Automorphisms of automatic shifts

TL;DR

This paper investigates the automorphism groups of constant length substitution shifts, revealing that roots of identity are letter exchanges and automorphisms are related to twisted compressions, with implications for their topological factors.

Contribution

It characterizes the automorphism groups of constant length substitution shifts, including roots of identity and their topological factors, advancing understanding of their structure.

Findings

Roots of identity are letter exchanging maps.

Nontrivial automorphisms are twisted compressions.

Topological factors are conjugate to substitution shifts.

Abstract

In this article we continue the study of automorphism groups of constant length substitution shifts and also their topological factors. We show that up to conjugacy, all roots of the identity map are letter exchanging maps, and all other nontrivial automorphisms arise from {\em twisted} compressions of another constant length substitution. We characterise the group of roots of the identity in both the measurable and topological setting. Finally, we show that any topological factor of a constant length substitution shift is topologically conjugate to a constant length substitution shift via a letter-to-letter code.

Tables

Table 1. Table 1. The local rule for the factor map in Example 9 .

	x	y	z
xxxx	y	z
xxxy	y	y
xyxx	y	y
yxxx		z
yxxy			z
yzxx	y	y
zxxx	z	z
zxxy			z
zyxx	y
zyxy			y

Table 2. Table 2. The local rule for the factor map in Example 25 .

	a	b	c
a	z	y	x
b	y	x	z
c	x	z	y

Table 3. Table 3. Local rule table for ϕ , italic-ϕ \phi, Example 50 . The entry in the row labelled with α 𝛼 \alpha and in the column labelled with β 𝛽 \beta is ϕ ​ ( α ​ β ) italic-ϕ 𝛼 𝛽 \phi(\alpha\beta) . The map ϕ italic-ϕ \phi need only be defined on the language of θ 𝜃 \theta .

	a	b	c	d	e	f	g	h
a						b		b
b	f		f	f	h	h
c		a
d		g				e
e		f		f
f	d	d			b		b	b
g						e
h		a				c

Table 4. Table 4. The local rule for the square root Φ Φ \Phi of the shift in Example 51 .

	A	B	C	D	E	F	G	H	I	J	K	L
A	J	J		F
B			I		I						L
C	H	H						D
D					C		E				E
E						B				B		A
F			C		C		E
G	J			F				F
H									G	K		G
I		H		D				D
J						K			G	K
K			I				L				L
L						B			A			A

Equations

$$\theta (a)={\theta }_0(a)\dots {\theta }_{r-1}(a);$$

$$h(\theta ):=\max \{n\ge 1:\gcd (n,r)=1,n|\gcd \{a:u_a=u_0\}\}.$$

$$T(x,i):=\{\begin{array}{ll}(x,i+1)& \text{ if }0\le i<h-1\\ ({\sigma }^{\prime }(x),0)& \text{ if }i=h-1\end{array}$$

$${\Psi }_i(x,j):=\{\begin{array}{ll}(\Psi (x),j+i)& \text{ if }j+i<h\\ (\Psi ({\sigma }^{\prime }(x)),j+i\mathrm{mod}h)& \text{ if }j+i\ge h.\end{array}$$

$${\theta }^{(\ell )}({\alpha }_1\dots {\alpha }_{\ell }):=(a_{k+1}\dots a_{k+\ell })(a_{k+2}\dots a_{k+\ell +1})\dots (a_{k+r}\dots a_{k+r+\ell -1}).$$

$${\pi }_Y(\Phi (x))=\kappa (\Phi )+{\pi }_X(x)$$

$$c=c(\theta ):=\underset{0\le j<r^k,k\in \mathbb{N}}{\min }\left|{\theta }^k(\mathcal{A}{)}_j\right|.$$

$${\theta }^{-c!p}\circ {\sigma }^{-k(1+r^p+r^{2p}+\dots +r^{(c!-1)p})}\circ \Phi \circ {\theta }^{c!p}=\Phi ,$$

$${\theta }^{-c!}\circ \Phi \circ {\theta }^{c!}=\Phi .$$

$$\theta (a)$$

$$\theta (b)$$

$$\theta (c)$$

$$\theta (d)$$

$$\tau (a)=x,\tau (b)=y,\tau (c)=x\text{mbox}and\tau (d)=z,$$

$$g(y)$$

$$g(z)$$

$$f({\theta }^{c!p}(x_{-1}\cdot x_0{)}_{i-1},{\theta }^{c!p}(x_{-1}\cdot x_0{)}_i)={\theta }^{c!p}(f(x_{-1},x_0)f(x_0,x_1){)}_{N+i},$$

