# Borel factors and embeddings of systems in subshifts

Nishant Chandgotia and Spencer Unger  Tata Institute of Fundamental Research,
Bengaluru, India<EMAIL_ADDRESS>University of Toronto,
Mississauga, Canada<EMAIL_ADDRESS>
###### Abstract.

In this paper we study the combinatorics of free Borel actions of the group
$\mathbb{Z}^{d}$ on Polish spaces. Building upon recent work by Chandgotia and
Meyerovitch, we introduce property $F$ on $\mathbb{Z}^{d}$-shift spaces $X$
under which there is an equivariant map from any free Borel action to the free
part of $X$. Under further entropic assumptions, we prove that any subshift
$Y$ (modulo the periodic points) can be Borel embedded into $X$. Several
examples satisfy property $F$ including, but not limited to, the space of
proper $3$-colourings, tilings by rectangles (under a natural arithmetic
condition), proper $2d$-edge colourings of $\mathbb{Z}^{d}$ and the space of
bi-infinite Hamiltonian paths. This answers questions raised by Seward, and
Gao-Jackson, and recovers a result by Weilacher and some results announced by
Gao-Jackson-Krohne-Seward.

###### Key words and phrases:

Borel dynamical systems, shift spaces, vertex colourings, edge colourings,
tilings, bi-infinite directed Hamiltonian paths, entropy, embeddings

###### 2020 Mathematics Subject Classification:

Primary: 37B10, 54H05, Secondary: 05C15

##  1\. Introduction

In this paper we focus on the following question: Under what hypothesis can
subshifts model all free $\mathbb{Z}^{d}$ dynamical systems? While there are
numerous contexts in which this question can be placed, we will work with free
Borel $\mathbb{Z}^{d}$ systems on Polish spaces. Our inspiration comes from
two recent groundbreaking results by Hochman: In his first paper [Hoc13],
Hochman showed that barring a _universally null set_ , that is, sets which
have measure zero for all invariant probability measures, any Borel
$\mathbb{Z}$ system (with small enough entropy) can be equivariantly embedded
into mixing shifts of finite type. It remained open to extend the embedding to
“dark matter”, a term coined by Mike Boyle, referring to part of the space
which does not have any invariant probability measures. For many interesting
examples “dark matter” can constitute the entire space. This was addressed in
a later paper by Hochman [Hoc19] where he showed that that such an embedding
exists even without leaving out a universally null set. Recently, in a private
communication, Seward and Hochman mentioned that they can construct an
embedding of this “dark matter” into the full shift for countable group
actions.

Fix $d\geq 2$. Our study begins with a recent paper [CM21] where the authors
defined a property called _flexibility_ : If a dynamical system is flexible,
the authors proved that free $\mathbb{Z}^{d}$ systems can be embedded in it
barring a universally null set under natural entropic restrictions. Numerous
dynamical systems like the space of proper $3$-colourings and domino tilings
(and many non symbolic examples) possess this property. In this paper we
consider a similar condition on subshifts called property $F$ (Definition 7.1)
which helps us prove similar results in the context of Borel dynamics. For
systems with property $F$ we showed the following results.

  1. (1)

All free Borel $\mathbb{Z}^{d}$ systems admit equivariant Borel maps to the
free part of a system with property $F$. (Theorem 7.2)

  2. (2)

All subshifts (under suitable restrictions on the entropy) can be Borel
embedded in a subshift with property $F$. (Theorem 9.1)

In this paper, we prove that many subshifts have property $F$:

  1. (1)

Hom-shifts: The space of graph homomorphisms from $\mathbb{Z}^{d}$ to a fixed
undirected graph with an odd cycle. This covers many interesting examples like
the space of proper $3$-colourings. (Section 10).

Gao and Jackson raised the following question in [GJ15]: What is the Borel
chromatic number of the free part of the full shift? Recently it has been
announced in [GJKS18] that Borel chromatic number is less than equal to $3$.
Our results confirm this announcement. In contrast it has been shown in
[GJKS18] that the continuous chromatic number of the free part of the full
shift is $4$.

  2. (2)

Rectangular tiling shifts: Tilings of $\mathbb{Z}^{d}$ by boxes which satisfy
a natural necessary and sufficient coprimality condition. This covers examples
like that of domino tilings (also known as perfect matchings). (Section 11)

It was a question by Gao and Jackson in [GJ15] and through personal
communication by Brandon Seward whether this condition was sufficient. The
resolution of this question in the case of domino tilings for $\mathbb{Z}^{2}$
actions was mentioned to us over personal communication by Brandon Seward and
is going to be part of their upcoming publication [GJKS]. In contrast it has
been shown that the free part of the full shift does not have any continuous
perfect matching [GJKS18].

  3. (3)

The space of directed bi-infinite Hamiltonian paths on $\mathbb{Z}^{d}$.
(Section 12)

This is an important example which does not precisely fall in the category of
subshifts (since it is not compact). Yet our techniques extend to this with
small modifications. It recovers, partially, a weak version of Dye’s theorem
[Dye59] which states that any free $\mathbb{Z}^{d}$ action is orbit equivalent
to a $\mathbb{Z}$-action (barring a null set of an invariant ergodic
probability measure). Our result proves the existence of such an orbit
equivalence where a null set need not be removed. In addition the orbit
equivalence maps the generators of the $\mathbb{Z}$ action to generators of
the $\mathbb{Z}^{d}$ action. This result for $\mathbb{Z}^{2}$ actions was also
mentioned to us by Brandon Seward over personal communication and that it will
be part of their upcoming publication [GJKS]. A similar orbit equivalence was
obtained in [DOWZ21] for $\mathbb{Z}^{2}$ dynamical systems barring a set of
measure zero for a given free ergodic probability measure.

  4. (4)

The space of colourings of edges of $\mathbb{Z}^{d}$ for $d\geq 2$ by $k$
colours for $k\geq 2d$. (Section 13)

It was asked in [GJKS18] to determine the Borel edge-chromatic number of
$\mathbb{Z}^{2}$ actions. Our theorem shows that the answer is $2d$ for
$\mathbb{Z}^{d}$ actions. In contrast [GJKS18] shows that the continuous
chromatic number of the free part of the full shift on $\mathbb{Z}^{2}$ is
$5$. This recovers results by Weilacher in [Wei21, Theorem 2]. Our result on
edge colorings is also a consequence of results in [BHT21].

###  1.1. Selected literature and general context

To put our results in context, we describe several closely related areas.

The ‘original’ context for such questions comes from Krieger’s generator
theorem [Kri75]. It states that under natural assumptions on the entropy any
free ergodic probability preserving $\mathbb{Z}$ system $(X,\mu,T)$ can be
embedded into mixing shifts of finite type. In other words mixing shifts of
finite type are ‘universal’. Naturally the question arose, which other systems
are ‘universal’? It has been found time and again that at the heart of the
matter are certain topological mixing properties: Quas and Soo proved
universality under ‘specification’ and some technical conditions [QS16]. These
technical conditions were loosened by Weiss and were finally gotten rid of by
Burguet [Bur20]. ‘Universality’ under similar mixing conditions can also be
found for $\mathbb{Z}^{d}$ symbolic systems where $d>2$, see Şahin and
Robinson [RŞ01] and Pavlov [Pav14]. In [CM21], Chandgotia and Meyerovitch
considered very general conditions which imply “almost Borel universality” for
various systems which don’t satisfy these mixing conditions but still have
sufficiently many orbit segments which can be ‘glued’ to one another. Further
the embeddings there are modulo a universally null set, a set with measure
zero with respect to all invariant probability measures, as opposed to modulo
a set of measure zero with respect to a specific one. One is immediately led
to the question: When can we embed a space which has no invariant probability
measure [Wei89]? In [Wei89], Weiss had shown that such spaces have countable
generators (later generalised to countable groups in [JKL02]). Tserunyan
showed that all such countable groups actions have a $32$ set generator under
the assumption of local compactness [Tse15] and for $\mathbb{Z}$ actions
Hochman showed that these spaces can be embedded into any mixing shift of
finite type [Hoc19]. This question is rather delicate and sets the context for
our paper: In our results we do not remove sets which admit no invariant
probability measures. While we do not prove the full universality of these
systems we make considerable progress towards it.

An important inspiration for our work also comes from the systematic study of
Borel graphs begun by Kechris, Solecki and Todorčević [KST99]. Identify
$\mathbb{Z}^{d}$ with its Cayley graph with standard generators. A free
$\mathbb{Z}^{d}$ action on a Polish space $X$ induces a natural graph
structure on $X$ which can be thought as a measurable collection of copies of
$\mathbb{Z}^{d}$. To understand what this graph is like, we can now try to
understand its Borel chromatic number, how can it be tiled and other similar
properties. We refer the reader to some comprehensive surveys by Kechris and
Kechris-Marks for more references and details [Kec, KM]. Kechris, Solecki and
Todorčević [KST99, Sections 4.8, 4.9] proved that the Borel chromatic number
and the Borel edge chromatic number of the $\mathbb{Z}^{d}$ full shift is
$2d+1$ and $4d-1$ respectively and asked what is the Borel chromatic number is
if periodic points were removed.

This was taken up by Gao and Jackson in [GJ15] where they proved many results
about the free part of the full shift on $\mathbb{Z}^{2}$ in both the Borel
and the continuous context. Modulo the periodic points, they proved that the
continuous chromatic number is less than and equal to $4$ and that it can be
continuously tiled by rectangles of the type $n\times n$, $n\times(n+1)$,
$(n+1)\times n$ and $(n+1)\times(n+1)$. Moreover they showed that these
results are sharp in the sense that if one were to remove one of the
rectangles or reduce the number of colours, then such continuous maps no
longer exist. Naturally we are lead to the question of the existence of a
Borel colouring or tiling. In [GJKS18], Gao, Jackson, Krohne and Seward
studied these problems very deeply in the continuous context and announced
that the free part of the full shift has Borel chromatic number $3$, can be
perfectly matched and asked about the edge chromatic number. Modulo a null set
for a given ergodic measure some of these questions were answered by Şahin
[Şah09], Prikhod’ko [Pri99], by Şahin and Robinson [Rc04] and by Chandgotia
and Meyerovitch [CM21]. Our results characterise when a given free Borel
action of $\mathbb{Z}^{d}$ can be Borel tiled by a given set of rectangles,
prove that the Borel edge chromatic number is $2d$ and recovers that the Borel
chromatic number is at most $3$. The result regarding the edge chromatic
number can be found in [Wei21, Theorem 2] and is also a consequence of
[BHT21]. Proof sketches for some of these results can be found in [GR21].

The motivation for Gao and Jackson came from a well-known question by Weiss
[Wei84] about hyperfiniteness for actions of countable amenable groups. A
Borel action of a countable group on a Polish space gives rise to the orbit
equivalence relation which is a countable Borel equivalence relation. Such a
relation is called _hyperfinite_ if it is an increasing union of finite Borel
equivalence relations. By versions of Rokhlin’s lemma [Roh48, Kak43, KW72,
OW87], it can be shown that the equivalence relations arising from the action
of amenable groups on Polish spaces are hyperfinite modulo a universally null
set. Weiss asked whether removing the universally null set is necessary. This
was answered positively for $\mathbb{Z}^{d}$ actions by Weiss, for finitely-
generated groups of polynomial growth by Jackson, Kechris, and Louveau
[JKL02], for abelian groups by Gao and Jackson [GJ15] and for locally
Nilpotent groups by Schneider and Seward [SS13]. Recently this has been
generalised to a large class of groups by Conley, Jackson, Marks, Seward and
Tucker-Drob [CJM+20]. Given that an action is hyperfinite, one can ask whether
the witness to hyperfiniteness, that is, the finite Borel equivalence
relations approximating them, are ‘nice’ and whether they ‘respect’ the
geometry of the acting group. This can be seen as a generalisation of a marker
lemma in the Borel context and can be found in [GJKS15] (look at [MU17,
Theorem 5.5] for a proof). The questions about the Borel chromatic numbers of
actions and Borel tilings makes use of these ‘nice’ witnesses to
hyperfiniteness.

We remark that the continuous version of our results are especially subtle and
lot remains to be explored. For instance we know that modulo the periodic
points the $2$-full shift and the proper $3$ colourings are Borel isomorphic
to one another [Hoc19, Hoc13] but these maps can’t be made continuous [Sal21].
Continuous embeddings for $\mathbb{Z}^{2}$ subshifts is known under strong
mixing conditions [Lig03, Lig04] and there are indications that it would fail
otherwise [CM21, Section 12]. Results by Gao and Jackson [GJ15] and Gao-
Jackson-Krohne-Seward [GJKS18] further illustrate the discrepancy between the
Borel and the continuous results.

Finally, in a recent paper Anton Bernshteyn in [Ber20] has pointed out
interesting relations between distributed computing, local algorithms and
questions in descriptive combinatorics. See also [BCG+21, GR21].

###  1.2. Organisation of the paper and a few words about the proofs

The paper is organized as follows. In Section 2 we state the main results of
the paper. In Section 3, we give some definitions that will be used
throughout. In Section 4, we give a proof of a theorem of Gao, Jackson, Krohne
and Seward using the Baire category theorem and use this to show that typical
constructions done in the context of invariant probability measures (some
consequences of the Rokhlin’s lemma) do not work in the Borel context. In
Section 5, we modify a different theorem of Gao, Jackson, Krohne and Seward to
suit our combinatorial constructions. This can be thought of as a weak version
of Rokhlin’s lemma where we prove some additional geometrical properties of
the towers/hyperfiniteness witness. In Section 6, we prove several
combinatorial results about tilings of the hyperfiniteness witness from
Section 5 by nice rectangular boxes. In Section 7, we introduce an extension
property (Property F) for patterns in $\mathbb{Z}^{d}$ as well as a version of
this property with markers and apply them to produce Borel maps. In
particular, we show that if a system $(Y,T)$ has property $F$, then there is
an equivariant Borel map from any free Borel $\mathbb{Z}^{d}$ system to
$(Y,T)$.

In Section 8, we introduce a marker lemma for subshifts with property $F$.
This is then used to show in Section 9 that if a shift space $(Y,T)$ satisfies
the version of Property $F$ with markers, then every subshift with low enough
entropy admits a Borel embedding into $(Y,T)$. In the remaining sections we
prove that the spaces from Theorem 2.1 satisfy Property $F$ and give entropy
bounds for the spaces in Theorem 2.2: Hom-shifts in Section 10, rectangular
tiling shifts in Section 11, directed bi-infinite Hamiltonian paths in Section
12 and proper edge colourings in Section 13.

