# Invariant measures of Toeplitz subshifts on non-amenable groups

Paulina Cecchi-Bernales  Departamento de Matemáticas, Facultad de Ciencias,
Universidad de Chile. Campus Juan Gómez Millas. Las Palmeras 3425, Ñuñoa,
Chile<EMAIL_ADDRESS>,  María Isabel Cortez  Facultad de Matemáticas,
Pontificia Universidad Católica de Chile. Edificio Rolando Chuaqui, Campus San
Joaquín. Avda. Vicuña Mackenna 4860, Macul, Chile<EMAIL_ADDRESS>and
Jaime Gómez  Facultad de Matemáticas, Pontificia Universidad Católica de
Chile. Edificio Rolando Chuaqui, Campus San Joaquín. Avda. Vicuña Mackenna
4860, Macul, Chile<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a countable residually finite group (for instance
${\mathbb{F}}_{2}$) and let $\overleftarrow{G}$ be a totally disconnected
metric compactification of $G$ equipped with the action of $G$ by left
multiplication. For every $r\geq 1$ we construct a Toeplitz $G$-subshift
$(X,\sigma,G)$, which is an almost one-to-one extension of
$\overleftarrow{G}$, having $r$ ergodic measures $\nu_{1},\cdots,\nu_{r}$ such
that for every $1\leq i\leq r$ the measure-theoretic dynamical systems
$(X,\sigma,G,\nu_{i})$ is isomorphic to $\overleftarrow{G}$ endowed with the
Haar measure. The construction we propose is general (for amenable and non
amenable residually finite groups), however, we point out the differences and
obstructions that could appear when the acting group is not amenable.

###### Key words and phrases:

Invariant measures, Toeplitz subshifts, residually finite group

###### 2010 Mathematics Subject Classification:

Primary: 37B05, 37B10

P. Cecchi-Bernales was supported by ANID/Proyecto Fondecyt Postdoctorado No.
3210746 and ANID/ECOS 210033

The research of M.I. Cortez was supported by proyecto Fondecyt Regular No.
1190538.

J. Gómez was supported by ANID/ Doctorado Nacional No. 21200054

##  1\. Introduction

A classical problem in topological dynamics is to describe the set of
invariant measures of a given dynamical system, which represents all possible
measure-theoretic dynamical systems housed by it. This set is known to be a
Choquet simplex whose extreme points correspond to the ergodic measures of the
system, and it is known to be non-empty provided the phase space is compact
and the acting group is amenable (see [17]). Indeed, amenable groups can be
characterized as those whose continuous actions on the Cantor set always admit
invariant measures [14]. Recently, in [13] has been shown that the amenability
of a group can be tested within the class of subshifts, i.e, the symbolic
continuous actions on the Cantor set. Thus, for non amenable groups it is
reasonable to wonder about the existence of invariant measures for specific
classes of subshifts.

The problem of describing the set of invariant measures of dynamical systems
given by the actions of amenable groups, has been extensively explored for the
class of Toeplitz subshifts. These subshifts were introduced by Jacobs and
Keane in the context of ${\mathbb{Z}}$-actions [20] and characterized as the
minimal symbolic almost 1-1 extensions of odometers in [11]. In [26], Williams
demonstrated that this class of ${\mathbb{Z}}$-minimal subshifts can have any
finite number of ergodic measures, or countably many, or uncountably many of
them. Later, Downarowicz [8] generalized William’s results by showing that any
Choquet simplex can be realized as the set of invariant measures of a Toeplitz
subshift in $\\{0,1\\}^{{\mathbb{Z}}}$, up to affine homeomorphism. Toeplitz
${\mathbb{Z}}$-subshifts have since been widely studied for the rich variety
of possible phenomena they can exhibit, both regarding topological and
ergodic-theoretic properties (see for instance [9], [19], [10], [16], [25],
[4]).

Toeplitz subshifts have been generalized from ${\mathbb{Z}}$-actions to more
general group actions in [5], [6], [21] and [22]. One of the consequences of
this generalization is the characterization of residually finite groups as
those ones that admit (non periodic) Toeplitz subshifts [6, 21]. In [6] the
authors broadened the characterization of Toeplitz subshifts as the minimal
symbolic almost 1-1 extensions of odometers to actions of residually finite
groups. Downarowicz’s result about the realization of Choquet simplices as
sets of invariant measures of Toeplitz subshifts was extended in [7] to the
class of residually finite amenable groups. This realization result has also
been proved in [3] for a special class of amenable groups, namely congruent
monotileable amenable groups, that contains as a non trivial subset the class
of residually finite groups. As the geometry of the set of invariant measures
of a topological dynamical system is an invariant under topological orbit
equivalence ([15]), the problem of realizing Choquet simplices among the
actions of a prescribed group, can be interpreted as a way to estimate the
size of the family of topological dynamical systems given by the actions of
this group up to orbit equivalence.

Results about the existence of invariant measures for continuous actions of
non amenable groups can be found, for example, in [18] and [12]. In this paper
we focus on the existence of invariant measures of Toeplitz subshifts given by
actions of countable residually finite groups which are not necessarily
amenable. This encompases all countable residually finite groups having an
isomorphic copy of the free group on $n\geq 2$ generators, ${\mathbb{F}}_{n}$.
To the best of our knowledge, the previous studies concerning the set of
invariant measures of Toeplitz subshifts, strongly rely on the structure of
$\mathbb{Z}$ or on the existence of Følner sequences, which do not exist in
the case of non amenable groups. The lack of Følner sequences is one of the
main challenges addressed in this work. Our main results are the following:

######  Theorem 1.1.

Let $G$ be a countable residually finite group and let $\overleftarrow{G}$ be
a totally disconnected metric compactification of $G$ equipped with the action
of $G$ by left multiplication. Then there exists a regular Toeplitz
$G$-subshift whose maximal equicontinuous factor is $\overleftarrow{G}$.

Theorem 1.2 is a direct consequence of Theorem 1.1.

######  Theorem 1.2.

Let $G$ be a countable residually finite group and let $\overleftarrow{G}$ be
a totally disconnected metric compactification of $G$ equipped with the action
of $G$ by left multiplication. Then there exists a uniquely ergodic Toeplitz
$G$-subshift $(X,\sigma,G)$ and an almost 1-1 factor map
$\pi:X\to\overleftarrow{G}$, such that if $\nu$ is the unique ergodic
probability measure of $(X,\sigma,G)$ then $\pi$ is a measure conjugacy
between $(X,\sigma,G,\nu)$ and $\overleftarrow{G}$ endowed with the Haar
measure.

######  Theorem 1.3.

Let $G$ be a countable residually finite group and let $\overleftarrow{G}$ be
a totally disconnected metric compactification of $G$ equipped with the action
of $G$ by left multiplication. For every integer $r>1$ there exists a Toeplitz
$G$-subshift $X\subseteq\\{1,\cdots,r\\}^{G}$ with at least $r$ ergodic
probability measures $\nu_{1},\cdots,\nu_{r}$, and whose maximal
equicontinuous factor is $\overleftarrow{G}$. Furthermore, for every $1\leq
i\leq r$, we have the following:

  1. (1)

$(X,\sigma,G,\nu_{i})$ is measure conjugate to $\overleftarrow{G}$ endowed
with the Haar measure.

  2. (2)

$\nu_{i}(\\{x\in X:x(1_{G})=i\\})\geq\mu(\\{x\in X:x(1_{G})=i\\})$ for every
invariant probability measure $\mu$.

The document is organized as follows. In Section 2 we give basic notions
concerning topological and measure-theoretic dynamical systems, as well as the
basic background on residually finite groups, $G$-odometers and Toeplitz
subshifts. Section 3 is devoted to prove the existence of uniquely ergodic
Toepliz subshifts for arbitrary residually finite groups (Theorem 1.2). In
particular, we give a necessary and sufficient condition for a Toeplitz array
to be regular. In Section 4 we define a sequence of periodic measures of the
full shift whose accumulation points are supported in a Toeplitz subshift $X$
that we introduce at the beginning of the section. We use this to show that
$X$ has at least $r$ ergodic measures. Finally, Section 5 is devoted to
complete the proof of Theorem 1.3.

##  2\. Preliminaries

###  2.1. Topological dynamical systems and invariant measures

Let $G$ be a countable discrete infinite group. We denote by $1_{G}$ the
identity of $G$. By a topological dynamical system we mean a continuous (left)
action $\phi$ of $G$ on a compact metric space $X$. We denote this topological
dynamical system by $(X,\phi,G)$. We say that $(X,\phi,G)$ is free if
$\phi^{g}x=x$ implies $g=1_{G}$, for every $x\in X$. The system is minimal if
for every $x\in X$ its orbit $O_{\phi}(x)=\\{\phi^{g}x:g\in G\\}$ is dense in
$X$. The system is equicontinuous if the collection of maps
$\\{\phi^{g}\\}_{g\in G}$ is equicontinuous. If $X$ is a Cantor set, we say
that the topological dynamical system is a Cantor system.

An invariant measure of the topological dynamical system $(X,\phi,G)$, is a
probability measure $\mu$ defined on the Borel sigma-algebra of $X$, that
verifies $\mu(\phi^{g}A)=\mu(A)$, for every $g\in G$ and every Borel set $A$.
An invariant measure $\mu$ is said to be ergodic if $\mu(A)\in\\{0,1\\}$
whenever $A$ is a Borel set verifying $\phi^{g}(A)=A$ for all $g\in G$. The
set of all invariant measures admitted by a system $(X,\phi,G)$ is denoted
${\mathcal{M}}(X,\phi,G)$. It is known to be a Choquet simplex whose extreme
points correspond to the ergodic measures of $(X,\phi,G)$ (We refer to [17,
Chapter 4] for details). When the set ${\mathcal{M}}(X,\phi,G)$ is a
singleton, the system is said to be uniquely ergodic. If $\mu$ is an invariant
measure of $(X,\phi,G)$, the quadruple $(X,\phi,G,\mu)$ is called a
probability-measure-preserving (p.m.p) dynamical system.

A factor map from $(X,\phi,G)$ to the topological dynamical system
$(Y,\varphi,G)$ is a continuous surjective map $\pi:X\to Y$ such that
$\pi(\phi^{g}x)=\varphi^{g}\pi(x)$, for every $x\in X$ and $g\in G$. In this
case we say that $(X,\phi,G)$ is an extension of $(Y,\varphi,G)$ and
$(Y,\varphi,G)$ is a factor of $(X,\phi,G)$. The factor map $\pi$ is almost
one to one (or almost 1-1) if the set of points in $Y$ having only one
preimage is residual. If the system $(Y,\varphi,G)$ is minimal, then this is
equivalent to the existence of $y\in Y$ such that $|\pi^{-1}\\{y\\}|=1$. If
$\pi$ is an almost 1-1 factor map, then we say that $(X,\phi,G)$ is an almost
1-1 extension of $(Y,\varphi,G)$.

An equicontinuous system $(Y,\varphi,G)$ is said to be the maximal
equicontinuous factor of $(X,\phi,G)$ if there is a factor map $\pi\colon X\to
Y$ such that for any other map $f\colon X\to Y^{\prime}$, with
$(Y,\varphi^{\prime},G)$ equicontinuous, there exists a factor map $q\colon
Y\to Y^{\prime}$ that satisfies $q\circ\pi=f$. Moreover, if $(X,\phi,G)$ is a
minimal almost $1$-$1$ extension of a minimal equicontinuous system
$(Y,\varphi,G)$, then $(Y,\varphi,G)$ is the maximal equicontinuous factor of
$(X,\phi,G)$ (See [22, Proposition 5.5.]).

Two p.m.p dynamical systems $(X,\phi,G,\mu)$ and $(Y,\varphi,G,\nu)$ are
measure conjugate if: (i) there exist conull sets $X^{\prime}\subseteq X$ and
$Y^{\prime}\subseteq Y$ satisfying $\phi^{g}(X^{\prime})\subseteq X^{\prime}$
and $\varphi^{g}(Y^{\prime})\subseteq Y^{\prime}$ for all $g\in G$, and (ii)
there is a bijective map $f:X^{\prime}\to Y^{\prime}$ which verifies
$f,f^{-1}$ are both measurable, $\nu(A)=\mu(f^{-1}(A))$ for every measurable
set $A\subseteq Y^{\prime}$ and $f(\phi^{g}(x))=\varphi^{g}(f(x))$ for all
$x\in X^{\prime}$ and $g\in G$. In this case, we say that $f$ is a measure
conjugacy.

###  2.2. $G$-odometers, residually finite groups and compactifications.

A countable group $G$ is said to be residually finite, if there exists a
nested sequence of finite index subgroups of $G$ with trivial intersection.
This is equivalent to the existence of a sequence of normal subgroups with the
same characteristics (see [2] for more details about residually finite
groups). Suppose that $G$ is an infinite residually finite group and let
$(\Gamma_{n})_{n\in\mathbb{N}}$ be a nested sequence of finite index subgroups
of $G$ such that $\bigcap_{n\in\mathbb{N}}\Gamma_{n}=\\{1_{G}\\}$. The
$G$-odometer associated to $(\Gamma_{n})_{n\in\mathbb{N}}$ is defined as

$\displaystyle\overleftarrow{G}:=\varprojlim(G/\Gamma_{n},\varphi_{n})=\\{(x_{n})_{n\in\mathbb{N}}\in\prod_{n\in\mathbb{N}}G/\Gamma_{n}:\varphi_{n}(x_{n+1})=x_{n},\mbox{
for every }n\in\mathbb{N}\\},$

where $\varphi_{n}\colon G/\Gamma_{n+1}\to G/\Gamma_{n}$ is the canonical
projection, for every $n\in\mathbb{N}$. The space $\overleftarrow{G}$ is a
Cantor set if we endow every $G/\Gamma_{n}$ with the discrete topology,
$\prod_{n\in\mathbb{N}}G/\Gamma_{n}$ with the product topology and
$\overleftarrow{G}$ with the induced topology. Observe that when the groups
$\Gamma_{n}$’s are normal, $\overleftarrow{G}$ is a subgroup of
$\prod_{n\in\mathbb{N}}G/\Gamma_{n}$, and $G$ can be seen as a dense subgroup
of $\overleftarrow{G}$ identifying $g\in G$ with
$(g\Gamma_{n})_{n\in\mathbb{N}}\in\overleftarrow{G}$. There is a natural
action $\phi$ of $G$ on $\overleftarrow{G}$ by coordinatewise left
multiplication. The topological dynamical system $(\overleftarrow{G},\phi,G)$
is a free equicontinuous minimal Cantor system which is also known as the
$G$-odometer associated to the sequence $(\Gamma_{n})_{n\in{\mathbb{N}}}$. It
is important to note that the $G$-odometers associated to subsequences of
$(\Gamma_{n})_{n\in\mathbb{N}}$ are conjugate as dynamical systems.

A $G$-odometer having a group structure (when the $\Gamma_{n}$’s are normal)
is a totally disconnected metric compactification of $G$, i.e, this is a
totally disconnected metric compact group $\overleftarrow{G}$ for which there
exists an injective homomorphism $i:G\to\overleftarrow{G}$ such that $i(G)$ is
dense in $\overleftarrow{G}$. Conversely, every totally disconnected metric
compactification of $G$ is a $G$-odometer, as the next lemma shows.

######  Lemma 2.1.

If $\overleftarrow{G}$ is a totally disconnected metric compactification of
$G$, then there exists a nested sequence $(\Gamma_{n})_{n\in\mathbb{N}}$ of
finite index normal subgroups of $G$ with trivial intersection, such that the
$G$-odometer associated to $(\Gamma_{n})_{n\in\mathbb{N}}$ is conjugate to
$\overleftarrow{G}$ equipped with the action of $G$ by left multiplication.
This implies that $G$ is residually finite if and only if $G$ has a totally
disconnected metric compactification.