$$g(u_{i-1},u_i,u_{i+1})=({\theta }^{c!}g(x_{-1},x_0,x_1){)}_i,$$

$$\theta (a)$$

$$\theta (b)$$

$$\theta (c)$$

$$\theta (d)$$

$$\bar{\theta }(x)$$

$$\bar{\theta }(y)$$

$$\theta (a)$$

$$\theta (b)$$

$$\theta (c)$$

$$\bar{\theta }(x)$$

$$\bar{\theta }(y)$$

$$p(\theta )=lcm\{p(a,b):a,b\text{ are a periodic pair}\},$$

$$a^{\prime }$$

$$={\theta }^{k_2-k_1}(a^{\prime }{)}_{j^{\prime }},$$

$$\theta ([a]{)}_j:=[{\theta }^{\prime }(a{)}_j]$$

$$\tau ([a]):=\tau (a).$$

$$(({\theta }^{\prime }{)}_j^k{)}^n(a_0)$$

$$(({\theta }^{\prime }{)}_j^k{)}^m(b_0)$$

$$(({\theta }^{\prime }{)}_j^k{)}^{nm}(a_0)$$

Full text

The authors were supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, Grant Agreement No 648132. The second author also thanks IRIF, Université Paris Diderot-Paris 7, for its hospitality and support.

1. Introduction

In this article we continue the study of the automorphism groups of constant length substitution shifts and complete some of our work in [5]. In particular, there we gave an algorithm for the computation of the automorphism group $\operatorname{Aut}(X_{\theta},\sigma)$ of a constant length substitutional shift. Here, we ask what are the possible elements of this group, and also, what kinds of factors these shifts can have. In both cases, the answer is a form of rigidity.

Our strategy in [5] was to attach a rational number $\kappa(\Phi)$ to each automorphism $\Phi$, which signifies by how much points are being shifted while being acted on by $\Phi$ (Theorem 6). In particular, a simple example of an automorphism whose $\kappa$-value equals zero is that coming from a permutation on letters which plays well with the substitution $\theta$, by which we mean that it commutes with some power of the substitution. These automorphisms are natural and indicate that there is a symmetry in the substitution rule, up to a shuffling of letters. Now one can construct examples of substitutions with automorphisms whose $\kappa$-value is zero, but which are not of this nature. However we show, in Theorem 32, that up to conjugacy, only the natural case manifests.

What remains to consider are the automorphisms with a nonzero rational $\kappa$-value $\frac{1}{k}$. It is not difficult to create such situations. Namely, one starts with a substitution $\eta$, and one takes its $k$-compression. This is the process whereby we rewrite a sequence as a concatenation of words of length $k$, and where the shift map moves from one word to the next. Cobham [3] used this terminology, and he showed that constant length substitutions are robust under this operation: a $k$-compression of a length $r$ substitution shift $(X_{\eta},\sigma)$ gives us another length $r$ substitution shift $(X_{\theta},\sigma)$. By construction, $\operatorname{Aut}(X_{\theta},\sigma)$ will contain a shadow of the shift on $X_{\eta}$, and this manifests as a $k$-th root of the shift map on $X_{\theta}$. Can automorphisms $\Phi$ with $\kappa(\Phi)=\frac{1}{k}$ arise in any other way? We discover, in Theorem 46, that up to conjugacy, the only other way that they can arise is if we twist after we compress $\eta$ (Definition 45). At some level, our results say that the automorphisms of our substitution shift $X_{\theta}$ are only constrained by fixed points, either of $\theta$, or of another substitution.

Can one hope that the letter exchanging maps of $\theta$ are related to those of $\eta$? In our world, this is generally not the case; we see this in Examples 51 and 52. There is a fundamental obstruction discussed in Remark 53. Hence we fall short of giving a general structure theorem. We could formulate one, but it would not be succinct.

We also characterise, in Theorem 37, the letter exchanging maps that define automorphisms of $X_{\theta}$, generalising results due to Lemanczyk and Mentzen in the bijective case [16]. In particular, they characterised the measurable essential centralizer of a bijective substitution as being the centraliser of the group generated by the columns of $\theta$; the topological statement is identical. To characterise the letter exchanging maps of an arbitrary constant length substitution, we work with minimal sets, a notion that was implicit in the algorithm of [5], but which was defined and used extensively by the first author in [20, Section 2]. We obtain characterisations in both the topological setting, in Theorem 37, and the measurable setting, in Theorem 44.

These are our principal results, but on our trip to discovering them, we obtained some other results of interest along the way. The first, which follows from Lemma 3 and Theorem 22, tells us that any topological shift factor of a constant length substitution shift $X_{\theta}$ is a letter-to-letter coding of a constant length substitution shift. This answers a question of Durand and Leroy in [8, Section 8]. It also suggests that the question that Coven, Dekking and Keane ask in [4], namely to list all constant-length substitutions whose shifts are conjugate to a given constant length substitution shift, is not constraining.