## Acknowledgments

This project began while the first author was attending a presentation by the
second author in the descriptive set theory seminar at the Hebrew University
of Jerusalem, where both the authors were postdocs. The vibrant academic
culture at the Hebrew University was a great inspiration to us. While a lot of
this work was disrupted by the COVID-19 pandemic, the first author would like
to extend special gratitude towards Shahar Mozes, Benjamin Weiss, Zemer
Kosloff and the Hebrew University of Jerusalem at large who ensured financial
stability and mental peace in difficult circumstances. In addition, we were
greatly encouraged and influenced by numerous questions and discussions with
Benjamin Weiss, Tom Meyerovitch, Brandon Seward, Anton Bernshteyn, Andrew
Marks, Omer Ben-Naria, Mike Boyle and Zemer Kosloff. The first author was
funded by ISF grants 1702/17, 1570/17 and ISF-Moked grants 2095/15 and
2919/19. The second author was partially supported by NSF grant DMS-1700425
and an NSERC Discovery Grant.

##  2\. Main results

###  2.1. Notation

A _Borel $\mathbb{Z}^{d}$ system_ is a pair $(X,T)$ where $X$ is a standard
Borel space and $T$ is an action of $\mathbb{Z}^{d}$ by Borel automorphisms.
Given a Borel $\mathbb{Z}^{d}$ system $(X,T)$ we let $\textit{Free}(X)=\\{x\in
X~{}:~{}\mbox{ for all }\gamma\in\mathbb{Z}^{d},T^{\gamma}(x)\neq x\\}.$ The
action is called _free_ if $\textit{Free}(X)=X$. If $X$ is compact and $T$ is
a $\mathbb{Z}^{d}$ action on $X$ by homeomorphisms, then the pair $(X,T)$ is
called a $\mathbb{Z}^{d}$ topological dynamical system.

An important action that we will focus on will be _the full-shift_
$(\mathcal{A}^{\mathbb{Z}^{d}},\sigma)$ where $\mathcal{A}$ is a finite set,
$\sigma$ is the $\mathbb{Z}^{d}$ action on $\mathcal{A}^{\mathbb{Z}^{d}}$
given by $(\sigma^{\gamma}(x))_{\delta}=x_{\gamma+\delta}.$ We give the set
$\mathcal{A}$ the discrete topology making $\mathcal{A}^{\mathbb{Z}^{d}}$
compact under the product topology and the action $\sigma$ is by
homeomorphisms. In this paper we will work with subsets
$X\subseteq\mathcal{A}^{\mathbb{Z}^{d}}$ which are invariant under the shift-
action; we will call them _symbolic systems_. If in addition they are compact
as well they will be called _subshifts_. Given a set
$B\subseteq\mathbb{Z}^{d}$ we denote the _language_ of $X$ on the set $B$ by
$\mathcal{L}(X,B)=\\{x|_{B}~{}:~{}x\in X\\}$ and the _language_ of $X$ by
$\mathcal{L}(X)=\cup_{B}\mathcal{L}(X,B)$ where the union is over all finite
sets $B$. A _pattern_ is an element of
$\mathcal{L}(\mathcal{A}^{\mathbb{Z}^{d}})$ while a _configuration_ is an
element of $\mathcal{A}^{\mathbb{Z}^{d}}$.

Elements of $\mathbb{Z}^{d}$ will typically be written as
$\alpha,\beta,\gamma,\delta,\ldots$ where for example
$\gamma=(\gamma_{1},\gamma_{2},\ldots,\gamma_{d})$. The standard generators of
$\mathbb{Z}^{d}$ will be denoted by $\epsilon^{i}$, that is,
$\epsilon^{i}_{j}=1$ if $i=j$ and $0$ otherwise. The other unit vectors are
denoted by $\epsilon^{i};d<i\leq 2d$ where $\epsilon^{i}=-\epsilon^{i-d}$.

###  2.2. Examples

Throughout the paper we will work with the undirected Cayley graph of
$\mathbb{Z}^{d}$ given by the standard generators, that is, the graph with
vertex set $\mathbb{Z}^{d}$ and edges of the form
$\\{\gamma,\gamma\pm\epsilon^{i}\\}$ for $i\leq d$.

####  2.2.1. Hom-shifts

Let $H$ be a finite undirected graph without multiple edges. We denote by
$\textit{Hom}(\mathbb{Z}^{d},H)$ the collection of graph homomorphisms
(adjacency preserving maps) from $\mathbb{Z}^{d}$ to $H$. These are called
_hom-shifts_. An important special case is proper $q$-colorings of
$\mathbb{Z}^{d}$, which can be written as $\textit{Hom}(\mathbb{Z}^{d},K_{q})$
where $K_{q}$ is the complete graph.

####  2.2.2. Rectangular tiling shifts

A _box_ in $\mathbb{Z}^{d}$ is a product of intervals. Given a finite set of
boxes $\mathcal{T}$, a _tiling_ of a set $B\subseteq\mathbb{Z}^{d}$ by
$\mathcal{T}$ is a function $a$ from $B$ to $\mathcal{T}$ such that for all
all $R\in\mathcal{T}$, $a^{-1}(R)$ is a disjoint union of translates of $R$.
The space of tilings $X(\mathcal{T})$ is the set of tilings of
$\mathbb{Z}^{d}$ by $\mathcal{T}$. We can assume that $\mathcal{T}$ consists
of boxes $R$ of the form $\prod_{i\leq d}[0,n_{i}^{R})$.

Our results will pertain to the so-called _coprime rectangular shifts_
$X(\mathcal{T})$ where for all $1\leq i\leq d$,
$\gcd(n_{i}^{R};R\in\mathcal{T})=1$. An important rectangular shift is that
_domino tilings_ , $X(\mathcal{T})$ where
$\mathcal{T}=\\{\\{\bar{0},\epsilon^{i}\\}\mid i\leq d\\}$ where we recall
that $\epsilon^{i}$ is the $i^{th}$ standard generator of $\mathbb{Z}^{d}$.
Domino tilings are sometimes also referred to as perfect matchings and dimer
tilings.

####  2.2.3. Directed bi-infinite Hamiltonian paths in $\mathbb{Z}^{d}$:

[DOWZ21]

Let $X_{bi}$ be the set of all functions
$x:\mathbb{Z}^{d}\to\\{\epsilon^{1},\dots,\epsilon^{2d}\\}$ such that the
graph on $\mathbb{Z}^{d}$ with directed edges of the form
$(\gamma,x_{\gamma}+\gamma)$ is a directed bi-infinite path.

####  2.2.4. Proper edge colourings in $\mathbb{Z}^{d}$:

For $t\in\mathbb{N}$ let $E_{t}$ be the set of injective functions from
$\\{\epsilon^{1},\dots,\epsilon^{2d}\\}$ to the set of colours
$\\{1,2,\ldots,t\\}$. We define _the set of proper $t$-edge colourings of
$\mathbb{Z}^{d}$_ as

$X^{(t)}=\\{x\in(E_{t})^{\mathbb{Z}^{d}}~{}:~{}\text{ if $\epsilon$ is a unit
vector and $\delta\in\mathbb{Z}^{d}$ then
$x_{\delta}(\epsilon)=x_{\delta+\epsilon}(-\epsilon)$}\\}.$

Equivalently we can think of them as colouring on edges such that edges
sharing a vertex are mapped to distinct colours. It is obvious that $X^{(t)}$
is non-empty if and only if $t\geq 2d$.

###  2.3. Main results

Our main results are the following:

######  Theorem 2.1.

Let $(X,S)$ be any of the following systems.

  1. (1)

$\textit{Hom}(\mathbb{Z}^{d},H)$ where $H$ is a graph which is not bipartite.

  2. (2)

A coprime rectangular tiling shift.

  3. (3)

The space of bi-infinite Hamiltonian paths.

  4. (4)

The set of proper $k$-edge colourings of $\mathbb{Z}^{d}$ where $k\geq 2d$ and
$d>1$.

For all free Borel $\mathbb{Z}^{d}$ actions $(Y,T)$ there exists an
equivariant Borel map $\phi:(Y,T)\to(\textit{Free}(X),S)$.

For a $\mathbb{Z}^{d}$ topological dynamical system $(Y,T)$ let $h_{top}(Y,T)$
denote the topological entropy of $(Y,T)$.

######  Theorem 2.2.

Let $(X,S)$ be any of the following systems.

  1. (1)

$\textit{Hom}(\mathbb{Z}^{d},H)$ where $H$ is a graph which is not bipartite.

  2. (2)

The space of domino tilings

  3. (3)

The set of proper $t$-edge colourings of $\mathbb{Z}^{d}$ where $t\geq 2d$ and
$d>1$

For all shift spaces $(Y,T)$, if $h_{top}(Y,T)<h_{top}(X,S)$ then there exists
an equivariant Borel embedding $\phi:(\textit{Free}{(Y)},T)\to(X,S)$.

######  Remark 1.

We note that for $d=1$, the space of proper $t$-edge colourings is equivalent
to the space of proper $t$-vertex colourings (which is covered by the space of
graph homomorphisms). Thus the same results for proper $t$-edge colourings
holds for $t\geq 3$ when $d=1$.

##  3\. Preliminaries

In this section, we give some definitions which connect the dynamics of our
systems to the finite combinatorics of their Cayley graphs. Let $(X,T)$ be a
system. We denote by $G_{T}$ its natural Cayley graph structure, meaning, that
the vertex set of $G_{T}$ is the space $X$ and there is an edge between $x$
and $y$ in $X$ if there is a standard generator $\gamma$ of $\mathbb{Z}^{d}$
such that $T^{\gamma}(x)=y$. We note that the connected components of $G_{T}$
are copies of the usual Cayley graph of $\mathbb{Z}^{d}$ given by the standard
generators. On the other hand, we note that Borel subsets $B$ of $X$ often
meet every connected component, so (for example) to define a Borel function on
$B$ we must work simultaneously in every connected component.

For a Borel set $B$, we will refer to the connected components of the induced
subgraph of $G_{T}$ with vertex set $B$ as the connected components of $B$.
Suppose that $P$ is a property of finite subsets of $\mathbb{Z}^{d}$ which is
invariant under translation. Let $C\subseteq X$ be contained in the orbit of a
point $x\in X$. We will say that $C$ has property $P$ if
$\\{\gamma\in\mathbb{Z}^{d}\mid T^{\gamma}(x)\in C\\}$ has property $P$. For
instance if $T$ is a free $\mathbb{Z}^{d}$ action on $X$ then
$T^{[-m,m]^{d}}(x)$ is a cube for all $x\in X$.

For a Borel set $B\subseteq X$, $m\in\mathbb{N}$ and
$F\subseteq\mathbb{Z}^{d}$, we write

$T^{F}(B)=\cup_{\gamma\in F}T^{\gamma}(B)\text{ and
}\partial^{m}B=T^{[-m,m]^{d}}(B)\setminus B;$

where the latter is the $m$-external boundary of $B$. We write $\partial B$
for $\partial^{1}B$. For $i\leq d$, we define $\partial_{i}B$ to be the inner
boundary in the direction $i$, that is, the set of $x\in B$ such that either
$T^{\epsilon^{i}}(x)\notin B$ or $T^{-\epsilon^{i}}(x)\notin B$. Similarly
$\partial_{i}^{+}(B)$ is the set of $x\in B$ such that
$T^{\epsilon^{i}}(x)\notin B$ while $\partial_{i}^{-}(B)$ is the set of $x\in
B$ such that $T^{-\epsilon^{i}}(x)\notin B$. All these sets are Borel. We will
use similar definitions for $B\subseteq\mathbb{Z}^{d}$.

######  Definition 3.1.

A set $B\subseteq\mathbb{Z}^{d}$ is a called a _ $k$-grid union_ if there
exists $C\subseteq k\mathbb{Z}^{d}$ and $\gamma\in\mathbb{Z}^{d}$ such that
$B=\gamma+C+[1,k]^{d}$ and both $B$ and $\mathbb{Z}^{d}\setminus B$ are
connected. We call $\gamma$ the _offset_ of $B$.

######  Definition 3.2.

A finite set $C\subseteq\mathbb{Z}^{d}$ is _coconnected_ if the induced
subgraph of the Cayley graph on $\mathbb{Z}^{d}\setminus C$ is connected.

##  4\. Is there a simple counterpart to Rokhlin’s lemma in Borel dynamics?

In this section, we give simple topological proofs of some theorems by Gao,
Jackson, Krohne and Seward from [GJKS15] and use them to explain why the
techniques of ergodic theory are not enough for the Borel context of our
paper.

We recall Rokhlin’s lemma which is the cornerstone of many constructions in
ergodic theory.

######  Theorem 4.1 (Rokhlin’s lemma [Roh48, Kak43] for $\mathbb{Z}$ actions
and [KW72] for $\mathbb{Z}^{d}$ actions).

Consider a free measure preserving action of $\mathbb{Z}^{d}$ on a probability
space $(X,\mu)$. For all $n\in\mathbb{N}$ and $\epsilon>0$, there exists a
Borel set $A\subseteq X$ such that $\mu(A)>1-\epsilon$ and the connected
components of $A$ are boxes with side length $n$.

Often in ergodic theory (for instance in many proofs of the Krieger’s
generator theorem [Kri75]), given a topological space $Y$, a function $f:X\to
Y$ is constructed in the following fashion. A sequence of constants
$\epsilon_{n}$ are chosen such that $\epsilon_{n}\to 0$ and sets $A_{n}$ are
chosen by Rokhlin’s Lemma (the connected components of $A_{n}$ are boxes with
side length $k_{n}$ large enough). Now for each $n$, a function
$f_{n}:A_{n}\to Y$ is constructed and the hope is that the sequence $f_{n}$
converges to a function $f$. To achieve this, $f_{n+1}$ is constructed using
$f_{n}$ by modifying it on a small set $B_{n}$ and giving an appropriate
definition to $f_{n+1}$ on $A_{n+1}\setminus A_{n}$. The connected components
of $B_{n}$ contain boxes of larger and larger sizes but they satisfy a
summability condition: $\sum_{n\in\mathbb{N}}\mu(B_{n})<\infty$. This allows
an application of the Borel-Cantelli lemma and ensures that $\mu$-almost every
$x\in X$ lies in at most finitely many $B_{n}$. In particular, there are major
modifications only finitely many times in the definition of the value of
function at $\mu$-almost every point in $X$. This implies the required
convergence.

Gao, Jackson, Krohne and Seward [GJKS15] have shown that under very mild
conditions this is not possible in the Borel setting. We will give a simple
proof for this result below.

