# On the Garden of Eden theorem
for $\mathscr{B}$-free subshifts

M. Lemańczyk, C. Richard, D. Sell

###### Abstract

We prove that on $\mathscr{B}$-free subshifts, with $\mathscr{B}$ satisfying
the Erdös condition, all cellular automata are determined by monotone sliding
block codes. In particular, this implies the validity of the Garden of Eden
theorem for such systems.

##  1 Introduction

###  1.1 Sets of multiples and $\mathscr{B}$-free sets

For a set $\mathscr{B}$ of natural numbers, consider its multiples
$\mathcal{M}_{\mathscr{B}}=\bigcup_{b\in\mathscr{B}}b\mathbb{Z}$. Interest in
sets of multiples arose in the 1930’s from abundant integers, i.e., integers
$n$ whose sum of proper divisors is larger than or equal to $|n|$. It is
readily seen that abundant integers are closed under multiplication by
integers, hence they indeed constitute a set of multiples. See [15, Ch. 11]
for further properties. One may restrict without loss of generality to
primitive $\mathscr{B}$, i.e., no element from $\mathscr{B}$ divides any other
element from $\mathscr{B}$. A rich class beyond the abundant integers is
provided by the case of $\mathscr{B}$ being Erdös where, in addition to
primitivity, $\mathscr{B}$ is assumed infinite, which excludes periodic sets
of multiples. Moreover, $\mathscr{B}$ is assumed coprime and thin, i.e.,
$\sum_{b\in\mathscr{B}}1/b<\infty$. The latter guarantees that the multiples
have a natural density. This motivates the attribution to Erdös, see the
discussion in Chapter 1 of [19].

Intimately related to sets of multiples are the sets
$\mathcal{F}_{\mathscr{B}}=\mathbb{Z}\setminus\mathcal{M}_{\mathscr{B}}$ of
$\mathscr{B}$-free integers. In particular, the abundant integers yield the
deficient integers. Over the last years, $\mathscr{B}$-free integers have been
intensely studied from a dynamical perspective, see [15] for a detailed
account. Historically, the guiding example seems square-free integers, which
is obtained from the Erdös set $\mathscr{B}=\\{p^{2}:p\text{ prime}\\}$. On
one side, the influential article by Sarnak [40] listed, among other results,
a number of dynamical properties of the subshift generated by the square-free
integers. On the other side square-free integers are pure point diffractive
[6, 39], in spite of their apparent irregularity. As mathematical diffraction
theory [2] was developing from the 1990’s onwards, a dynamical perspective on
diffraction emerged, since pure point diffractivity had been shown to be
equivalent to discrete dynamical spectrum of the subshift generated by the
structure [5], when equipped with its natural pattern frequency measure. In
fact, squarefree integers [36, Ch. 10] and general $\mathscr{B}$-free integers
are examples of Meyer’s weak model sets, and this viewpoint was very fruitful
not only for analysing the $\mathscr{B}$-free subshift dynamics [21, 23, 24],
but also for developing a dynamical structure theory of general weak model
sets [25, 27, 26, 24]. Moreover, $\mathscr{B}$-free subshifts admit natural
multidimensional generalizations to lattices and number fields [3, 8, 14],
some of which may serve as useful models in physics.

Despite their exceptional naturality, $\mathscr{B}$-free subshifts display an
impressive dynamical diversity varying from proximality in the Erdös case to
minimality, which leads to the classical theory of Toeplitz subshifts [15]. In
both cases their entropy can be both zero or positive. In this paper we deal
with the proximal case, with initial focus on Erdös sets. In this latter case,
on one hand, the associated dynamical systems exhibit properties
characteristic of the full shift: their entropy is positive, their set of
invariant measures is the Poulsen simplex [32, 29], and they are intrinsically
ergodic [31]. Moreover, they are hereditary: if $x$ belongs to the subshift
and if $y\leq x$ coordinatewise, then $y$ belongs to the subshift [1]. On the
other hand, the unique measure of maximal entropy is not Gibbs [30], a rather
surprising property that was first discovered for the square-free subshift
[38]. Moreover, their automorphism group is trivial, see below.

###  1.2 Endomorphisms of subshifts

In recent years, some effort has been made in order to understand the
endomorphism group ${\rm End}(X,S)$ and the automorphism group ${\rm
Aut}(X,S)$ of a subshift $(X,S)$. In the proximal case, unless $|X|=1$, ${\rm
End}(X,S)$ is larger than ${\rm Aut}(X,S)$, as the map sending each point to
the fixed point of $S$ is a member of ${\rm End}(X,S)$. Clearly, both ${\rm
End}(X,S)$ and ${\rm Aut}(X,S)$ are valuable topological conjugacy invariants.
For Erdös $\mathscr{B}$-free subshifts $(X,S)$, Mentzen’s theorem [35] states
that ${\rm Aut}(X,S)$ is trivial, i.e., it consists only of the powers of the
shift. Whereas Mentzen’s proof was based on arithmetic arguments, an extension
beyond the Erdös case by dynamical methods was recently given in [24, Thm.
4.1]. See also [4] for a multidimensional generalisation. In the Toeplitz
case, only partial results are available [13, 16].

In this article we study endomorphisms of proximal $\mathscr{B}$-free
subshifts. We adopt here language from the theory of cellular automata [17]
and call an endomorphism of a subshift $(X,S)$ a Cellular Automaton (CA). By
the classical Curtis-Lyndon-Hedlund (CLH) theorem [34, Thm. 6.2.9], each CA is
given by a (sliding block) code
$\phi:\\{0,1\\}^{\\{-\ell,\ldots,\ell\\}}\to\\{0,1\\}$ for some $\ell\geq 0$
via the formula111It is implicit here that we consider only those blocks
$B\in\\{0,1\\}^{\\{-\ell,\ldots,\ell\\}}$ that appear in some $x\in X$.

(1) $\tau(x)(n)=\phi(x(n-\ell,n+\ell))\text{ for all }x\in X\text{ and
}n\in{\mathbb{Z}}.$

It follows that ${\rm End}(X,S)$ is countable, which directed a lot of effort
towards classifying all possible countable groups that can appear as
invertible CA of a subshift [9, 11]. Let us restrict to hereditary subshifts.
In that case, if a code $\phi$ is monotone, i.e., $\phi(B)\leq B(0)$ for each
block $B\in\\{0,1\\}^{\\{-\ell,\ldots,\ell\\}}$, then the induced map $\tau$
is an endomorphism. (Note that monotone codes for hereditary
$\mathscr{B}$-free subshifts are already mentioned in the preprint version of
[13].) The full shift shows that the converse does not hold: it is hereditary,
but there are CA, considered modulo the powers of the shift, which do not
arise from monotone codes. The following two questions are natural:

  * (i)

When is every CA given by a monotone code, up to a power of the shift?

  * (ii)

How to read off non-invertibility properties of a CA from its code?

###  1.3 Results of the paper

Let $\tau$ be a CA on $(X,S)$. The elements of $X\setminus\tau(X)$ are called
Garden of Eden (GoE) configurations in the theory of cellular automata, see
[7] for a recent review. The name arises from the fact that such
configurations can only occur at time $0$, when considering iterates
$\tau^{k}(X_{0})$ of an initial set $X_{0}\subset X$. A surjective CA does not
admit any GoE configuration. In that context, a crucial property of a CA is a
weakened form of injectivity: A CA is pre-injective if it is injective on
every collection of configurations that mutually differ in at most finitely
many coordinates. The Garden of Eden theorem holds if for any CA on some
subshift, surjectivity is equivalent to pre-injectivity. The implication that
surjectivity implies pre-injectivity is also called the Moore property, the
reverse implication is called the Myhill property. It is known that the GoE
holds for irreducible subshifts of finite type [7, Thm. 7.15] and that any
strongly irreducible subshift satisfies the Myhill property [7, Thm. 7.10]. In
our setting, which is not subsumed by the above results, the GoE theorem holds
for the class of Erdös $\mathscr{B}$-free subshifts. In fact, our main result
even answers the two questions raised above.

######  Theorem 1.1.

The $\mathscr{B}$-free subshift of any Erdös set $\mathscr{B}$ satisfies the
GoE theorem. In fact, every CA is given by a monotone code, modulo a power of
the shift. If the CA is onto, then it equals a power of the shift.

In particular, our result states a very strong form of the Moore property,
which implies Mentzen’s theorem that the automorphism group of an Erdös
$\mathscr{B}$-free subshift is trivial. We will obtain Theorem 1.1 as a
particular case of Theorem 3.12 below.