Finally, we investigate the relationship between $\operatorname{Aut}(X,\sigma)$ and the existence of a uniform-to-one factor for $X$. This has been studied by Herning [12] in the case where $X=X_{\theta}$ and the factor is metrically a rotation. In Theorem 24, we show that $\operatorname{Aut}(X,\sigma)$ contains a square root of the identity if and only if $X$ has a two-to-one factor. In Example 25, we show that there is no possibility of extending this result.

2. Preliminaries

Let $\mathcal{A}$ be a finite alphabet, with the discrete topology. We endow $\mathcal{A}^{\mathbb{Z}}$ with the product topology, and let $\sigma:\mathcal{A}^{\mathbb{Z}}\rightarrow\mathcal{A}^{\mathbb{Z}}$ denote the (left) shift map. We consider only infinite minimal shifts $(X,\sigma)$, i.e. those such that the $\sigma$-orbit of any point in $X$ is dense in $X$. The language $\mathcal{L}_{X}$ of a shift $(X,\sigma)$ is the set of all finite words $w=w_{1}\ldots w_{n}$ that appear as a subword of some $x$ in $X$, i.e. $w=x_{j}\ldots x_{j+n-1}$ for some $j$.

2.0.1. Automorphism groups

A factor map from $(X,\sigma)$ to $(Y,\sigma)$ is a map $\Phi:X\rightarrow Y$ which is continuous and commutes with $\sigma$. If $X=Y$, $\Phi$ is called an endomorphism. If $\Phi$ is bijective, it is called a conjugacy, and if $\Phi$ is a self-conjugacy then it is called an automorphism. We say $\Phi:X\rightarrow X$ has (finite)order m if $m$ is the least positive integer such that $\Phi^{m}$ is the identity map $\operatorname{Id}$, and we say that $\Phi$ is a $k$-th root of the shift if $\Phi^{k}=\sigma$. Let ${\operatorname{Aut}}(X,\sigma)$ denote the automorphism group of $(X,\sigma)$; it contains the (normal) subgroup generated by the shift, denoted by $\langle\sigma\rangle$.

Let $X\subset\mathcal{A}^{\mathbb{Z}}$ and let $Y\subset\mathcal{B}^{\mathbb{Z}}$. By the Curtis–Hedlund–Lyndon theorem, for any factor map $\Phi:X\rightarrow Y$, there exist $\ell$, $r$, and a map $f\colon\mathcal{A}^{\ell+r+1}\to\mathcal{B}$ with the property that $(\Phi(x))_{n}=f(x_{n-\ell},\ldots,x_{n},\ldots,x_{n+r})$ for all $x$ and $n$. Let $\ell,r$ be the smallest possible integers so that such an $f$ exists. We call $\ell$ and $r$ the left and right radius of $\Phi$ respectively. We say $\Phi$ has radius $R$ if its left radius and right radius are both at most $R$.

2.0.2. Constant length substitutions

A substitution is a map from $\mathcal{A}$ to the set of nonempty finite words on $\mathcal{A}$. We use concatenation to extend $\theta$ to a map on finite and infinite words from $\mathcal{A}$. We say that $\theta$ is primitive if there is some $k\in\mathbb{N}$ such that for any $a,a^{\prime}\in\mathcal{A}$, the word $\theta^{k}(a)$ contains at least one occurrence of $a^{\prime}$. By iterating $\theta$ on any fixed letter in $\mathcal{A}$, we obtain one-sided right-infinite points $u=u_{0}\ldots$ such that $\theta^{j}(u)=u$ for some natural $j$. Similarly we can obtain one-sided left-infinite points $v=\ldots v_{-1}$ such that $\theta^{j}(v)=v$. A bi-infinite periodic point for $\theta$ is a concatenation of a left-infinite periodic point $v=\ldots v_{-1}$ and a right-infinite periodic point $u=u_{0}\ldots$ provided that $v_{-1}u_{0}$ appears in some word $\theta^{k}(a)$. The pigeonhole principle implies that $\theta$-periodic points always exist, and, for primitive substitutions, we define $X_{\theta}$ to be the shift orbit closure of any one of these bi-infinite $\theta$-periodic points and call $(X_{\theta},\sigma)$ a substitution shift. The substitution $\theta$ has (constant) length r if for each $a\in\mathcal{A}$, $\theta(a)$ is a word of length $r$.

Let $\theta$ be a primitive length $r$ substitution such that $X_{\theta}$ is infinite. We say that $\theta$ is recognizable if any $x\in X_{\theta}$ can be written in a unique way as $x=\sigma^{k}(\theta(y))$ where $y\in X_{\theta}$ and $0\leq k<r$. Our substitutions are recognizable, see [13] for a proof in the injective case, which is all we will need.