###### Proof.

If $G$ is residually finite, then any $G$-odometer is a totally disconnected
metric compactification of $G$. Now, suppose that $\overleftarrow{G}$ is a
totally disconnected metric compactification of $G$. We can identify $G$ with
a dense subgroup of $\overleftarrow{G}$. Since $\overleftarrow{G}$ is compact
and totally disconnected, $\overleftarrow{G}$ is profinite (see for example
[24, Theorem 1.1.12]). The metrizability of $\overleftarrow{G}$ implies that
$\overleftarrow{G}$ is a countable inverse limit of finite groups ([24,
Corollary 1.1.13 and Remark 2.6.7]), namely
$\overleftarrow{G}=\varprojlim(G_{n},\tau_{n})$. For every $n\in\mathbb{N}$,
let $\pi_{n}:\overleftarrow{G}\to G_{n}$ be the natural projection. The group
$\Gamma_{n}={\rm Ker}(\pi_{n})\cap G$ is a finite index subgroup of $G$.
Indeed, the map $\phi_{n}:G_{n}\to G/\Gamma_{n}$ given by
$\phi_{n}(a)=\\{(g_{j})_{j\in\mathbb{N}}\in G:g_{n}=a\\}$ is a well defined
isomorphism. Since $\varphi_{n}\circ\phi_{n+1}=\phi_{n}\circ\tau_{n}$, for
every $n\in\mathbb{N}$, we deduce that $\overleftarrow{G}$ is conjugate to the
odometer associated with the sequence $(\Gamma_{n})_{n\in\mathbb{N}}$. ∎

###  2.3. Toeplitz $G$-subshifts

Let $\Sigma$ be a finite set with at least two elements. The set

$\Sigma^{G}=\\{x=(x(g))_{g\in G}:x(g)\in\Sigma,\mbox{ for every }g\in G\\}$

is a Cantor set if we endow $\Sigma$ with the discrete topology and
$\Sigma^{G}$ with the product topology. The (left) shift action $\sigma$ of
$G$ on $\Sigma^{G}$ is defined as

$\sigma^{g}x(h)=x(g^{-1}h),\mbox{ for every }h,g\in G\mbox{ and
}x\in\Sigma^{G}.$

The topological dynamical system $(\Sigma^{G},\sigma,G)$ is a Cantor system
known as the full G-shift. If $X\subseteq\Sigma^{G}$ is a closed
$\sigma$-invariant set, we say that $X$ is a subshift. The system
$(X,\sigma|_{X},G)$, given by the restriction of $\sigma$ on $X$, is also
called a subshift (see for example [2], for more details).

The definitions and statements written in the rest of this section can be
found in [6]. We include some of the proofs for the sake of completeness.

Let $x\in\Sigma^{G}$ and let $\Gamma\subseteq G$ be a subgroup of finite
index. We define

$\displaystyle{\rm Per}(x,\Gamma,\alpha)$ $\displaystyle=\\{g\in G\colon
x(\gamma g)=\alpha\mbox{ for every }\gamma\in\Gamma\\},\mbox{ for every
}\alpha\in\Sigma.$ $\displaystyle{\rm Per}(x,\Gamma)$
$\displaystyle=\bigcup_{\alpha\in\Sigma}{\rm Per}(x,\Gamma,\alpha).$

The elements of ${\rm Per}(x,\Gamma)$ are those belonging to some coset
$\Gamma g$ for which $x$ restricted to $\Gamma g$ is constant.

######  Remark 2.2.

Observe that if $\Gamma$ is normal then $g\in{\rm Per}(x,\Gamma)$ if and only
if $x(g)=x(\gamma g)=x(g\gamma)$ for every $\gamma\in\Gamma$.

An element $\eta\in\Sigma^{G}$ is a Toeplitz array or a Toeplitz element, if
for every $g\in G$ there exists a finite index subgroup $\Gamma$ of $G$ such
that $g\in{\rm Per}(\eta,\Gamma)$. The finite index subgroup $\Gamma$ is a
group of periods of $\eta$ if ${\rm Per}(\eta,\Gamma)\neq\emptyset$. Observe
that for every $g\in G$, for every $\alpha\in\Sigma$ and every group of
periods $\Gamma$ of $\eta$ we have $g{\rm Per}(\eta,g^{-1}\Gamma
g,\alpha)={\rm Per}(\sigma^{g}\eta,\Gamma,\alpha)$. A group of periods
$\Gamma$ of $\eta$ is an essential group of periods of $\eta$ if ${\rm
Per}(\eta,\Gamma,\alpha)\subseteq{\rm Per}(\sigma^{g}\eta,\Gamma,\alpha)$ for
every $\alpha\in\Sigma$, implies $g\in\Gamma$.

######  Lemma 2.3.

Let $\eta\in\Sigma^{G}$ be a Toeplitz array. For every group of periods
$\Gamma$ of $\eta$, there exists an essential group of periods $K$ of $\eta$
such that ${\rm Per}(\eta,\Gamma,\alpha)\subseteq{\rm Per}(\eta,K,\alpha)$,
for every $\alpha\in\Sigma$.

###### Proof.

Let $\Gamma$ be a group of periods of $\eta$. There exists a normal finite
index subgroup $H$ of $G$ such that $H\subseteq\Gamma$ (see for example [6,
Lemma 1]). Observe that ${\rm Per}(\eta,\Gamma,\alpha)\subseteq{\rm
Per}(\eta,H,\alpha)$, for every $\alpha\in\Sigma$. Since $w\in{\rm
Per}(\eta,H,\alpha)$ if and only if $Hw\subseteq{\rm Per}(\eta,H,\alpha),$ the
set ${\rm Per}(\eta,H,\alpha)$ is a disjoint union of equivalence classes in
$G/H$. Namely, ${\rm Per}(\eta,H,\alpha)=Hw_{1}\cup\cdots\cup Hw_{m}$. Thus
for every $g\in G$, the set $g{\rm Per}(\eta,H,\alpha)$ is also a disjoint
union of the same number of equivalence classes. This implies that if ${\rm
Per}(\eta,H,\alpha)\subseteq g{\rm Per}(\eta,g^{-1}Hg,\alpha)=g{\rm
Per}(\eta,H,\alpha)$ then ${\rm Per}(\eta,H,\alpha)=g{\rm
Per}(\eta,H,\alpha)$. From this it follows that $K$, the set of all $g\in G$
such that ${\rm Per}(\eta,H,\alpha)\subseteq g{\rm Per}(\eta,g^{-1}Hg,\alpha)$
for every $\alpha\in\Sigma$, is a group. Since $K$ contains $H$, $K$ is a
finite index subgroup. Furthermore, if $w\in{\rm Per}(\eta,H,\alpha)$ then
$gw\in{\rm Per}(\eta,H,\alpha)$, for every $g\in K$, which implies that ${\rm
Per}(\eta,H,\alpha)\subseteq{\rm Per}(\eta,K,\alpha)$, for every
$\alpha\in\Sigma$, because $\alpha=\eta(w)=\eta(gw)$, for every $g\in K$. On
the other hand, if ${\rm Per}(\eta,K,\alpha)\subseteq g{\rm
Per}(\eta,g^{-1}Kg,\alpha)$, then for $w\in{\rm Per}(\eta,H,\alpha)$ we have
$g^{-1}w\in{\rm Per}(\eta,g^{-1}Kg,\alpha)$, which implies
$\alpha=\eta(g^{-1}w)=\eta(g^{-1}kw)$, for every $k\in K$. Since $H\subseteq
K$ is a normal subgroup, in particular we have $\alpha=\eta(hg^{-1}w)$, for
every $h\in H$. This implies $g^{-1}w\in{\rm Per}(\eta,H,\alpha)$ and then
$g\in K$. This shows that $K$ is an essential group of periods. ∎

If $\eta\in\Sigma^{G}$ is a Toeplitz array, then there exists a period
structure $(\Gamma_{n})_{n\in\mathbb{N}}$ of $\eta$, that is, a nested
sequence of essential periods of $\eta$ such that
$G=\bigcup_{n\in\mathbb{N}}\Gamma_{n}$ (see [6, Corollary 6]).

A Toeplitz $G$-subshift or simply a Toeplitz subshift, is the subshift
generated by the closure of the $\sigma$-orbit of a Toeplitz array.

Let $\eta\in\Sigma^{G}$ be a Toeplitz array and let
$(\Gamma_{n})_{n\in\mathbb{N}}$ be a period structure of $\eta$. Let
$X=\overline{O_{\sigma}(\eta)}$ be the associated Toeplitz subshift. For each
$n\in\mathbb{N}$, we define

$C_{n}=\\{x\in X:{\rm Per}(x,\Gamma_{n},\alpha)={\rm
Per}(\eta,\Gamma_{n},\alpha),\mbox{ for every }\alpha\in\Sigma\\}.$

Using the fact that every $\Gamma_{n}$ is an essential group of periods, it is
possible to verify that $\sigma^{g}\eta\in C_{n}$ if and only if
$g\in\Gamma_{n}$. This implies following result.

######  Lemma 2.4.

([6, Lemma 8 and Proposition 6]) For every $n\in\mathbb{N}$, the set $C_{n}$
is closed. In addition, for every $g,h\in G$ the following statements are
equivalent:

  1. (1)

$\sigma^{g}C_{n}\cap\sigma^{h}C_{n}\neq\emptyset.$

  2. (2)

$\sigma^{g}C_{n}=\sigma^{h}C_{n}.$

  3. (3)

$g\Gamma_{n}=h\Gamma_{n}$.

From this we get that the collection $\\{\sigma^{g^{-1}}C_{n}:g\in D_{n}\\}$
is a clopen partition of $X$.

###### Proof.

Let denote
$X_{n}=\overline{\\{\sigma^{\gamma}\eta\colon\gamma\in\Gamma_{n}\\}}$, for
each $n\in\mathbb{N}$. It is straightforward to check that ${\rm
Per}(\eta,\Gamma_{n},\alpha)\subseteq{\rm Per}(x,\Gamma_{n},\alpha)$, for
every $x\in X_{n}$ and $\alpha\in\Sigma$. Let $H\subseteq\Gamma_{n}$ be a
normal finite index subgroup of $G$. From [1, Theorem 1.13], for every $x\in
X$ the action of $H$ on $\overline{\\{\sigma^{h}x\colon h\in H\\}}$ is
minimal. Since $H$ is a finite index subgroup of $\Gamma_{n}$, applying again
[1, Theorem 1.13], we get that for every $x\in X$ the action of $\Gamma_{n}$
on $\overline{\\{\sigma^{\gamma}x\colon\gamma\in\Gamma_{n}\\}}$ is minimal. In
particular, the action of $\Gamma_{n}$ on $X_{n}$ is minimal. From this we
have ${\rm Per}(x,\Gamma_{n},\alpha)\subseteq{\rm
Per}(\eta,\Gamma_{n},\alpha)$ for every $x\in X_{n}$ and $\alpha\in\Sigma$,
which implies that $X_{n}\subseteq C_{n}$. On the other hand, let $x\in C_{n}$
and $(g_{i})_{i\in\mathbb{N}}$ be a sequence in $G$ such that
$(\sigma^{g_{i}}x)_{i\in\mathbb{N}}$ converges to $\eta$. Since $\Gamma_{n}$
is a finite index subgroup, taking subsequences, there exists $w\in G$ such
that $g_{i}=w\gamma_{i}$ for some $\gamma_{i}\in\Gamma_{n}$, for every
$i\in\mathbb{N}$. This implies that $\sigma^{w^{-1}}\eta$ is in
$Y=\overline{\\{\sigma^{\gamma}x:\gamma\in\Gamma_{n}\\}}$ and then ${\rm
Per}(\eta,\Gamma_{n},\alpha)\subseteq{\rm
Per}(\sigma^{w^{-1}}\eta,\Gamma_{n},\alpha)$ for every $\alpha\in\Sigma$. The
property of being essential implies that $w\in\Gamma_{n}$ and then $\eta\in
X_{n}\cap Y$. Since $Y$ is a minimal component with respect to the action of
$\Gamma_{n}$, we deduce that $Y=X_{n}$ and then $C_{n}=X_{n}$. Observe that
$X$ is a finite disjoint union of minimal components for the action of
$\Gamma_{n}$ (If $w_{1},\cdots,w_{k}$ are representatives of each coset in
$\\{g\Gamma_{n}:g\in G\\}$, then $X$ is the union of the minimal components
that contain the points $\sigma^{w_{1}}\eta,\cdots,\sigma^{w_{k}}\eta$), this
implies that $X_{n}=C_{n}$ is clopen. From the property of being an essential
period, it follows that $\sigma^{g}\eta\in C_{n}$ implies $g\in\Gamma_{n}$.
From the minimality of the action of $G$ and the fact that $C_{n}$ is open, we
can infer that if $\sigma^{g}x\in C_{n}$, then $g\in\Gamma_{n}$, for every
$x\in C_{n}$. This implies the results of the Lemma. ∎

######  Remark 2.5.

Observe that the stabilizer of $x\in X$ is the group
$\bigcap_{n\in\mathbb{N}}v_{n}\Gamma_{n}v_{n}^{-1}$, where $v_{n}\in G$ is
such that $x\in\sigma^{v_{n}}C_{n}$. Thus if the groups $\Gamma_{n}$ are
normal subgroups of $G$, the Toeplitz subshift $(X,\sigma,G)$ is free if and
only if $\bigcap_{n\in\mathbb{N}}\Gamma_{n}=\\{1_{G}\\}$.

######  Proposition 2.6.

([6, Proposition 5, Theorem 2 and Proposition 7]) Let $\eta\in\Sigma^{G}$ be a
Toeplitz array and let $X=\overline{\\{\sigma^{g}\eta:g\in G\\}}$. Suppose
that $(X,\sigma|_{X},G)$ is free and $(\Gamma_{n})_{n\in\mathbb{N}}$ is a
period structure of $\eta$. Let $\overleftarrow{G}$ be the $G$-odometer
associated to $(\Gamma_{n})_{n\in\mathbb{N}}$. The map
$\pi:X\to\overleftarrow{G}$ given by

$\pi(x)=(g_{n}\Gamma_{n})_{n\in\mathbb{N}},\mbox{ where
}x\in\sigma^{g_{n}}C_{n},\mbox{ for every }n\in\mathbb{N}$

is an almost 1-1 factor map (then $\overleftarrow{G}$ is the maximal
equicontinuous factor of $X$). Moreover,

$\\{x\in X:x\mbox{ is a Toeplitz array
}\\}=\pi^{-1}\\{y\in\overleftarrow{G}:|\pi^{-1}\\{y\\}|=1\\}.$

###  2.4. Good sequences of tiles for residually finite groups

In [7] it was shown that amenable residually finite groups have Følner
sequences with additional properties. We prove in Lemmas 2.7 and 2.8 that when
the residually finite group $G$ is not amenable, it has sequences of finite
subsets having similar properties, without being Følner.

######  Lemma 2.7.

Let $G$ be a residually finite group, and let $(\Gamma_{n})_{n\in\mathbb{N}}$
be a strictly decreasing sequence of finite index subgroups of $G$, such that
$\bigcap_{n\in\mathbb{N}}\Gamma_{n}=\\{1_{G}\\}.$ There exists an increasing
sequence $(n_{i})_{i\in\mathbb{N}}\subseteq\mathbb{N}$ and
$(D_{i})_{i\in\mathbb{N}}$ a sequence of finite subsets of $G$ such that for
every $i\in\mathbb{N}$

  1. (1)

$1_{G}\in D_{1}$ and $D_{i}\subseteq D_{i+1}$.

  2. (2)

$D_{i}$ is a fundamental domain of $\Gamma_{n_{i}}$.

  3. (3)

$G=\bigcup_{i\in\mathbb{N}}D_{i}$.

  4. (4)

$D_{i+1}=\bigcup_{v\in D_{i+1}\cap\Gamma_{n_{i}}}vD_{i}$.