Subsequently, the paper discusses automorphism groups beyond the Erdös case.
There is a natural hierarchy ranging from general proximal $\mathscr{B}$-free
subshifts, characterised by the condition that $\mathscr{B}$ contains an
infinite coprime subset [15, Thm. B], to general hereditary $\mathscr{B}$-free
subshifts, which extend the Erdös case. An example of a hereditary (hence
proximal) $\mathscr{B}$-free shift that is not Erdös is given by the deficient
integers.

Using arithmetic arguments, we can extend Mentzen’s theorem from the Erdös
case to general hereditary $\mathscr{B}$-free shifts. Our result contains [24,
Thm. 4.1] as a special case. In the general proximal case the situation seems
more subtle. We can reduce the analysis of the automorphism group to that of
an auxiliary hereditary subsystem, which is also a
$\mathscr{B}^{\prime}$-subshift, without changing the density. Recalling that
$\mathscr{B}$ is Behrend if the density of $\mathscr{B}$-free integers is
zero, it seems appropriate to call this reduction the “relative Behrend” case.
Behrend $\mathscr{B}$-free subshifts are proximal and have zero entropy, and
these two properties fully characterize the Behrend case.

From a dynamical point of view, the Erdös case is “very arithmetic” in the
sense that the $\mathscr{B}$-free subshift coincides with the so-called
$\mathscr{B}$-admissible subshift, compare Section 2. In the general proximal
case, we might not have such a coincidence (an example is given by the
deficient integers). In fact, if the subshift has positive topological
entropy, then the above coincidence characterizes the Erdös case. Indeed, as
shown in [22], the coincidence implies that $\mathscr{B}$ is coprime and, in
the coprime case, the entropy is positive if and only if $\mathscr{B}$ is thin
[1]. Moreover, there are primitive Behrend sets for which we do have
coincidence [22].

###  1.4 Plan of the paper

The paper is organized follows. In a preparatory Section 2 we recall some
basic theory, concentrating on various equivalences between arithmetic
properties of $\mathscr{B}$ and topological properties of the corresponding
$\mathscr{B}$-free subshift. In Section 3 we prove Theorem 3.12. In Section 4
we show that the heredity of a $\mathscr{B}$-free subshift forces any of its
automorphisms to be trivial. The result applies in particular to the Erdös
case, and more generally to the proximal case, where $\mathscr{B}$
additionally obeys the so-called tautness property. In Section 5 we briefly
discuss CA on general proximal $\mathscr{B}$-free subshifts and formulate some
open problems.

##  2 $\mathscr{B}$-dynamical systems

Here we collect properties of $\mathscr{B}$-free and $\mathscr{B}$-admissible
subshifts for later use, which are mainly taken from [15]. For more general
background we refer to the monographs [19] and [34, 41, 12].

###  2.1 Some generalities on subshifts

Consider the compact space $\\{0,1\\}^{{\mathbb{Z}}}$ of bi-infinite
$0-1$-sequences, equipped with the usual product metric $d$. The support of
$x\in\\{0,1\\}^{{\mathbb{Z}}}$ is the set
${supp}(x)=\\{j\in{\mathbb{Z}}:\>x(j)=1\\}$. The (left) shift
$S:\\{0,1\\}^{{\mathbb{Z}}}\to\\{0,1\\}^{{\mathbb{Z}}}$, defined by
$(Sx)(j)=x(j+1)$ for all $j\in{\mathbb{Z}}$, is a homeomorphism. If
$X\subset\\{0,1\\}^{{\mathbb{Z}}}$, which we always assume to be non-empty, is
closed and $S$-invariant then the resulting dynamical system $(X,S)$ is called
a subshift. $X$ is called hereditary, if $y\in\\{0,1\\}^{{\mathbb{Z}}}$
belongs to $X$ whenever there is some $x\in X$ such that $y\leq x$
coordinatewise. Denote by $\widetilde{X}\subset\\{0,1\\}^{{\mathbb{Z}}}$ the
smallest hereditary set containing $X$. Given a subshift $(X,S)$, we call the
subshift $(\widetilde{X},S)$ its hereditary closure. We call $x,y\in X$
asymptotic if $d(S^{r}x,S^{r}y)\to 0$ for $|r|\to\infty$. It is easily seen
that $x,y$ are asymptotic if and only if they differ in at most finitely many
coordinates. If $d(S^{r_{k}}x,S^{r_{k}}y)\to 0$ along a sequence $(r_{k})$
that tends to infinity, then $x,y$ are called proximal. A subshift $(X,S)$ is
called proximal if any two elements from $X$ are proximal.

Denote by $M(X,S)$ the simplex of $S$-invariant probability measures on
$(X,S)$, which is non-empty by the Krylov-Bogolioubov theorem, see e.g. [41,
Cor. 6.9.1]. Each $\nu\in M(X,S)$ is determined by its values $\nu(B)$ on all
finite blocks or words $B\in\bigcup_{m=1}^{\infty}\\{0,1\\}^{m}$. More
precisely, we identify any block $B\in\\{0,1\\}^{m}$, where $|B|=m$ is the
length of $B$, with the corresponding cylinder set $\\{x\in
X:\>x(0,m-1)=B\\}$, and thus the set of all finite blocks generates the Borel
$\sigma$-algebra on $X$. Given $\nu\in M(X,S)$ and a sequence $(N_{k})$, the
element $x\in X$ is generic along $(N_{k})$ for $\nu$ if

(2) $\int f\,d\nu=\lim_{k\to\infty}\frac{1}{N_{k}}\sum_{n\leq N_{k}}f(S^{n}x)$

for each $f\in C(X)$. If $(N_{k})=(k)$, then $x$ is called generic for $\nu$.
Recall that by the variational principle, see e.g. [12, Sec. 7.4] or [41, Thm.
8.6], the topological entropy $h(X,S)$ is the supremum of the measure-
theoretic entropies $h_{\nu}(X,S)$ with $\nu\in M(X,S)$. Note that for a
subshift the supremum is in fact attained, see e.g. [12, Fact 7.2.4] or [41,
Thms. 8.2, 8.7]. A subshift $(X,S)$ is called intrinsically ergodic [42] if
there is precisely one measure of maximal entropy.

###  2.2 $\mathscr{B}$-free subshifts and their hereditary closures

Assume in the remainder of this article that $\mathscr{B}\subset{\mathbb{N}}$
does not contain 1 and is primitive, if not stated otherwise. Consider the set
$\mathcal{F}_{\mathscr{B}}={\mathbb{Z}}\setminus\mathcal{M}_{\mathscr{B}}$ of
$\mathscr{B}$-free integers, which is nonempty by assumption. By the
Davenport-Erdös theorem, see e.g. [19, Thm. 0.2], $\mathscr{B}$-free integers
possess the logarithmic density

$\delta({\cal F}_{\mathscr{B}})=\lim_{N\to\infty}\frac{1}{\log
N}\sum_{n\in{\cal F}_{\mathscr{B}}\cap\\{1,\ldots,N\\}}\frac{1}{n}\ ,$

which coincides with its natural upper density. In the Erdös case, the natural
density exists, see e.g. [19, Sec. 1.1]. Let $\eta=\mathbbm{1}_{{\cal
F}_{\mathscr{B}}}\in\\{0,1\\}^{\mathbb{Z}}$ denote the characteristic function
of $\mathcal{F}_{\mathscr{B}}$ and define
$X_{\eta}=\overline{\\{S^{r}\eta:\>r\in{\mathbb{Z}}\\}}\subset\\{0,1\\}^{{\mathbb{Z}}}$.
The system $(X_{\eta},S)$ is the corresponding $\mathscr{B}$-free subshift.
Due to its particular arithmetic origin, the point $\eta$ enjoys the following
maximality property222Maximality arguments have been used before e.g. in [3,
p. 186].. As this property is crucial in later arguments, we provide its short
proof.

######  Lemma 2.1.

Let $(X_{\eta},S)$ be a $\mathscr{B}$-free subshift. If $\eta\leq y\in
X_{\eta}$ then $\eta=y$.