Let $\theta$ be a length $r$ substitution. We write

[TABLE]

with this notation we see that for each $0\leq i\leq r-1$, we have a map $\theta_{i}:\mathcal{A}\rightarrow\mathcal{A}$ where $\theta_{i}(a)$ is the $(i+1)$-st letter of $\theta(a)$. Similarly, for any $k$ we write $(\theta^{k})_{i}$ to denote the map which sends $a\in\mathcal{A}$ to the $(i+1)$-st letter of $\theta^{k}(a)$.

Let $\theta$ be a primitive length $r$ substitution with fixed point $u$, and with $(X_{\theta},\sigma)$ infinite. The height $h=h(\theta)$ of $\theta$ is defined as

[TABLE]

If $h>1$, this means that $\mathcal{A}$ decomposes into $h$ disjoint subsets: $\mathcal{A}_{1}\cup\ldots\cup\mathcal{A}_{h}$, where a symbol from $\mathcal{A}_{i}$ is always followed by a symbol from $\mathcal{A}_{i+1}$.

If $\theta$ has a nontrivial height $h\geq 2$, then the shift $(X_{\theta},\sigma)$ is topologically conjugate to a constant height suspension of a height one substitution shift $(X_{\theta^{\prime}},\sigma)$, where $\theta^{\prime}$ is called the pure base of $\theta$. In this case $(X_{\theta},\sigma)\cong(X_{\theta^{\prime}}\times\{0,\ldots,h-1\},T)$ where

[TABLE]

We refer the reader to [6, Remark 9, Lemmas 17 and 19] for details.

For every $\Psi\in\operatorname{Aut}(X_{\theta^{\prime}},\sigma^{\prime})$ and $0\leq i\leq h-1$ we can define $\Psi_{i}\in\operatorname{Aut}(X_{\theta^{\prime}}\times\{0,\ldots,h-1\},T)$ as

[TABLE]

The following is proved in [5, Proposition 3.5].

Proposition 1**.**

Let $\theta$ be a primitive, length $r$ substitution, of height $h$, and such that $(X_{\theta},\sigma)$ is infinite. Let $\theta^{\prime}$ be a pure base of $\theta$. Then $\operatorname{Aut}(X_{\theta},\sigma)=\{\Psi_{j}:\Psi\in\operatorname{Aut}(X_{\theta^{\prime}},\sigma^{\prime})\mbox{ and }0\leq j<h\}.$

Proposition 1 tells us that we need only study automorphism groups of shifts $(X_{\theta},\sigma)$ where $\theta$ has height one, and henceforth this will almost always be a standing assumption.

2.0.3. The maximal equicontinuous factor of a constant length substitution shift

The maximal equicontinuous factor of a dynamical system $(X,\sigma)$ encodes its continuous eigenvalues. For a precise definition, see [5]. The following is shown by Dekking [6], with partial results by Kamae [15] and Martin [18].

Theorem 2**.**

Let $\theta$ be a primitive, length $r$ substitution, of height $h$, and such that $(X_{\theta},\sigma)$ is infinite. Then the maximal equicontinuous factor of $(X_{\theta},\sigma)$ is $(\mathbb{Z}_{r}\times\mathbb{Z}/h\mathbb{Z},+(1,1))$.

2.0.4. Automatic shifts and sliding block code representations

Let $\theta:\mathcal{A}\rightarrow\mathcal{A}^{r}$ be a primitive constant length substitution. If $\tau:\mathcal{A}\rightarrow\mathcal{B}$ and $Y:=\tau(X_{\theta})$, then $Y$ is closed and $\sigma$-invariant, and we call $(Y,\sigma)$ an $r$-automatic shift, specifying that it is generated by $(\theta,\tau)$ when necessary. We use this terminology since for $u$ one-sided and $\theta$-fixed, $\tau(u)$ is known as an $r$-automatic sequence. Note that $(Y,\sigma)$ is a factor of $(X_{\theta},\sigma)$ via a radius zero factor map.

It can happen that $X_{\theta}$ is infinite, but $\tau(X_{\theta})$ is finite. We are not interested in this case and will always require that the automatic shifts we consider are infinite. If $(X_{\theta},\sigma)$ is minimal and $Y=\tau(X_{\theta})$, then $(Y,\sigma)$ is minimal.