###### Proof.

Observe that if $S\subseteq G$ is a finite set, then there exists $l\geq 1$
such that if $g_{1}$ and $g_{2}$ are two different elements in $S$, then
$g_{1}\Gamma_{l}\neq g_{2}\Gamma_{l}$. Indeed, suppose there exists a finite
subset $S$ of $G$ such that for every $l\geq 1$ there are two different
elements $g_{l}$ and $h_{l}$ in $S$ such that
$g_{l}\Gamma_{l}=h_{l}\Gamma_{l}$. Since $S$ is finite, there exists a
subsequence $(\Gamma_{l_{i}})_{i\in\mathbb{N}}$ such that $g_{l_{i}}=g$ and
$h_{l_{i}}=h$, for every $i\in\mathbb{N}$, where $g,h\in G$. This implies that
$h^{-1}g\in\bigcap_{i\in\mathbb{N}}\Gamma_{l_{i}}=\bigcap_{i\geq
0}\Gamma_{i}=\\{1_{G}\\}$, which is a contradiction with the fact that $g\neq
h$.

Suppose $G=\\{g_{1},g_{2},\ldots\\}$ with $1_{G}=g_{1}$. We will construct a
sequence $(F_{i})_{i\in\mathbb{N}}$ of finite subsets of $G$ verifying points
(1), (2) and (3): choose $F_{1}$ as a fundamental domain of $G/\Gamma_{1}$
such that $g_{1}\in F_{1}$. Let $\Gamma_{n_{1}}=\Gamma_{1}$. Let
$S_{2}=F_{1}\cup\\{g_{1},g_{2}\\}$. By the previous discussion, there exists
$\Gamma_{n_{2}}\subseteq\Gamma_{n_{1}}$ satisfying $g\Gamma_{n_{2}}\neq
h\Gamma_{n_{2}}$ for every pair of distinct elemetns $g,h\in S_{2}$. Thus, we
can take $F_{2}$ such that $S_{2}\subseteq F_{2}$. Suppose we have defined
$\Gamma_{n_{k-1}}$ and $F_{k-1}$ such that
$\Gamma_{n_{k-1}}\subseteq\Gamma_{n_{k-2}}$ and $F_{k-1}$ is a fundamental
domain of $\Gamma_{n_{k-1}}$ containing the set
$F_{k-2}\cup\\{g_{1},\cdots,g_{k-1}\\}$. Now, consider
$S_{k}=\\{g_{1},g_{2},\ldots,g_{k}\\}\cup F_{k-1}$. By the previous
discussion, we can choose $\Gamma_{n_{k}}\subseteq\Gamma_{n_{k-1}}$ such that
the elements in $S_{k}$ are in different equivalence classes of
$G/\Gamma_{n_{k}}$. Therefore, we can get $F_{k}$ satisfying $S_{k}\subseteq
F_{k}$.

Thus, we obtain (1) and (2). Additionally,

$\displaystyle
G=\bigcup_{i=1}^{\infty}S_{i}\subseteq\bigcup_{i=1}^{\infty}F_{i},$

which implies (3).

Now we will apply [7, Lemma 3] in order to get a sequence
$(D_{i})_{i\in\mathbb{N}}$ as desired. Define $n_{i_{1}}=n_{1}$ and
$D_{1}=F_{1}$. Let $j>1$ and suppose that we have defined
$n_{i_{1}}<\cdots<n_{i_{j-1}}$ and $D_{j-1}$ a fundamental domain of
$G/\Gamma_{n_{i_{j-1}}}$. Since
$G=\bigcup_{v\in\Gamma_{n_{i_{j-1}}}}vD_{j-1}$, there exists $i_{j}>i_{j-1}$
such that

$\displaystyle F_{i_{j-1}}\subseteq\bigcup_{v\in
F_{i_{j}}\cap\Gamma_{n_{i_{j-1}}}}vD_{j-1}.$

Let define $D_{j}:=\bigcup_{v\in F_{i_{j}}\cap\Gamma_{n_{i_{j-1}}}}vD_{j-1}$.
By construction, the sequence $(D_{j})_{j\in\mathbb{N}}$ verifies (3).
Applying [7, Lemma 3] we get that $(D_{j})_{j\in\mathbb{N}}$ satisfies (1),
(2) and (4) for the subsequence $(\Gamma_{n_{i_{j}}})_{j\in\mathbb{N}}$. ∎

The next Lemma follows by induction applying Lemma 2.7.

######  Lemma 2.8.

Let $G$ be a residually finite group, and let $(\Gamma_{n})_{n\in\mathbb{N}}$
be a strictly decreasing sequence of finite index subgroups of $G$, such that
$\bigcap_{n\in\mathbb{N}}\Gamma_{n}=\\{1_{G}\\}.$ There exists an increasing
sequence $(n_{i})_{i\in\mathbb{N}}\subseteq\mathbb{N}$ and
$(D_{i})_{i\in\mathbb{N}}$ a sequence of finite subsets of $G$ such that for
every $i\in\mathbb{N}$

  1. (1)

$\\{1_{G}\\}\subseteq D_{i}\subseteq D_{i+1}$

  2. (2)

$D_{i}$ is a fundamental domain of $G/\Gamma_{n_{i}}$.

  3. (3)

$G=\bigcup_{i=1}^{\infty}D_{i}$.

  4. (4)

$D_{j}=\bigcup_{v\in D_{j}\cap\Gamma_{n_{i}}}vD_{i}$, for each $j>i\geq 1$.

##  3\. Uniquely ergodic case: regular Toeplitz subshifts

In this section we prove Theorem 1.2.

###  3.1. Factor maps and measures.

Let $(X,\phi,G)$ and $(Y,\varphi,G)$ be two topological dynamical systems such
that $(Y,\varphi,G)$ admits an invariant measure $\nu$. Let $\beta(X)$ and
$\beta(Y)$ the Borel sigma-algebras of $X$ and $Y$ respectively. If $\pi:X\to
Y$ is a factor map then the collection of sets
$\pi^{-1}\beta(Y)=\\{\pi^{-1}(A):A\in\beta(Y)\\}$ is a sigma-algebra contained
in $\beta(X)$. Moreover, if $A\in\pi^{-1}\beta(Y)$ then
$\phi^{g}(A)\in\beta(Y)$, for every $g\in G$. The function
$\mu:\pi^{-1}\beta(Y)\to[0,1]$ defined by $\mu(\pi^{-1}(A))=\nu(A)$, for every
$A\in\beta(Y)$, is an invariant probability measure on $\pi^{-1}\beta(Y)$.

######  Remark 3.1.

If $\omega$ is an invariant probability measure of $(X,\phi,G)$, then
$\nu^{*}:\beta(Y)\to[0,1]$ defined as $\nu^{*}(A)=\omega(\pi^{-1}(A))$ for
every $A\in\beta(Y)$, is an invariant probability measure of $(Y,\varphi,G)$.
Thus, if $(Y,\varphi,G)$ is uniquely ergodic then $\nu^{*}=\nu$, which implies
that all the invariant probability measures of $(X,\phi,G)$ coincide on
$\pi^{-1}\beta(Y)$.

Since

$\\{y\in Y:|\pi^{-1}(y)|=1\\}=\bigcap_{n\geq 1}\left\\{y\in Y:{\rm
diam}(\pi^{-1}(y))<\frac{1}{n}\right\\},$

the set $\\{y\in Y:|\pi^{-1}(y)|=1\\}$ is a $G_{\delta}$-set (see [9]). Thus,
the set

$\\{y\in Y:|\pi^{-1}\\{y\\}|>1\\}$

belongs to $\beta(Y)$.

######  Lemma 3.2.

If $\nu(\\{y\in Y:|\pi^{-1}\\{y\\}|>1\\})=0$, then $\mu$ extends to a unique
invariant probability measure on $\beta(X)$.

###### Proof.

Let $A\subseteq X$ be a closed set. Then $\pi(A)\in\beta(Y)$, because $\pi(A)$
is compact. Thus $B=\pi^{-1}(\pi(A))$ belongs to $\pi^{-1}\beta(Y)$. Observe
that $A\subseteq B$ and

$B\setminus A\subseteq\pi^{-1}(\\{y\in Y:|\pi^{-1}\\{y\\}|>1\\}).$

This implies that $B\setminus A$ is a negligible set (which implies that
$B\setminus A$ is in the completion of $\pi^{-1}\beta(Y)$ with respect to
$\mu$), and since $A=B\setminus(B\setminus A)$, we get that $A$ is in the
completion of $\pi^{-1}\beta(Y)$. It follows that every open set is in the
completion of $\pi^{-1}\beta(Y)$, from which we get that $\beta(X)$ is
contained in the completion of $\pi^{-1}\beta(Y)$. This implies that $\mu$ can
be extended in a unique way to $\beta(X)$. It is straightforward to check that
$\mu$ is invariant on $\beta(X)$. ∎

###  3.2. Regular Toeplitz $G$-subshifts.

Let $\Sigma$ be a finite set with at least two elements. Let
$\eta\in\Sigma^{G}$ be a Toeplitz array, and suppose that the decreasing
sequence of finite index subgroups $(\Gamma_{n})_{n\in\mathbb{N}}$ is a period
structure of $\eta$. The Toeplitz $G$-subshift $X=\overline{O_{\sigma}(\eta)}$
is an almost 1-1 extension of the $G$-odometer $\overleftarrow{G}$ associated
to $(\Gamma_{n})_{n\in\mathbb{N}}$ (see Proposition 2.6). If
$\pi:X\to\overleftarrow{G}$ is the almost 1-1 factor map, then

$\mathcal{T}=\\{x\in X:x\mbox{ is a Toeplitz
array}\\}=\pi^{-1}\\{y\in\overleftarrow{G}:|\pi^{-1}\\{y\\}|=1\\}.$

In other words, the set of Toeplitz arrays in $X$ is exactly the pre-image of
the set of elements in $\overleftarrow{G}$ having exactly one pre-image (see
Proposition 2.6). The set of Toeplitz arrays is invariant under the action of
$G$. Indeed, if $x\in\mathcal{T}$ and $g,h\in G$, there exists a finite index
subgroup $\Gamma$ of $G$ such that $x(g^{-1}h)=x(\gamma g^{-1}h)$, for every
$\gamma\in\Gamma$. From this we get

$\sigma^{g}x(h)=x(g^{-1}h)=x(\gamma g^{-1}h)=x(g^{-1}g\gamma
g^{-1}h)=\sigma^{g}x(g\gamma g^{-1}h)\quad\forall\gamma\in\Gamma,$

which implies that $h\in{\rm Per}(\sigma^{g}x,g\Gamma g^{-1})$. This shows
that $\sigma^{g}x\in\mathcal{T}.$ Since $\mathcal{T}$ is invariant, its image
$\pi(\mathcal{T})$ is invariant and measurable in $\overleftarrow{G}$ (see for
example [17, Theorem 2.8]). Thus, if $\nu$ is the unique ergodic measure of
$\overleftarrow{G}$, then $\nu(\pi(\mathcal{T}))\in\\{0,1\\}$.

######  Proposition 3.3.

The following statements are equivalent:

  1. (1)

$\nu(\pi(\mathcal{T}))=1$.

  2. (2)

There exists an invariant probability measure $\mu$ of $(X,\sigma,G)$ such
that $\mu(\mathcal{T})=1$.

  3. (3)

There exists a unique invariant probability measure $\mu$ of $(X,\sigma,G)$
and $\mu(\mathcal{T})=1$.

###### Proof.

If $\nu(\\{y\in\overleftarrow{G}:|\pi^{-1}\\{y\\}|=1\\})=1$ then the measure
$\mu$ defined on $\pi^{-1}\beta(\overleftarrow{G})$, extends to a unique
invariant probability measure on $\beta(X)$ (see Lemma 3.2). Moreover,
$\mu(\mathcal{T})=1$. From this we get (1) implies (3). The rest of the proof
is obvious. ∎

######  Definition 3.4.

We say that the Toeplitz element $\eta\in\Sigma^{G}$ or the Toeplitz
$G$-subshift $(X,\sigma,G)$ is regular, if any of the statements of
Proposition 3.3 is satisfied.

######  Remark 3.5.

In [23], the notion of regularity over Toeplitz subshifts is defined for the
case where the acting group is amenable. Using the Lemma below, it can be
shown that the definition presented here coincides with that used in [23] when
$G$ is amenable. Note that, by definition, regular Toeplitz $G$-subshifts are
uniquely ergodic.

Let $(D_{i})_{i\in{\mathbb{N}}}$ be a sequence of fundamental domains of $G$
as in Lemma 2.8 (in the language of Lemma 2.8, after taking subsequences, we
can assume that $n_{i}=i$). For each $i\in{\mathbb{N}}$, let define

$d_{i}=\frac{|D_{i}\cap{\rm Per}(\eta,\Gamma_{i})|}{|D_{i}|}.$

Since $|D_{i+1}\cap\Gamma_{i}||D_{i}|=|D_{i+1}|$ and ${\rm
Per}(\eta,\Gamma_{i})\subseteq{\rm Per}(\eta,\Gamma_{i+1})$, we have

$d_{i+1}=\frac{|D_{i+1}\cap{\rm
Per}(\eta,\Gamma_{i+1})|}{|D_{i+1}|}\geq\frac{|D_{i+1}\cap\Gamma_{i}||D_{i}\cap{\rm
Per}(\eta,\Gamma_{i})|}{|D_{i+1}|}=d_{i}.$

This implies that the sequence $(d_{i})_{i\in\mathbb{N}}$ is increasing,
therefore, there exists $d\in[0,1]$ such that $\lim_{i\to\infty}d_{i}=d$.

######  Lemma 3.6.

For every Toeplitz array $x\in X$, we have $G=\bigcup_{n\in\mathbb{N}}{\rm
Per}(x,\Gamma_{n})$.

###### Proof.

Let $x\in X$ be a Toeplitz array and let $\\{H_{n}\\}_{n\in\mathbb{N}}$ be a
period structure of $x$. Since $X$ is minimal, we have
$X=\overline{\mathcal{O}_{\sigma}(x)}$. Thus, Proposition 2.6 implies that
there exists an almost 1-1 factor map $\rho\colon X\to\overleftarrow{H}$,
where $\overleftarrow{H}=\varprojlim(G/H_{n},p_{n})$, and $p_{n}\colon
G/H_{n+1}\to G/H_{n}$ is the canonical projection for every $n\in\mathbb{N}$.
By [22, Proposition 5.5], we get that $\overleftarrow{H}$ is the maximal
equicontinuous factor of $X$ and consequently, we obtain that
$\overleftarrow{G}$ and $\overleftarrow{H}$ are conjugate with conjugacy
$\overline{\rho}\colon\overleftarrow{G}\to\overleftarrow{H}$. Let
$\rho\colon\overleftarrow{G}\to\overleftarrow{H}$ be defined by
$\rho(\overline{g})=\overline{\rho}(\overline{g})\overline{\rho}(e_{\overleftarrow{G}})$,
where $\overline{g}\in\overleftarrow{G}$, $e_{\overleftarrow{G}}$ and
$e_{\overleftarrow{H}}$ denote the class of the neutral element $1_{G}$ in
$\overleftarrow{G}$ and in $\overleftarrow{H}$, respectively. Since
$\overline{\rho}$ is a conjugacy, it implies that $\rho$ is so as well.
Moreover, $\rho(e_{\overleftarrow{G}})=e_{\overleftarrow{H}}$. Applying [6,
Lemma 2], we get that for every $n\in\mathbb{N}$, there exists
$k_{n}\in\mathbb{N}$ such that $\Gamma_{k_{n}}\subseteq H_{n}$ and
consequently, ${\rm Per}(x,\Gamma_{n_{k}})\supseteq{\rm Per}(x,H_{n})$. Hence,

$\displaystyle G=\bigcup_{n\in\mathbb{N}}{\rm
Per}(x,H_{n})\subseteq\bigcup_{k\in\mathbb{N}}{\rm
Per}(x,\Gamma_{n_{k}})\subseteq\bigcup_{n\in\mathbb{N}}{\rm
Per}(x,\Gamma_{n}).$

This completes the proof. ∎

######  Lemma 3.7.

Let $\eta\in\Sigma^{G}$ be a Toeplitz array and let
$(\Gamma_{n})_{n\in\mathbb{N}}$ be a period structure of $\eta$ such that
every $\Gamma_{n}$ is normal in $G$. Then $\eta$ is regular if and only if
$d=1$.