###### Proof.

Assume for a contradiction that $\eta\neq y$. Then ${\rm supp}(y)$ contains
some integer $m$ that is not $\mathscr{B}$-free. Hence $m=bn$ for some
$b\in\mathscr{B}$ and some $n\in{\mathbb{Z}}$. Since $\eta\leq y$, we have in
particular $|{\rm supp}(\eta)\text{ mod }b|<|{\rm supp}(y)\text{ mod }b|$.
This is contradictory, as there is a finite word $w$ in $y$ such that $|{\rm
supp}(w)\text{ mod }b|=|{\rm supp}(y)\text{ mod }b|$. Indeed, for every
$r\in{\rm supp}(y)\text{ mod }b$ choose first a position $j\in
r+b{\mathbb{Z}}$ such that $y(j)=1$. Then take a finite word in $y$ containing
all $j$. As $y\in X_{\eta}$, the word $w$ is also contained in $\eta$. Hence
$|{\rm supp}(\eta)\text{ mod }b|\geq|{\rm supp}(w)\text{ mod }b|$.

A distinguished measure on $(X_{\eta},S)$ is the so-called Mirsky measure
$\nu_{\eta}$. The point $\eta$ is generic along a subsequence for
$\nu_{\eta}$, see [15, Prop. E]. In fact, $\eta$ is a generic point for the
Mirsky measure in the logarithmic sense, i.e., if we replace in (2) the Cesàro
summation by the logarithmic one [15]. In the Erdös case, $\eta$ is generic in
the usual sense. The Mirsky measure is the push-forward of the Haar measure on
an odometer canonically associated to $\mathscr{B}$. In particular, the system
$(X_{\eta},\nu_{\eta},S)$ is a zero entropy system, which is ergodic and has
discrete spectrum [15, Thm. F].

For a general $\mathscr{B}$-free subshift, no formula for its topological
entropy is known. However, a non-trivial upper bound can be given using the
description as a weak model set [20, Thm. 4.5]. The situation changes if we
pass to the hereditary closure $\widetilde{X}_{\eta}$ of a $\mathscr{B}$-free
subshift. This can then be used to obtain a formula for the case of a proximal
$\mathscr{B}$-free subshift, as shown in Corollary 2.7 below. In that case the
upper bound from [20] is attained. The following results appear as Proposition
K, Theorem J and Remark 10.2 in [15]. The proofs rely on earlier results for
the Erdös case [31] and for squarefree integers [38].

######  Theorem 2.2 ([31],[15]).

Consider a $\mathscr{B}$-free subshift $(X_{\eta},S)$. Then:

(3) $h(\widetilde{X}_{\eta},S)=d\cdot\log 2\ ,$

where $d=\nu_{\eta}(1)$ is the logarithmic density of 1’s in $\eta$, and

(4) $\mbox{$(\widetilde{X}_{\eta},S)$ is intrinsically ergodic},$

with the measure of maximal entropy $\nu_{\eta}\ast
B(\frac{1}{2},\frac{1}{2})$, where $B(\frac{1}{2},\frac{1}{2})$ stands for the
$(\frac{1}{2},\frac{1}{2})$-Bernoulli measure on $\\{0,1\\}^{{\mathbb{Z}}}$
and the multiplicative convolution above is the image of the product measure
$\nu_{\eta}\otimes B(\frac{1}{2},\frac{1}{2})$ via the coordinatewise
multiplication map.333Note that the image of this map is contained in
$\widetilde{X}_{\eta}$.

###  2.3 Tautness in $\mathscr{B}$-free subshifts

The notion of tautness, which arises naturally in the theory of sets of
multiples [19, Def. 0.11], also turned out to be important for
$\mathscr{B}$-free sets. Recall that $\mathscr{B}$ is taut if

$\delta({\cal F}_{\mathscr{B}})<\delta({\cal
F}_{\mathscr{B}\setminus\\{b\\}})\text{ for each }b\in\mathscr{B}\ .$

It is easy to see that $\mathscr{B}$ thin implies $\mathscr{B}$ taut. In
particular, the Erdös case is taut. From a dynamical point of view, the role
of tautness is seen in the following three results. The first one combines
[30, Cor. 1.11] with [23, Thm. 2], which is based on a measure-theoretic
characterisation of tautness in terms of weak model sets [21, Thm. A].

######  Theorem 2.3 ([23],[30]).

$\mathscr{B}$ is taut if and only if ${\rm supp}(\nu_{\eta})=X_{\eta}$.

The second result is Theorem C in [15].

######  Theorem 2.4 ([15]).

For any $\mathscr{B}$ there exists a unique taut $\mathscr{B}^{\prime}$ such
that $\eta^{\prime}\leq\eta$, $X_{\eta^{\prime}}\subset\widetilde{X}_{\eta}$,
$\nu_{\eta}=\nu_{\eta^{\prime}}$,
$M(\widetilde{X}_{\eta},S)=M(\widetilde{X}_{\eta^{\prime}},S)$, in particular,
$h(\widetilde{X}_{\eta},S)=h(\widetilde{X}_{\eta^{\prime}},S)$.

The third result combines [15, Thm. B] and [23, Thm. 3]. It relates
proximality, which is an invariant of topological conjugacy, to heredity,
which serves as a useful tool for analysis of $\mathscr{B}$-free sets, but
lacks invariance in general.

######  Theorem 2.5 ([23]).

Assume that $\mathscr{B}$ is taut. Then $(X_{\eta},S)$ is proximal if and only
if $(X_{\eta},S)$ is hereditary.

###  2.4 Proximal $\mathscr{B}$-free subshifts and Behrend subshifts

Several characterizations of proximality of $\mathscr{B}$-free subshifts can
be found in [15, Thms. 3.7, 3.12] and [21, Thm. C]. We mention here that
proximality is equivalent to the property that the all zero sequence
$0^{{\mathbb{Z}}}$ belongs to $X_{\eta}$, or to the existence of an infinite
coprime subset of $\mathscr{B}$. Let us call $\mathscr{B}$ proximal if the
induced $\mathscr{B}$-free subshift is proximal. A particular class of
proximal systems is obtained when we consider Behrend sets [19, Def. 0.6],
i.e., sets that satisfy $\delta({\cal F}_{\mathscr{B}})=0$. It is easily seen
that in the Behrend case the Mirsky measure $\nu_{\eta}$ is just the Dirac
measure $\delta_{0^{{\mathbb{Z}}}}$. In fact, in this case, $(X_{\eta},S)$ is
proximal and uniquely ergodic [15, Rem. 6.3]. In particular, it has zero
entropy.

Let us recall the following result, which is implicit in the proof of Eqn.
(25) on page 13 in [30]. Its proof is a rather direct consequence of Theorem
2.3.

######  Proposition 2.6 ([30]).

For any $\mathscr{B}$ and the unique taut $\mathscr{B}^{\prime}$ as in Theorem
2.4, we have $X_{\eta^{\prime}}\subset X_{\eta}$.

This can be used to obtain the following new result about proximal
$\mathscr{B}$-free systems. The proximal case will further be studied in
Section 5.

######  Corollary 2.7.

If $(X_{\eta},S)$ is proximal, then

$h(X_{\eta},S)=d\cdot\log 2\ ,$

where $d=\nu_{\eta}(1)$. In particular, $h(X_{\eta},S)=0$ if and only if
$\mathscr{B}$ is Behrend. Moreover, each proximal $\mathscr{B}$-free subshift
is intrinsically ergodic, and the measure of maximal entropy is
$\nu_{\eta}\ast B(\frac{1}{2},\frac{1}{2})$.

###### Proof.

Let $\mathscr{B}^{\prime}$ be the unique taut as in Theorem 2.4. Then

$h(\widetilde{X}_{\eta^{\prime}},S)=h(\widetilde{X}_{\eta},S)\ .$

Now, $X_{\eta^{\prime}}\subset X_{\eta}$ in view of Proposition 2.6, so
$(X_{\eta^{\prime}},S)$ is, as a subsystem of a proximal system, also
proximal. But $\mathscr{B}^{\prime}$ is taut, so by Theorem 2.5,
$(X_{\eta^{\prime}},S)$ is hereditary. Now, by Theorem 2.2 (used twice) and
remembering that $\nu_{\eta}(1)=\nu_{\eta^{\prime}}(1)$, we obtain

$d\cdot\log 2=h(X_{\eta^{\prime}},S)\leq h(X_{\eta},S)\leq
h(\widetilde{X}_{\eta},S)=d\log 2$

and the first claim follows. The intrinsic ergodicity in the proximal case
follows from Theorems 2.4, 2.5, 2.2 and Proposition 2.6.