Given a constant length substitution $\theta:\mathcal{A}\rightarrow\mathcal{A}^{r}$, $\ell\in\mathbb{N}$ and $0\leq k<r$, we define $\theta^{(\ell,k)}:\mathcal{A}^{\ell}\rightarrow(\mathcal{A}^{(\ell)})^{r}$, the $k$-shifted $\ell$-sliding block representation of $\theta$, which appears in [21, Section 5.4]. Namely if $(\alpha_{1}\ldots\alpha_{\ell})\in\mathcal{A}^{\ell}$, and $\theta(\alpha_{1}\ldots\alpha_{\ell})=a_{1}\ldots a_{\ell r}$, let

[TABLE]

If $k=0$ we will write $\theta^{(\ell)}$ for $\theta^{(\ell,0)}$. It is straightforward to show that for each $\ell\geq 1$ and each $0\leq k<r$, $(X_{\theta^{(\ell,k)}},\sigma)$ is topologically conjugate to $(X_{\theta},\sigma)$. The $\ell$-sliding block representation is particularly useful as it allows us to change the radius of a factor map.

Lemma 3**.**

Let $(Y,\sigma)$ be a shift and let $\Phi:X_{\theta}\rightarrow Y$ be a shift-commuting continuous map with left radius $\ell$ and right radius $r$. Then there is a shift commuting map $\Phi^{(\ell+r+1)}:X_{\theta^{(\ell+r+1)}}\rightarrow Y$, with left and right radius zero.

2.1. Maximal equicontinuous factors of automatic shifts

Let $(Y,\sigma)$ be an automatic shift generated by $(\theta,\tau)$. As $(Y,\sigma)$ is a topological factor of $(X_{\theta},\sigma)$, its maximal equicontinuous factor must be a factor of that of $(X_{\theta},\sigma)$, which is $(\mathbb{Z}_{r}\times\mathbb{Z}/h\mathbb{Z},+(1,1))$. In principle, $(\mathbb{Z}_{r},+1)$ may have smaller non-finite factors, in particular when $r$ is not prime. In the next lemmas we show that as long as $(Y,\sigma)$ is not finite it always has $(\mathbb{Z}_{r},+1)$ as an equicontinuous factor. We briefly describe the required set up, referring the reader to [6] and [11] for more detail. Recall that a partition $\{P_{1},\ldots,P_{k}\}$ of a shift $(X,\sigma)$ is cyclic if $\sigma({P_{i}})=P_{i+1\bmod k}$. A partition is $\sigma^{n}$-minimal if each of its elements are $\sigma^{n}$-minimal, i.e. each $(P_{i},\sigma^{n})$ is a minimal shift. For each $n$, any minimal shift has a cyclic $\sigma^{n}$-minimal cyclic partition, which by minimality is unique up to cyclic permutation of its members. Let $\gamma(n)$ be the cardinality of the cyclic $\sigma^{n}$-minimal partition. In the case that we have a substitutional shift $(X_{\theta},\sigma)$, Dekking [6, Lemma II.7] shows that $\gamma(r^{n})=r^{n}$, and that if $P_{0}:=\theta^{n}(X_{\theta})$, then $P_{0}$ generates a $\sigma^{r^{n}}$-cyclic partition.

The proof of the following lemma is very similar to that of [6, Lemma 7].

Lemma 4**.**

If $(Y,\sigma)$ is an infinite primitive $r$-automatic shift, then $\gamma(r^{n})=r^{n}$.

Proof.

First we show that there are at most $C+C^{2}(\gamma(r^{n})-1)$ words of length $r^{n}$ in the language $\mathcal{L}_{Y}$ defined by $Y$. If $Y=\tau(X_{\theta})$, with $\theta$ defined on $\mathcal{A}$ of cardinality $C$, let $v=\tau(u)$ for any $u$ bi-infinite and $\theta$-fixed. Now $v\in\tau(\theta^{n}(X_{\theta}))$ for each $n$. Therefore $\sigma^{k\gamma(r^{n})}(v)\in\tau(\theta^{n}(X_{\theta}))$, so that $v$ is composed of overlapping blocks of the form $\theta^{n}(a)$ spaced at intervals $\gamma(r^{n})$. As there are at most $C$ words $\theta^{n}(a)$, and at most $C^{2}$ words $\theta^{n}(a)\theta^{n}(b)$, this implies that there are at most $C+C^{2}(\gamma(r^{n})-1)$ words of length $r^{n}$ in $\mathcal{L}_{Y}$. Now if $\gamma(r^{n})<r^{n}$, then by [6, Lemma 3(vi)], we have $\frac{\gamma(r^{n})}{r^{n}}\rightarrow 0$. This implies that for $n$ large, $\mathcal{L}_{Y}$ contains fewer than $r^{n}$ words of length $r^{n}$, contradicting the assumption that $Y$ is infinite. ∎

We note that for each $k$, $\tau(\theta^{k}(X_{\theta}))$ is a closed, minimal $\sigma^{r^{k}}$-invariant set. This follows from the fact $\theta^{k}(X_{\theta})$ satisfies these properties. Also, $\tau(\theta^{k}(X_{\theta}))$ must generate a $\sigma^{r^{k}}$-invariant partition.