###### Proof.

For every $n\in\mathbb{N}$, let $C_{n}$ be as defined in Setion 2.3. We define

$\displaystyle H_{n}=\bigcup_{h\in{\rm
Per}(\eta,\Gamma_{n})}\sigma^{h^{-1}}C_{n}\hskip 8.53581pt\mbox{ and }\hskip
8.53581ptH=\bigcup_{n\in\mathbb{N}}H_{n}.$

If $x\in H$, then there exist $n\in\mathbb{N}$ and $h\in{\rm
Per}(\eta,\Gamma_{n})$ such that $x\in\sigma^{h^{-1}}C_{n}$. This implies that
${\rm Per}(\sigma^{h}x,\Gamma_{n},\alpha)={\rm Per}(\eta,\Gamma_{n},\alpha)$,
for every $\alpha\in\Sigma$. Moreover, since $h\in{\rm Per}(\eta,\Gamma_{n})$,
we have $h\in{\rm Per}(\sigma^{h}x,\Gamma_{n})$, which means that
$x(1_{G})=x(h^{-1}\gamma h)$, for every $\gamma\in\Gamma_{n}$, (recall that
the $\Gamma_{n}$’s are normal). Therefore, $1_{G}$ is periodic in $x$ with
respect to a finite index subgroup of $G$. We have shown that
$H_{n}\subseteq\\{x\in X\mid 1_{G}\in{\rm Per}(x,\Gamma_{n})\\}$.

Now, let $x\in X$ such that $1_{G}\in{\rm Per}(x,\Gamma_{n})$. Since
$\\{\sigma^{h^{-1}}C_{n}:h\in D_{n}\\}$ is a partition of $X$, there exists
$h\in D_{n}$ such that $x\in\sigma^{h^{-1}}C_{n}$. It follows that ${\rm
Per}(x,\Gamma_{n})={\rm Per}(\sigma^{h^{-1}}\eta,\Gamma_{n})$. Since
$1_{G}\in{\rm Per}(x,\Gamma_{n})={\rm Per}(\sigma^{h^{-1}}\eta,\Gamma_{n})$,
we can deduce that $h\in{\rm Per}(\eta,\Gamma_{n})$ and $x\in H_{n}$.
Therefore, we have $H_{n}=\\{x\in X\mid 1_{G}\in{\rm Per}(x,\Gamma_{n})\\}$.

From above we deduce that $\bigcap_{g\in G}\sigma^{g}H\subseteq\mathcal{T}$.
Next, let $x\in\mathcal{T}$. From Lemma 3.6, we get that
$G=\bigcup_{n\in\mathbb{N}}{\rm Per}(x,\Gamma_{n})$. For each $g\in G$, we
have that $\sigma^{g^{-1}}x$ is also a Toeplitz array. Thus, $1_{G}\in{\rm
Per}(\sigma^{g^{-1}}x,\Gamma_{i})$ for some $i\geq 1$, which implies
$\sigma^{g^{-1}}x\in H$. Hence, $x\in\sigma^{g}H$ for every $g\in G$.
Consequently, $x\in\bigcap_{g\in G}\sigma^{g}H$. We have shown
$\mathcal{T}\subseteq\bigcap_{g\in G}\sigma^{g}H$ and with this,
$\mathcal{T}=\bigcap_{g\in G}\sigma^{g}H$.

Suppose $d=1$. Since
$C_{n}=\pi^{-1}\\{(y_{j})_{j\in\mathbb{N}}\in\overleftarrow{G}:y_{n}=\Gamma_{n}\\}$,
we have

$H_{n}=\pi^{-1}\left(\bigcup_{h\in{\rm
Per}(\eta,\Gamma_{n})}\\{(y_{j})_{j\in\mathbb{N}}\in\overleftarrow{G}:y_{n}=h^{-1}\Gamma_{n}\\}\right)$

and

$H=\pi^{-1}\left(\bigcup_{n\in\mathbb{N}}\bigcup_{h\in{\rm
Per}(\eta,\Gamma_{n})}\\{(y_{j})_{j\in\mathbb{N}}\in\overleftarrow{G}:y_{n}=h^{-1}\Gamma_{n}\\}\right)$

This implies that $\nu(\pi(H_{n}))=\dfrac{|D_{n}\cap{\rm
Per}(\eta,\Gamma_{n})|}{|D_{n}|}=d_{n}$ and $\nu(\pi(H))=d=1$. On the other
hand,

$\pi\left(\bigcap_{g\in G}\sigma^{g}H\right)=\bigcap_{g\in
G}\phi^{g}\pi(H)\subseteq\pi(\mathcal{T}).$

Since $\nu$ is invariant, we deduce that $\nu(\pi(\mathcal{T}))=1$.

Conversely, suppose that $d<1$. Since
$\nu(\pi(\mathcal{T}))\leq\nu(\pi(H))=d<1$ and $\pi(\mathcal{T})$ is
$G$-invariant, we obtain $\nu(\pi(\mathcal{T}))=0$. Thus, we conclude $\eta$
is not regular. ∎

######  Proposition 3.8.

Let $G$ be a residually finite group. Let $(\Gamma_{n})_{n\in\mathbb{N}}$ be a
decreasing sequence of finite index normal subgroups of $G$ with trivial
intersection. Let $\overleftarrow{G}$ be the $G$-odometer associated to the
sequence $(\Gamma_{n})_{n\in\mathbb{N}}$. Then, there exists a free regular
Toeplitz $G$-subshift $X\subseteq\\{0,1\\}^{G}$ which is an almost 1-1
extension of $\overleftarrow{G}$.

###### Proof.

We will show that the example given in [6, Theorem 5] is regular. Let
$(\Gamma_{n})_{n\in\mathbb{N}}$ be the sequence given in the statement.
Consider $(D_{n})_{n\in\mathbb{N}}$ as in Lemma 2.8. Taking subsequences if it
necessary, we can assume that $D_{i}$ is a fundamental domain of
$G/\Gamma_{i}$ and $\frac{|D_{i-1}|}{|D_{i}|}<\frac{1}{i}$, for every
$i\in\mathbb{N}$. Define the sequence $(S_{n})_{n\geq 0}$ of subsets of $G$
inductively: Let $S_{0}:=\\{1_{G}\\}$. Consider $v_{1}\in
D_{1}\setminus\\{1_{G}\\}$ and let $S_{1}:=\\{v_{1}\\}$. For $n>1$, let
$S_{n}=(v_{n-1}\Gamma_{n-1}\cap D_{n})\setminus D_{n-1}$ and let $v_{n}\in
S_{n}$. The sequence $\eta\in\\{0,1\\}^{G}$ is defined as follows:

$\displaystyle\eta(w)=\begin{cases}0,&\mbox{ if }w\in\bigcup_{n\geq
0}S_{2n}\Gamma_{2n+1},\\\ 1,&\mbox{ else.}\end{cases}$

The subshift $X=\overline{O_{\sigma}(\eta)}$ is a Toeplitz $G$-subshift which
is an almost 1-1 extension of $\overleftarrow{G}$. Moreover,
$(\Gamma_{n})_{n\in\mathbb{N}}$ is a periodic structure of $\eta$ (see [6]).

It remains to prove that $\eta$ is a regular Toeplitz array. Let
$n\in\mathbb{N}$ and $g\in D_{2n}\setminus S_{2n}$. We have $g\Gamma_{2n}\cap
v\Gamma_{2n}=\emptyset$, for every $v\in S_{2n}.$ Since
$v\Gamma_{2n+1}\subseteq v\Gamma_{2n}$, then

$g\Gamma_{2n}\cap v\Gamma_{2n+1}=\emptyset\mbox{ for every }v\in S_{2n}.$

For $m>n$, observe that $S_{2m}\subseteq
v_{2n}\Gamma_{2n}\cdots\Gamma_{2m-1}\subseteq v_{2n}\Gamma_{2n}.$ Thus

$S_{2m}\Gamma_{2m+1}\subseteq v_{2n}\Gamma_{2n}\Gamma_{2m+1}\subseteq
v_{2n}\Gamma_{2n}.$

Since $g\neq v_{2n}\in D_{2n}$, then $g\Gamma_{2n}\cap
v_{2n}\Gamma_{2n}=\emptyset.$ This implies

$g\Gamma_{2n}\cap S_{2m}\Gamma_{2m+1}=\emptyset\mbox{ for every }m\geq n.$

If $g\Gamma_{2n}\cap S_{2m}\Gamma_{2m+1}=\emptyset$ for every $m<n$, then
$g\Gamma_{2n}\cap\bigcup_{m\geq 0}S_{2m}\Gamma_{2m+1}=\emptyset$. This implies
that $\eta(g\Gamma_{2n})=1$ and then $g\in{\rm Per}(\eta,\Gamma_{2n}).$ If
there exists $m<n$ such that $g\Gamma_{2n}\cap
S_{2m}\Gamma_{2m+1}\neq\emptyset$, then since
$\Gamma_{2n}\subseteq\Gamma_{2m+1}$, we have that $g\in{\rm
Per}(\eta,\Gamma_{2n},0)$. This implies that $D_{2n}\setminus
S_{2n}\subseteq{\rm Per}(\eta,\Gamma_{2n})\cap D_{2n}.$

Thus,

$\displaystyle\lim_{n\to\infty}\dfrac{|D_{2n}\setminus
S_{2n}|}{|D_{2n}|}\leq\lim_{n\to\infty}\dfrac{|D_{2n}\cap{\rm
Per}(\eta,\Gamma_{2n})|}{|D_{2n}|}.$

Note that

$|v_{2n-1}\Gamma_{2n-1}\cap D_{2n}|=|\Gamma_{2n-1}\cap
D_{2n}|=\frac{|D_{2n}|}{|D_{2n-1}|}.$

Then,

$|D_{2n}\setminus
S_{2n}|=|D_{2n}|-|S_{2n}|=|D_{2n}|-\left(\frac{|D_{2n}|}{|D_{2n-1}|}-|D_{2n-1}|\right).$

Therefore,

$\displaystyle\lim_{n\to\infty}\dfrac{|D_{2n}\setminus
S_{2n}|}{|D_{2n}|}=1-\lim\limits_{n\to\infty}\dfrac{1}{|D_{2n}|}+\lim\limits_{n\to\infty}\dfrac{|D_{2n-1}|}{|D_{2n}|}=1.$

Consequently, $\lim\limits_{n\to\infty}\dfrac{|D_{n}\cap{\rm
Per}(\eta,\Gamma_{n})|}{|D_{n}|}=1$, i.e., $\eta$ is a regular Toeplitz array.

∎

###### Proof of Theorem 1.2.

This is direct from Propositions 3.3, 3.8, 2.6 and Lemma 2.1. ∎

##  4\. $G$-subshifts with at least $r$ ergodic measures

Let $r>1$ be an integer. In this section we prove that for any residually
finite group $G$, there exists a Toeplitz $G$-subshift having at least $r$
ergodic measures. In the case where $G$ is also amenable, the Toeplitz
$G$-subshift has exactly $r$ ergodic measures and we recover a result in [7]
about the realization of Choquet simplices as sets of invariant measures of
subshifts of residually finite amenable groups, for the case of finite
dimensional Choquet simplices (see Remark 4.14).

Through this section, $G$ will be a residually finite group,
$(\Gamma_{i})_{i\in\mathbb{N}}$ a strictly decreasing sequence of finite index
normal subgroups of $G$ with trivial intersection, and
$(D_{i})_{i\in\mathbb{N}}$ a sequence as in Lemma 2.8. Taking subsequences if
necessary, we can assume that $D_{i}$ is a fundamental domain of
$G/\Gamma_{i}$ and $[G:\Gamma_{i}]$, $[\Gamma_{i}:\Gamma_{i+1}]\geq 3$, for
every $i\in\mathbb{N}$.

###  4.1. Construction of the Toeplitz array.

Consider the alphabet $\Sigma=\\{1,2,\ldots,r\\}$. Inspired by the ideas in
[26], we will construct a Toeplitz element $\eta\in\Sigma^{G}$ such that
$X=\overline{O_{\sigma}(\eta)}$ has at least $r$ ergodic measures.

Fix the sequence $(\alpha_{i})_{i\geq 0}$ in $\Sigma$ given by $\alpha_{i}=j$
whenever $j\equiv i(\mbox{mod}\;r)$. We will define the Toeplitz sequence
$\eta$ in several steps:

Step 1: We define $J(0)=\\{1_{G}\\}$ and $\eta(g)=\alpha_{1}$, for every
$g\in\Gamma_{1}$.

Step 2: We define

$\displaystyle J(1)=D_{1}\setminus\Gamma_{1}.$

We put $\eta(hg)=\alpha_{2}$ for every $h\in J(1)$ and $g\in\Gamma_{2}$.

Note that $J(1)$ is the set of elements in $D_{1}$ which were not defined in
the previous step. Thus, $\eta$ is defined in the whole set $D_{1}$ in this
second step.

Step m+1:  We define

$J(m)=D_{m}\setminus\bigcup_{i=0}^{m-1}J(i)\Gamma_{i+1}.$

Define $\eta(hg)=\alpha_{m+1}$ for every $h\in J(m)$ and $g\in\Gamma_{m+1}$.

Observe that $J(m)$ is the set of elements in $D_{m}$ which were not defined
in the previous steps. Thus $\eta$ is defined in the whole set $D_{m}$ during
the step $m+1$.

This construction produces a non-periodic Toeplitz sequence. Indeed, we have
that $G$ is equal to the disjoint union of the sets $\\{J(i)\Gamma_{i+1}:i\geq
0\\}$, and by construction, $i+1$ is the smallest $k\geq 1$ such that
$J(i)\Gamma_{i+1}$ is contained in ${\rm Per}(\eta,\Gamma_{k})$, for every
$i\geq 0$.

######  Proposition 4.1.

$(\Gamma_{n})_{n\in\mathbb{N}}$ is a period structure for $\eta$.

###### Proof.

Let $i\geq 1$. Observe that

(1) ${\rm Per}(\eta,\Gamma_{i})=\bigcup_{l=0}^{i-1}J(l)\Gamma_{l+1}.$

If $g\in G$ is such that ${\rm Per}(\eta,\Gamma_{i})\subseteq{\rm
Per}(\sigma^{g}\eta,\Gamma_{i})$, then $g^{-1}w\in{\rm Per}(\eta,\Gamma_{i})$,
for every $w\in{\rm Per}(\eta,\Gamma_{i})$. By Equation (1), this implies that

(2)
$\bigcup_{l=0}^{i-1}J(l)\Gamma_{l+1}\subseteq\bigcup_{l=0}^{i-1}gJ(l)\Gamma_{l+1}.$

We will prove inductively that $\Gamma_{i}$ is an essential period of $\eta$.

For $i=1$, let $g\in G$ be such that ${\rm Per}(\eta,\Gamma_{1})\subseteq{\rm
Per}(\sigma^{g}\eta,\Gamma_{1})$. From equation (2) we get that
$\Gamma_{1}\subseteq g\Gamma_{1}$, which implies that $g\in\Gamma_{1}$.