###  2.5 $\mathscr{B}$-admissible subshifts

Recall that $A\subset{\mathbb{Z}}$ is $\mathscr{B}$-admissible if, for every
$b\in\mathscr{B}$, there is some residue class modulo $b$ that does not
intersect $A$. Analogously, a word is called $\mathscr{B}$-admissible if its
support is $\mathscr{B}$-admissible. Then a subset of $\\{0,1\\}^{\mathbb{Z}}$
is admissible if and only if all its words are admissible. Also note that
admissibility is stable under shifting. Clearly, any $\mathscr{B}$-free set is
$\mathscr{B}$-admissible, as 0 is the missing residue class mod $b$ for each
$b\in\mathscr{B}$. The shift invariant and closed space

$X_{\mathscr{B}}:=\\{x\in\\{0,1\\}^{{\mathbb{Z}}}:\>{\rm supp}(x)\text{ is
}\mathscr{B}\text{-admissible}\\}$

was introduced in [40]. The subshift $(X_{\mathscr{B}},S)$ is called the
$\mathscr{B}$-admissible subshift [15], see also the discussion of
admissibility in [15, Sec. 2.8]. As $X_{\mathscr{B}}$ is hereditary by
definition, we have

$X_{\eta}\subset\widetilde{X}_{\eta}\subset X_{\mathscr{B}}\ ,$

where the above inclusions are strict in general, e.g. Example 2.42 in [15].
We have $X_{\eta}=X_{\mathscr{B}}$ in the Erdös case [1, Cor. 4.2]. Note that
the equality $X_{\eta}=X_{\mathscr{B}}$ implies that $\mathscr{B}$ is coprime
[22]. Hence positive entropy characterises the Erdös case. Moreover, there are
Behrend coprime sets for which the above equality holds [22].

##  3 Proof of the Garden of Eden theorem

###  3.1 $\mathscr{B}$-admissible subshifts and the Moore property

####  3.1.1 Monotonicity in $\mathscr{B}$-admissible subshifts

######  Lemma 3.1.

Assume that $\mathscr{B}$ is coprime and infinite. Let $u$ be a
$\mathscr{B}$-admissible word. Let $N_{u}:=\rvert\operatorname{supp}(u)\lvert$
denote the cardinality of its support. Let $b_{0}\in\mathscr{B}$ be such that
$b_{0}>2\lvert u\rvert$. Then there exist a $\mathscr{B}$-admissible word
$w_{u}$ and a set $M=\\{m_{1},\ldots,m_{b_{0}-N_{u}}\\}$ of positions in
$w_{u}$ with $w_{u}(m_{i}+1,m_{i}+\lvert u\rvert)=u$ for every $m_{i}\in M$,
such that $M\bmod
b_{0}=\\{0,\ldots,b_{0}-1\\}\setminus(-\operatorname{supp}(u))$ holds.
Moreover, we have $\operatorname{supp}(w_{u})\bmod
b_{0}=\\{1,\ldots,b_{0}-1\\}$.

###### Proof.

First we note that, because of $b_{0}>\lvert u\rvert$, the elements of
$\operatorname{supp}(u)\subseteq\\{1,\ldots,\lvert u\rvert\\}$ are pairwise
different modulo $b_{0}$. Moreover, $b_{0}$ is coprime to
$P:=\prod_{b\in\mathscr{B}\,\setminus\\{b_{0}\\},\,b\leq L}b$, where we choose
$L\geq N_{u}(b_{0}-N_{u})$ so that $P>|u|$. We define

$M=\\{m_{1},\ldots,m_{b_{0}-N_{u}}\\}\subseteq\\{nP:n=0,\ldots,b_{0}-1\\}$

to be those $(b_{0}-N_{u})$-many numbers that satisfy $m_{i}\bmod
b_{0}\notin-\operatorname{supp}(u)$. Consequently, none of the sets
$m_{i}+\operatorname{supp}(u)$ intersects the zero residue class modulo
$b_{0}$. Now, we define $w_{u}$ as the word that is given by
$\operatorname{supp}(w_{u}):=\bigcup\nolimits_{i=1}^{b_{0}-N_{u}}(m_{i}+\operatorname{supp}(u))$.
In particular, $w_{u}$ consists of $b_{0}-N_{u}$ shifted copies of $u$ at
positions $m_{i}$, which lie in all residue classes modulo $b_{0}$ except
$-\operatorname{supp}(u)$. Now, we check that $w_{u}$ is
$\mathscr{B}$-admissible:

  * •

Since none of the sets $m_{i}+\operatorname{supp}(u)$ intersects the zero
residue class modulo $b_{0}$, also $\operatorname{supp}(w_{u})$ is not
intersecting the zero residue class modulo $b_{0}$.

  * •

Let $b\in\mathscr{B}\,\setminus\\{b_{0}\\}$ with $b\leq L$. Since $u$ is
$\mathscr{B}$-admissible, there is a residue class modulo $b$ that
$\operatorname{supp}(u)$ does not intersect. Since all positions of
$\operatorname{supp}(u)$ in $\operatorname{supp}(w_{u})$ differ by a multiple
of $b$, there is a residue class modulo $b$ that $\operatorname{supp}(w_{u})$
does not intersect.

  * •

Finally, let $b>L$. Since $\operatorname{supp}(w_{u})$ is the union of
$(b_{0}-N_{u})$-many copies of $\operatorname{supp}(u)$, we obtain
$\rvert\operatorname{supp}(w_{u})\lvert\leq(b_{0}-N_{u})N_{u}<b$. Hence,
$\operatorname{supp}(w_{u})$ cannot intersect all residue classes modulo $b$.

To show the second assertion, i.e. that $\operatorname{supp}(w_{u})$
intersects all residue classes modulo $b_{0}$ except the zero class note that
for every $j\in\operatorname{supp}(u)$, the set $j+M$ intersects by
construction all residue classes except those in $j-\operatorname{supp}(u)$.
Therefore, if $\rvert\operatorname{supp}(u)\lvert=1$, we are done (as
$j-\operatorname{supp}(u)=\\{0\\}$). For
$\rvert\operatorname{supp}(u)\lvert>1$, let $j_{1}$ and $j_{N_{u}}$ denote
respectively the smallest and the largest element of $\operatorname{supp}(u)$.
Note that all elements in $j_{1}-\operatorname{supp}(u)$ are non-positive,
while all elements in $j_{N_{u}}-\operatorname{supp}(u)$ are non-negative.
Because of $b_{0}>2\lvert u\rvert$, the zero residue class is the only residue
class that all sets $\bigcup\nolimits_{i=1}^{b_{0}-N_{u}}(j+m_{i})$ miss.

######  Proposition 3.2.

Assume that $\mathscr{B}$ is coprime and infinite. Let $\tau\colon
X_{\mathscr{B}}\to X_{\mathscr{B}}$ be a CA, given by the code
$\phi\colon\\{0,1\\}^{\\{-\ell,\ldots,\ell\\}}\to\\{0,1\\}$. Then there exists
$t\in\\{-\ell,\ldots,\ell\\}$ such that $\tau(x)\leq S^{t}x$ holds for all
$x\in X_{\mathscr{B}}$.

###### Proof.

If $\phi$ maps every word of length $2\ell+1$ to $0$, then
$\tau(x)=0^{\mathbb{Z}}$ for every $x\in X_{\mathscr{B}}$ and the claim is
trivially true. Thus, we may assume that the set
$U:=\phi^{-1}(1)\subseteq\\{0,1\\}^{\\{-\ell,\ldots,\ell\\}}$ is non-empty.
For every $u\in U$, we write
$J_{u}:=\operatorname{supp}(u)\subseteq\\{-\ell,\ldots,\ell\\}$ for the
support and $N_{u}:=\rvert J_{u}\lvert$ for its cardinality.

We will prove $\bigcap\nolimits_{u\in U}J_{u}\neq\varnothing$. Then the
assertion follows easily: let $t\in\bigcap\nolimits_{u\in U}J_{u}$. If
$\tau(x)(j)=1$ for some $x\in X_{\mathscr{B}}$ and $j\in\mathbb{Z}$, then
$x(j-\ell,j+\ell)\in U$ holds, which implies $x(j+t)=1$. Thus, we obtain
$\tau(x)\leq S^{t}x$.