Let $T$ be an action of a countable group $\Gamma$ on a compact space $X$ by
homeomorphisms. We say that $(X,T)$ is _minimal_ if each orbit of the action
is dense. Equivalently, for all open sets $U$ there is a finite set
$F\subseteq\Gamma$ such that $T^{F}(U)=X$. A _complete section_ is a
measurable set which meets all the orbits of the action.

######  Theorem 4.2.

[GJKS15, Theorem 3.2] Let $(X,T)$ be a free minimal action of a countable
group $\Gamma$. Let $B_{n}\subseteq X$ be a sequence of Borel sets such that
for all finite $F\subseteq\Gamma$ and for all sufficiently large $n$, the set
$\\{x\in X\mid T^{\gamma}(x)\in B_{n}$ for all $\gamma\in F\\}$ is a complete
section of a comeager set. Then the set $\\{x\in X~{}:~{}x\mbox{ belongs to
}B_{n}\mbox{ for infinitely many }n\\}$ is comeager.

######  Remark 2.

We make a few remarks:

  1. (1)

Gao, Jackson, Krohne and Seward remark that such points can be found in the
free part of the full shift $(\mbox{Free}(\\{0,1\\}^{\Gamma}),\sigma)$. This
uses an addional theorem by Gao, Jackson and Seward [GJS09] that there are
minimal subsystems contained in $(\mbox{Free}(\\{0,1\\}^{\Gamma},\sigma)$. We
refer the reader to [ABT18] for an elementary proof of this fact using the
Lovász local lemma.

  2. (2)

In [Pri99] Prikhod´ko and in [Şah09] Şahin proved that if
$R_{i}\subseteq\mathbb{Z}^{d};i\in\mathbb{N}$ is an infinite sequence of
rectangles which are coprime then for any ergodic measure preserving
$\mathbb{Z}^{d}$ system $(X,\mu,T)$ there is a Borel partition of a set of
measure one: $B_{i};i\in\mathbb{N}$ such that the shape of the connected
components of the set $B_{i}$ are $R_{i}$. In fact given a positive
probability vector $p_{i};i\in\mathbb{N}$ they could ensure that the sets
$\mu(B_{i})=p_{i}$ as well. By ergodicity we have that the sets $B_{i}$ are
complete sections for a set of measure $1$. This is no longer true in the
Borel setting due to Theorem 4.2 if the minimum side length of $R_{i}$ is
unbounded. This shows that results in ergodic theory are not necessarily true
in the Borel context. Also have a look at the discussion in Section 14.1.

A key fact that we will use in the proof is that if $B\subseteq X$ is a
complete section of a comeager set, then it cannot be meager. If it were
meager it would contradict Baire’s theorem since it is a section of the set
$C=\bigcup_{\gamma\in\Gamma}T^{\gamma}(B)$ and the set $X\setminus C$ is
meager. In addition, we will use a couple of standard facts: Borel sets $A$
have the Baire property, that is, there is an open set such that its symmetric
difference with $A$ is meager. Secondly if $A$ is a set with the Baire
property and for all open sets $O$, $O\cap A$ is not meager then $A$ is
comeager.

###### Proof.

Let $A=\\{x\in X~{}:~{}x\mbox{ belongs to }B_{n}\mbox{ for infinitely many
}n\\}$. Given a finite set $F\subseteq\Gamma$ we write

$B_{n,F}=\\{x\in X~{}:~{}T^{\gamma^{-1}}(x)\in B_{n}\mbox{ for all }\gamma\in
F\\}.$

By assumption, we have that the set $B_{n,F}$ is a complete section of a
comeager set for large enough $n$ and hence cannot be meager.

Let $U\subseteq X$ be an open set. Choose a finite set $F\subseteq\Gamma$ such
that $\bigcup_{\gamma\in F}T^{\gamma}(U)=X$. This implies that

$\bigcup_{\gamma\in F}T^{\gamma}(U)\cap B_{n,F}=B_{n,F}.$

Hence there is $\gamma\in F$ such that $T^{\gamma}(U)\cap B_{n,F}$ is not a
meager set. But if $x\in T^{\gamma}(U)\cap B_{n,F}$ then
$T^{\gamma^{-1}}(x)\in U\cap B_{n}$.

Thereby we have that

$T^{\gamma^{-1}}\left(T^{\gamma}(U)\cap B_{n,F}\right)\subseteq U\cap B_{n}$

is not meager in $U$ for all sufficiently large $n$.

It follows that for all $N\in\mathbb{N}$

$U\cap(\bigcup_{n\geq N}B_{n})$

is not meager. Since this is true for an arbitrary open set $U$ and the sets
$\bigcup_{n\geq N}B_{n}$ have the Baire property we have that the set
$\bigcup_{n\geq N}B_{n}$ is comeager. Finally we have that the set

$A=\bigcap_{N\in\mathbb{N}}(\bigcup_{n\geq N}B_{n})$

is comeager. This completes the proof. ∎

As an application, we have another theorem of Gao, Jackson, Krohne and Seward
[GJKS15] which shows that if we have an action of $\mathbb{Z}^{d}$ where the
action of each generator is minimal with a sequence $B_{n}$ of Borel complete
sections whose connected components are rectangles with side lengths going to
infinity, then there is a comeager set of points which lie on the boundary of
infinitely many $B_{n}$. In the setting of ergodic theory, this issue does not
arise provided we restrict our attention to a conull set (many such statements
can be found in [CM21, Section 7]).

######  Corollary 4.3.

[GJKS15, Theorem 5.1] Let $d\geq 2$ and $(X,T)$ be a minimal $\mathbb{Z}^{d}$
system such that subsystem with respect to the $\mathbb{Z}\times\\{0\\}^{d-1}$
is also minimal. Given a sequence of Borel sets $B_{n}\subseteq X$ with the
following properties:

  1. (1)

$B_{n}$ is a complete section.

  2. (2)

The connected components of $B_{n}$ are finite rectangles such that if $v_{n}$
is the minimum side length of a rectangle in $B_{n}$, then
$\lim_{n\to\infty}v_{n}=\infty$.

Then the set

$\\{x\in X~{}:~{}x\mbox{ belongs to }\partial B_{n}\mbox{ for infinitely many
}n\\}$

is comeager.

######  Remark 3.

As with Theorem 4.2, this corollary was also proved for the free part of the
full shift. This follows immediately from our result because there are many
subshifts satisfying the hypothesis of this theorem which are contained in the
free part of the full shift. Here is an easy way to construct such an example.
Take irrational numbers
$\alpha_{1},\alpha_{2},\ldots,\alpha_{d}\in\mathbb{R}/\mathbb{Z}$ which are
rationally independent and consider the $\mathbb{Z}^{d}$ action $T$ on
$\mathbb{R}/\mathbb{Z}$ given by

$T^{\gamma}(x)=x+\sum_{k=1}^{d}\gamma_{k}\alpha_{k}$

and equivariant map

$\phi:(\mathbb{R}/\mathbb{Z},T)\to(\\{0,1\\}^{\mathbb{Z}^{d}},\sigma)$

given by

$(\phi(x))_{\gamma}=1_{[0,1/2)}(T^{\gamma}(x))\mbox{ for
}\gamma\in\mathbb{Z}^{d}.$

The closure of $\phi(\mathbb{R}/\mathbb{Z})$ gives an example of such a shift
space. It is a simple exercise to check that the shift space is minimal for
the $\mathbb{Z}\times\\{0\\}^{d-1}$ subaction.

###### Proof.

To apply Theorem 4.2 we need Borel sets $C_{n}\subseteq\partial B_{n}$ such
that

  1. (1)

The sets $C_{n}$ are complete section for a comeager subset of $X$ for the
subaction $\mathbb{Z}\times\\{0\\}^{d-1}$.

  2. (2)

The connected components of $C_{n}$ with respect to the subaction
$\mathbb{Z}\times\\{0\\}^{d-1}$ are intervals of size at least $v_{n}$ where
$v_{n}\to\infty$.

Once we prove this we are done. Let

$C^{\prime}_{n}=\\{x\in\partial B_{n}~{}:~{}T^{v\epsilon^{1}}\in\partial
B_{n}\text{ for }1\leq v\leq v_{n}\\}.$

The orbit of each connected component of $B_{n}$ intersects $C^{\prime}_{n}$
so we know that the sets $C^{\prime}_{n}$ are complete sections for the entire
$\mathbb{Z}^{d}$ action. Let

$C_{n}=\bigcup_{v=1}^{v_{n}}T^{v\epsilon^{1}}(C^{\prime}_{n}).$

Since the sets $C_{n}$ are Borel (hence have the Baire property), there exist
open sets $U_{n}$ such that $U_{n}\Delta C_{n}$ is a meager set. In addition
since they are complete sections for the entire action, the sets $C_{n}$ are
not meager and hence $C_{n}\setminus(U_{n}\Delta C_{n})$ is also not meager.
Because of the minimality of the subaction we have that

$T^{\mathbb{Z}\times\\{0\\}^{d-1}}U_{n}=X$

and hence

$X\setminus(T^{\mathbb{Z}\times\\{0\\}^{d-1}}(C_{n}))\subseteq(T^{\mathbb{Z}\times\\{0\\}^{d-1}}(U_{n}\Delta
C_{n}))$

is meager. Thus the sets $C_{n}$ are complete sections for the subaction for
the comeager set

$X\setminus\cup_{n\in\mathbb{N}}(T^{\mathbb{Z}\times\\{0\\}^{d-1}}(U_{n}\Delta
C_{n})).$

This completes the proof. ∎

##  5\. Hyperfiniteness

In this section, we formulate a witness to hyperfiniteness which reduces the
constructions of the maps from our main theorems to finite combinatorics.

######  Definition 5.1.

A finite subset $F$ of $\mathbb{Z}^{d}$ is $\alpha$-_almost cube_ if there are
cubes $S$ and $S^{\prime}$ of side lengths $s$ and $\alpha s$ respectively
with the same center such that $S\subseteq F\subseteq S^{\prime}$. In this
case, we say that $F$ has side length $s$.

We reformulate Theorem 5.5 of [MU17] (announced in Gao, Jackson, Krohne and
Seward [GJKS15]).

######  Theorem 5.2.

Let $(X,T)$ be a free Borel $\mathbb{Z}^{d}$ system with $d>1$ and let
$0<\delta<1$. If $1<r_{1}<r_{2}\dots$ is a sequence of natural numbers
satisfying $12\sum_{j<k}r_{j}<\delta r_{k}$, then there is a sequence of Borel
sets $B_{1},B_{2},\dots$ such that

  1. (1)

the connected components $C$ of $B_{j}$ are coconnected $(1+\delta)$-almost
cubes of side length $r_{j}$

  2. (2)

for all $x\in X$, there is $k\in\mathbb{N}$ such that $x\in B_{k}$ and

  3. (3)

if $C,D$ are connected components of $B_{l}$ and $B_{m}$ respectively with
$l\leq m$, then $d(\partial C,\partial D)>r_{l}$.

For all $x\in X$ and for all $\gamma$ in $\mathbb{Z}^{d}$, there is $k$ such
that $x,T^{\gamma}(x)$ are in the same connected component of $B_{k}$. Indeed
if $A$ is a connected component of $B_{k}$ for some $k$ then by conditions (2)
and (3), $A\cup\partial A\subseteq B_{l}$ for some $l>k$. By repeating this
argument $m$ times, we get that there exists $l^{\prime}$ such that

$T^{[-m,m]^{d}}(A)\subseteq B_{l^{\prime}}.$

The proof is a routine modification of the proof in [MU17]. We sketch it
noting that our sets $B_{i}$ are essentially $\bigcup D_{i}$ from the proof
there.

###### Proof.

Let $1<r_{1}<\dots$ be such that $12\sum_{j<k}r_{j}<\delta r_{k}$. Let $C_{i}$
be a sequence of Borel maximal $6r_{i}$-discrete sets (See [KST99] Theorem 4.2
or [MU17] Lemma A.2 for details). Suppose that we have defined $B_{j}$ for all
$j<i$ such that additionally the connected components of $B_{j}$ are $4r_{j}$
apart.

We define $A^{i}_{0}=\mathcal{B}_{r_{i}/2}(C_{i})$ where $\mathcal{B}_{r}(C)$
is defined to be $\\{T^{\gamma}(x)\mid x\in C$ and $|\gamma|_{\infty}\leq
r\\}$. We define $A^{i}_{j}$ by induction on $j\leq i$ by the formula
$A^{i}_{j}=\mathcal{B}_{2r_{i-j}}(A^{i}_{j-1},B_{i-j})$ where
$\mathcal{B}_{r}(A,B)=A\cup\mathcal{B}_{r}(Z)$ where

$Z=\\{x\in B\mid\text{ the connected component $C$ of $x$ in $B$ satisfies
$d(A,C)\leq r$}\\}.$

Let $B_{i}=A^{i}_{i-1}$. It is straightforward to show by induction that
$A^{i}_{j}\subseteq\mathcal{B}_{r_{i}/2+6\sum_{k<j}r_{i-k}}(C_{i})$. It is
immediate that the connected components of $B_{i}$ are $(1+\delta)$-almost
cubes of side length $r_{i}$ with pairwise distance at least $4r_{i}$.

Next suppose that $0<j\leq i$ and consider a connected component $C$ of
$B_{i}$ and $D$ of $B_{i-j}$. Then by our construction either
$\mathcal{B}_{r_{i-j}}(D)\subseteq C$ or $d(A^{i}_{j},D)>2r_{i-j}$ (See [MU17]
Lemma A.5). In the second case, it follows that
$d(C,D)>2r_{i-j}-\sum_{k<i-j}6r_{k}>r_{i-j}$.

As in [MU17] we make the connected components $C$ of the $B_{i}$ coconnected
by adding the finite connected components of $X\setminus B_{i}$ to $B_{i}$. It
is straightforward to see that this retains the essential properties of the
$B_{i}$. ∎

We make use of the following consequence of Theorem 5.2.

######  Corollary 5.3.

Let $(X,T)$ be a free Borel $\mathbb{Z}^{d}$ system with $d>1$, $0<\delta<1$
and $n,M\in\mathbb{N}$. If $r_{1}<r_{2}\dots$ is a sequence of natural numbers
satisfying $24\sum_{j<k}r_{k}<\delta r_{k}$ and $r_{1}>2n+12M/\delta$, then
there is sequence of Borel sets $B_{1},B_{2},\dots$ such that

  1. (1)

the connected components $C$ of $B_{j}$ are coconnected $M$-grid unions which
are $(1+\delta)$-almost cube of side length $r_{j}$.

  2. (2)

for all $x\in X$, there is $i\in\mathbb{N}$ such that $x\in B_{i}$ and

  3. (3)

if $C,D$ are connected components of $B_{i}$ and $B_{j}$ respectively, then
$\partial^{n}C\cap\partial^{n}D=\emptyset$.