Suppose the result is true for $i\geq 1$. We will prove it for $i+1$. Let
$g\in G$ be such that ${\rm Per}(\eta,\Gamma_{i+1},\alpha)\subseteq{\rm
Per}(\sigma^{g}\eta,\Gamma_{i+1},\alpha)$, for every $\alpha\in\Sigma$. Since
$\Gamma_{i+1}\subset\Gamma_{i}$, we have ${\rm
Per}(\eta,\Gamma_{i},\alpha)\subseteq{\rm Per}(\eta,\Gamma_{i+1},\alpha)$.
Consider $h\in{\rm Per}(\eta,\Gamma_{i},\alpha)$ and
$\gamma_{i}\in\Gamma_{i}$. Let $d\in D_{i+1}$ and
$\gamma_{i+1}\in\Gamma_{i+1}$ be such that $\gamma_{i}=d\gamma_{i+1}$. Then

$h\gamma_{i}\gamma_{i+1}^{-1}=hd\in{\rm
Per}(\eta,\Gamma_{i},\alpha)\subseteq{\rm
Per}(\eta,\Gamma_{i+1},\alpha)\subseteq{\rm
Per}(\sigma^{g}\eta,\Gamma_{i+1},\alpha).$

Thus
$\alpha=\sigma^{g}\eta(hd)=\sigma^{g}\eta(hd\gamma_{i+1})=\sigma^{g}\eta(h\gamma_{i})$.
Since $\gamma_{i}\in\Gamma_{i}$ was arbitrarily taken, we deduce that
$h\in{\rm Per}(\sigma^{g}\eta,\Gamma_{i},\alpha)$. Because this is true for
every $h\in{\rm Per}(\eta,\Gamma_{i},\sigma)$, the hypothesis implies
$g\in\Gamma_{i}$.

Hence, we get $J(l)\Gamma_{l+1}=gJ(l)\Gamma_{l+1}$, for every $0\leq l\leq
i-1$. By Equation (2) this implies that $J(i)\Gamma_{i+1}\subseteq
gJ(i)\Gamma_{i+1}.$ From this we get that for $u\in J(i)\subseteq D_{i}$,
there exist $v\in J(i)\subseteq D_{i}$ and $\gamma\in\Gamma_{i+1}$ such that
$u=gv\gamma$. Since $g\in\Gamma_{i}$ and $\Gamma_{i}$ is normal, there exists
$g^{\prime}\in\Gamma_{i}$ such that $u=gv\gamma=vg^{\prime}\gamma\in
v\Gamma_{i}$. From $u$ and $v$ are in $D_{i}$, we can deduce $v=u$ and then
$g^{\prime}=\gamma^{-1}\in\Gamma_{i+1}$. Finally, $gv=vg^{\prime}\in
v\Gamma_{i+1}=\Gamma_{i+1}v$, which implies that $g\in\Gamma_{i+1}$. ∎

###  4.2. Construction of periodic measures on $\Sigma^{G}$.

In this section we will define a sequence of periodic invariant measures
defined on $\Sigma^{G}$. Then we will show that the accumulation points of
this sequence are supported on $X=\overline{O_{\sigma}(\eta)}$. Unlike the
non-amenable case, when $G$ is amenable it is guaranteed that the measures are
supported on $X$ (see Remark 4.4 below).

For every $n\geq 1$ we define $\eta_{n}\in\Sigma^{G}$ as

$\eta_{n}(\gamma D_{n})=\eta(D_{n}),\mbox{ for every }\gamma\in\Gamma_{n},$

that is, $\eta_{n}(\gamma g)=\eta(g)$ for every $\gamma\in\Gamma_{n}$ and
$g\in D_{n}$. Thus we have $\sigma^{\gamma}(\eta_{n})=\eta_{n}$, for every
$\gamma\in\Gamma_{n}$, which implies that

$O_{\sigma}(\eta_{n})=\\{\sigma^{u^{-1}}(\eta_{n}):u\in D_{n}\\}.$

We define the following probability $\sigma$-invariant Borel measure on
$\Sigma^{G}$,

$\mu_{n}=\frac{1}{|D_{n}|}\sum_{u\in
D_{n}}\delta_{\sigma^{u^{-1}}(\eta_{n})}.$

Let $i\in\\{1,\cdots,r\\}$ and $[i]$ be the subset of all $x\in\Sigma^{G}$
such that $x(1_{G})=i$.

For every $n$ such that $n+1\equiv i(\mod r)$ we have

$\mu_{n}([i])=\frac{|J(n)|+|{\rm Per}(\eta,\Gamma_{n},i)\cap
D_{n}|}{|D_{n}|}\geq\frac{|J(n)|}{|D_{n}|}=1-\frac{|D_{n}\cap{\rm
Per}(\eta,\Gamma_{n})|}{|D_{n}|},$ $\mu_{n}([j])=\frac{|{\rm
Per}(\eta,\Gamma_{n},j)\cap D_{n}|}{|D_{n}|}\leq\frac{|D_{n}\cap{\rm
Per}(\eta,\Gamma_{n})|}{|D_{n}|},\mbox{ for
}j\in\\{1,\cdots,r\\}\setminus\\{i\\}.$

Recall that in Section 3 it was shown that $d_{n}=\frac{|D_{n}\cap{\rm
Per}(\eta,\Gamma_{n})|}{|D_{n}|}$ defines an increasing sequence converging to
some $d\in[0,1]$. Using the same argument, it is possible to show that
$d_{n,j}=\frac{|D_{n}\cap{\rm Per}(\eta,\Gamma_{n},j)|}{|D_{n}|}$ also defines
an increasing sequence in $[0,1]$. This implies that every accumulation point
$\mu$ of $(\mu_{i+k_{s}r-1})_{s\geq 0}$ satisfies

$\mu([i])=1-d+\lim_{n\to\infty}\frac{|{\rm Per}(\eta,\Gamma_{n},i)\cap
D_{n}|}{|D_{n}|}\geq 1-d,$

and

$\mu([j])=\lim_{n\to\infty}\frac{|{\rm Per}(\eta,\Gamma_{n},j)\cap
D_{n}|}{|D_{n}|}=t_{j}\leq d\mbox{ for every
}j\in\\{1,\cdots,r\\}\setminus\\{i\\}.$

######  Proposition 4.2.

For the Toeplitz array $\eta$ defined above we have

$1-d_{n+1}=\left(1-\frac{1}{|D_{1}|}\right)\prod_{j=1}^{n}\left(1-\frac{|D_{j}|}{|D_{j+1}|}\right),\mbox{
for every }n\in\mathbb{N}.$

This implies that $d<1-d$.

###### Proof.

Since $({\rm Per}(\eta,\Gamma_{n+1})\setminus{\rm Per}(\eta,\Gamma_{n}))\cap
D_{n+1}=D_{n}\setminus D_{n}\cap{\rm Per}(\eta,\Gamma_{n})$, we have

$\displaystyle d_{n+1}=\frac{|D_{n+1}\cap{\rm
Per}(\eta,\Gamma_{n+1})|}{|D_{n+1}|}$ $\displaystyle=$
$\displaystyle\frac{|D_{n}\cap{\rm
Per}(\eta,\Gamma_{n})||D_{n+1}\cap\Gamma_{n}|+(|D_{n}|-|D_{n}\cap{\rm
Per}(\eta,\Gamma_{n})|)}{|D_{n+1}|}$ $\displaystyle=$
$\displaystyle\frac{|D_{n}\cap{\rm
Per}(\eta,\Gamma_{n})|}{|D_{n}|}+\frac{|D_{n}|}{|D_{n+1}|}\left(1-\frac{|D_{n}\cap{\rm
Per}(\eta,\Gamma_{n})|}{|D_{n}|}\right)$ $\displaystyle=$ $\displaystyle
d_{n}+\frac{|D_{n}|}{|D_{n+1}|}(1-d_{n}).$

The previous equation also implies that

$\displaystyle 1-d_{n+1}=(1-d_{n})-\frac{|D_{n}|}{|D_{n+1}|}(1-d_{n})$
$\displaystyle=$
$\displaystyle(1-d_{n})\left(1-\frac{|D_{n}|}{|D_{n+1}|}\right)$
$\displaystyle=$
$\displaystyle(1-d_{n-1})\left(1-\frac{|D_{n-1}|}{|D_{n}|}\right)\left(1-\frac{|D_{n}|}{|D_{n+1}|}\right)$
$\displaystyle=$
$\displaystyle(1-d_{1})\prod_{j=1}^{i}\left(1-\frac{|D_{j}|}{|D_{j+1}|}\right).$

Since $d_{1}=\frac{1}{|D_{1}|}$ we get the equality. Taking a subsequence
$(\Gamma_{n_{j}})_{j\in\mathbb{N}}$ of $(\Gamma_{n})_{n\in\mathbb{N}}$ such
that $\frac{|D_{n_{j}}|}{|D_{n_{j+1}}|}$ is small enough, we obtain that
$d<1-d$. ∎

######  Remark 4.3.

From Proposition 4.2, and taking a subsequence of
$(\Gamma_{n})_{n\in\mathbb{N}}$ if necessary, we get that there are at least
$r$ different accumulation points of $(\mu_{n})_{n\in\mathbb{N}}$, namely
$\nu_{1},\cdots,\nu_{r}$, where $\nu_{j}$ is an accumulation point of
$(\mu_{j+k_{s}r-1})_{s\geq 0}$ for every $i\neq j\in\\{1,\cdots,r\\}$.
Furthermore, if $\mu$ is an accumulation point of $(\mu_{n})_{n\in\mathbb{N}}$
then there exists $i\in\\{1,\cdots,r\\}$ such that

$(\mu([1]),\cdots,\mu([r]))=(\nu_{i}([1]),\cdots,\nu_{i}([r]))=(t_{1},\cdots,t_{i-1},1-d+t_{i},t_{i+1},\cdots,t_{r})=\vec{t}_{i},$

where

$t_{j}=\lim_{n\to\infty}\frac{|{\rm Per}(\eta,\Gamma_{n},j)\cap
D_{n}|}{|D_{n}|}\mbox{ for every }j\in\\{1,\cdots,r\\}.$

Taking subsequences of $(\Gamma_{n})_{n\in\mathbb{N}}$ we can assume that
$(\mu_{j+k_{s}r-1})_{s\geq 0}$ converges to $\nu_{j}$, for every $1\leq j\leq
r$. In other words, we can assume that $\nu_{1},\cdots,\nu_{r}$ are the unique
accumulation points of $(\mu_{n})_{n\in\mathbb{N}}$.

######  Remark 4.4.

Suppose that $G$ is amenable. Then we can assume that the sequence
$(D_{n})_{n\in\mathbb{N}}$ is Følner (see [7]). Let $x$ be any element in
$\Sigma^{G}$ and define the measures $\mu_{n}$ as before, taking $\eta=x$. The
accumulation points of $(\mu_{n})_{n\in\mathbb{N}}$ are supported on
$\overline{O_{\sigma}(x)}$. Indeed, if $U\subseteq\Sigma^{G}$ is a cylinder
set given by fixing the coordinates of its points in some finite set
$F\subseteq G$, then

$\mu_{n}(U)=\frac{|\\{v\in\partial_{F}D_{n}:\sigma^{v^{-1}}\eta_{n}\in
U\\}|}{|D_{n}|}+\frac{|\\{v\in
D_{n}\setminus\partial_{F}D_{n}:\sigma^{v^{-1}}\eta_{n}\in U\\}|}{|D_{n}|},$

where $\partial_{F}D_{n}=\\{v\in D_{n}:vF\not\subseteq D_{n}\\}$. If
$(D_{n})_{n\in\mathbb{N}}$ is Følner then the first term of the sum goes
always to zero with $n$. This implies that $\nu_{i}(U)>0$ for some $1\leq
i\leq r$, only if $U$ intersects the orbit of $x$, from which we deduce that
the measure $\nu_{1},\cdots,\nu_{r}$ are supported on
$\overline{O_{\sigma}(x)}$.

When the group $G$ is non-amenable, the supports of the accumulation points of
$(\mu_{n})_{n\in\mathbb{N}}$ depend on the choice of $x$. Indeed, if $x$ is an
element of a subshift without invariant measures (subshifts are a test family
for amenability, see [13]) then the accumulation points of
$(\mu_{n})_{n\in\mathbb{N}}$ can not be supported on
$\overline{O_{\sigma}(x)}$.

In the sequel, we will show that the accumulation points of
$(\mu_{n})_{n\in\mathbb{N}}$ are supported on $X$ (regardless of whether $G$
is amenable or not).

######  Lemma 4.5.

Let $n\geq 1$. For every $m\geq n+2$ there exists

(3) $\gamma\in(\Gamma_{n+1}\cap
D_{m})\setminus(D_{n+1}\Gamma_{n+2}\cup\cdots\cup D_{m-1}\Gamma_{m}).$

Moreover,

$\left|(\Gamma_{n+1}\cap D_{m})\setminus(D_{n+1}\Gamma_{n+2}\cup\cdots\cup
D_{m-1}\Gamma_{m})\right|\geq\frac{|D_{m}|}{|D_{n+1}|}\prod_{l=1}^{m-n-1}\left(1-\frac{|D_{n+l}|}{|D_{n+l+1}|}\right).$

Furthermore, if $\gamma$ satisfies (3), then

$\gamma D_{n+1}\subseteq D_{m}\setminus(D_{n+1}\Gamma_{n+2}\cup\cdots\cup
D_{m-1}\Gamma_{m}).$

###### Proof.

For $m=n+2$, observe that $\gamma\in(\Gamma_{n+1}\cap
D_{n+2})\setminus\\{1_{G}\\}$ satisfies the property. Indeed, if
$\gamma=u\gamma^{\prime}$ for $u\in D_{n+1}$ and
$\gamma^{\prime}\in\Gamma_{n+2}$, then $u=1_{G}$. This implies
$\gamma=\gamma^{\prime}\in\Gamma_{n+2}$, but since $\gamma\in D_{n+2}$, this
is only possible if $\gamma=1_{G}$.

We will continue by induction on $m\geq n+2$. Suppose there exists

$\gamma\in(\Gamma_{n+1}\cap D_{m})\setminus(D_{n+1}\Gamma_{n+2}\cup\cdots\cup
D_{m-1}\Gamma_{m}).$

Let $\gamma_{0}\in(\Gamma_{m}\cap D_{m+1})\setminus\\{1_{G}\\}$. Since
$\gamma\in\Gamma_{n+1}\cap D_{m}$ and

$D_{m+1}=\bigcup_{\gamma^{\prime}\in\Gamma_{m}\cap
D_{m+1}}\gamma^{\prime}D_{m},$

we have that $\gamma_{0}\gamma\in\Gamma_{n+1}\cap D_{m+1}.$ Suppose there
exist $n+1\leq s\leq m-1$, $u\in D_{s}$ and $\gamma^{\prime}\in\Gamma_{s+1}$
such that $\gamma_{0}\gamma=u\gamma^{\prime}$. Since $s+1\leq m$ and
$\gamma_{0}\in\Gamma_{m}$, this implies that $\gamma\in D_{s}\Gamma_{s+1}$
which is a contradiction with the choice of $\gamma$. On the other hand, since
$\gamma_{0}\gamma\in D_{m+1}$ the only way that $\gamma_{0}\gamma\in
D_{m}\Gamma_{m+1}$ is having $\gamma_{0}\gamma\in D_{m}$, which is only
possible if $\gamma_{0}=1_{G}$ (because $\gamma\in D_{m}$ and
$\gamma_{0}\in\Gamma_{m}$). We have shown that

$\gamma_{0}\gamma\in(\Gamma_{n+1}\cap
D_{m+1})\setminus(D_{n+1}\Gamma_{n+2}\cup\cdots\cup D_{m}\Gamma_{m+1}).$

This implies that

$N_{m+1,n}\geq\left(\frac{|D_{m+1}|}{|D_{m}|}-1\right)N_{m,n},$

where

$N_{m+1,n}=|(\Gamma_{n+1}\cap
D_{m+1})\setminus(D_{n+1}\Gamma_{n+2}\cup\cdots\cup D_{m}\Gamma_{m+1})|.$

From this we get

$N_{m+1,n}\geq\prod_{l=1}^{m-n}\left(\frac{|D_{n+l+1}|}{|D_{n+l}|}-1\right)=\frac{|D_{m+1}|}{|D_{n+1}|}\prod_{l=1}^{m-n}\left(1-\frac{|D_{n+l}|}{|D_{n+l+1}|}\right).$

Finally, let $\gamma$ be an element in $\Gamma_{n+1}$ satisfying relation (3)
and let $u\in D_{n+1}$. Suppose there exist $n+1\leq s\leq m-1$, $v\in D_{s}$
and $\gamma^{\prime}\in\Gamma_{s+1}$ such that $\gamma u=v\gamma^{\prime}$. We
can write $v=\gamma^{\prime\prime}u^{\prime}$, with $u^{\prime}\in D_{n+1}$
and $\gamma^{\prime\prime}\in\Gamma_{n+1}\cap D_{s}$. The equation $\gamma
u=\gamma^{\prime\prime}u^{\prime}\gamma^{\prime}$ implies $u=u^{\prime}$ and
then $\gamma\in D_{s}\Gamma_{s+1}$, which is not possible. This finishes the
proof. ∎

Let $n\geq 1$. We define

(4) $U_{n}=\\{x\in\Sigma^{G}:x(D_{n+1})=\eta_{n}(D_{n+1})\\}.$

Observe that $U_{n}$ is the set of all $x\in\Sigma^{G}$ such that $x(\gamma
D_{n})=\eta(D_{n})$, for every $\gamma\in D_{n+1}\cap\Gamma_{n}$.