To prove that the intersection is non-empty, we proceed by contradiction, so
let us assume $\bigcap\nolimits_{u\in U}J_{u}=\varnothing$. We fix
$b_{0}\in\mathscr{B}$ with $b_{0}>4\ell+2$. Thus, $b_{0}$ is large enough so
that Lemma 3.1 can be applied to words of length $2\ell+1$. For every $u\in
U$, let $w_{u}$ denote the $\mathscr{B}$-admissible word constructed in Lemma
3.1, which contains $b_{0}-N_{u}$ occurrences of $u$, starting in pairwise
different residue classes modulo $b_{0}$. Now, we arrange all the $w_{u}\\!$’s
in a single element $x\in X_{\mathscr{B}}$. Then we show that the support of
$\tau(x)$ intersects every residue class modulo $b_{0}$. Hence $\tau(x)$ is
not $\mathscr{B}$-admissible, which yields a contradiction.

We write $\mathscr{B}\,\setminus\\{b_{0}\\}=\\{b_{1},b_{2},b_{3},\ldots\\}$,
where the elements $b_{k}$ are in increasing order. Let $K$ be such that
$b_{K+1}>\max(\rvert U\lvert\cdot(b_{0}-(2\ell+1))(2\ell+1),b_{0})$ holds. For
every $u\in U$, the associated word $w_{u}\in\\{0,1\\}^{\\{1,\ldots,\lvert
w_{u}\rvert\\}}$ is $\mathscr{B}$-admissible. Hence, for every
$b_{i}\in\mathscr{B}$ there is at least one residue class $r_{u,i}$ modulo
$b_{i}$ that $\operatorname{supp}(w_{u})$ misses. By the Chinese Remainder
Theorem, there exists $s_{u}\in\mathbb{N}$ which solves $s_{u}+r_{u,i}\equiv
0\bmod b_{i}$ simultaneously for $i=0,1,\ldots,K$. If we define
$x(k_{u}\cdot\prod_{i=0}^{K}b_{i}+s_{u}+1,k_{u}\cdot\prod_{i=0}^{K}b_{i}+s_{u}+\lvert
w_{u}\rvert):=w_{u}$ for some $k_{u}\in\mathbb{N}$, then the support of $x$ in
this interval misses the zero residue class modulo $b_{i}$ for all
$i\in\\{0,\ldots,K\\}$. We do this for every $u\in U$, so that $x$ contains an
occurrence of each $w_{u}$. The sequence $x$ is $\mathscr{B}$-admissible,
since its support misses the zero residue class for $b_{0},\ldots,b_{K}$ by
construction and misses some residue class for each $b\geq b_{K+1}>\rvert
U\lvert\cdot(b_{0}-(2\ell+1))(2\ell+1)$ due to having at most $\rvert
U\lvert\cdot(b_{0}-(2\ell+1))(2\ell+1)$ elements.

By the construction in Lemma 3.1, $w_{u}\in\\{0,1\\}^{\\{1,\ldots,\lvert
w_{u}\rvert\\}}$ contains copies of $u$ centered at positions
$\\{m_{u,1},\ldots,m_{u,b_{0}-N_{u}}\\}=:M$ with $M\bmod
b_{0}=\\{0,\ldots,b_{0}-1\\}\setminus-J_{u}$ (note that $m_{u,i}$ now denotes
the center, rather than the starting point, since we use
$J_{u}\subseteq\\{-\ell,\ldots,\ell\\}$, compared to
$J_{u}\subseteq\\{1,\ldots,\lvert u\rvert\\}$ in Lemma 3.1). Moreover,
$\operatorname{supp}(w_{u})$ misses only the zero residue class modulo
$b_{0}$, which implies $s_{u}\equiv 0\bmod b_{0}$. Since $x$ contains $w_{u}$
starting at position $k_{u}\cdot\prod_{i=0}^{K}b_{i}+s_{u}+1$, it contains $u$
centered at positions $k_{u}\cdot\prod_{i=0}^{K}b_{i}+s_{u}+m_{u,i}$, that is,
in the residue classes $M\bmod b_{0}$. Because of $u\in U$, if follows that
$\tau(x)$ has (mod $b_{0}$) $1\\!$’s in
$\\{0,\ldots,b_{0}-1\\}\setminus(-J_{u})$. Since we assumed
$\bigcap\nolimits_{u\in U}J_{u}=\varnothing$ and $b_{0}>2\ell+1$, the support
of $\tau(x)$ intersects every residue class modulo $b_{0}$.

####  3.1.2 Moore property on $\mathscr{B}$-admissible subshifts

######  Proposition 3.3.

Assume that $X_{\eta}=X_{\mathscr{B}}$. Let $\tau\colon X_{\eta}\to X_{\eta}$
be a surjective CA, given by the code
$\phi\colon\\{0,1\\}^{\\{-\ell,\ldots,\ell\\}}\to\\{0,1\\}$. Then there exists
$t\in\\{-\ell,\ldots,\ell\\}$ with $\tau=S^{t}$. In particular, $(X_{\eta},S)$
satisfies the Moore property.

###### Proof.

Since $\tau$ is surjective, there exists $z\in X_{\eta}$ with $\tau(z)=\eta$.
Since $X_{\eta}=X_{\mathscr{B}}$, the set $\mathscr{B}$ is coprime by [22].
Moreover, $\mathscr{B}$ is infinite, since otherwise the periodicity of $\eta$
would imply a finite set $X_{\eta}$, while $X_{\mathscr{B}}$ is always
uncountable. Hence Proposition 3.2 yields $\eta=\tau(z)\leq S^{t}z$ for some
$t\in\\{-\ell,\ldots,\ell\\}$. Since $\eta$ is a maximal element in $X_{\eta}$
by Lemma 2.1, we obtain $\eta=S^{t}z$. Moreover, $\tau$ commutes with the
shift, which yields $\tau(\eta)=\tau(S^{t}z)=S^{t}\tau(z)=S^{t}\eta$. Since
$\eta$ is a transitive point, the result follows.

######  Remark 3.4.

The above corollary applies to the Erdös case, where the equality
$X_{\eta}=X_{\mathscr{B}}$ is established [1, Cor. 4.2]. But there are also
coprime Behrend sets for which we have this equality [22].

###  3.2 Hereditary subshifts and the Myhill property

####  3.2.1 A discussion of pre-injectivity

The following weakening of injectivity due to Gromov is standard in the theory
of cellular automata [7].

######  Definition 3.5.

A CA $\tau$ on a subshift $(X,S)$ is pre-injective if, for any $x,y\in X$, the
property $\tau(x)=\tau(y)$ implies that either $x=y$ or that $x$ differs from
$y$ at infinitely many coordinates.

In order to get acquainted with the notion of pre-injectivity, consider the
following two auxiliary results.

######  Lemma 3.6.

Consider some shift invariant hereditary set
$X_{0}\subset\\{0,1\\}^{\mathbb{Z}}$ of $0-1$-sequences that differ from the
all zero sequence $0^{\mathbb{Z}}$ in at most finitely many coordinates.
Assume that $\tau:X_{0}\to X_{0}$ commutes with the shift, is monotone and
injective. Then $\tau$ is the identity map.

###### Proof.

Note first that $0^{\mathbb{Z}}$ is the only shift-invariant element in
$X_{0}$. As $\tau$ commutes with the shift, this yields
$\tau(0^{\mathbb{Z}})=0^{\mathbb{Z}}$. Next, take any $x\in X_{0}$ and assume
$\tau(x)<x$. As $\tau(x)\leq x$, by induction we obtain $n\in{\mathbb{N}}$
such that $\tau^{n}(x)=0^{\mathbb{Z}}$. Hence
$\tau^{n}(x)=\tau^{n}(0^{\mathbb{Z}})$, and injectivity yields
$x=0^{\mathbb{Z}}$, a contradiction. Thus, we get $\tau(x)=x$ for all $x\in
X_{0}$.

The next result is probably well-known. As we were not able to trace a
reference which applies to our setting, we also provide its proof. We say that
$\tau:X\to Y$ is bounded-to-one if there exists a finite constant $B$ such
that $|\tau^{-1}(\\{y\\})|\leq B$ for all $y\in Y$.

######  Lemma 3.7.

Let $(X,S)$ be a hereditary subshift with CA $\tau$. If $\tau$ is pre-
injective, then $\tau$ is bounded-to-one.

###### Proof.

Assume for the moment that the following claim holds for some $\delta>0$:

> Consider distinct $x,y\in X$ such that $\tau(x)=\tau(y)$. Then
> $d(S^{r}x,S^{r}y)\geq\delta$ for all but finitely many $r\in\mathbb{N}$ or
> $d(S^{-r}x,S^{-r}y)\geq\delta$ for all but finitely many $r\in\mathbb{N}$.

