a
nonzero element $w\in\ell^{1}(\mathbb{H},\mathbb{R})$ such that
$wh=hw=(1-u_{3})^{2}$ (cf. [Goell1]*Theorem 4.2). As in Subsection 5.2.1, the
map
$\bar{\rho}^{w}\colon\ell^{\infty}(\mathbb{H},\mathbb{Z})\to\ell^{\infty}(\mathbb{H},\mathbb{R})$
is continuous in the weak∗-topology on closed, bounded subsets of
$\ell^{\infty}(\mathbb{H},\mathbb{Z})$, and the map
$\xi\coloneqq\eta\circ\bar{\rho}^{w}\colon\ell^{\infty}(\mathbb{H},\mathbb{Z})\to\mathbb{T}^{\mathbb{H}}$
satisfies that $\xi(\\{-1,0,1\\}^{\mathbb{H}})=X_{h}$ (cf. [Goell1]*Theorem
5.1). It follows that $\xi(\mathbb{Z}\mathbb{H})\subset\Delta^{1}(X_{h})$.
Hence $\Delta^{1}(X_{h})$ is dense in $X_{h}$ and $\lambda_{h}$ is
intrinsically ergodic.

Is there any way to use an argument similar to Proposition 6.9 to prove the
existence of summable homoclinic points for this and other ‘asymmetric’
elements of $h\in\mathbb{Z}\mathbb{H}$ ?

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