We fix the maximal equicontinuous factor map that we consider henceforth. First suppose we consider a substitutional shift $(X_{\theta},\sigma)$. We write $\Lambda_{r^{n}}(x)=i$ if $x\in\sigma^{i}\theta^{n}(X_{\theta})$. Note that if $\Lambda_{n}(x)=i$, then $\Lambda_{n+1}(x)\equiv i\mod r^{n}$. Using this we can define $\pi(x):=\ldots x_{2}\,x_{1}\,x_{0}$ where for each $n\in\mathbb{N}$, $\Lambda_{n}(x)=\sum_{i=0}^{n-1}r^{i}x_{i}$.

If $\theta$ has trivial height, we have defined a maximal equicontinuous factor map. If $\theta$ has positive height $h$, then $X_{\theta}$ also has a $\sigma^{h}$-cyclic partition. Here we choose an arbitrary but fixed base, and use it to extend $\pi$ to a maximal equicontinuous factor map.

Finally if we consider an automatic shift $(Y,\sigma)$ generated by $\theta$, the convention we use is that we take the image of these partitions by $\tau$.

Now by Lemma [6, Lemma 11], $(X_{\theta},\sigma)$ has no continuous irrational eigenvalues, and so neither does $(Y,\sigma)$. Thus the maximal equicontinuous factor of $(Y,\sigma)$ is generated by the $\sigma^{n}$-minimal cyclic partitions [9]. On the one hand, the maximal equicontinuous factor of $(Y,\sigma)$ must be a factor of $(\mathbb{Z}_{r}\times\mathbb{Z}/h\mathbb{Z},+(1,1))$. On the other hand Lemma 4 tells us that each $(\mathbb{Z}/r^{n}\mathbb{Z},+1)$ is a factor of $(Y,\sigma)$. We have proved

Theorem 5**.**

If $(Y,\sigma)$ is an infinite $r$-automatic shift generated by $(\theta,\tau)$ where $\theta$ is primitive and of height $h$, then $(\mathbb{Z}_{r}\times\mathbb{Z}/\bar{h}\mathbb{Z},+(1,1))$ is the maximal equicontinuous factor of $(Y,\sigma)$ for some $\bar{h}$ dividing $h$. Furthermore, we can take the maximal equicontinuous factor maps $\pi_{X}$ and $\pi_{Y}$ to satisfy $\pi_{X}=\pi_{Y}\circ\tau$.

2.2. Automorphisms of constant length substitution shifts

We recall key ingredients that will be essential in our study of automorphism groups of automatic shifts.

Let $\operatorname{End}(X,\sigma)$ be the set of endomorphisms of $(X,\sigma)$. Given two shifts $(X,\sigma)$ and $(Y,\sigma)$, and let $\operatorname{Fac}(X,Y)$ denote the set of factor maps from $(X,\sigma)$ to $(Y,\sigma)$. The following is proved in [5, Theorem 3.3].

Theorem 6**.**

Let $(X,\sigma)$ and $(Y,\sigma)$ be infinite minimal shifts. Suppose that the group rotation $(G,R)$ is the maximal equicontinuous factor of both $(X,\sigma)$ and $(Y,\sigma)$ and let $\pi_{X}$ and $\pi_{Y}$ be the respective factor maps. Then there is a map $\kappa:\operatorname{Fac}(X,Y)\rightarrow G$ such that

[TABLE]

for all $x\in X$ and $\Phi\in\operatorname{Fac}(X,Y)$. Also

1. (1)

if $(Z,\sigma)$ is another shift which satisfies the assumptions on $(X,\sigma)$, then $\kappa(\Psi\circ\Phi)=\kappa(\Psi)+\kappa(\Phi)$ for $\Phi\in\operatorname{Fac}(X,Y)$, $\Psi\in\operatorname{Fac}(Y,Z)$, and 2. (2)

if $\min_{g\in G}|\pi_{X}^{-1}(g)|=\min_{g\in G}|\pi_{Y}^{-1}(g)|=c<\infty$, then $\kappa$ is at most $c$-to-one. In particular, if $\pi_{X}$ and $\pi_{Y}$ are somewhere one-to-one, then $\kappa$ is an injection.

A constant-length substitution is called injective if $\theta(i)\neq\theta(j)$ for any distinct $i$ and $j$ in $\mathcal{A}$. It will be convenient to deal with injective substitutions in what follows. The following theorem of Blanchard, Durand and Maass allows us to restrict our attention to that situation.