The above claim implies the statement of the lemma by the following argument.
Fix any $x\in X$ and assume that $\tau^{-1}(\\{x\\})$ contains $N$ elements
$\\{x_{1},\ldots,x_{N}\\}$. As $X$ is compact, we find
$\\{z_{1}^{+},z_{1}^{-},\ldots,z_{N}^{+},z_{N}^{-}\\}\subset X$ and an
increasing sequence of integers $(r_{k})$ such that $S^{r_{k}}x_{i}\to
z_{i}^{+}$ and $S^{-r_{k}}x_{i}\to z_{i}^{-}$ for all $i$. This may be seen by
an iterative argument taking subsequences. Now, for any $i\neq j$ the claim
implies $d(z_{i}^{+},z_{j}^{+})\geq\delta$ or
$d(z_{i}^{-},z_{j}^{-})\geq\delta$, by continuity of the metric. Thus by the
compactness of $X$, the number $N$ cannot be arbitrarily large and is bounded
uniformly in $\in X$. Hence $|\tau^{-1}(\\{x\\})|$ can be bounded by some
finite number that does not depend on the choice of $x$.

As the map $\tau$ is a CA, it can be described by a code of length $\ell$ as
in (1). Choose $\delta_{\ell}>0$ sufficiently small such that for $x,y\in X$
we have $x(-4\ell,4\ell)=y(-4\ell,4\ell)$ whenever $d(x,y)<\delta_{\ell}$. We
prove the above claim for $\delta=\delta_{\ell}$ by an indirect argument.
Assume there exist distinct $x,y\in X$ such that $\tau(x)=\tau(y)$. In order
to argue for contradiction, assume there exists an infinite increasing
sequence of integers $(r_{k})$ such that
$d(S^{r_{k}}x,S^{r_{k}}y)<\delta_{\ell}$. By our choice of $\delta_{\ell}$, we
then have

(5) $x(r_{k}-4\ell,r_{k}+4\ell)=y(r_{k}-4\ell,r_{k}+4\ell)\ .$

As $\tau$ is assumed to be pre-injective, $x$ and $y$ differ at infinitely
many positions. In particular, we may select $k_{0}<k_{1}$ such that

(6) $x(r_{k_{0}}-4\ell,r_{k_{1}}+4\ell)\neq
y(r_{k_{0}}-4\ell,r_{k_{1}}+4\ell)\ .$

Let $D=\\{r_{k_{0}}-4\ell,\ldots,r_{k_{1}}+4\ell\\}$ and define
$x^{\prime},y^{\prime}\in\\{0,1\\}^{{\mathbb{Z}}}$ by
$x^{\prime}=x\cdot\mathbbm{1}_{D}$ and $y^{\prime}=y\cdot\mathbbm{1}_{D}$, the
dot denoting coordinatewise multiplication. Then $x^{\prime}$ and $y^{\prime}$
differ due to (6), but they differ at finitely many positions only. Moreover,
$x^{\prime},y^{\prime}\in X$ by heredity. On the other hand,
$\tau(x^{\prime})=\tau(y^{\prime})$ which contradicts the pre-injectivity of
$\tau$. To see the latter, consider $n\in\\{r_{k_{0}}-2\ell,\ldots
r_{k_{1}}+2\ell\\}$ and note $\tau(x^{\prime})(n)=\tau(x)(n)$ and
$\tau(y^{\prime})(n)=\tau(y)(n)$ as the code has length $\ell$. Thus
$\tau(x^{\prime})(n)=\tau(y^{\prime})(n)$ due to $\tau(x)=\tau(y)$. For
$n\notin\\{r_{k_{0}}-2\ell,\ldots,r_{k_{1}}+2\ell\\}$, note that
$\tau(x^{\prime})(n)$ and $\tau(y^{\prime})(n)$ are both coded by the same
interval by (5). Hence $\tau(x^{\prime})(n)=\tau(y^{\prime})(n)$. Altogether,
we obtain $\tau(x^{\prime})=\tau(y^{\prime})$.

####  3.2.2 Myhill property on classes of hereditary subshifts

The above two lemmata are crucial ingredients for a proof of the Myhill
property. Our first result is based on Lemma 3.6 and uses monotonicity.

######  Proposition 3.8.

Assume that $\mathscr{B}\subset{\mathbb{N}}\setminus\\{1\\}$ is primitive and
infinite. If $X_{\eta}=X_{\mathscr{B}}$, then the $\mathscr{B}$-free subshift
$(X_{\eta},S)$ has the Myhill property.

###### Proof.

Denote by $X_{0}\subset X_{\mathscr{B}}$ the collection of all $0-1$-sequences
in $X_{\mathscr{B}}$ that are asymptotic to the all zero sequence
$0^{\mathbb{Z}}$. Clearly, $X_{0}$ is dense in $X_{\mathscr{B}}$. As
$X_{\eta}=X_{\mathscr{B}}$, it is also dense in $X_{\eta}$. As $\tau\circ
S^{-t}$ is pre-injective whenever $\tau$ is pre-injective, the result follows
from Proposition 3.2, Lemma 3.6 and the continuity of $\tau$.

The previous result applies in particular to the Erdös case. We have the
following version for hereditary subshifts, where we use Lemma 3.7, which does
not resort to monotonicity.

######  Corollary 3.9.

Let $(X,S)$ be a hereditary subshift which is intrinsically ergodic and whose
measure of maximal entropy has full topological support. Then $(X,S)$ has the
Myhill property. Let $\tau$ be a CA on $(X,S)$. If $\tau$ is pre-injective,
then $\tau$ is surjective.

###### Proof.

Let $\tau$ be a pre-injective CA on $(X,S)$ and suppose $Y:=\tau(X)\subset X$
with $Y\neq X$. Then $(Y,S)$ is a subsystem of $(X,S)$ satisfying
$h(Y,S)=h(X,S)$. The latter statement uses the variational principle and the
fact that all fibers $\tau^{-1}(\\{x\\})$ have finite cardinality by Lemma
3.7. As $(Y,S)$ is a subshift, we may take a measure $\nu$ on $Y$ of maximal
entropy. Then $\nu$ is also a measure of maximal entropy for $(X,S)$. But by
assumption $\nu$ has full support on $X$, which is in conflict with $Y\neq X$.

Now, notice that if $\nu_{\eta}$ has full support, then also $\nu_{\eta}\ast
B(\frac{1}{2},\frac{1}{2})$ has full support in $\widetilde{X}_{\eta}$ (cf.
[30]) and therefore, in view of Corollary 3.9 and Theorems 2.2 and 2.3, we
obtain the following.

######  Corollary 3.10.

The Myhill property holds for $(\widetilde{X}_{\eta},S)$ whenever
$\mathscr{B}$ is taut. In particular, it holds in the Erdös case.

######  Remark 3.11.

Apart from the Erdös case, there are more examples of hereditary systems to
which the above applies, see [31]; for instance, it applies to the hereditary
closure of Sturmian sequences.

###  3.3 The GoE theorem for $\mathscr{B}$-free subshifts

The following result contains Theorem 1.1 from the introduction as a special
case. Indeed, in the Erdös case we have $X_{\eta}=X_{\mathscr{B}}$, see [1,
Cor. 4.2].

######  Theorem 3.12.

Assume that a $\mathscr{B}$-free subshift $(X_{\eta},S)$ satisfies
$X_{\eta}=X_{\mathscr{B}}$. Then it satisfies the GoE theorem. In fact, every
CA is given by a monotone code, modulo a power of the shift. If the CA is
onto, then it equals a power of the shift.

###### Proof.

The Moore property has been proved in Proposition 3.3. The Myhill property and
the remaining statements follow from Proposition 3.8.