######  Lemma 4.6.

Let $n\geq 1$ and $m>n$ be such that $m\equiv n(\mod r)$. If
$\gamma\in\Gamma_{n+1}\cap D_{m}$ satisfies relation (3), then
$\sigma^{\gamma^{-1}}(\eta)\in U_{n}$. This implies that $U_{n}\cap
O_{\sigma}(\eta)\neq\emptyset.$

###### Proof.

Let $m>n$ be such that $m\equiv n(\mod r)$ and let
$\gamma_{0}\in\Gamma_{n+1}\cap D_{m}$ an element of the group satisfying the
relation (3). From the choice of $\gamma_{0}$, if $g\in D_{n+1}$ then
$\eta(\gamma_{0}g)$ has been defined in step $k\in\\{1,\cdots,m\\}$ if and
only if $g\in{\rm Per}(\eta,\Gamma_{n+1})$. Observe this implies

(5) $\eta(\gamma_{0}D_{n})=\eta(D_{n})$

and

(6) $\eta(\gamma_{0}\gamma u)=\eta(u),\forall u\in D_{n}\cap{\rm
Per}(\eta,\Gamma_{n})\mbox{ and }\gamma\in D_{n+1}\cap\Gamma_{n}.$

If $u\in D_{n}\setminus{\rm Per}(\eta,\Gamma_{n})=J(n)$ then $\gamma_{0}\gamma
u\in J(m)$, which implies that

(7) $\eta(\gamma_{0}\gamma u)=\alpha_{m+1}=\alpha_{n+1}=\eta(u).$

From (5), (6) and (7) we get

$\eta(\gamma_{0}\gamma D_{n})=\eta(D_{n}),\mbox{ for every }\gamma\in
D_{n+1}\cap\Gamma_{n},$

which implies that

$\eta(\gamma_{0}D_{n+1})=\eta_{n}(D_{n+1}),$

and then $\sigma^{\gamma_{0}^{-1}}(\eta)\in U_{n}$. ∎

######  Proposition 4.7.

The accumulation points of $(\mu_{n})_{n\in\mathbb{N}}$ are supported on
$X=\overline{O_{\sigma}(\eta)}$. Therefore, $X$ has at least $r$ different
invariant probability measures.

###### Proof.

Let $\nu$ be an accumulation point of $(\mu_{n})_{n\in\mathbb{N}}$. We have
that $\nu=\nu_{i}$ for some $i\in\\{1,\cdots,r\\}$.

Let $C\subseteq\Sigma^{G}$ be a clopen set such that $\nu(C)=\varepsilon>0$.
We can assume that $C=\\{y\in\Sigma^{G}:y(D_{n})=P\\}$, where $P$ is some
element in $\Sigma^{D_{n}}$, for some fixed $n\geq 1$. Since $\nu(C)>0$, we
have that $\mu_{i+rs-1}(C)>0$, for infinitely many $s$’s. This implies that
$O_{\sigma}(\eta_{m})\cap C\neq\emptyset$ for infinitely many $m$’s which are
equal $\mod r$. From this we get there exists $u_{m}\in D_{m}$ such that
$\eta_{m}(u_{m}D_{n})=P$. We can always assume that $D_{m}\cdot D_{m}\subseteq
D_{m+1}$, which implies that $u_{m}D_{m}\subseteq D_{m+1}$. From Lemma 4.6, it
follows there exists $g\in G$ such that $\sigma^{g}(\eta)\in C$. Therefore
$\nu$ is supported on $\overline{O_{\sigma}(\eta)}$.

From Remark 4.3 we deduce that $X$ has at least $r$ different invariant
probability measures. ∎

###  4.3. Lower bound for the number of ergodic measures.

In this section we show that $X$ has at least $r$ ergodic measures. First we
need a couple of Lemmas.

######  Lemma 4.8.

For every $i\geq 0$, we have

$J(i+1)=\bigcup_{\gamma\in(D_{i+1}\cap\Gamma_{i})\setminus\\{1_{G}\\}}\gamma
J(i).$

###### Proof.

If $u\in D_{i+1}$ then $u=\gamma v$ for some $\gamma\in D_{i+1}\cap\Gamma_{i}$
and $v\in D_{i}$. Thus if $u\in J(i+1)=D_{i+1}\setminus{\rm
Per}(\eta,\Gamma_{i+1})$, then $\gamma\neq 1_{G}$. Furthermore, $v\notin{\rm
Per}(\eta,\Gamma_{i})$, otherwise ${u\in{\rm Per}(\eta,\Gamma_{i})}$, which is
not possible. This implies
$J(i+1)\subseteq\bigcup_{\gamma\in(D_{i+1}\cap\Gamma_{i})\setminus\\{1_{G}\\}}\gamma
J(i)$.

If $v\in J(i)$ then $u\notin{\rm Per}(\eta,\Gamma_{i})$. If $u\in{\rm
Per}(\eta,\Gamma_{i+1})$ the only possibility is $u\in J(i)\Gamma_{i+1}\cap
D_{i+1}=J(i).$ From this we get
$\bigcup_{\gamma\in(D_{i+1}\cap\Gamma_{i})\setminus\\{1_{G}\\}}\gamma
J(i)\subseteq J(i+1).$ ∎

######  Lemma 4.9.

For every $i\geq 1$ and $\gamma\in\Gamma_{i}$, there exists $l\geq i$ such
that $\gamma J(i)\subseteq J(l)\Gamma_{l+1}$.

###### Proof.

Since $J(i)=D_{i}\setminus{\rm Per}(\eta,\Gamma_{i})$, then $\gamma
J(i)\cap{\rm Per}(\eta,\Gamma_{i})=\emptyset$. This implies that

$\gamma J(i)\subseteq\bigcup_{l\geq i}J(l)\Gamma_{l+1}.$

Let $l=\min\\{k\geq i:\gamma J(i)\cap J(k)\Gamma_{k+1}\neq\emptyset\\}$. Let
$u\in J(i)$ be such that $\gamma u=v_{l}\gamma_{l+1}$, for some $v_{l}\in
J(l)$ and $\gamma_{l+1}\in\Gamma_{l+1}$. Since $v_{l}\in D_{l}$, there exist
$v\in D_{i}$ and $\gamma^{\prime}\in\Gamma_{i}\cap D_{l}$ such that
$v_{l}=\gamma^{\prime}v$. The relation $\gamma u=v_{l}\gamma_{l+1}$ implies
$v=u$ and $\gamma=\gamma^{\prime}\gamma_{l+1}^{\prime}$, for some
$\gamma_{l+1}^{\prime}\in\Gamma_{l+1}$. Thus if $s\in J(i)$ then $\gamma
s=\gamma^{\prime}\gamma_{l+1}^{\prime}s=\gamma^{\prime}s\gamma_{l+1}^{\prime\prime}$,
for some $\gamma_{l+1}^{\prime\prime}\in\Gamma_{l+1}$. This implies that
$\gamma s\in D_{l}\Gamma_{l+1}\subseteq
J(l)\Gamma_{l+1}\cup\bigcup_{k=0}^{l-1}J(k)\Gamma_{k+1}$. The choice of $l$
implies that $\gamma s\in J(l)\Gamma_{l+1}$, and then $\gamma J(i)\subseteq
J(l)\Gamma_{l+1}$. ∎

######  Corollary 4.10.

For every $i\geq 0$ and $\gamma\in\Gamma_{i}$, there exists $\alpha\in\Sigma$
such that

$\eta(g)=\alpha\mbox{ for every }g\in\gamma J(i).$

###### Proof.

The case $i=0$ is trivial. Suppose that $i\geq 1$ and $\gamma\in\Gamma_{i}$.

From Lemma 4.9, there exists $l\geq i$ such that $\gamma J(i)\subseteq
J(l)\Gamma_{l+1}$. By the definition of $\eta$ we get $\eta(g)=\alpha_{l+1}$,
for every $g\in\gamma J(i)$. ∎

####  4.3.1. Partitions and invariant measures

Recall that $\\{\sigma^{v^{-1}}C_{n}:v\in D_{n}\\}$ is a clopen partition of
$X$, where

$C_{n}=\\{x\in X:{\rm Per}(x,\Gamma_{n},\alpha)={\rm
Per}(\eta,\Gamma_{n},\alpha),\mbox{ for every }\alpha\in\Sigma\\}.$

For every $1\leq i\leq r$, let

$C_{n,i}=\\{x\in C_{n}:x(g)=i\mbox{ for every }g\in J(n)\\}.$

Corollary 4.10 implies $\\{C_{n,i}:1\leq i\leq r\\}$ is a covering of $C_{n}$,
therefore,

${\mathcal{P}}_{n}=\\{\sigma^{v^{-1}}C_{n,i}:1\leq i\leq r,v\in D_{n}\\}$

is a clopen partition of $X$.

######  Lemma 4.11.

For every $n\geq 1$ and $1\leq j\leq r$, we have

  1. (1)

$C_{n+1}\subseteq C_{n,\alpha_{n+1}}$, and

  2. (2)

$\sigma^{\gamma^{-1}}C_{n+1,j}\subseteq C_{n,j}$, for every
$\gamma\in(\Gamma_{n}\cap D_{n+1})\setminus\\{1_{G}\\}$.

###### Proof.

Since ${\rm Per}(\eta,\Gamma_{n})\subseteq{\rm Per}(\eta,\Gamma_{n+1})$, we
have $C_{n+1}\subseteq C_{n}$. Furthermore, $D_{n}\subseteq{\rm
Per}(\eta,\Gamma_{n+1})$, which implies that $x(D_{n})=\eta(D_{n})$, for every
$x\in C_{n+1}$. In particular, $x(g)=\eta(g)=\alpha_{n+1}$, for every $g\in
J(n)$. From this we get $C_{n+1}\subseteq C_{n,\alpha_{n+1}}$.

Using that $C_{n+1,j}\subseteq C_{n}$, we get that for every
$\gamma\in\Gamma_{n}$, $\sigma^{\gamma^{-1}}C_{n+1,j}\subseteq C_{n}$. On the
other hand, if $\gamma\in(\Gamma_{n}\cap D_{n+1})\setminus\\{1_{G}\\}$ and
$y\in C_{n+1,j}$, then Lemma 4.8 implies that
$\sigma^{\gamma^{-1}}(y)(g)=y(\gamma g)=j$, for every $g\in J(n)$. This shows
that $\sigma^{\gamma^{-1}}C_{n+1,j}\subseteq C_{n,j}$. ∎

Let $\triangle$ be the convex generated by the vectors
$\\{\vec{t}_{1},\cdots,\vec{t}_{r}\\}\in\mathbb{R}^{r}$ (see definition in
Remark 4.3). That is,

$\triangle=\left\\{\sum_{i=1}^{r}\alpha_{i}\vec{t}_{i}:\sum_{i=1}^{r}\alpha_{i}=1,\alpha_{1},\cdots,\alpha_{r}\geq
0\right\\}.$

Since the vectors $\vec{t}_{1},\cdots,\vec{t}_{r}$ are linearly independent,
the convex $\triangle$ is a simplex.

######  Proposition 4.12.

There is an affine surjective map $p$ from the space of invariant probability
measures of $X$ to $\triangle$. Furthermore, $p(\nu_{i})=\vec{t}_{i}$ for
every $1\leq i\leq r$. This implies that $X$ has at least $r$ ergodic
measures.

###### Proof.

For every $n\in\mathbb{N}$ and $i\in\\{1,\cdots,r\\}$ we set
$a_{n,i}=|D_{n}\cap{\rm Per}(\eta,\Gamma_{n},i)|$.

Let $C_{0,i}=[i]\cap X$. We have that

$\displaystyle C_{0,i}=\bigcup_{g\in
J(n)}\sigma^{g^{-1}}C_{n,i}\cup\bigcup_{g\in{\rm Per}(\eta,\Gamma_{n},i)\cap
D_{n}}\sigma^{g^{-1}}C_{n}.$

Therefore, for every invariant probability measure $\mu$ of $X$ we have

$\displaystyle\mu(C_{0,i})$ $\displaystyle=$
$\displaystyle|J(n)|\mu(C_{n,i})+a_{n,i}\mu(C_{n})$ $\displaystyle=$
$\displaystyle(|J(n)|+a_{n,i})\mu(C_{n,i})+a_{n,i}\sum_{j\neq i}\mu(C_{n,j})$
$\displaystyle=$
$\displaystyle\frac{|J(n)|+a_{n,i}}{|D_{n}|}\mu(C_{n,i})|D_{n}|+\frac{a_{n,i}}{|D_{n}|}\sum_{j\neq
i}\mu(C_{n,j})|D_{n}|.$

Taking a subsequence $(n_{k})_{k}$ in order that
$\lim_{k\to\infty}\mu(C_{n_{k},j})|D_{n_{k}}|=\alpha_{j}\in[0,1]$, we get that

$\mu(C_{0,i})=\sum_{j=1}^{r}\vec{t}_{j}(i)\alpha_{j},$

with $\sum_{j}\alpha_{j}=1.$ This implies that
$(\mu(C_{0,1}),\cdots,\mu(C_{0,r}))$ belongs to $\triangle$. The map
$\mu\mapsto(\mu(C_{0,1}),\cdots,\mu(C_{0,r}))$ is an affine map from the set
of invariant probabilty measures of $X$ to $\triangle$, sending an
accumulation point $\nu_{i}$ of $(\mu_{i+kr-1})_{k\in\mathbb{N}}$ to
$\vec{t}_{i}$. We will call this map as $p$. Furthermore, we have that $p$ is
surjective and then the set of invariant probability measures of $X$ has at
least $r$ extreme points (observe that if $n<r$, then the linear maps from a
$n$-dimensional vector space to a $r$-dimensional one, cannot be surjective).

∎

######  Lemma 4.13.

Let $p$ be the surjective affine map from the space of invariant probability
measures of $X$ to $\triangle$ introduced in Proposition 4.12. If $\mu$ and
$\nu$ are two invariant measures of $X$ such that $p(\mu)=p(\nu)$, then
$\mu|_{\mathcal{P}_{n}}=\nu|_{\mathcal{P}_{n}}$, for every $n\in\mathbb{N}$.

###### Proof.

For every $n\geq 0$, let $A_{n}$ be the $r$-dimensional integer matrix given
by

$A_{n}(i,j)=\left\\{\begin{array}[]{ll}\frac{|D_{n+1}|}{|D_{n}|}-1&\mbox{ if
}i=j\neq\alpha_{n+1}\\\ \frac{|D_{n+1}|}{|D_{n}|}&\mbox{ if
}i=j=\alpha_{n+1}\\\ 0&\mbox{ if }i\neq j\mbox{ and }i\neq\alpha_{n+1}\\\
1&\mbox{ if }i\neq j\mbox{ and }i=\alpha_{n+1}.\end{array}\right.,$

where $|D_{0}|=1$.