###### Proof.

Let $n,M\in\mathbb{N}$ and $\delta>0$. Let $\delta^{\prime}>0$ be such that
$\delta/2<\delta^{\prime}<\delta$ be such that
$\delta^{\prime}+2M/r_{1}<\delta$. Let $B_{i}^{\prime}$ witness Theorem 5.2
for $\delta^{\prime}$.

We can choose a Borel set $A_{i}\subseteq B_{i}^{\prime}$ which meets every
connected component of $B_{i}^{\prime}$ exactly once. For all finite sets
$F\subseteq\mathbb{Z}^{d}$ let $\alpha_{F}$ denote the minimal element of $F$
according to the lexicographic order. The lexicographic order induces a total
order on orbits of elements of $X$.

The elements of $B_{i}^{\prime}$ whose component has shape $F$ and are minimal
in their connected component is given by

$(\cap_{\alpha\in
F}T^{-\alpha+\alpha_{F}}(B_{i}^{\prime}))\cap(\cap_{\alpha\in\partial
F}T^{-\alpha+\alpha_{F}}(X\setminus B_{i}^{\prime})).$

The union over all such sets $F$ gives us a Borel subset of $A_{i}\subseteq
B_{i}^{\prime}$ which meets every connected component of $B_{i}^{\prime}$
exactly once.

For $\gamma\in M\mathbb{Z}^{d}$ let $A_{i}^{\gamma}$ be the set of $x\in
A_{i}$ such that $T^{\gamma+[0,M)^{d}}(x)$ intersects the connected component
of $x$ in $B_{i}^{\prime}$. Then let $B_{i}=\bigcup_{\gamma\in
M\mathbb{Z}^{d}}T^{\gamma+[0,M)^{d}}(A_{i}^{\gamma})$. It is clear that the
connected components of $B_{i}$ consist of $M$-grid unions.

Next, we have that $B_{i}\subseteq\mathcal{B}_{M}(B_{i}^{\prime})$ and the
connected components of $B_{i}^{\prime}$ are contained in boxes of side length
$(1+\delta^{\prime})r_{i}$, so the connected components of $B_{i}$ are
contained in boxes with side length
$(1+\delta^{\prime})r_{i}+2M=(1+\delta^{\prime}+2M/r_{i})r_{i}<(1+\delta)r_{i}$.
Finally, it easy to see that if $C$ and $D$ connected components of $B_{l}$
and $B_{m}$ respectively with $l\leq m$, then $d(\partial C,\partial
D)>r_{l}-4M\geq r_{1}-4M>2n$.

As in the proof of Theorem 5.2 we can enlarge each $B_{i}$ so that the
connected components are coconnected while retaining their essential
properties. ∎

##  6\. Tiling complements

Anticipating the definition of property F, we show that certain differences of
$k$-grid unions can be tiled by “nice” rectangles. To start we prove a lemma
about tiling sets of the form $B\setminus A$ where $A$ is an $Nk$-grid union
and $B$ is a slightly larger $k$-grid union with a different offset. These
tilings will be used to show that most of our spaces have property F. We will
need some special properties for directed bi-infinite Hamiltonian paths in
Section 12. These properties appear in Corollary 6.5.

A _$k$ -box_ is a box whose sides are of length $k$. An _almost $k$-box_ is a
box such that all its sides are of length between $k$ and $2k$ and at most one
of its sides has length different from $k$ .

For $N,k\in\mathbb{N}$, let $R_{Nk}$ be the interval $[0,Nk)$ in $\mathbb{Z}$
and $S_{Nk}$ be the interval $[-3k,(N+3)k)$ in $\mathbb{Z}$.

######  Lemma 6.1.

Let $N,k\geq 10$ and let $L\subseteq Nk\mathbb{Z}^{d}$ and
$\gamma\in\mathbb{Z}^{d}$ with $0\leq\gamma_{i}<k$ for all $i\leq d$. If we
set $A=\bigcup_{\beta\in L}\beta+(R_{Nk})^{d}$ and $B=\bigcup_{\beta\in
L}\beta+\gamma+(S_{Nk})^{d}$, then $B\setminus A$ can be tiled by almost
$k$-boxes.

We record a definition for use later. For $i\leq d$, we define an operation
$S_{i}((a,b),\prod_{j\leq d}[\beta_{j},\beta^{\prime}_{j}))$ to be the set
obtained by replacing $[\beta_{i},\beta^{\prime}_{i})$ by $(a,b)$ in the given
product. Further, if $R$ is a union of products of the form $\prod_{j\leq
d}[\beta_{j},\beta^{\prime}_{j})$, then we define $S_{i}((a,b),R)$ to be the
union of the sets $S_{i}((a,b),\prod_{j\leq d}[\beta_{j},\beta^{\prime}_{j}))$
over the products which make up $R$.

###### Proof.

We define a sequence of sets $F_{i}$ for $i\leq d$ by

$F_{i}=\bigcup_{\beta\in L}\big{(}\prod_{j\leq
i}\beta_{j}+\gamma_{j}+S_{Nk}\big{)}\times\big{(}\prod_{j>i}\beta_{j}+R_{Nk}\big{)}$

It is straightforward to see that $F_{0}=A$ and $F_{d}=B$. For $i>0$, the
definitions of $F_{i-1}$ and $F_{i}$ only differ on coordinate $i$ and on this
coordinate we see that $\beta_{i}+R_{Nk}\subseteq\beta_{i}+\gamma_{i}+S_{Nk}$
since $0\leq\gamma_{i}<k$. It follows that $F_{i-1}\subseteq F_{i}$ and hence
$A\subseteq B$. Now $B\setminus A$ can be written as $\bigcup_{0<i\leq
d}F_{i}\setminus F_{i-1}$ where we note that the sets in the union are
disjoint.

So to prove the lemma it suffices to prove the following claim:

######  Claim 6.2.

For $i>0$, $F_{i}\setminus F_{i-1}$ can be tiled by almost $k$-boxes.

We note that it is enough to tile the connected components of $F_{i}\setminus
F_{i-1}$ by appropriate boxes. We start by proving that for every $i$, $F_{i}$
can be tiled with $k$-boxes. Let $\gamma\upharpoonright i$ be the vector given
by the first $i$ elements of $\gamma$ followed by $d-i$ zeros. In particular
$\gamma\upharpoonright 0$ is the vector of all zeros. It is clear that $F_{i}$
can be written as the union of sets of the form $\gamma\upharpoonright
i+\prod_{j\leq d}[\alpha_{j}k,(\alpha_{j}+1)k)$ where
$\alpha\in\mathbb{Z}^{d}$. This forms a tiling of $F_{i}$ by $k$-boxes.

For a connected component $R$ of $\partial_{i}F_{i-1}$, the tiling of
$F_{i-1}$ by $k$-boxes induces a tiling $\mathcal{T}$ of $R$ by boxes with
side length $1$ in the $i$ direction and $k$ in all the other directions.
Using the fact that $N>6$, the connected components of $F_{i}\setminus
F_{i-1}$ are of the form $S_{i}((a_{R},b_{R}),R)$ where $R$ is a connected
component of $\partial_{i}F_{i-1}$ and $(a_{R},b_{R})$ is an interval of
length at least $2k$. It is immediate from the definition of $S_{i}$ and the
tiling $\mathcal{T}$ that $S_{i}((a_{R},b_{R}),R)$ is tiled by boxes of the
form $S_{i}((a_{R},b_{R}),T)$ for $T\in\mathcal{T}$ which can be tiled by
almost $k$-boxes. This completes the proof. ∎

We use the previous lemma to tile differences between a connected component
$C$ of $B_{i}$ and connected components $D$ of $B_{j}$ for $j<i$ with
$D\subseteq C$.

######  Lemma 6.3.

Let $N,k\geq 10$. Suppose that $B_{1},B_{2},\dots$ witnesses Corollary 5.3
with $M=Nk$ and $n=10k$. Then for every $i$ and every connected component $C$
of $B_{i}$, there is a tiling of $C\setminus\bigcup_{D\in Z}D$ with almost
$k$-boxes where $Z$ is the set $\\{D\mid D$ is a connected component of
$B_{j}$ for $j<i$ and $D\subseteq C\\}$.

This lemma will be an immediate consequence of the following non-Borel
version.

######  Proposition 6.4.

Let $N,k\geq 10$. Let $Z$ be a finite collection of $Nk$-grid unions such that
the collection $D+[-10k,10k]^{d};D\in Z$ are disjoint and contained in a
larger $Nk$-grid union $C$. Then $C\setminus\cup_{D\in Z}D$ can be tiled by
almost $k$-boxes.

###### Proof.

We enlarge each $D\in Z$ to $D^{\prime}$ such that $C\setminus\bigcup_{D\in
Z}D^{\prime}$ can be tiled with $k$-boxes. The regions $D^{\prime}\setminus D$
will be handled by Lemma 6.1.

By shifting everything we can assume that for some $L^{*}\subseteq
k\mathbb{Z}^{d}$ we have that

$C=\bigcup_{\beta\in L^{*}}\beta+[0,k)^{d}.$

Each $D$ is of the form $\bigcup_{\beta\in L}\beta+\gamma+[0,Nk)^{d}$ for some
$\gamma\in\mathbb{Z}^{d}$ and $L\subseteq Nk\mathbb{Z}^{d}$. Let $\gamma_{D}$
be a vector of minimal length so that $\gamma+\gamma_{D}+L\subseteq
k\mathbb{Z}^{d}$. Note that $\|\gamma_{D}\|<k$.

Let

$D^{\prime}=\bigcup_{\beta\in L}\beta+\gamma+\gamma_{D}+[-3k,(N+3)k)^{d}.$

By Lemma 6.1, $D^{\prime}\setminus D$ can be tiled by almost $k$-boxes. We
note also that $C\setminus D^{\prime}$ can also be tiled by $k$-boxes since
$C$ has such a tiling and $D^{\prime}$ can be written as a union of (a subset
of) these tiles.

Clearly, any element of $D^{\prime}$ is distance at most $4k$ from an element
of $D$. Hence if $D,E\in Z$, then $D^{\prime}$ and $E^{\prime}$ are disjoint.
Finally, $C\setminus\bigcup_{D\in Z}D^{\prime}$ can be tiled by $k$-boxes,
since the tiling of $C$ which was used to tile $C\setminus D^{\prime}$ is
independent of $D$. ∎

To apply these results in Section 12 we will need some special properties of
the tilings. To state these properties we need some more definitions. The
_faces_ of a box $B$ are the sets $\partial_{i}^{+}(B)$ and
$\partial_{i}^{-}(B)$ for $1\leq i\leq d$. We say that two disjoint boxes $A$
and $B$ _meet_ if are vectors $\alpha\in A$ and $\beta\in B$ such that their
difference is a standard generator of $\mathbb{Z}^{d}$. Given two disjoint
boxes we say that they meet _face to face_ if one has a face which is a
translate of a face of the other by a standard generator. A tiling $c$ of a
set $B\subseteq\mathbb{Z}^{d}$ is called _face to face_ if all boxes in $c$
which meet, meet face to face.

Given two disjoint boxes $A$ and $B$ we say that they meet _almost face to
face_ if they have faces $A^{\prime}$ and $B^{\prime}$ such that there is a
standard generator $\epsilon$ for which $A^{\prime}\cap(B^{\prime}+\epsilon)$
has cardinality at least $1/(12)^{d}$ fraction of the minimum of the volume of
$A^{\prime}$ and $B^{\prime}$. Given a tiling, we regard the boxes in the
tiling as vertices of a graph which are connected by an edge if they meet
almost face to face. We say that a tiling is _almost face to face_ if the
associated graph is connected. We call this graph the _almost face to face
graph_.

######  Corollary 6.5.

Let $C$ and $D;D\in Z$ be as in Proposition 6.4 where $k\geq 100^{d}$. There
is a tiling of $C\setminus\bigcup_{D\in Z}D$ by almost $k$-boxes. The tiling
of $C\setminus\bigcup_{D\in Z}D$ has the following three properties:

  1. (1)

For each $D\in Z$, the tiling of $C\setminus\bigcup_{D\in Z}D$ meets the
tiling of $D$ by $k$-boxes, face to face on $\partial_{1}(D)$.

  2. (2)

The tiling of $C\setminus\bigcup_{D\in Z}D$ agrees with the tiling of $C$ by
$k$-boxes on $\partial C$.

  3. (3)

The tiling of $C\setminus\bigcup_{D\in Z}D$ is almost face to face.

###### Proof.

For this proof we will trace the proof of Proposition 6.4 making necessary
observations as we go along.

First the sets $D$ were expanded slightly to get sets $D^{\prime}$ where the
sets $C\setminus\bigcup_{D\in Z}D^{\prime}$ can be tiled with $k$-boxes (which
proves property $(2)$) while the regions $D^{\prime}\setminus D$ (which
contain the boundary of $D$) were handled by Lemma 6.1. In the proof of Lemma
6.1 we wrote the set corresponding to $D^{\prime}\setminus D$ as a union of
sets $F_{i}\setminus F_{i-1}$ each of which were tiled separately. In the
construction the set $F_{1}\setminus F_{0}$ contained $\partial_{1}(D)$ and we
have that the tiling of $F_{1}\setminus D$ is obtained by taking the tiling of
$D$ by $k$-boxes and extending it perpendicular to $\partial_{1}D$. It follows
that the tiling of $D^{\prime}\setminus D$ that we constructed meets the
natural tiling of $D$ by $k$-boxes face to face on $\partial_{1}(D)$. This
proves Property $(1)$.

Now we will prove Property $(3)$. Let $B$ be a translate of $[1,k]^{d}$
centered in $[0,1]^{d}$. Consider the set

$C^{\prime}:=\\{\alpha~{}:~{}\alpha+B\subseteq C\setminus\bigcup_{D\in
Z}D\\}.$

$C^{\prime}$ is a connected set (since it is just the set
$C\setminus\bigcup_{D\in Z}D$ with a small portion of its boundary removed).
Given any two boxes $B_{1},B_{2}$ from the tiling, we can find a path $P$ in
$C^{\prime}$ with terminal vertices $\alpha,\beta\in C^{\prime}$ contained in
$B_{1}$ and $B_{2}$ respectively. Now suppose that
$\gamma,\gamma+\epsilon^{i}\in P$ such that $\gamma$ is in the tile
$B^{\prime}$ and $\gamma+\epsilon^{i}$ is in another tile $B^{\prime\prime}$.
Now all the tiles $\tilde{B}$ which meet $B^{\prime}$ on
$\partial_{i}^{+}(B^{\prime})$ are connected in the almost face to face graph.
In addition there has to be at least one tile among these tiles $\tilde{B}$
which meets $B^{\prime}$ almost face to face. Consequently we have that
$B^{\prime}$ and $B^{\prime\prime}$ are in the same component of the almost
face to face graph. Thus by following the path $P$ we must have that $B_{1}$
and $B_{2}$ are also in the same component and that the tiling is almost face
to face. ∎

We end the section with a lemma which is used to ensure points that we
construct are aperiodic.