Theorem 7** ([10]).**

Let $\theta$ be a constant-length substitution such that $X_{\theta}$ is infinite. Then there exists an injective constant length substitution $\theta^{\prime}$ such that $(X_{\theta},\sigma)$ is topologically conjugate to $(X_{\theta^{\prime}},\sigma)$.

We say that $\theta$ *has column number $c$ * if

[TABLE]

In other words, for some $k\in\mathbb{N}$, and some $(i_{1},\ldots,i_{k})$, $|\theta_{i_{1}}\circ\ldots\circ\theta_{i_{k}}(\mathcal{A})|=c$ and $c$ is the least such number.

The following proposition, [5, Proposition 3.10] tells us that up to a shift, every automorphism has a small radius.

Proposition 8**.**

Let $\theta$ be an injective primitive, length $r$ substitution of height one, and such that $(X_{\theta},\sigma)$ is infinite. If $\Phi\in\operatorname{Aut}(X_{\theta},\sigma)$ has the property that $\kappa(\Phi)\in\mathbb{Z}_{r}\setminus\mathbb{Z}$ is periodic, then $\Phi$ has right radius zero and left radius at most 1.

Further, if $\kappa(\Phi)=k/(1-r^{p})$ for $0<k<r^{p}-1$, one has

[TABLE]

where $c$ is the column number.

If $\kappa(\Phi)=0$, then $\Phi$ has left and right radius at most 1 and $\Phi$ satisfies

[TABLE]

Example 9**.**

In this example we show that for automatic shifts, it is no longer the case that automorphisms with a zero $\kappa$-value have radius at most one, as in Proposition 8. Consider the substitution $\theta$

[TABLE]

Now $X_{\theta}$ has an automorphism $\Phi$ whose $\kappa$-value equals zero, given by the local rule $f$ defined by the permutation $(ab)(cd)$. Next we consider the coding $\tau:\{a,b,c,d\}\to\{x,y,z\}$ defined by

[TABLE]

and let $Y$ be the automatic shift defined by $(\theta,\tau)$. One can verify that $\tau$ is injective, so there exists an automorphism of $Y$ that corresponds to $\Phi$, whose local rule is given by $g$, partially given by

[TABLE]

However, the local rule is a bit more complicated when considering the image of $x$ under $g$, as we need to consider the preceeding and following $2$ elements, and the rule is given in Table 1. This table contains all appearing blocks of length $5$ such that the middle element is $x$. To be able to define this local rule, it is crucial that every appearence of such a block uniquely defines the preimage of the middle element.

The next result, [5, Proposition 3.21] gives us conditions to check whether a local rule defines an element of $\operatorname{Aut}(X_{\theta},\sigma)$. If $\Phi$ has local rule $f$, left radius $\ell$ and right radius $r$, we write $\Phi=\Phi^{f}_{\ell,r}$.

Proposition 10**.**

Let $\theta$ be an injective primitive, length $r$ substitution on $\mathcal{A}$, with column number $c$, of height one, and such that $(X_{\theta},\sigma)$ is infinite. Let $p\geq 1$ and $0<k<r^{p}-1$.

Let $f\colon\mathcal{A}^{2}\to\mathcal{A}$. Then $\Phi:=\Phi^{f}_{1,0}$ satisfies $\Phi\in\operatorname{Aut}(X_{\theta},\sigma)$ and $\kappa(\Phi)=k/(1-r^{p})$ if and only if:

1. (1)

if $x_{0}x_{1}x_{2}\in\mathcal{L}_{X_{\theta}}$, then $f(x_{0},x_{1})f(x_{1},x_{2})\in\mathcal{L}_{X_{\theta}}$; and 2. (2)

For each $x_{-1}x_{0}x_{1}\in\mathcal{L}(X_{\theta})$ and $0\leq i<r^{c!p}$

[TABLE]

*where $N=k(1+r^{p}+\ldots+r^{(c!-1)p})$ and $x_{-1}\cdot x_{0}$ denotes a finite segment of a bi-infinite sequence with $x_{-1}$ in the *$-1$st coordinate and $x_{0}$ in the [math]-th coordinate.

Similarly given $g\colon\mathcal{A}^{3}\to\mathcal{A}$, $\Phi^{g}_{1,1}$ satisfies $\Phi^{g}_{1,1}\in\operatorname{Aut}(X_{\theta},\sigma)$ and $\kappa(\Phi^{g}_{1,1})=0$ if and only if

1. (3)

$g(x_{0},x_{1},x_{2})g(x_{1},x_{2},x_{3})\in\mathcal{L}_{X_{\theta}}$* whenever $x_{0}x_{1}x_{2}x_{3}\in\mathcal{L}_{X_{\theta}}$; and* 2. (4)

For any $x_{-1}x_{0}x_{1}\in\mathcal{L}_{X_{\theta}}$ and $0\leq i<r^{c!}$,

[TABLE]

where $u=u_{-r^{c!}}u_{-r^{c!}+1}\ldots u_{-1}\cdot u_{0}\ldots u_{2r^{c!}-1}$ is the word obtained by applying $\theta^{c!}$ to the word $x_{-1}\cdot x_{0}x_{1}$.