From Lemma 4.11, for every $1\leq i\leq r$ we have

$C_{n,i}=\left\\{\begin{array}[]{ll}\bigcup_{\gamma\in(D_{n+1}\cap\Gamma_{n})\setminus\\{1_{G}\\}}\sigma^{\gamma^{-1}}C_{n+1,i}&\mbox{
if }i\neq\alpha_{n+1}\\\
\bigcup_{\gamma\in(D_{n+1}\cap\Gamma_{n})\setminus\\{1_{G}\\}}\sigma^{\gamma^{-1}}C_{n+1,\alpha_{n+1}}\cup\bigcup_{j=1}^{r}C_{n+1,j}&\mbox{
if }i=\alpha_{n+1}\\\ \end{array}\right.$

Thus if $\mu$ is an invariant probability measure of $X$, then

$\mu(C_{n,i})=\left\\{\begin{array}[]{ll}\left(\frac{|D_{n+1}|}{|D_{n}|}-1\right)\mu(C_{n+1,i})&\mbox{
if }i\neq\alpha_{n+1}\\\
\left(\frac{|D_{n+1}|}{|D_{n}|}-1\right)\mu(C_{n+1,\alpha_{n+1}})+\sum_{j=1}^{r}\mu(C_{n+1,j})&\mbox{
if }i=\alpha_{n+1}\\\ \end{array}\right.$

In other words, we have $A_{n}\mu^{(n+1)}=\mu^{(n)}$, where
$\mu^{(n)}=(\mu(C_{n,1}),\cdots,\mu(C_{n,r}))$. Since the matrices $A_{n}$ are
invertible (the columns are linearly independent), we have

$\mu^{(n+1)}=A_{n}^{-1}\cdots A_{0}^{-1}\mu^{(0)}.$

Using that $\mu^{(0)}=p(\mu)$, we conclude. ∎

######  Remark 4.14.

If $G$ is amenable, then the sequence $(D_{n})_{n\in\mathbb{N}}$ can be chosen
Følner. In this case, Lemma 4.13 implies immediately that the affine map $p$
is a bijection, because the set of points that
$(\mathcal{P}_{n})_{n\in\mathbb{N}}$ do not separate, has zero measure with
respect to any invariant measure (See [3, Lemma 17]). When the sequence
$(D_{n})_{n\in\mathbb{N}}$ is not Følner, that set could be a priori a full
measure set.

##  5\. Measure-theoretic conjugacy.

In this section we study the properties of the measures $\nu_{i}$ constructed
in Section 4. We start by proving some technical lemmas which allow us to
conclude that for any invariant measure $\mu\in\mathcal{M}(X,\sigma)$ which
projects on $\nu_{i}$ for some $1\leq i\leq r$, the p.m.p. dynamical system
$(X,\sigma,\mu)$ is measure conjugate to the odometer with its unique
invariant measure (Proposition 5.8). As a consequence of that, we obtain that
any invariant measure which projects on some $\nu_{i}$, coincides with
$\nu_{i}$ (Corollary 5.9) and that the $\nu_{i}^{\prime}s$ are ergodic and the
unique measures that maximize the measures of symbol cylinders (Theorem 1.3).
In the rest of this section, $U_{n}$ is the set defined in (4), i.e,

$U_{n}=\\{x\in\Sigma^{G}:x(D_{n+1})=\eta_{n}(D_{n+1})\\}.$

Additionally, for every $1\leq i\leq r$, we fix $\nu_{i}$ the limit of
$(\mu_{i+kr-1})_{k\in\mathbb{N}}$ (see Remark 4.3).

######  Lemma 5.1.

Let $1\leq i\leq r$. For every $n\geq 1$ such that $n+1\equiv i(\mod r)$, we
have

$\nu_{i}(U_{n})\geq\lim_{s\to\infty}\frac{1}{|D_{n+1}|}\prod_{l=1}^{sr-1}\left(1-\frac{|D_{n+l}|}{|D_{n+l+1}|}\right)$

and

$\nu_{i}(\bigcup_{v\in
D_{n+1}}\sigma^{v^{-1}}U_{n})\geq\lim_{s\to\infty}\prod_{l=1}^{sr-1}\left(1-\frac{|D_{n+l}|}{|D_{n+l+1}|}\right).$

###### Proof.

Let $k\geq 2$. From Lemma 4.5 we have

$N_{n+k,n}\geq\prod_{l=1}^{k-1}\left(\frac{|D_{n+l+1}|}{|D_{n+l}|}-1\right)=\frac{|D_{n+k}|}{|D_{n+1}|}\prod_{l=1}^{k-1}\left(1-\frac{|D_{n+l}|}{|D_{n+l+1}|}\right).$

On the other hand, if $k=sr$ for some $s\geq 1$, then Lemma 4.6 implies that
for every $\gamma\in(\Gamma_{n+1}\cap
D_{n+sr})\setminus(D_{n+1}\Gamma_{n+2}\cup\cdots\cup D_{n+sr-1}\Gamma_{n+sr})$
we have

$\eta_{n}(D_{n+1})=\eta(\gamma D_{n+1})=\eta_{n+sr}(\gamma D_{n+1}).$

Thus we have $\sigma^{\gamma^{-1}}\eta_{n+sr}\in U_{n}$ and $\sigma^{(\gamma
v)^{-1}}\eta_{n+sr}\in\sigma^{v^{-1}}U_{n}$ for $v\in D_{n+1}$, which implies

$\mu_{n+sr}(U_{n})\geq
N_{n+sr,n}\frac{1}{|D_{n+sr}|}\geq\frac{1}{|D_{n+1}|}\prod_{l=1}^{sr-1}\left(1-\frac{|D_{n+l}|}{|D_{n+l+1}|}\right),$

and

$\mu_{n+sr}(\bigcup_{v\in D_{n+1}}\sigma^{v^{-1}}U_{n})\geq
N_{n+sr,n}\frac{|D_{n+1}|}{|D_{n+sr}|}\geq\prod_{l=1}^{sr-1}\left(1-\frac{|D_{n+l}|}{|D_{n+l+1}|}\right).$

Since $\nu_{i}$ is the limit of $(\mu_{n+sr})_{s\in\mathbb{N}}$ and
$\lim_{s\to\infty}\prod_{l=1}^{sr-1}(1-\frac{|D_{n+l}|}{|D_{n+l+1}|})$ exists,
we conclude. ∎

For every $1\leq i\leq r$ and $k\geq 1$, we denote

$Y_{i,k}=\bigcap_{\gamma\in\Gamma_{i+kr-1}\cap
D_{i+kr}}\sigma^{\gamma}C_{i+kr-1,i}.$

######  Lemma 5.2.

Let $1\leq i\leq r$ and $k\geq 1$. Then

$U_{i+kr-1}\cap\overline{O_{\sigma}(\eta)}=Y_{i,k}.$

###### Proof.

If $x\in Y_{i,k}$ then $x(\gamma a)=\eta(a)$ and $x(\gamma b)=i$, for every
$a\in{\rm Per}(\eta,\Gamma_{i+kr-1})$, $\gamma\in\Gamma_{i+kr-1}\cap D_{i+kr}$
and $b\in J(i+kr-1)$. This implies that
$\sigma^{\gamma^{-1}}x(D_{i+kr-1})=\eta(D_{i+kr-1})$ for every
$\gamma\in\Gamma_{i+kr-1}\cap D_{i+kr}$, which means that $Y_{i,k}\subseteq
U_{i+kr-1}\cap\overline{O_{\sigma}(\eta)}$.

Let $x\in U_{i+kr-1}\cap\overline{O_{\sigma}(\eta)}$. There exist $v\in
D_{i+kr}$ and $1\leq j\leq r$ such that $x\in\sigma^{v^{-1}}C_{i+kr,j}$. Let
$y\in C_{i+kr,j}$ be such that $x=\sigma^{v^{-1}}y$.

Since $x\in U_{i+kr-1}$, we have

(8) $x(\gamma D_{i+kr-1})=\eta(D_{i+kr-1}),\mbox{ for every
}\gamma\in\Gamma_{i+kr-1}\cap D_{i+kr}.$

Since $y\in C_{i+kr,j}\subseteq C_{i+kr}$, we have $y(g)=\eta(g)$ for every
$g\in{\rm Per}(\eta,\Gamma_{i+kr})$. In particular,

(9) $y(D_{i+kr-1})=\eta(D_{i+kr-1}).$

Finally, the relation $x=\sigma^{v^{-1}}y$ implies

(10) $y(vD_{i+kr-1})=x(D_{i+kr-1})=\eta(D_{i+kr-1}).$

We will show that $v\in\Gamma_{i+kr-1}$. Suppose that $v\notin\Gamma_{i+kr-1}$
and let $0\leq n<i+kr-1$ be the biggest $n$ verifying $v\in\Gamma_{n}$ (here
we assume $\Gamma_{0}=G$ and $D_{0}=\\{1_{G}\\}$). Let
$\gamma\in\Gamma_{n+1}\cap D_{i+kr-1}$ and $u\in D_{n+1}$ be such that
$v=\gamma u$. Since $n+1\leq i+kr-1$, from Equation (10) we have

(11) $y(vD_{n+1})=y(\gamma uD_{n+1})=\eta(D_{n+1}).$

Using that $G/\Gamma_{n+1}$ is a group, we know that there exists $w\in
D_{n+1}$ such that $uw\in\Gamma_{n+1}$. Since $u\in\Gamma_{n}$, we have
$w\in\Gamma_{n}\cap D_{n+1}$, which implies $wD_{n}\subseteq D_{n+1}$. Then
from Equation (11) we get

$y(vwD_{n})=\eta(wD_{n}).$

Equations (11) and (9), together with the fact that $\gamma uw\in\Gamma_{n+1}$
and $D_{n}\subseteq{\rm Per}(y,\Gamma_{n+1})$, imply

$y(vwD_{n})=y(\gamma uwD_{n})=y(D_{n})=\eta(D_{n}).$

From the last two equations we deduce that $\eta(D_{n})=\eta(wD_{n})$.
However, since $w\in\Gamma_{n}\cap D_{n+1}$, the definition of $\eta$ requires
that $w=1_{G}$. Indeed, if $w\neq 1_{G}$ then
$\eta(wg)=\alpha_{n+2}\neq\eta(g)=\alpha_{n+1}$, for every $g\in J(n)$. But if
$w=1_{G}$ then $v\in\Gamma_{n+1}$, which contradicts the choice of $n$. This
shows that $v\in\Gamma_{i+kr-1}$.

Since $y(g)=\eta(g)$ for every $g\in{\rm Per}(\eta,\Gamma_{i+kr})\supseteq{\rm
Per}(\eta,\Gamma_{i+kr-1})$, we have that $x(g)=y(vg)=y(g)=\eta(g)$ for every
$g\in{\rm Per}(\eta,\Gamma_{i+kr-1})$. This implies that $x\in C_{i+kr-1}$,
which means that for every $\gamma\in\Gamma_{i+kr-1}$ there exists $1\leq
j\leq r$ such that $\sigma^{\gamma^{-1}}x\in C_{i+kr-1,j}$. From Equation (8)
we get that for every $\gamma\in\Gamma_{i+kr-1}\cap D_{i+kr}$ the index $j$ is
equal to $i$. This shows that $x\in Y_{i,k}$.

∎

For every $1\leq i\leq r$ and $k\geq 0$, define

(12) $\displaystyle Z_{i,k}=\bigcup_{v\in D_{i+kr}}\sigma^{v^{-1}}C_{i+kr,i}.$

######  Lemma 5.3.

Let $1\leq i\leq r$ and $k\geq 1$. Then

$\bigcup_{v\in D_{i+kr}}\sigma^{v^{-1}}Y_{i,k}\subseteq
Z_{i,k}\cup\bigcup_{j\neq i}\bigcup_{v\in
D_{i+kr-1}}\sigma^{v^{-1}}C_{i+kr,j}.$

###### Proof.

The inclusion $\bigcup_{v\in
D_{i+kr}}\sigma^{v^{-1}}C_{i+kr,i}\subseteq\bigcup_{v\in
D_{i+kr}}\sigma^{v^{-1}}Y_{i,k}$ is direct from Lemma 4.11.

Suppose that $x\in\bigcup_{v\in D_{i+kr}}\sigma^{v^{-1}}Y_{i,k}$. Let $v\in
D_{i+kr}$, $\gamma\in D_{i+kr}\cap\Gamma_{i+kr-1}$ and $u\in D_{i+kr-1}$ be
such that $\sigma^{v}(x)\in Y_{i,k}$ and $v=\gamma u$. Then
$x\in\sigma^{u^{-1}}C_{i+kr-1,i}$. On the other hand, if $w\in D_{i+kr}$,
$\gamma^{\prime}\in\Gamma_{i+kr-1}\cap D_{i+kr}$, $u^{\prime}\in D_{i+kr-1}$
and $1\leq j\leq r$ are such that $x\in\sigma^{w^{-1}}C_{i+kr,j}$ and
$w=\gamma^{\prime}u^{\prime}$, then Lemma 4.11 implies that
$x\in\sigma^{u^{\prime-1}}C_{i+kr-1,j}$ if $\gamma^{\prime}\neq 1_{G}$. From
this we deduce that for every $j\neq i$,

$\left(\bigcup_{v\in
D_{i+kr}}\sigma^{v^{-1}}Y_{i,k}\right)\cap\left(\bigcup_{v\in
D_{i+kr}\setminus D_{i+kr-1}}\sigma^{v^{-1}}C_{i+kr,j}\right)=\emptyset.$

This shows the result.

∎

######  Lemma 5.4.

If

$\lim_{n\to\infty}\lim_{k\to\infty}\prod_{l=1}^{k-1}\left(1-\frac{|D_{n+l}|}{|D_{n+l+1}|}\right)=1,$

then for every $1\leq i\leq r$

$\lim_{k\to\infty}\nu_{i}\left(Z_{i,k}\right)=1.$

###### Proof.

We have

$\displaystyle\nu_{i}\left(Z_{i,k}\right)+\frac{|D_{i+kr-1}|}{|D_{i+kr}|}$
$\displaystyle\geq$
$\displaystyle\nu_{i}\left(Z_{i,k}\right)+\nu_{i}\left(\bigcup_{j\neq
i}\bigcup_{w\in D_{i+kr-1}}\sigma^{w^{-1}}C_{i+kr,j}\right)$ $\displaystyle=$
$\displaystyle\nu_{i}\left(Z_{i,k}\cup\bigcup_{j\neq i}\bigcup_{w\in
D_{i+kr-1}}\sigma^{w^{-1}}C_{i+kr,j}\right)$

The previous equation together with Lemmas 5.1, 5.2 and 5.3 imply

$\nu_{i}\left(Z_{i,k}\right)+\frac{|D_{i+kr-1}|}{|D_{i+kr}|}\geq\nu_{i}\left(\bigcup_{v\in
D_{i+kr}}\sigma^{v^{-1}}Y_{i,k}\right)\geq\lim_{s\to\infty}\prod_{l=1}^{sr-1}\left(1-\frac{|D_{i+kr-1+l}|}{|D_{i+kr+l}|}\right).$

Then we have

$1+\frac{|D_{i+kr-1}|}{|D_{i+kr}|}\geq\nu_{i}\left(Z_{i,k}\right)+\frac{|D_{i+kr-1}|}{|D_{i+kr}|}\geq\lim_{s\to\infty}\prod_{l=1}^{sr-1}\left(1-\frac{|D_{i+kr-1+l}|}{|D_{i+kr+l}|}\right).$

From this we get

$\lim_{k\to\infty}\nu_{i}(Z_{i,k})=1.$

∎

######  Lemma 5.5.

For every $1\leq i\leq r$ and $k\geq 0$, we have

$Z_{i,k}\subseteq Z_{i,k+1}\cup\bigcup_{v\in
D_{i+(k+1)r-1}}\sigma^{v^{-1}}C_{i+(k+1)r},$

and

(13) $Z_{i,k}\subseteq\left(\bigcap_{l\geq k}Z_{i,l}\right)\cup\bigcup_{l\geq
k+1}\bigcup_{v\in D_{i+lr-1}}\sigma^{v^{-1}}C_{i+lr}.$