######  Lemma 6.6.

Let $N,k>100$. Suppose that $B_{1},B_{2},\dots$ witnesses Corollary 5.3 with
$M=Nk$ and $n=10k$ and sequence $r_{i}$ for $i\in\mathbb{N}$. Then for every
$i$ and $\beta\in\mathbb{Z}^{d}$ with $r_{i}/4\leq|\beta|\leq r_{i}/2$ there
is a Borel set $E_{i}\subseteq B_{i}$ such that each connected component of
$E_{i}$ is a cube $E$ of side length $k$ contained in a connected component
$C$ of $B_{i}$ such that

  1. (1)

$\partial^{k}E,\partial^{k}\beta+E\subseteq C\setminus\bigcup_{D\in Z}D$ and

  2. (2)

$\partial^{k}E,\partial^{k}\beta+E$ and $\partial^{k}D$ for $D\in Z$ are
disjoint

where $Z=\\{D\mid D$ is a connected component of $B_{j}$ for $j<i$ and
$D\subseteq C\\}$.

###### Proof.

Let $C$ and $Z$ be as in the lemma. Since the sets $\partial^{10k}(C)$ and
$\partial^{10k}(D)$ for $D\in Z$ are disjoint it follows that the set

$\bar{C}=C\setminus(\partial^{3k}(\mathbb{Z}^{d}\setminus C)\cup\bigcup_{D\in
Z}(D\cup\partial^{3k}D))$

is connected.

Since $\bar{C}$ is bounded, there is $x\in\bar{C}$ such that
$T^{\beta}(x)\notin\bar{C}$. Let us now see why $\bar{C}\cap
T^{-\beta}(\bar{C})$ is non-empty. $C$ contains a cube of length $r_{i}-4k$
where $r_{i}-4k>|\beta|$ which is at distance $4k$ away from the boundary of
$C$. Since it intersects its translate by $-\beta$ the same must occur for its
inner $4k$ boundary as well. Thus we have that

$\partial^{4k}(\mathbb{Z}^{d}\setminus
C)\setminus\partial^{3k}(\mathbb{Z}^{d}\setminus C)$

must also intersect its translate by $-\beta$. But this set is disjoint from
$D\cup\partial^{3k}D$ for all $D\in Z$. Thus we have that $\bar{C}\cap
T^{-\beta}(\bar{C})$ is non-empty. Let $y\in\bar{C}\cap T^{-\beta}(\bar{C})$.
Since $\bar{C}$ is connected there is a path $p$ from $x$ to $y$ contained in
$\bar{C}$. Let $w$ be the first vertex on the path $p$ for which
$T^{\beta}(w)\in\bar{C}$. Let $E$ be a cube of side length $k$ centered at
$w$. It is clear that $E$ satisfies the conditions of the lemma. Since there
are only finitely many choices of $E$ given the shapes of various connected
components involved it follows that there exists a Borel set $E_{i}$ with the
required properties. ∎

##  7\. Property F

In this section, we isolate a mixing condition called property $F$ and prove
that any free Borel $\mathbb{Z}^{d}$ system admits a Borel factor map into the
free part of a symbolic system with property F. A _family of patterns_ is a
collection $(P_{C}\subseteq\mathcal{L}(X,C);\text{$C$ is a $k$-grid union})$
such that $P_{C}=\sigma^{\alpha}(P_{C-\alpha})$ for all
$\alpha\in\mathbb{Z}^{d}$.

######  Definition 7.1.

A family of patterns $(P_{C};\text{$C$ is a $k$-grid union})$ has _property
$F$ with gap $m$ _if the following two properties are satisfied.

  1. (1)

For every

    1. (a)

$k$-grid unions $C,C_{1},C_{2},\ldots,C_{r}$ and

    2. (b)

patterns $c_{i}\in P_{C_{i}}$,

such that $C_{1}+[-m,m]^{d},C_{2}+[-m,m]^{d},\ldots,C_{r}+[-m,m]^{d}$ are
disjoint and contained in $C$, there exists $c\in P_{C}$ such that
$c|_{C_{j}}=c_{j}\text{ for }1\leq j\leq r$.

  2. (2)

If $x$ is a configuration such that there is an increasing sequence of
$k$-grid unions $C_{k}$ which cover $\mathbb{Z}^{d}$ and $x|_{C_{k}}\in
P_{C_{k}}\text{ for all }k$ then $x\in X$.

We note that each $k$-grid union $C,C_{1},C_{2},\dots$ may use a different
offset $\gamma$. We also note that property $(2)$ is satisfied by every
subshift. We will need this to prove that the space of bi-infinite Hamilton
paths has Property F.

######  Theorem 7.2.

If $(Y,T)$ is a free Borel $\mathbb{Z}^{d}$ system and $(X,S)$ is a symbolic
system satisfying Property F, then there is an equivariant Borel map from $Y$
to the free part of $X$.

###### Proof.

Suppose that $(X,S)$ is a symbolic system contained in
$\\{1,2,\ldots,l\\}^{\mathbb{Z}^{d}}$ which has property $F$ relative to
$k$-grid unions with gap $m$. Let $P_{C}$ for $k$-grid unions $C$ be the
witnessing set of patterns. Let $B_{1},B_{2}\dots$ witness Corollary 5.3 for
$(Y,T)$ with $M=n=10k$, $\delta=1/2$ and any sequence $r_{1}<r_{2}<\dots$
satisfying the hypothesis of the corollary.

We define a function $\Phi$ using property $F$ as follows. Let $\beta_{i}$ for
$i\in\mathbb{N}$ be a sequence of vectors such that

  1. (1)

for all $i\in\mathbb{N}$, $r_{i}/4\leq|\beta_{n}|\leq r_{i}/2$ and

  2. (2)

for all $\gamma\in\mathbb{Z}^{d}$, there is $j\in\mathbb{N}$ such that
$j\gamma=\beta_{i}$ for some $i\in\mathbb{N}$.

Such a sequence is easily constructed by induction. We define a function
$\Phi$ which captures our applications of property F. The inputs are
$(1+\delta)$-almost cube $M$-grid unions $C$ of side length $r_{i}$ in
$\mathbb{Z}^{d}$ and patterns $d_{1},\dots d_{r}$ over $(1+\delta)$-almost
cubes $D_{1},\dots D_{r}$ of side lengths $r_{j}$ for various $j<i$ such that
$d_{i}\in P_{D_{i}}$ and $\partial^{10k}D_{1},\dots\partial^{10k}D_{r}$ are
disjoint subsets of $C$. Given such an input, we can apply Lemma 6.6 with
$\beta=\beta_{i}$, to find a cube $E$ of side length $k$ as in the lemma. We
then choose $e\in P_{E}$ and $e^{\prime}\in P_{\beta_{i}+E}$ such that
$e^{\prime}\neq\sigma^{\beta_{i}}(e)$. Finally we let
$d=\cup_{i=1}^{r}d_{i}\cup e\cup e^{\prime}$ and $\Phi(C,d)$ be a pattern
obtained by applying property F to the patterns $d_{1},\dots
d_{r},e,e^{\prime}$ inside $C$. We can assume that $\Phi$ commutes with the
shift operation.

We will now define a sequence of Borel functions
$f_{n}:\cup_{i=1}^{n}B_{i}\to\\{1,\dots l\\}$ by induction.

To initialize the construction we let $B_{0}=\emptyset$ and $f_{0}$ be the
trivial function. Suppose that we have constructed $f_{n}$ for some $n$. Let
$y\in B_{n+1}\setminus\cup_{i=1}^{n}B_{i}$. We define the following:

  1. (1)

$C_{y}=\\{\gamma\mid y,T^{\gamma}(y)$ are in the same connected component of
$B_{n+1}\\}$,

  2. (2)

$D_{y}=\\{\gamma\in C_{y}\mid T^{\gamma}(y)\in B_{i}$ for some $i\leq n\\}$

  3. (3)

$d_{y}$ is the function with domain $D_{y}$ such that
$d_{y}(\gamma)=f_{n}(T^{\gamma}(y))$.

Let $f_{n+1}(y)=c_{y}(\bar{0})$ where $c_{y}=\Phi(C_{y},d_{y})$. For
$y\in\cup_{i=1}^{n}B_{i}$ we define $f_{n+1}(y)=f_{n}(y)$. Since $C_{y},D_{y}$
and $d_{y}$ are Borel functions on $B_{n+1}\setminus\cup_{i=1}^{n}B_{i}$ with
finite range and $\Phi$ is a map between finite sets it follows that $f_{n+1}$
is Borel. Moreover, since the assignments of $C_{y},D_{y}$ and $c_{y}$ are
equivariant and the operation $\Phi$ commutes with the shift, we have that for
a connected component $C$ of $B_{n+1}$, the pattern given by fixing $y\in C$
and taking $\gamma\mapsto(\cup_{i=1}^{n+1}f_{i})(T^{\gamma}(y))$ for $\gamma$
such that $T^{\gamma}(y)\in C$ is in $P_{C}$. Further by our choices of $e$
and $e^{\prime}$ above we have that for all $y\in B_{n+1}$ there exists
$\gamma$ such that

$f_{n+1}(T^{\gamma}(y))\neq f_{n+1}(T^{\gamma+\beta_{n+1}}(y)).$

Thus the map $\hat{f}:Y\to X$ given by

$\hat{f}(y)(\gamma)=f_{n}(T^{\gamma}(y))\text{ for $n\in\mathbb{N}$ and all
$y\in B_{n}$}$

is equivariant and maps to the free part of $X$. ∎

In Section 9 we will prove that if $(Y,T)$ is a symbolic space with
appropriate entropy then we can actually find an embedding.

##  8\. Markers

The entropy of the collection $\mathcal{C}=(C_{B};B\text{ is a $k$-grid
union})$ is defined by

$h(\mathcal{C})=\liminf\frac{1}{(nk)^{d}}\log{|C_{[1,nk]^{d}}|}.$

The limit exists under the assumption of property $F$ (but we will not need
this fact) and is non-zero (except in the trivial case of each $C_{B}$ being a
singleton). The proof follows from the proof of [CM21, Lemma 5.3].

We say that patterns $a,b\in\mathcal{L}(X)$ clash if there is a site $\gamma$
such that $a_{\gamma}\neq b_{\gamma}$. A collection
$\mathcal{M}=(M_{n}\subseteq\mathcal{L}(X,[1,nk]^{d});n\in\mathbb{N})$ is said
to have _the marker property_ with tolerance $r$ if for all $a,b\in M_{n}$ and
$|\gamma|<nk-r$, we have that $\sigma^{\gamma}(a)$ clashes with $b$ unless
$\gamma=\bar{0}$ and $a=b$.

We say that the collection $\mathcal{M}$ has _the marker property $F$_ if it
has the marker property witnessed by $M_{n}$ for $n\in\mathbb{N}$ and there
exists a collection $\mathcal{C}=(C_{B};B\text{ is a $k$-grid union})$ with
property $F$ such that $M_{n}\subseteq C_{[1,nk]^{d}}\text{ for all
}n\in\mathbb{N}$.

######  Theorem 8.1.

If $X$ is a shift space which has a collection $\mathcal{C}$ with property $F$
then it is has a collection $\mathcal{M}$ with marker property $F$ and
$h(\mathcal{C})=h(\mathcal{M})$.

We will follow the strategy as in [CM21]. The situation is much simpler here
because we are dealing with shift spaces, property $F$ is much stronger than
flexibility (being used in that context) and the dimension $d\geq 2$. Marker
constructions have a long history in ergodic theory and similar ideas can be
found throughout the literature improvised according to context.

We will need the following lemma.

######  Lemma 8.2 (No small periods).

Let $X$ be a shift space, $\mathcal{C}=(C_{n};n\in\mathbb{N})$ where
$C_{n}\subseteq\mathcal{L}(X,[1,nk]^{d})$ be a collection with entropy
$h(\mathcal{C})>0$. Let $r\in\mathbb{N}$. There exists a collection
$\mathcal{C}^{\prime}=(C_{n}^{\prime};n\in\mathbb{N})$ where
$C_{n}^{\prime}\subseteq C_{n}$ such that
$h(\mathcal{C}^{\prime})=h(\mathcal{C})$ and for all $a\in C_{n}^{\prime}$ and
$\gamma\in[-r,r]^{d}\setminus\\{\bar{0}\\}$, $a$ clashes with
$\sigma^{\gamma}(a)$.

This lemma is very similar to [Hoc10, Proposition 12] with essentially the
same proof. However, we cannot use the statement directly, so we repeat the
argument.

###### Proof.

Let $\gamma\in[-r,r]^{d}\setminus\\{\bar{0}\\}$. Consider the set of patterns

$P_{n,\gamma}=\\{a\in\mathcal{C}_{n}~{}:~{}\text{$a$ does not clash with
$\sigma^{\gamma}(a)$}\\}.$

The elements of $P_{n,\gamma}$ are periodic with period $\gamma$ and thus if
$a,b\in P_{n,\gamma}$ agree on the set $[1,nk]^{d}\setminus[3r,nk-3r]^{d}$,
then $a=b$.

Hence there is an injection from $P_{n,\gamma}$ to
$\mathcal{L}(X,[1,nk]^{d}\setminus[3r,nk-3r]^{d})$ and it follows that
$|P_{n,\gamma}|<c\exp^{c^{\prime}n^{d-1}}$ for some constants $c,c^{\prime}>0$
which only depend on $r$.

Thus the collection $\mathcal{C}^{\prime}=(C_{n}^{\prime};n\in\mathbb{N})$
given by

$C_{n}^{\prime}=\\{a\in C_{n}~{}:~{}a\notin P_{n,\gamma}\text{ for
}\gamma\in[-r,r]^{d}\setminus\\{\bar{0}\\}\\}$

satisfies the conclusion of the lemma.

∎

We are now prepared for the construction of the markers.