##  4 The automorphism group is trivial for hereditary $\mathscr{B}$-free
systems

###  4.1 Some facts about hereditary subshifts

Assume that $X\subset\\{0,1\\}^{{\mathbb{Z}}}$ is a hereditary subshift.
Assume additionally that $(X,S)$ is transitive and let the orbit
$(S^{n}z)_{n\in{\mathbb{Z}}}$ of $z\in X$ be dense in $X$. We will constantly
assume that $z$ has infinite support (otherwise the dynamics of $(X,S)$ is
very simple: it consists of finitely many orbits attracted by a fixed point).
Given a non-negative $a\in{\mathbb{Z}}$ and a block
$B\in\\{0,1\\}^{\\{-a,\ldots,a\\}}$, we say that this block appears centrally
in $z$ at position $m$ if $z(m+n)=B(n)$ for each $n\in[-a,a]$. It follows that
the set of those $m$ such that $B$ appears centrally in $z$ at position $m$
must be infinite. Indeed, let $z(m+n)=B(n)$ for each $n\in[-a,a]$. Since $z$
has infinite support, there exists $j\geq 1$ such that for some
$s\in[-(a+j),a+j]\setminus[-a,a]$, we have $z(m+s)=1$. Now define a new block
$C\in\\{0,1\\}^{\\{-(a+j),\ldots,a+j\\}}$ arising from $z(m-(a+j),m+a+j)$ by
replacing all the 1’s at positions in $[m-(a+j),m+a+j]\setminus[-a,a]$ by the
0’s. By the heredity of $X$, it follows that the block $C$ has to appear
(centrally) in $z$ at a certain position $m^{\prime}$ with necessarily
$m^{\prime}\neq m$. Now, repeat the same reasoning with $B$ replaced by $C$,
etc. In this way, we obtain an infinite sequence of different positions of
(central) appearances of the original block $B$ in $z$.

Assume additionally that the point $z\in X$ is maximal, that is: for each
$x\in X$, if $z\leq x$ then $z=x$. Note that whenever $z$ is a transitive
maximal point and $\tau\in{\rm Aut}(X,S)$ satisfies $\tau(x)\leq x$ for each
$x\in X$ then $\tau=Id$. Indeed, $z\leq\tau^{-1}(z)$, which means that
$\tau(z)=z$ and $\tau=Id$ follows from the fact that $z$ is a transitive
point.

Note also that the sequence $0^{\mathbb{Z}}$ is a fixed point belonging to
$X$. Unless $(X,S)$ is the full shift, $0^{\mathbb{Z}}$ is the only fixed
point for $S$. Hence, assuming that $(X,S)$ is not the full shift, we must
have

(7) $\tau(0^{\mathbb{Z}})=0^{\mathbb{Z}}\text{ for each }\tau\in{\rm
Aut}(X,S).$

Given $\tau\in{\rm Aut}(X,S)$, let
$\phi:\\{0,1\\}^{\\{-\ell,\ldots,\ell\\}}\to\\{0,1\\}$ be the corresponding
code

(8) $\tau(x)(n)=\phi(x(n-\ell,n+\ell))\text{ for each }n\in{\mathbb{Z}}.$

By an abuse of notation, we will also write that $\tau(x)=\phi(x)$ (or even
$\phi(C)$, where $C$ is a block). It follows from (7) and (8) that

(9) $\phi(0^{\\{-\ell,\ldots,\ell\\}})=0.$

###  4.2 The result and its proof

The following result extends Mentzen’s theorem beyond the Erdös case. It also
extends the result from Theorem 3.12 in the hereditary case
$X_{\eta}=X_{\mathscr{B}}$.

######  Theorem 4.1.

Let $(X_{\eta},S)$ be a $\mathscr{B}$-free subshift. If $X_{\eta}$ is
hereditary then ${\rm Aut}(X_{\eta},S)$ is trivial, that is, it equals
$\\{S^{n}:\>n\in{\mathbb{Z}}\\}$.

######  Corollary 4.2.

Consider some $\mathscr{B}$-free subshift $(X_{\eta},S)$ such that
$\mathscr{B}$ is taut and proximal. Then ${\rm Aut}(X_{\eta},S)$ is trivial.

######  Remark 4.3.

The result contained in Corollary 4.2 was already proved by Keller [24, Thm.
4.1] using so called generic dynamics tools. For a discussion, see the
following section.

To prove Theorem 4.1, we will need two lemmata which follow the strategy of
the proof of Mentzen’s theorem [35]. Let $u\in\\{0,1\\}^{{\mathbb{Z}}}$ be
defined as $u(n)=0$ for each $n\in{\mathbb{Z}}$ except for $n=0$. We write
$u=0^{-{\mathbb{N}}}10^{\mathbb{N}}$. By heredity, $u\in X_{\eta}$.

######  Lemma 4.4.

There exists $k\in{\mathbb{Z}}$ such that $\tau(u)=S^{k}u$. Equivalently, the
code $\phi:\\{0,1\\}^{\\{-\ell,\ldots,\ell\\}}\to\\{0,1\\}$ has the property
that it takes value 1 on exactly one block of length $2\ell+1$ having one
element support.

###### Proof.

Note that $\phi$ cannot take value 0 on all blocks of length $2\ell+1$ with
one element support as otherwise $\tau(u)=0^{\mathbb{Z}}$ and $\tau$ would not
be invertible. Suppose that the support of $\tau(u)$ has at least two
elements. So let $C_{1},C_{2}$ be two different blocks of length $2\ell+1$
having one-element support and $\phi(C_{1})=\phi(C_{2})=1$. Let
$C_{1}(m_{1})=C_{2}(m_{2})=1$ and set $s:=m_{1}-m_{2}$. Choose
$b\in\mathscr{B}$ so that

(10) $\gcd(s,b)=1.$

Such a $b$ exists as $X_{\eta}$ is hereditary, whence $(X_{\eta},S)$ is
proximal and therefore $\mathscr{B}$ contains an infinite coprime subset. Let,
with $r$ maximal, $m_{1},\ldots,m_{r}$ be such that $\eta(m_{i})=1$ for
$i=1,\ldots,r$ and $m_{i}$ mod $b$ are pairwise different (we recall that on
$\eta$ at least the zero residue class of $b$ does not appear, so $r<b$ and
all $m_{i}$ are different from zero modulo $b$). We claim that, with no loss
of generality, we can assume that the distances $|m_{i}-m_{j}|$ for $i\neq j$
are as large as we want. Indeed, as we have seen, the heredity of $X_{\eta}$
implies that $\eta$ is recurrent. This means that for infinitely many
$n\in{\mathbb{Z}}$ the elements $m_{1}+n,\ldots,m_{r}+n$ satisfy the
aforementioned properties of $m_{1},\ldots,m_{r}$ (the sets
$\\{m_{1},\ldots,m_{r}\\}$ and $\\{m_{1}+n,\ldots,m_{r}+n\\}$ must be equal
modulo $b$ by the maximality of $r$). Now, proceed in the following way: take
such a “good” $n_{2}$ arbitrary large and among elements
$m_{1}+n_{2},\ldots,m_{r}+n_{2}$ find a one, say $m_{j}+n_{2}$ such that

$m_{j}+n_{2}=m_{2}\text{ mod }b$

and replace $m_{2}$ by $m_{j}+n_{2}$. In this way $m_{1}$ and $m_{2}$ are well
separated. Then choose another “good” $n_{3}$ (arbitrary large) so that

$m_{j}+n_{3}=m_{3}\text{ mod }b$

for some $j$ and replace $m_{3}$ by $m_{j}+n_{3}$. We will obtain our claim in
$r$ steps.

Having the elements $m_{1},\ldots,m_{r}$ well separated, we consider a large
interval $I$ in ${\mathbb{Z}}$ containing all these elements. Then let $C$ be
the block whose positions are indexed by $I$ so that the only positions of 1’s
are given by $m_{1},\ldots,m_{r}$. By heredity (and the fact that $\eta$ is
transitive), it follows that the block $C$ appears in $\eta$ at a certain
position $n$. Now, $\tau(\eta)\in X_{\eta}$ and we consider $\phi(\eta)$ or,
more precisely, we look at the code $\phi(C)$ of the block $C$. It follows
that for some $n^{\prime}\in{\mathbb{Z}}$, we have

$\tau(\eta)(m_{i}+n^{\prime})=1=\tau(\eta)(m_{i}+n^{\prime}+s)\text{ for
}i=1,\ldots,r.$

It follows that the set $Z$ consisting of $m_{i}+n^{\prime}$ mod $b$,
$i=1,\ldots,r$, is a subset contained in the support of $\eta$ modulo $b$ and
so is the set $Z+s$ mod $b$. Since the cardinality of $Z$ is maximal, $Z+s=Z$
which is conflict with (10) .

In what follows, we replace $\tau\in{\rm Aut}(X_{\eta},S)$ by $S^{-k}\tau$,
and, by slightly enlarging the length $\ell$ if necessary, we can assume that
the only block of length $2\ell+1$ with the one element support whose code is
1 is the block that has 1 at the central position. We continue the proof of
Theorem 4.1, but now we have to show that $\tau=Id$.