###### Proof.

Lemma 4.11 implies that

$Z_{i,k}\subseteq\bigcup_{v\in D_{i+kr+s}}\sigma^{v^{-1}}C_{i+kr+s,i}\mbox{
for every }1\leq s<r.$

Lemma 4.11 also implies that

$\bigcup_{v\in D_{i+kr+r-1}}\sigma^{v^{-1}}C_{i+kr+r-1,i}\subseteq
Z_{i,k+1}\cup\bigcup_{j\neq i}\bigcup_{v\in
D_{i+(k+1)r-1}}\sigma^{v^{-1}}C_{i+(k+1)r,j}.$

Combining the two previous equations we get

$Z_{i,k}\subseteq Z_{i,k+1}\cup\bigcup_{v\in
D_{i+(k+1)r-1}}\sigma^{v^{-1}}C_{i+(k+1)r},$

which implies that

$\displaystyle Z_{i,k}$ $\displaystyle=$ $\displaystyle
Z_{i,k}\cap\left(Z_{i,k+1}\cup\bigcup_{v\in
D_{i+(k+1)r-1}}\sigma^{v^{-1}}C_{i+(k+1)r}\right)$ $\displaystyle\subseteq$
$\displaystyle(Z_{i,k}\cap Z_{i,k+1})\cup\bigcup_{v\in
D_{i+(k+1)r-1}}\sigma^{v^{-1}}C_{i+(k+1)r}$

Using an induction argument, we get that for every $m>k$

$\displaystyle Z_{i,k}$ $\displaystyle\subseteq$
$\displaystyle\bigcap_{l=k}^{m}Z_{i,l}\cup\bigcup_{l=k+1}^{m}\bigcup_{v\in
D_{i+lr-1}}\sigma^{v^{-1}}C_{i+lr}.$ $\displaystyle\subseteq$
$\displaystyle\bigcap_{l=k}^{m}Z_{i,l}\cup\bigcup_{l\geq k+1}\bigcup_{v\in
D_{i+lr-1}}\sigma^{v^{-1}}C_{i+lr}.$

From this we get (13). ∎

######  Lemma 5.6.

For every $1\leq i\leq r$, for every $k\geq 0$ and for every invariant
probability measure $\mu$ of $X$ we have

$\mu\left(\bigcap_{l\geq
k}Z_{i,l}\right)\leq\mu(Z_{i,k})\leq\mu\left(\bigcap_{l\geq
k}Z_{i,l}\right)+\sum_{l={k+1}}^{\infty}\frac{|D_{i+lr-1}|}{|D_{i+lr}|}.$

Moreover, if
$\lim_{k\to\infty}\sum_{l={k+1}}^{\infty}\frac{|D_{i+lr-1}|}{|D_{i+lr}|}=0$
and $p(\mu)=p(\nu_{i})$ then

$\lim_{k\to\infty}\mu\left(\bigcap_{l\geq k}Z_{i,l}\right)=1$

and $\mu(A_{i})=1$, where

$A_{i}=\bigcap_{g\in G}\bigcup_{k\geq 0}\bigcap_{l\geq k}\sigma^{g}(Z_{i,l}).$

###### Proof.

If $p(\mu)=p(\nu_{i})$ then Lemma 4.13 implies $\mu(Z_{k,i})=\nu_{i}(Z_{k,i})$
for every $k$. Then Lemmas 5.4 and 5.5 imply that
$\lim_{k\to\infty}\mu\left(\bigcap_{l\geq k}Z_{i,l}\right)=1$ and this in turn
implies that $\mu\left(\bigcup_{k\geq 0}\bigcap_{l\geq k}Z_{i,l}\right)=1$.
Since $\mu$ is invariant, we get that $\mu(A_{i})=1.$ ∎

######  Lemma 5.7.

For $1\leq i\leq r$, let $A_{i}$ be the set defined in Lemma 5.6. The factor
map $\pi\colon X\to\overleftarrow{G}$ from $X$ to its associated $G$-odometer
is injective when restricted to $A_{i}$.

###### Proof.

Let $x,y\in A_{i}$ such that $\pi(x)=\pi(y)$. We have that for each $g\in G$
there exists $k_{g}\geq 0$ such that $\sigma^{g^{-1}}x,\sigma^{g^{-1}}y\in
Z_{i,l}$, for every $l\geq k_{g}$. Consider $k\geq k_{g}$. Since
$\pi(x)=\pi(y)$, we get that $x,y\in\sigma^{v^{-1}_{i+kr}}C_{i+kr}$ for some
$v_{i+kr}\in D_{i+kr}$ and hence
$\sigma^{g^{-1}}x,\sigma^{g^{-1}}y\in\sigma^{(v_{i+kr}g)^{-1}}C_{i+kr}$. If
$v\in D_{i+kr}$ is such that $v_{i+kr}g\in v\Gamma_{i+kr}$, then
$\sigma^{g^{-1}}x,\sigma^{g^{-1}}y\in\sigma^{v^{-1}}C_{i+kr}$. Since
$\sigma^{g^{-1}}x$ and $\sigma^{g^{-1}}y$ are in $Z_{i,k}$, we get
$\sigma^{g^{-1}}x,\sigma^{g^{-1}}y\in\sigma^{v^{-1}}C_{i+kr,i}$. Thus, there
are $w,z\in C_{i+kr,i}$ such that $\sigma^{g^{-1}}x=\sigma^{v^{-1}}w$ and
$\sigma^{g^{-1}}y=\sigma^{v^{-1}}z$. Since $w,z\in C_{i+kr,i}$, we have
$w|_{D_{i+kr}}=z|_{D_{i+kr}}$, which in turn implies that $w(v)=z(v)$.
Consequently, we get

$\displaystyle
x(g)=\sigma^{g^{-1}}x(1_{G})=\sigma^{v^{-1}}w(1_{G})=w(v)=z(v)=\sigma^{v^{-1}}z(1_{G})=\sigma^{g^{-1}}y(1_{G})=y(g).$

Since this is true for every $g\in G$, we can conclude that $x=y$. ∎

######  Proposition 5.8.

Let $\mu$ be an invariant probability measure of $X$. If $\mu(A_{i})=1$, for
some $1\leq i\leq r$, then $(X,\sigma,\mu)$ and $(\overleftarrow{G},\phi,\nu)$
are measure conjugate.

###### Proof.

Suppose there exists $i$ such that $\mu(A_{i})=1$. By Lemma 5.7,
$\pi|_{A_{i}}\colon A_{i}\to\pi(A_{i})$ is bijective. Using [17, Theorem 2.8]
we get that $\pi(A_{i})$ is measurable. Moreover, using that $A_{i}$ is
$G$-invariant and $\pi\colon X\to\overleftarrow{G}$ is a closed factor map we
can conclude that $\pi|_{A_{i}}$ is an isomorphism of measurable spaces such
that $\pi|_{A_{i}}(\sigma^{g}x)=\phi^{g}\pi|_{A_{i}}(x)$. Note that
$\overleftarrow{G}$ is uniquely ergodic and for this reason,
$\mu(\pi|_{A_{i}}^{-1}(B))=\nu(B)$, for every measurable subset $B$ of
$\pi(A_{i})$. This implies that $\pi|_{A_{i}}$ is a measure conjugacy. ∎

######  Corollary 5.9.

Let $p$ be the surjective affine map from ${\mathcal{M}}(X,\sigma)$ to
$\triangle$ introduced in Proposition 4.12. Let $\mu$ be an invariant measure
of $(X,\sigma)$. If $p(\mu)=p(\nu_{i})$ for some $i\in\\{1,2,\cdots,r\\}$,
then $\mu=\nu_{i}$

###### Proof.

Suppose $p(\mu)=p(\nu_{i})$ for some $1\leq i\leq r$. Let $A_{i}$ be defined
as in Lemma 5.6. By Lemmas 5.6 and 5.8, $(X,\mu)$ and
$(\overleftarrow{G},\nu)$ are measure conjugate and $(X,\nu_{i})$ and
$(\overleftarrow{G},\nu)$ are also measure conjugate with measure conjugacy
$\pi|_{A_{i}}$ as in the proof of Lemma 5.8. Let $O\subseteq X$ be an open
set. Since $O\cap A_{i}$ is an open set in $A_{i}$ and $\pi\mid_{A_{i}}$ is a
measure conjugacy, $\pi\mid_{A_{i}}(O\cap A_{i})$ is a measurable set in
$\overleftarrow{G}$. Let $V=\pi\mid_{A_{i}}(O\cap A_{i})$. Since
$\mu(\pi|_{A_{i}}^{-1}(B))=\nu_{i}(\pi|_{A_{i}}^{-1}(B))=\nu(B)$, for every
measurable subset $B$ of $\pi(A_{i})$, we obtain

$\displaystyle\mu(O)=\mu(O\cap
A_{i})=\mu(\pi|_{A_{i}}^{-1}(V))=\nu(V)=\nu_{i}(\pi|_{A_{i}}^{-1}(V))=\nu_{i}(O\cap
A_{i})=\nu_{i}(O).$

Since the open set $O$ was arbitrarily taken, we conclude that $\mu=\nu_{i}$.

∎

###### Proof of Theorem 1.3.

Let $1\leq i\leq r$. Suppose that $\nu_{i}$ is not an extreme point of
${\mathcal{M}}(X,\sigma,G)$. This implies that there exist two distinct
invariant measures $\mu_{1}$, $\mu_{2}$, and $0<t<1$ such that

$\nu_{i}=t\mu_{1}+(1-t)\mu_{2}.$

Thus,

$p(\nu_{i})=tp(\mu_{1})+(1-t)p(\mu_{2}).$

If $p(\mu_{1})=p(\mu_{2})$, then $p(\nu_{i})=p(\mu_{2})$, which by Corollary
5.9 implies that $\mu_{1}=\nu_{i}$, and then $\nu_{i}=\mu_{1}=\mu_{2}$, a
contradiction. Thus, $p(\mu_{1})$ and $p(\mu_{2})$ are two distinct elements
in $\triangle$ and $p(\nu_{i})$ lies between them. But this is not possible
since $p(\nu_{i})=\vec{t_{i}}$ is an extreme point in $\triangle$. We conclude
that $\nu_{i}$ must be an extreme point of ${\mathcal{M}}(X,\sigma,G)$, i.e.,
an ergodic measure. From Lemma 5.6 and Proposition 5.8 we get (1) of Theorem
1.3.

For the second part of the statement, note that for every
$\mu\in{\mathcal{M}}(X,\sigma,G)$, we have that there exist
$\alpha_{j}\in[0,1]$ such that $\sum_{j=1}^{r}\alpha_{j}=1$ and
$p(\mu)=\sum_{j=1}^{r}\alpha_{j}\vec{t_{j}}$. In particular,

$\displaystyle\mu([i])=\sum_{j=1}^{r}\alpha_{j}\nu_{j}([i])=\alpha_{i}(1-d)+t_{i}=\alpha_{i}\nu_{i}([i])+t_{i}=\alpha_{i}\nu_{i}([i])+(1-\alpha_{i})t_{i}.$

Since $t_{i}<\nu_{i}([i])$, we have that $\mu([i])=\nu_{i}([i])$ if and only
if $\alpha_{i}=1$.

Finally, Lemma 2.1 allows to write the statement in terms of totally
disconnected metrisable compactification of $G$. ∎

### Acknowledgements

The authors warmly thank Raimundo Briceño and Samuel Petite for many fruitful
discussions.

## References

  * [1] Auslander, J.; Minimal flows and their extensions, North-Holland Mathematics Studies, 153, North-Holland, Amsterdam, 1988.
  * [2] Ceccherini-Silberstein, T., Coornaert, M.; Cellular automata and groups, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010.
  * [3] Cecchi-Bernales, P., Cortez, M.I.; Invariant measures for actions of congruent monotileable amenable groups, Groups Geom. Dyn. 13 (2019), no. 3, 821–839.
  * [4] Cecchi-Bernales, P., Donoso, S.; Strong orbit equivalence and superlinear complexity, to appear in Israel Journal of Mathematics.
  * [5] Cortez, M.I., ${\mathbb{Z}}^{d}$-Toeplitz arrays, Discrete and Continuous Dynamical Systems 15 (2006) no. 3, 859-881.
  * [6] Cortez, M. I., Petite, S.; $G$-odometers and their almost one-to-one extensions, J. Lond. Math. Soc. (2) 78 (2008), no. 1, 1–20.
  * [7] Cortez, M. I., Petite, S.; Invariant measures and orbit equivalence for generalized Toeplitz subshifts, Groups, Geometry and Dynamics (8), 2014.
  * [8] Downarowicz, T.; The Choquet simplex of invariant measures for minimal flows, Israel Journal of Mathematics (74), 1991.
  * [9] Downarowicz, T.; Survey of odometers and Toeplitz flows, Algebraic and topological dynamics, 7–37, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, 2005.
  * [10] Downarowicz, T., Lacroix, Y.; A non-regular Toeplitz flow with preset pure point spectrum; Studia Math. 120 (1996), 235-246.
  * [11] Downarowicz, T., Lacroix, Y.; Almost 1-1 extensions of Furstenberg-Weiss type and applications to Toeplitz flows, Studia Math. (2) 130 (1998), 149–170.
  * [12] Elek, G.; Free minimal actions of countable groups with invariant probability measures. Ergodic Theory Dynam. Systems 41 (2021), no. 5, 1369–1389.
  * [13] Frisch, J., Kechris, A., Shinko, F., Vidnyánszky, Z.; Realizations of countable Borel equivalence relations, arXiv preprint arXiv:2109.12486 (2021)
  * [14] Giordano, T., de la Harpe, P.; Moyennabilité des groupes dénombrables et actions sur les espaces de Cantor. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 11, 1255–1258.
  * [15] Giordano, T., Matui, H., Putnam, I. F., Skau, C. F.; Orbit equivalence for Cantor minimal $\mathbb{Z}^{d}$-systems. Invent. Math. 179 (2010), no. 1, 119–158.
  * [16] Gjerde, R., Johansen, Ø.; Bratteli-Vreshik models for Cantor minimal systems: applications to Toeplitz flows, Ergodic Theory and Dynamical Systems 20 (2000), 1687–1710.
  * [17] Glasner, E.; Ergodic Theory via Joinings, Mathematical Surveys and Monographs, 101. American Mathematical Society, Providence, RI, 2003. Zbl 1038.37002 MR 1958753.
  * [18] Hjorth, G., Molberg, M.; Free continuous actions on zero-dimensional spaces. Topology Appl. 153 (2006), no. 7, 1116–1131.
  * [19] Iwanik, A.; Toeplitz flows with pure point spectrum, Studia Math. 118 (1996), 27-35.
  * [20] Jacobs, K., Kean, M.; 0-1 sequences of Toeplits type, Z. Wahrsch. Verw. Gebeite 13 (1969), 123-131.
  * [21] Krieger, F.;Sous-décalages de Toeplitz sur les groupes moyennables résiduellement finis. J. Lond. Math. Soc. (2) 75 (2007), no. 2, 447–462.
  * [22] Krieger, F.; Toeplitz subshifts and odometers for residually finite groups. École de Théorie Ergodique, 147–161, Sémin. Congr., 20, Soc. Math. France, Paris, 2010.
  * [23] Ła̧cka, M., Straszak, M.; Quasi-uniform convergence in dynamical systems generated by an amenable group action, J. Lond. Math. Soc. (2) 98 (2018), no. 3, 687–707.
  * [24] Ribes, L., Zalesskii, P.; Profinite groups, second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 40, Springer-Verlag, Berlin, 2010.
  * [25] Sugisaki, F.; Toeplitz flows, ordered Bratteli diagrams and strong orbit equivalence, Ergodic Theory and Dynamical Systems 21 no. 6 (2001), 1867-1881.
  * [26] Williams, S.; Toeplitz minimal flows which are not uniquely ergodic, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete (67), 1984, 95-107.