###### Proof of Theorem 8.1.

Let $\mathcal{C}$ be a collection with property $F$ and gap $m$. By Lemma 8.2
there exists $n\in\mathbb{N}$ and $a^{\prime}\in C_{[1,nk]^{d}}$ such that
$a^{\prime}$ clashes with $\sigma^{\gamma}(a^{\prime})$ for for all
$\gamma\in[-1,1]^{d}\setminus\\{\bar{0}\\}$. The collection
$\mathcal{M}=(M_{t};t\in\mathbb{N})$ with the marker property will consist of
patterns whose boundaries are densely packed by elements of $a^{\prime}$ but
with different phases. This is to ensure that given two patterns from $M_{t}$
for some $t$ if their boundaries intersect substantially then there must be a
clash because of the various phases.

First we divide the support of the elements of $M_{t}\subseteq C_{[1,tk]^{d}}$
into thin annuli and a large central region.

  1. (1)

For $t$ large enough we consider concentric annuli in $[1,tk]^{d}$ with width
$100(m+nk)$ labelled $A_{\gamma}$ for $\gamma\in[0,m+nk]^{d}$ starting from
the outside towards the inside and leaving gaps of width $3(m+nk)$ in-between.

  2. (2)

The complement of (the union of) the annuli $A_{\gamma}$ is a box of the form
$[1+\theta k,(t-\theta)k]^{d}$ for some constant $\theta\in\mathbb{N}$ which
independent of $t$.

Having defined these regions, we also define some patterns that will be used
in the definition of $M_{t}$. Using Lemma 8.2, we find $a^{\prime\prime}\in
C_{[1,sk]^{d}}$ for some $s$ such that for all $\gamma$ in
$[-2k\theta,2k\theta]$, $\sigma^{\gamma}(a^{\prime\prime})$ clashes with
$a^{\prime\prime}$. Next we consider a configuration $x$ given by
$\sigma^{\gamma}(x)|_{[1,nk]^{d}}=a^{\prime}$ for all
$\gamma\in(2m+n)k\mathbb{Z}^{d}$.

Now define $M_{t}$ as the set of all patterns $b$ in $C_{[1,tk]^{d}}$ with the
following properties:

  1. (i)

$b|_{A_{\gamma}}=\sigma^{\gamma}(x)|_{A_{\gamma}}$ for all
$\gamma\in[0,m+nk]^{d}$ with the additional condition that copies of
$a^{\prime}$ in $\sigma^{\gamma}(x)$ which lie on the boundary of $A_{\gamma}$
are included.

  2. (ii)

$b|_{[1+\theta k,(t-\theta)k]^{d}}$ is a translate of a pattern $c\in
C_{[1,(t-2\theta)k]^{d}}$ such that $c$ extends
$\sigma^{\bar{m}}(a^{\prime\prime})$ where $\bar{m}$ is the constant $m$
vector.

By property F, a straightforward entropy calculation shows that
$h(\mathcal{M})=h(\mathcal{C})$. To finish we will show that for all distinct
$e,f\in M_{t}$, $f$ clashes with $\sigma^{\gamma}(e)$ for all
$\gamma\in[-(t-2\theta)k,(t-2\theta)k]^{d}$. There are two cases. Firstly if
$|\gamma|_{\infty}\leq 2\theta k$, then Condition (ii) from the definition of
$M_{t}$ ensures that $f$ clashes with $\sigma^{\gamma}(e)$. Specifically, each
of $f$ and $\sigma^{\gamma}(e)$ contain copies of $a^{\prime\prime}$ which
clash by the choice of $a^{\prime\prime}$.

Secondly, if $|\gamma|_{\infty}>2\theta k$, then we work with Condition (i)
find a clash between $f$ and $\sigma^{\gamma}(e)$. For each $i\leq d$, let
$\alpha_{i}$ be the remainder of $\gamma_{i}$ when divided by $m+nk$. Next
find $\delta$ with $|\delta|_{\infty}<m+nk$ such that
$|\delta+\alpha-\gamma|_{\infty}=1$. We consider the restriction of $e$ to the
annulus $A_{\delta}$ and note that by our assumption on $|\gamma|_{\infty}$,
$A_{\delta}$ intersects $\sigma^{\gamma}(A_{\delta})$ in a rectangle of side
length $100(m+nk)$. Moreover, on this rectangle $\sigma^{\gamma}(e)$ is equal
to
$\sigma^{\gamma}(\sigma^{\delta}(x))=\sigma^{\alpha}(\sigma^{\delta}(x))=\sigma^{\alpha+\delta}(x)$.
By the choice of $\delta$ and the size of the intersection of $A_{\delta}$ and
$\sigma^{\gamma}(A_{\delta})$, there are copies of $a^{\prime}$ in $f$ and
$\sigma^{\gamma}(e)$ which are shifts of each other by a vector of norm $1$.
So $f$ and $\sigma^{\gamma}(e)$ clash our the choice of $a^{\prime}$. ∎

##  9\. Constructing the embedding

We will now strengthen Theorem 7.2 in the case of shift spaces.

######  Theorem 9.1.

Let $(X,S)$ be a symbolic space with a collection $\mathcal{C}$ with Property
F and $(Y,T)$ is a shift space such that $h(Y,T)<h(\mathcal{C})$. Then there
is an equivariant Borel embedding from the free part of $Y$ to $X$.

Suppose $X$ is a symbolic space with entropy $h(X)$ and $Y$ is a shift space
with a collection
$\mathcal{C}=(C_{n};n\in\mathbb{N});C_{n}\subseteq\mathcal{L}(X,[1,nk]^{d})$
with marker property $F$, entropy $h$, gap and tolerance $r/3$ (for some $r\in
3\mathbb{N}$). We can assume that each $C_{n}$ satisfies Lemma 8.2 for shifts
of norm at most $r$.

Since $h(Y,T)<h(\mathcal{C})$, by a standard calculation involving the
topological entropy, there are an $n\in\mathbb{N}$ and an injective function

$\Psi:\mathcal{L}(X,[1,nk+2r]^{d})\cup\mathcal{L}(X,[1,nk+2r+1]^{d})\to\sigma^{\bar{r}}(C_{n})$

where $\bar{r}$ is the constant $r$ vector. For ease of notation below, we can
extend $\Psi$ to all shifts of the domain in a way that commutes with the
shift operation.

We note that $\sigma^{\bar{r}}(C_{n})$ is a set of patterns on
$[r+1,nk+r]^{d}$ and $\partial^{r}[r+1,nk+r]^{d}$ is contained in
$[1,nk+2r]^{d}$ (and hence in $[1,nk+2r+1]^{d}$ as well).

Let $B_{1},B_{2},\dots$ be a sequence of Borel sets witnessing Corollary 5.3
with parameters appropriate to apply the proof of Theorem 2.1 for the space of
tilings using tiles $[1,nk+2r]^{d}$ and $[1,nk+2r+1]^{d}$. Let $H:X\to
X(\\{[1,nk+2r]^{d},[1,nk+2r+1]^{d}\\})$ be the Borel factor map obtained from
the theorem. In particular $[1,nk+2r]^{d}$ and $[1,nk+2r+1]^{d}$ tile $B_{n}$
for all $n$.

Let $R_{x}\subseteq\mathbb{Z}^{d}$ be the tile of $\bar{0}$ in $H(x)$. Let
$\gamma_{x}\in\mathbb{Z}^{d}$ be such that $R_{x}$ is either
$\gamma_{x}+[1,nk+2r]^{d}$ or $\gamma_{x}+[1,nk+2r+1]^{d}$. Let $B_{0}$ be the
set of $x$ such that $\bar{0}\in\gamma_{x}+[r+1,nk+r]^{d}$.

We define a function $f_{0}$ on $B_{0}$ by $f_{0}(x)=c(\bar{0})$ where
$c=\Psi(x|_{R_{x}})$. Since the connected components of $B_{0}$ are at least
$2r$ apart and at least $2r/3$ distance from $\partial B_{n}$ for all $n$, by
using property $F$ we can extend $f_{0}$ defined on $B_{0}$ to $f_{n}$ defined
on $\cup_{m=0}^{n}B_{m}$ for all $n$ as in the proof of Theorem 7.2. In fact
it is simpler here because we do not need to use Lemma 6.6. At the end of the
construction, we let $f=\bigcup_{j\in\mathbb{N}}f_{j}$ and $\hat{f}$ be given
by $\hat{f}(y)(\gamma)=f(T^{\gamma}(y))$. We have as in Theorem 7.2 that the
resulting maps $f$ and $\hat{f}$ are Borel.

We would like to prove that $\hat{f}$ is an embedding of $Y$ into $X$. Suppose
that $y,y^{\prime}\in free(Y)$. We associate to $y$ an infinite pattern
$p=\gamma\mapsto f_{0}(T^{\gamma}(y))$ for $\gamma$ such that
$T^{\gamma}(y)\in B_{0}$ and similarly associate $p^{\prime}$ to $y^{\prime}$.
If $\mathop{\mathrm{dom}}(p)=\mathop{\mathrm{dom}}(p^{\prime})$, then
$\hat{f}(y)=\hat{f}(y^{\prime})$ implies $p=p^{\prime}$ which implies
$y=y^{\prime}$, since the map $\Phi$ is injective.

So we can assume that
$\mathop{\mathrm{dom}}(p)\neq\mathop{\mathrm{dom}}(p^{\prime})$. We fix a
connected component $R$ of $\mathop{\mathrm{dom}}(p)$ which is not equal to
any connected component of $\mathop{\mathrm{dom}}(p^{\prime})$. Since sets of
the form $R^{\prime}\cup\partial^{r+1}R^{\prime}$ for connected components
$R^{\prime}$ of $\mathop{\mathrm{dom}}(p^{\prime})$ cover $\mathbb{Z}^{d}$,
there is some connected component $R^{\prime}$ at distance at most $r+1$ from
the center of $R$. It follows that $R$ is a shift of $R^{\prime}$ by some
$\gamma$ of norm at most $r+1+nk/2<nk-r$. By the tolerance of the set of
markers we have that $p|_{R}$ clashes with $p^{\prime}|_{R^{\prime}}$ and
hence $\hat{f}(y)\neq\hat{f}(y^{\prime})$.

##  10\. Hom-shifts have property $F$

In this section we will prove that hom-shifts have property $F$. The argument
we present is essentially contained in [CM21, Section 9] but needs some
restatement because of our setting.

######  Theorem 10.1.

Let $\text{Hom}(\mathbb{Z}^{d},H)$ be a hom-shift where $H$ is a connected
graph which is not bipartite. Then it has property $F$ for a sequence of
patterns with entropy equal to the topological entropy of the hom-shift.

For this proof we denote the interior boundary of a set $C$ by

$\partial_{\text{int}}C=\\{\alpha\in C~{}:~{}\alpha\text{ is adjacent to a
vertex in $\mathbb{Z}^{d}\setminus C$}\\}.$

Further by parity of a coordinate $\alpha\in\mathbb{Z}^{d}$ we mean the parity
of the sum of its coordinates, which we denote as
$\mathop{\mathrm{parity}}(\alpha)$.

###### Proof.

Choose vertices $v,w\in H$ which form an edge. For every connected set $C$ fix
a site $\alpha_{C}\in C$ such that

$\alpha_{\beta+C}=\beta+\alpha_{C}.$

Consider the set of patterns

$P_{C}=\left\\{a\in\textit{Hom}(C,H)~{}:~{}\text{ for all
}\beta\in\partial_{\text{int}}C,a_{\beta}=\begin{cases}v&\text{ if
$\mathop{\mathrm{parity}}(\alpha_{C})=\mathop{\mathrm{parity}}(\beta)$}\\\
w&\text{ otherwise }.\end{cases}\right\\}.$

We proved in [CM21, Proposition 9.2] that the entropy of this collection of
patterns is equal to the entropy of hom-shift. We are left to prove property
$F$ for the collection. Since $H$ is connected and not bipartite, we can fix
an even integer $N$ so that there is a path of length $N$ from $v$ to $w$ and
some $N^{\prime}>N$. Let $v_{0}\dots v_{N}$ be such a path with $v_{0}=v$ and
$v_{N}=w$.

Now let $C,C_{1},C_{2},\ldots,C_{r}$ be connected subsets such that
$C_{1}+[-N^{\prime},N^{\prime}]^{d},C_{2}+[-N^{\prime},N^{\prime}]^{d},\ldots,C_{r}+[-N^{\prime},N^{\prime}]^{d}$
are disjoint and contained in $C$. Further let $c_{i}\in P_{C_{i}}$ be given.
We need to find $c\in P_{C}$ such that $c|_{C_{i}}=c_{i}$.

Consider connected subsets $C_{i}^{\prime}=\partial^{N}C_{i}$ for all $1\leq
i\leq r$. Given $1\leq i\leq r$, we define patterns
$c^{\prime}_{i}\in\mathcal{L}(\textit{Hom}(C_{i}^{\prime},H))$ extending
$c_{i}$ depending on the parity of $\alpha_{C}$ and $\alpha_{C_{i}}$. If
$\alpha_{C}$ and $\alpha_{C_{i}}$ have the same parity, then for $\alpha\in
C^{\prime}_{i}\setminus C_{i}$ we set $c^{\prime}_{i}(\alpha)=v$ if
$\mathop{\mathrm{parity}}(\alpha_{C})=\mathop{\mathrm{parity}}(\alpha)$ and
$c_{i}^{\prime}(\alpha)=w$ otherwise. If $\alpha_{C}$ and $\alpha_{C_{i}}$
have different parity, then to define $c^{\prime}_{i}(\alpha)$ we let $t$ be
such that $\alpha\in\partial^{t}C_{i}\setminus\partial^{t-1}C_{i}$ and let
$c_{i}^{\prime}(\alpha)=v_{t-1}$ if
$\mathop{\mathrm{parity}}(\alpha)=\mathop{\mathrm{parity}}(\alpha_{C_{i}})$
and $c_{i}^{\prime}(\alpha)=v_{t}$ otherwise.

It follows that for the patterns $c^{\prime}_{i}$ we must have that for all
$\alpha\in\partial_{\text{int}}(C^{\prime}_{i})$

$c^{\prime}_{i}(\alpha)=\begin{cases}\text{a vertex adjacent to $w$}&\text{ if
$\mathop{\mathrm{parity}}(\alpha_{C})=\mathop{\mathrm{parity}}(\alpha)$}\\\
w&\text{otherwise}.\end{cases}$

and the pattern $c\in\mathcal{L}(\textit{Hom}(\mathbb{Z}^{d},H))$ extending
the $c_{i}^{\prime}$ defined on $\alpha\in C\setminus\cup_{1\leq i\leq
r}C_{i}^{\prime}$ by $c(\alpha)=v$ if
$\mathop{\mathrm{parity}}(\alpha)=\mathop{\mathrm{parity}}(\alpha_{C})$ and
$c(\alpha)=w$ otherwise is what we set out to obtain.∎

##  11\. Rectangular tilings have property $F$

In this section we will show that coprime rectangular shifts have property
$F$. The argument comes from [CM21, Section 10].