######  Lemma 4.5.

For an arbitrary block $C$ of length $2\ell+1$ with the zero at the central
position, we have $\phi(C)=0$.

###### Proof.

Suppose that for some $C$ as above, $\phi(C)=1$. Let $C$ appear (centrally) in
$\eta$ at position $s$. We have $\eta(s)=0$, so there exists $b\in\mathscr{B}$
such that $s=mb$ for some $m\in{\mathbb{Z}}$. Choose positions
$m_{1},\ldots,m_{r}$ so that the distances between these elements are as large
as we want (by the proof of Lemma 4.4), and mod $b$ they “realize” the maximal
number $r$ of non-zero residue classes in $\eta$. In particular, the $m_{i}$
are all different from $s$. Take a large block $E$ containing the following
positions: $s$ (with the whole block $C$) and $m_{1},\ldots,m_{r}$. Replace
all 1’s that appear in it at all other positions than those given by $C$ and
$m_{1},\ldots,m_{r}$ by 0’s, to obtain the block $F$. By heredity, the block
$F$ appears in $\eta$ at a certain position $r^{\prime}$. It easily follows
that the support of $\tau(\eta)$ will now contain a certain translation of the
set $\\{s,m_{1},\ldots,m_{r}\\}$, and since this set mod $b$ is of cardinality
$\geq r+1$, we obtain $\tau(\eta)\notin X_{\eta}$ and hence a contradiction.

Proof of Theorem 4.1 It follows from Lemma 4.5 that for each $x\in X_{\eta}$,
we have $\tau(x)\leq x$. It follows that $\eta\leq\tau^{-1}(\eta)$, and since
Lemma 2.1 holds in $X_{\eta}$, the result follows.

###  4.3 The property $\tau(0^{-{\mathbb{N}}}10^{\mathbb{N}})\neq
0^{\mathbb{Z}}$

The following shows that $\tau(0^{-{\mathbb{N}}}10^{\mathbb{N}})\neq
0^{\mathbb{Z}}$ is a very strong condition.

######  Lemma 4.6.

Consider a hereditary $\mathscr{B}$-free subshift $(X_{\eta},S)$. Assume that
a CA $\tau$ on $(X_{\eta},S)$ is surjective and that
$\tau(0^{-{\mathbb{N}}}10^{\mathbb{N}})\neq 0^{\mathbb{Z}}$. Then $\tau$ is a
power of a shift. In particular, $(X_{\eta},S)$ satisfies the Moore property.

###### Proof.

Write $x_{0}=0^{\mathbb{Z}}$ and $x_{1}=0^{-{\mathbb{N}}}10^{\mathbb{N}}$ and
note that indeed $x_{0},x_{1}\in X_{\eta}$ by heredity. Note first that there
exists $r\in{\mathbb{Z}}$ such that $\tau(x_{1})=S^{r}x_{1}$ by the same proof
as in Lemma 4.4, where $\tau(0^{-{\mathbb{N}}}10^{\mathbb{N}})\neq
0^{\mathbb{Z}}$ is now true by assumption and not as a consequence of
injectivity. As $\tau$ is surjective if and only if $\tau\circ S^{-r}$ is
surjective, by redefining $\tau$ we may assume without loss of generality that
$\tau(x_{1})=x_{1}$. Now, Lemma 4.5 yields $\tau(x)\leq x$ for each $x\in
X_{\eta}$. As $\tau$ is assumed surjective, there exists $z\in X_{\eta}$ such
that $\tau(z)=\eta$. But $\eta=\tau(z)\leq z$, which is only possible if
$\eta=z$ by the maximality property of $\eta$ stated in Lemma 2.1. Since
$\eta$ is a transitive point in $X_{\eta}$, we get $\tau=Id$.

Also note the following simple consequence of $0^{\mathbb{Z}}$ and
$0^{-{\mathbb{N}}}10^{\mathbb{N}}$ being asymptotic.

######  Lemma 4.7.

Let $(X,S)$ be a subshift such that $0^{\mathbb{Z}}\in X$ and
$0^{-{\mathbb{N}}}10^{\mathbb{N}}\in X$. If $\tau$ is a pre-injective CA on
$(X,S)$, then $\tau(0^{-{\mathbb{N}}}10^{\mathbb{N}})\neq 0^{\mathbb{Z}}$.

Noting that for taut $\mathscr{B}$, the proximality and the heredity of
$(X_{\eta},S)$ are equivalent notions, the previous two lemmata can be
combined into the following result.

######  Corollary 4.8.

Let $(X_{\eta},S)$ be a $\mathscr{B}$-free subshift, where $\mathscr{B}$ is
assumed taut and proximal. Let $\tau$ be a surjective CA on $(X_{\eta},S)$.
Then $\tau$ is pre-injective if and only if
$\tau(0^{-{\mathbb{N}}}10^{\mathbb{N}})\neq 0^{\mathbb{Z}}$.

The latter result shows that Proposition 1 in [24] implicity uses the Moore
property that surjectivity implies pre-injectivity. The Moore property holds
in the Erdös case, but seems open in the more general case, where
$\mathscr{B}$ is taut and proximal.

##  5 Proximal $\mathscr{B}$-free subshifts and open questions

It is not clear to us to which extent Theorem 4.1 is true in the (general)
proximal case. We have however the following reduction of the problem.

######  Corollary 5.1.

Let $(X_{\eta},S)$ be a proximal $\mathscr{B}$-free subshift. Let
$\mathscr{B}^{\prime}$ be the unique taut as in Theorem 2.4. Assume that
$\tau\in{\rm Aut}(X_{\eta},S)$. Then, $X_{\eta^{\prime}}={\rm
supp}(\nu_{\eta})={\rm supp}(\nu_{\eta^{\prime}})$ and
$\tau(X_{\eta^{\prime}})=X_{\eta^{\prime}}$ with
$\tau|_{X_{\eta^{\prime}}}=S^{k}$ for some $k\in{\mathbb{Z}}$.

###### Proof.

Since $\mathscr{B}^{\prime}$ is taut, by Theorem 2.3, $X_{\eta^{\prime}}={\rm
supp}(\nu_{\eta^{\prime}})={\rm supp}(\nu_{\eta})$. Moreover, by Corollary
2.7, $(X_{\eta},S)$ is intrinsically ergodic with the measure of maximal
entropy $\nu_{\eta^{\prime}}\ast B(\frac{1}{2},\frac{1}{2})$. Hence this
latter measure must be preserved by $\tau$. In particular, $\tau$ preserves
its support. It is not hard to see that the support of
$\nu_{\eta^{\prime}}\ast B(\frac{1}{2},\frac{1}{2})$ is $X_{\eta^{\prime}}$
and the result now follows from Theorem 4.1.

In other words, in the general proximal case we reduced the problem of
determining ${\rm Aut}(X_{\eta},S)$ to the “relative Behrend” situation. This
yields an intriguing question: What is the automorphism group of a
$\mathscr{B}$-free system in the Behrend case itself?444As shown in the recent
article [16], in the minimal case, the group of automorphisms can be non-
trivial. It would be interesting to know whether the GoE theorem holds in this
case. It would also be interesting to know whether, in the hereditary case,
all codes given by elements of ${\rm End}(X_{\eta},S)$ are monotone. We were
not able to decide whether, in the general case of a $\mathscr{B}$-free
system, $\tau$ preserves the Mirsky measure, i.e.
$\tau_{\ast}(\nu_{\eta})=\nu_{\eta}$, for all $\tau\in{\rm Aut}(X_{\eta},S)$.

It also seems natural to consider questions about a measure-theoretic GoE
theorem. We give some samples of that. Assume that a CA $\tau$ preserves the
Mirsky measure. Let $\tau$ be called $\nu_{\eta}$-pre-injective if it is
injective on any collection of $0-1$-sequences that differ in a set of zero
density. Is this notion equivalent to $\tau$ being onto $\nu_{\eta}$-almost
surely? If yes, is there an analogous result for measures different from the
Mirsky measure? May GoE proofs be more transparent when using isomorphism to
the maximal equicontinuous generic factor [24] or to the Kronecker factor?

Acknowledgments Research of the first author is supported by Narodowe Centrum
Nauki grant UMO-2019/33/B/ST1/00364.

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Faculty of Mathematics and Computer Science,
Nicolaus Copernicus University, Chopin street 12/18, 87-100 Toruń, Poland
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Department of Mathematics,
University of Erlangen-Nürnberg, Cauerstraße 11, 91058 Erlangen, Germany
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