######  Theorem 11.1.

Let $X(\mathcal{T})$ be a coprime rectangular shift. Then it has property $F$.
If $X(\mathcal{T})$ is the set of domino tilings of $\mathbb{Z}^{2}$ then it
has property $F$ for a sequence of patterns with entropy equal to the
topological entropy of $X(\mathcal{T})$.

###### Proof.

Let $k$ be the product of the side lengths of all the boxes in $\mathcal{T}$.
For every $10k$-grid union $C$ we let

$P_{C}=\\{\text{tilings of $C$ by elements of $\mathcal{T}$}\\}.$

In the case of domino tilings of $\mathbb{Z}^{2}$ in the proof of [CM21,
Theorem 10.3] we showed by following [Kas61, CKP01] that the entropy of this
collection of patterns is the topological entropy of the space of domino
tilings. We are now left to prove property $F$ for this collection.

Let $C,C_{1},C_{2},\ldots,C_{r}$ be $10k$-grid unions such that
$C_{1}+[-10k,10k]^{d},C_{2}+[-10k,10k]^{d},\ldots,C_{r}+[-10k,10k]^{d}$ are
disjoint and contained in $C$ with offsets
$\alpha,\alpha_{1},\ldots,\alpha_{d}\in[0,10k)^{d}$. Let $a_{i}\in P_{C_{i}}$
for all $1\leq i\leq r$. We will now construct $a\in P_{C}$ such that
$a|_{C_{i}}=a_{i}$ for all $1\leq i\leq r$.

Now we write

$C_{i}=\bigcup_{j=1}^{r_{i}}\alpha_{i}+\gamma_{i,j}+[0,10k)^{d}$

for some set

$\\{\gamma_{i,j}~{}:~{}1\leq j\leq r_{i}\\}\subseteq 10k\mathbb{Z}^{d}.$

Choose $\alpha_{i}^{\prime}$ such that $\alpha-\alpha^{\prime}_{i}\in
k\mathbb{Z}^{d}$ and $\alpha^{\prime}_{i}\in\alpha_{i}+[0,k)^{d}$. We now
consider the sets

$C^{\prime}_{i}=\bigcup_{j=1}^{r_{i}}\alpha^{\prime}_{i}+\gamma_{i,j}+[-3k,13k)^{d}.$

The sets $C^{\prime}_{i}$ are disjoint (since the sets $C_{i}+[-10k,10k]^{d}$
are disjoint). By Lemma 6.1, $C^{\prime}_{i}\setminus C_{i}$ can be tiled by
boxes each of whose side length is at least $k$ and at most one side different
from $k$. Further by our choice of $\alpha^{\prime}_{i}$ we have that

$C\setminus\bigcup C^{\prime}_{i}$

can be tiled by boxes all of whose sides lengths are $k$. Since each side
length of a box in $\mathcal{T}$ is a factor of $k$ it follows that
$\mathcal{T}$ can tile $C\setminus\bigcup C^{\prime}_{i}$. Fix such a tiling
$c^{\prime}$ on $C\setminus\bigcup C^{\prime}_{i}$. We are left to show that
if $B$ is a box such that each of whose side length at least $k$ and at most
one side different from $k$ then it can be tiled by elements of $\mathcal{T}$.
This is an easy consequence of the solution to the well-known Diophantine
Frobenius problem or the coin problem. Assume that $B=[0,N)\times[0,k)^{d-1}$
where $N\geq k$. Let

$\mathcal{T}=\\{T_{1},T_{2},\ldots,T_{s}\\}$

and $l_{i}$ be the length of $T_{i}$ in the the $\epsilon^{1}$ direction.
Since the highest common factor of $l_{i};1\leq i\leq s$ is $1$ it follows
that $N$ can be written as a sum of multiples of $l_{i}$ and thus $B$ can be
tiled by layers perpendicular to the direction $\epsilon^{1}$ where each layer
(of width a suitable multiple of $l_{i}$) is tiled by a single element of
$\mathcal{T}$. From this we obtain a tiling of $C^{\prime}_{i}\setminus C_{i}$
by elements of $\mathcal{T}$ which we call $a^{\prime}_{i}$. The tilings
$a^{\prime}_{i};1\leq i\leq r$ and $c^{\prime}$ give us a tiling of

$C\setminus\bigcup_{i=1}^{r}C_{i}.$

Together with the tilings $a_{i}$ of $C_{i}$, we get a tiling $a$ of $C$ which
we were looking for. ∎

##  12\. Property $F$ For Directed Bi-infinite Hamiltonian paths

###  12.1. The setup

In this section we prove that directed bi-infinite Hamiltonian paths have
property $F$. For the purpose of this section, we think of a Hamiltonianian
cycle (path) $c$ as a function where the domain is the set of vertices of the
cycle (path) and a directed edge is drawn from $\alpha$ to $\beta$ if
$\alpha+c(\alpha)=\beta$.

Our basic method for connecting Hamiltonian cycles is given in the following
proposition.

######  Proposition 12.1.

Let $C$ and $D$ be finite subsets of $\mathbb{Z}^{d}$ and $c_{1}$ and $c_{2}$
be directed Hamiltonian cycles on them. Suppose there is an edge
$(\alpha,\beta)$ in $c_{1}$ and an edge $(\gamma,\delta)$ in $c_{2}$ such that

  1. (1)

$\alpha$ is adjacent to $\delta$.

  2. (2)

$\beta$ is adjacent to $\gamma$.

Then by replacing $(\alpha,\beta)$ by $(\alpha,\delta)$ and $(\gamma,\delta)$
by $(\gamma,\beta)$ we get a Hamiltonian cycle on $C\cup D$.

The proof is obvious and left to the reader.

######  Theorem 12.2.

The space of bi-infinite directed Hamiltonian paths has property $F$.

Let $k>1000\times 100^{d}$ be an even integer and $C$ be a $100k$-grid union.
With any such grid union, there is a unique tiling of $C$ with $k$-boxes.
Choose a $k$-box $C^{\prime}$ from this tiling such that
$\partial^{+}_{1}(C^{\prime})\subseteq\partial^{+}_{1}(C)$. We let $P_{C}$ to
be the set of directed Hamiltonian paths which start at a vertex $\alpha$ and
end at vertex $\alpha+\epsilon^{2}$ such that
$\alpha,\alpha+16\epsilon^{2}\in\partial^{+}_{1}(C^{\prime})\text{ but
}\alpha+17\epsilon^{2}\notin\partial^{+}_{1}(C^{\prime}).$

We will prove that this collection of patterns satisfy property $F$. For this
let $C,C_{1},C_{2},\ldots,C_{r}$ be $100k$-grid unions and patterns $c_{i}\in
P_{C_{i}}$ such that $C_{1}+[-10k,10k]^{d}$, $C_{2}+[-10k,10k]^{d}$,
…,$C_{r}+[-10k,10k]^{d}$ are disjoint and contained in $C$. We will extend the
$c_{i}$ to a path $c\in P_{C}$.

To begin we will consider some very special directed Hamiltonian paths on two
dimensional boxes. These paths are constructed especially so that they can be
easily put together to form directed Hamiltonian paths on almost $k$-boxes and
that directed Hamiltonian paths on adjacent almost $k$-boxes can be connected.
These connections will be made using Proposition 12.1.

######  Lemma 12.3.

Let $l,m>90d$ such that $lm$ is an even integer. There is a directed
Hamiltonian cycle $c_{l,m}$ on $[1,l]\times[1,m]$ with the following features.

  1. (1)

The Hamiltonian cycle moves in the clockwise direction on all the edges of the
boundary except one.

  2. (2)

At distance greater than $4$ from the boundary the edges of cycle are moving
either in the $\epsilon^{1}$ direction if they are at even distance from the
top edge or the $-\epsilon^{1}$ direction if they are at odd distance from the
top edge.

To aid the reader, various cases (according to the parity of $l$ and $m$) have
been illustrated in Figure 1.

###### Proof.

We begin by drawing a clockwise oriented cycle on the boundary of
$[1,l]\times[1,m]$. Now we delete one of these edges to move the interior of
box. The path is extended inside using a path that goes back and forth across
the rectangle in the $\epsilon^{1}$ or $-\epsilon^{1}$ direction working from
top to bottom. This works well except when $m$ is odd. In this case some extra
‘teeth’ like structures of length $1$ are added near the boundary in the
$\epsilon^{2}$ direction. ∎

Figure 1. Hamiltonian cycles $c_{l,m}$ on a $17\times 18$, $16\times 17$ and
$16\times 16$ box respectively. Notice that if we choose a vertex at distance
greater than $4$ from the boundary then the Hamiltonian cycle will just
consist of lines in the $\epsilon^{1}$ direction and on the boundary the cycle
moves in the clockwise direction.

###  12.2. Hamiltonian cycles in two dimensions

With Lemma 12.3 we can readily prove property $F$ for $d=2$. By Corollary 6.5,
we can tile $C\setminus\cup_{i}C_{i}$ by almost $k$-boxes such that

  1. (1)

For each $1\leq i\leq r$, the tiling of $C\setminus\cup_{i}C_{i}$ meets the
tiling of $C_{i}$ by $k$ boxes, face to face on $\partial_{1}(C_{i})$.

  2. (2)

The tiling of $C\setminus\cup_{i}C_{i}$ agrees with the $k$-box tiling of $C$
on $\partial C$.

  3. (3)

The tiling of $C\setminus\cup_{i}C_{i}$ is almost face to face.

We start by constructing a Hamiltonian path on $C\setminus\cup_{i}C_{i}$.
Recall the graph structure on almost $k$-tilings. We regard the boxes in the
tiling as vertices of a graph where we place an edge between two boxes if they
meet in some face which is at least $1/12^{d}$ of the minimum volume of over
all faces of the two boxes.

By using property (3) above, we fix an ordering on the almost $k$-boxes in the
tiling of $C\setminus\cup_{i}C_{i}$ such that each box (after the first) is
almost face to face with some earlier box. We start the construction by
placing a Hamiltonian cycle $c_{l,m}$ for appropriate $l$ and $m$ on each tile
in the almost $k$-box tiling $C\setminus\cup_{i}C_{i}$. Using the ordering
above and Proposition 12.1, it is straightforward to iteratively connect the
Hamiltonian cycles on each almost $k$-box into one large Hamiltonian cycle.
Note that we use that each $c_{l,m}$ is essentially a clockwise cycle on the
boundary.

By property (1) above the tiling of $C\setminus\cup_{i}C_{i}$ meets the tiling
of $C_{i}$ by $k$ boxes face to face on $\partial_{1}(C_{i})$. Hence we can
connect each $c_{i}$ to the Hamiltonian cycle on $C\setminus\cup_{i}C_{i}$.
Finally since the tiling of $C\setminus\cup_{i}C_{i}$ agrees with the $k$-box
tiling of $C$ on $\partial C$ (property (2)), we can open an edge along
$\partial^{+}_{1}(C)$ to get the required Hamiltonian path with the prescribed
entry and exit points.

###  12.3. Hamiltonian cycles in dimensions greater than two

For $d>2$, the situation is much more complicated. The main complication is an
issue of parity. Any box can be divided into several two dimensional layers in
the direction $\epsilon^{1}\times\epsilon^{2}$. However we cannot place the
Hamiltonian cycles $c_{l,m}$ in each of these layers as it is; we have to
alternate the orientation between clockwise and anticlockwise otherwise it
will be difficult to connect them (using Proposition 12.1). Keeping this in
mind we define Hamiltonian cycles $d_{l,m}$ where the edges are reversed.
Specifically $(\alpha,\beta)$ is a (directed) edge for $c_{l,m}$ if and only
if $(\beta,\alpha)$ is an edge for $d_{l,m}$.

It is easy to see that the Hamiltonian cycles $d_{l,m}$ have analogous
properties to the ones in Lemma 12.3. In particular clockwise is replaced with
counterclockwise in (1) and the roles of $\epsilon^{1}$ and $-\epsilon^{1}$
are reversed in (2).

Let $R=\prod_{t}[l_{t},m_{t}]$ be a box such that $(m_{1}-l_{1})(m_{2}-l_{2})$
is even. The box can be partitioned into two dimensional layers as follows.

$\bigsqcup_{\alpha\in\prod_{t>2}[l_{t},m_{t}]}[l_{1},m_{1}]\times[l_{2},m_{2}]\times\\{\alpha\\}$

If $\alpha$ is an even vector we place a copy of $c_{m_{1}-l_{1},m_{2}-l_{2}}$
on $[l_{1},m_{1}]\times[l_{2},m_{2}]\times\\{\alpha\\}$ and a copy of
$d_{m_{1}-l_{1},m_{2}-l_{2}}$ if $\alpha$ is an odd vector. Clearly if
$\alpha,\beta$ differ by a unit vector then the corresponding Hamiltonian
cycles have opposite orientation and can be connected to each other by opening
an edge in the $\epsilon^{1}$ direction at distance greater than $4$ from the
boundary using Proposition 12.1. There is a natural graph structure on
$\prod_{t>2}[l_{t},m_{t}]$ which connects vectors differing by a unit vector.
By finding a spanning tree for this graph and connecting Hamiltonian cycles on
adjacent two dimensional layers we obtain a Hamiltonian cycle on the entire
box $R$. This Hamiltonian cycle will be called the _even type Hamiltonian
cycle_ on the box. The _odd type_ Hamiltonian cycle will be the one obtained
by reversing all the edges.

The set $C\setminus\cup_{i}C_{i}$ can be tiled by almost $k$-boxes such that
they satisfy the conclusion of Corollary 6.5. As in the two dimensional case,
the graph of almost $k$-boxes (where two such boxes are considered adjacent if
they meet almost face to face) is connected. As before we order them so that
each almost $k$-box in the tiling meets some earlier almost $k$-box almost
face to face. Now we would like to iteratively connect Hamiltonian cycles on
each of the almost $k$-boxes. This works as before with the change that at
each step we need to choose either an even type or an odd type Hamiltonian
cycle so that we can connect them in accordance with Proposition 12.1.

Suppose that $R$ meets almost face to face with $R^{\prime}$ using the faces
$\partial_{t}(R)$ and $\partial_{t}(R^{\prime})$. For the moment, consider the
unique undirected cycles on $R$ and $R^{\prime}$. If $t=1$, then the
undirected cycles have parallel edges in the $\pm\epsilon^{2}$ direction on
$\partial_{t}(R)$ and $\partial_{t}(R^{\prime})$. If $t\geq 2$, then the
undirected cycles have parallel edges in the $\pm\epsilon^{1}$ direction on
$\partial_{t}(R)$ and $\partial_{t}(R^{\prime})$. In either case, for each
choice of even or odd type cycle on $R$, there will be a choice of even or odd
type cycle on $R^{\prime}$ which satisfies the hypothesis of Proposition 12.1.
As before we open an edge on $\partial_{1}^{+}C$ as required by the definition
of $P_{C}$.

The remaining issue is to connect the Hamiltonian path $\tilde{c}$ constructed
on $C\setminus\cup_{i}C_{i}$ with the existing paths $c_{i}$ on each $C_{i}$.
By property (1) of Corollary 6.5, the tiling of $C\setminus\cup_{i}C_{i}$ is
face to face with the tiling of $C_{i}$ on $\partial_{1}(C_{i})$. Let $\alpha$
be the starting vertex of $c_{i}$ and $\beta$ be the ending vertex. From the
definition of $P_{C_{i}}$ and the definition of the Hamiltonian cycles on
almost $k$-boxes above, $\tilde{c}$ has an edge parallel to
$\\{\alpha,\beta\\}$, but the direction of the edge may not allow us to apply
Proposition 12.1 to connect them. Suppose that $R$ is the almost $k$-box in
the tiling of $C\setminus\cup_{i}C_{i}$ which is adjacent to $\alpha$ and
$\beta$. As stated in Corollary 6.5, $R$ is a translate of the box
$[1,l]\times[1,k]^{d-1}$ for some $l\geq k$.

We revise the definition of $\tilde{c}$ on $R$ to reverse the edge parallel to
$\\{\alpha,\beta\\}$. For simplicity we assume that the Hamiltonian cycle we
used on $R$ is of even type. Partition $R$ into two pieces
$\partial_{1}^{-}(R)$ and $R\setminus\partial^{-}_{1}(R)$. Note that we can
place a Hamiltonian cycle $e$ of even type on $R\setminus\partial^{-}_{1}(R)$.
This ensures that all the previous connections made through $R$ in $\tilde{c}$
can still be made with the new cycle. Next we construct a Hamiltonian cycle on
$\partial^{-}_{1}(R)$ which can be connected to both $c_{i}$ and $e$.

Recall that $\partial_{1}^{-}(R)$ is a translate of
$\\{1\\}\times[1,k]^{d-1}$. We partition $\partial_{1}^{-}(R)$ into layers
parallel to $\epsilon^{2}\times\epsilon^{3}$. Let $L$ be the layer adjacent to
$\alpha$ and $\beta$. We will place a Hamiltonian cycle on $L$ which can be
connected to both $c_{i}$ and $e$. First rotate $c_{k,k}$ and $d_{k,k}$ to the
$\epsilon^{2}\times\epsilon^{3}$ plane so that edges at distance more than $4$
from the boundary are either in the $\epsilon^{2}$ direction or in the
$-\epsilon^{2}$ direction. Choose one among these orientations so that it
connects with $e$ and finally modify it (as in Figure 2) to be able to connect
it with $c_{i}$.

Figure 2. We see that original choice of the Hamiltonian cycle on the left do
not allow the designated entry and exit points. We first perturb the
Hamiltonian cycle to ensure the entry and exit point. Keeping the upper part
of the Hamiltonian cycle exactly the same we introduce teeth below the
perturbed region to reach all the vertices below.

To finish we need to extend to the rest of the layers of
$\partial^{-}_{1}(R)$. As in previous arguments, we order the layers starting
with $L$ such that any layer has distance $1$ to some previous layer (recall
that $\partial^{-}_{1}(R)$ is just a translate of $\\{1\\}\times[1,k]^{d-1}$).
Then we iteratively extend our Hamiltonian cycle on $L$ to the other layers
using the rotated versions of $c_{k,k}$ and $d_{k,k}$ (given above)
alternatively. This finishes the modification of the Hamiltonian cycle on $R$
and with it the proof that the space of Hamiltonian cycles satisfies extension
portion of property F.

###  12.4. The compactness criterion

Suppose that $x$ is a configuration such that for an increasing union of
$100k$-grid unions $B_{i}$ we have that $x|_{B_{i}}\in P_{B_{i}}$ for all $i$.
To complete the proof of Property $F$ we need to show that $x$ is bi-infinite
directed Hamiltonian path. For this it is enough to prove that for all
$\alpha,\beta\in\mathbb{Z}^{d}$ there is a directed path along $x$ which goes
from $\alpha$ to $\beta$ or vice versa and that every
$\gamma\in\mathbb{Z}^{d}$ has a successor and the predecessor. For this, take
$i$ large enough such that $\alpha,\beta\in B_{i}$. Then immediately we have
that there is a directed path along $x$ of length less than or equal to
$|B_{i}|$ connecting the two. Finally take $i^{\prime}$ large enough such that
$\gamma$ and its neighbours are in $B_{i^{\prime}}$. Then we have by
construction that $\gamma$ must have both a predecessor and a successor in
$B_{i^{\prime}}$.

This completes the proof.

##  13\. Proper edge colourings have property $F$

Recall that $E_{t}$ denotes the set of injective maps from the set of unit
vectors to the set of colours $\\{1,2,\ldots,t\\}$ and $X^{(t)}$ denotes the
space of proper $t$-edge colourings of $\mathbb{Z}^{d}$. We can think of the
set of proper $t$-colourings both as maps from edges of $\mathbb{Z}^{d}$ to
the set of colours and as maps from $\mathbb{Z}^{d}$ to $E_{t}$. We will use
both these (equivalent) perspectives interchangeably.

######  Theorem 13.1.

Let $t\geq 2d$. The space of proper $t$-colourings $X^{(t)}$ has property $F$
for a sequence of patterns with entropy equal to the topological entropy of
the $X^{(t)}$.

######  Remark 4.

This shows that Borel edge chromatic number of any free Borel $\mathbb{Z}^{d}$
action is $2d$.

###### Proof.

Recall that given a set $C\subseteq\mathbb{Z}^{d}$ we defined
$\partial_{i}^{+}(C)$ as the set of $\alpha\in C$ such that
$\alpha+\epsilon^{i}\notin C$ while $\partial_{i}^{-}(C)$ was defined as the
set of $\alpha\in C$ such that $\alpha-\epsilon^{i}\notin C$. Fix $t\geq 2d$.
Let $C\subseteq\mathbb{Z}^{d}$ be connected. For $d<i\leq 2d$ let
$\epsilon^{i}=-\epsilon^{i-d}$. Let

$\displaystyle P_{C}=\left\\{a\in\mathcal{L}(X^{(t)},C)~{}:~{}\text{ the
outward edges normal to the faces $\partial_{i}^{+}(C)$ and
$\partial_{i}^{-}(C)$ are coloured $i$ in $a$}\right\\}.$

This boundary condition is amenable to certain natural reflections which we
will describe next.

Let $a$ be a proper $t$-edge colouring of a box $B$ and let
$F=\partial_{i}^{+}(B)$. Then we can reflect $a$ along $F+1/2\epsilon^{i}$ to
get a proper $t$-edge colouring of the union of the box $B$ and its reflection
along $F$. This symmetry is remarkably useful and very similar to the symmetry
available to us from graph homomorphisms. In particular it allows, as in
[CM21, Section 9], application of “reflection positivity” which gives us that
the entropy of these patterns is equal to the entropy of the subshift. Since
it is essentially the same argument, we do not repeat it here. We are left to
prove property $F$ for these patterns. For this we will use Proposition 6.4 in
a crucial manner.

We fix $N,k=10$. Let $C,C_{1},C_{2},\ldots,C_{r}$ be $Nk$-grid unions with
offsets $\bar{0},\alpha^{1},\alpha^{2},\ldots,\alpha^{r}\in[1,nk]^{d}$ such
that $C_{1}+[-100,100]^{d}$, $C_{2}+[-100,100]^{d}$, …, $C_{r}+[-100,100]^{d}$
are disjoint and contained in $C$. By Proposition 6.4, we know that

$C\setminus\cup_{i=1}^{r}C_{i}$

can be tiled by almost $100$-boxes, that is, boxes all of whose side lengths
are at least $100$ and at most one of them is different from $100$. Thus to
complete the proof we need to show that $P_{B}$ is non-empty for all almost
$100$-boxes $B$. If all the sides of $B$ are even then the construction is
easy: The edges parallel to the direction $\epsilon^{i}$ are coloured $i$ and
$i+d$ alternatively. This would ensure that the outward edges normal to
$\partial_{i}^{+}(B)$ and $\partial_{i}^{-}(B)$ are coloured $i$.

Now suppose (without the loss of generality) that the side in the direction
$\epsilon^{1}$ is odd and the rest are even. The edges in the directions
$\epsilon^{i}$ are coloured $i$ and $i+d$ alternately for $i\geq 3$. The edges
in the direction $\epsilon^{1}$ are coloured (from smaller to larger first
coordinate value) $1$, $2+d$, $1+d$ followed by $1$ and $1+d$ alternatively.
Order the lines in the $\epsilon^{2}$ direction according the value of the
first coordinate on the line. The ones with the smallest value of the first
coordinate are coloured $2$ and $1+d$ alternatively. The lines with the next
highest value of the first coordinate are coloured $2$ and $1$ alternatively.
The rest of the lines are labelled $2$ and $2+d$ alternatively. It completes
the required colouring and hence the proof. ∎

##  14\. Discussions and Open questions

###  14.1. From results in ergodic theory to the Borel setting

We expand here on a comment and question by Anton Bernsthyn. It is a natural
question to ask if results regarding embeddings and factorings that have been
proved in the presence of an ergodic measure can be automatically upgraded to
the Borel setting. In other words, in ergodic theory the following situation
often arises. Given the action of the group $\mathbb{Z}^{d}$ on a standard
probability space $(X,\mu)$, there is a map from $X$ to a finite set
$\\{1,2,\ldots,l\\}$ such that the image of the equivariant induced map to
$\\{1,2,\ldots,l\\}^{\mathbb{Z}^{d}}$ satisfies certain local constraints
$\mu$ almost everywhere. The question is whether a probability measure is
really required in this context. If we take the alphabet to be infinite then
the answer is no due to Theorem 4.2 (look at Remark 2).

###  14.2. From the Borel to the continuous setting

It will be exciting to know more about the continuous analogue of our results.
We mentioned many related results in the introduction (Section 1) and direct
the reader there for the references. Here we will focus on the main questions
and subtleties in the subject.

It is clear that our techniques do not automatically carry forward to the
continuous settings. Theorem 2.10 of [GJKS18] showed that a continuous
(clopen) version of Theorem 5.2 fails for the free part of the full shift
$2^{\mathbb{Z}^{d}}$.

Since Theorem 5.2 is the mechanism for constructing the Borel maps in our
paper, Theorem 2.10 of [GJKS18] shows that there is no straightforward
modification of our methods to give continuous maps. In the case of colorings,
Corollary 4.4 of [GJKS18] is equivalent to the statement that there is no
continuous factor map from the free part of $2^{\mathbb{Z}^{2}}$ to the space
of proper $3$-colorings. So our result that there is a Borel such map cannot
be improved to a continuous one. Further interesting conditions were found in
[GJKS18] (given by the twelve tiles theorem) which give equivalent
formulations of whether the free part of $2^{\mathbb{Z}^{2}}$ can be mapped
continuously and equivariantly to a given shift of finite type. These
equivalent formulations may be difficult to check. For instance, it would be
nice to have an answer to the following question: When can the free part of
the full shift be tiled continuously by a given set of rectangles? From
results by Gao-Jackson-Krohne-Seward we know that there is no continuous
perfect matching (domino tiling) of the free part of the full shift. This and
the fact that there is no continuous $3$-colouring are related to the fact
that perfect matchings have a non-trivial cohomology while the full shift
doesn’t [Sch95]) On the other hand, for $k>0$ there are continuous tilings by
rectangles of the form $n\times n$, $n\times(n+1)$, $(n+1)\times n$ and
$(n+1)\times(n+1)$.

Problems about continuous embeddings appear to be subtle. Results by Lightwood
[Lig03, Lig04] say that the strong mixing condition are enough when we talk
about embeddings of subshifts without periodic points. On the other hand
results by Salo [Sal21] indicate that embeddings of the free part of subshifts
can be naturally extended to an embedding of the subshifts in certain cases.
This is very problematic because little is known about continuous embeddings
between $\mathbb{Z}^{d}$-subshifts for $d\geq 2$. For $d=1$, a necessary and
sufficient condition for finding embeddings into a shift of finite type is
given by Krieger’s embedding theorem [Kri82].

###  14.3. Full universality

One of our main results says that if $Y$ is a subshift with a collection
$\mathcal{C}$ which has property $F$ and entropy $h(\mathcal{C})$ then the
free part of any subshift with entropy lower than $h(\mathcal{C})$ can be
Borel embedded in it (Theorem 9.1). On the other hand, results in [CM21] give
Borel embeddings of any system modulo a universally null set into systems with
a property similar to property $F$ called flexibility.

The Gurevic entropy of a Borel $\mathbb{Z}^{d}$ system $(X,T)$ is given by the
supremum of the Kolmogorov-Sinai entropy of the invariant probability measures
on the space.

We conjecture that if $(Y,S)$ is a subshift which has a collection
$\mathcal{C}$ with property $F$ and entropy $h(\mathcal{C})$ and $(X,T)$ is
any free Borel $\mathbb{Z}^{d}$ system whose Gurevic entropy is strictly
smaller that $h(\mathcal{C})$ then $(X,T)$ can be embedded into $(Y,S)$. For
mixing shifts of finite type in one dimension (which are automatically
strongly irreducible subshifts) this is known due to Hochman [Hoc13, Hoc19].
Along with some arguments by Tom Meyerovitch, this conjecture follows from an
upcoming result by Mike Hochman and Brandon Seward.

###  14.4. Property $F$ for the non-symbolic setting

In [CM21] a property called flexibility was defined for $\mathbb{Z}^{d}$
topological dynamical systems and it was shown (modulo a universally null set)
that under appropriate entropy constraints any Borel $\mathbb{Z}^{d}$ system
can be embedded into it. We could not come with an analogue of flexibility for
general $\mathbb{Z}^{d}$ topological dynamical systems. In particular we would
like to know whether the following is true. Let $(X,T)$ be a $\mathbb{Z}$
topological dynamical systems with a strong mixing property like non-uniform
specification (look at [CM21] for the definition). Does there exist an
equivariant map from the free part of any Borel $\mathbb{Z}$ system to the
free part of $(X,T)$? By [CM21] the answer is yes modulo sets of universal
measure zero.

## References

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