Remark 11**.**

Proposition 8 implies that if $\Phi$ is a (topological) automorphism, then it is also a measure preserving (measurable) automorphism of $(X_{\theta},\sigma,\mu)$, where $\mu$ is the unique $\sigma$-invariant measure on $X_{\theta}$. Conversely, Host and Parreau [14] show that if $\Phi$ is a measure preserving (measurable) automorphism of $(X_{\theta},\sigma,\mu)$, with the additional conditions that $\theta$ has column number at least two, and that $\theta$ is reduced, then $\Phi$ is almost everywhere defined by a sliding block code of radius one, and that it satisfies a version of (2) or (3). Now Proposition 10 tells us that $\Phi$ is therefore (almost everywhere equal to) a topological automorphism. We note that reduced substitutions (defined in [14]) are always strongly injective, a notion we define and study in Section 4.

3. Topological factors of substitution shifts

In this section we study automorphism groups of factors of constant-length substitution dynamical systems. We show in particular that we can both produce new automorphisms, and lose automorphisms, when we pass to a factor. Moreover, we show that any shift factor is indeed topologically conjugate to a constant length substitution dynamical system, via a code, i.e. a letter-to letter map, i.e. it is an automatic shift, so that we can explicitly describe its automorphism group. In the last part we consider the special case when the factor map is uniform-to-$1$.

3.1. Automorphism groups and factors

Our motivation for the main results in Section 3 arose as we noticed that in general there is no relationship between the automorphism group of a substitution shift $(X_{\theta},\sigma)$ and that of one of its topological factors $(Y,\sigma)=(\tau(X_{\theta}),\sigma)$. We summarise this with the following two examples.

Example 12**.**

An example where ${\bf\operatorname{Aut}(X_{\theta},\sigma)\backslash\operatorname{Aut}(Y,\sigma)}$ is nonempty. Define $\theta$ as

[TABLE]

This substitution has column number 2 and the fixed points, namely $b.a$, $c.a$, $a.b$, $d.b$, comprise the only $\pi$-fibre with more than two elements, where $\pi:X_{\theta}\rightarrow\mathbb{Z}_{2}$ is the maximal equicontinuous factor map. Now $X_{\theta}$ has a nontrivial automorphism $\Phi\in\operatorname{Aut}(X_{\theta},\sigma)$ corresponding to the local rule given by the letter-exchanging map $(ab)(cd)$. Note that $\Phi$ has $\kappa$-value zero, and it acts by permuting elements within the fibres of $X_{\theta}$.

Now consider the projection $\tau:\{a,b,c,d\}\to\{x,y\}$ given by $\tau(a)=\tau(b)=x$, $\tau(c)=\tau(d)=y$. We find that $Y=X_{\bar{\theta}}$ where

[TABLE]

where $\theta$ has two fixed points $x.x$ and $y.x$. Since $\theta$ has a coincidence, i.e. $c(\theta)=1$, it does not have any nontrivial automorphisms with a zero $\kappa$-value. In other words, The map $\tau$ is everywhere two-to-one, and it collapses the fibres for $X_{\theta}$. Note that if $\Phi$ is sent to an automorphism via $\tau$, it must be an automorphism which also permutes within fibres. There is no non-trivial $\Phi^{\prime}\in\operatorname{Aut}(Y,\sigma)$ such that $\kappa(\Phi^{\prime})=0$ as $\bar{\theta}$ has a coincidence. In other words, here both the identity map and $\Phi$ are collapsed by $\tau$ to the identity.

It is appropriate to put the next example here, even though our claims use ideas from Section 4.

Example 13**.**

An example where ${\bf\operatorname{Aut}(Y,\sigma)\backslash\operatorname{Aut}(X_{\theta},\sigma)}$ is nonempty. We consider the following primitive substitution $\theta$

[TABLE]

Note that this substitution is strongly injective (Definition 30), and so if there exists $\Phi\in\operatorname{Aut}(X_{\theta})$ with $\kappa(\Phi)=0$ then by Theorem 37, it must have radius [math], i.e. it must exchange letters. Now the fixed points of $\theta$ are $a.a,c.a,a.b,c.b$ and so if there is a non-trivial automorphism it must send $a$ to $b$ and $a$ to $c$, a contradiction. Thus $\Phi=\operatorname{Id}$. However, now we consider the coding $\tau(a)=x$, $\tau(b)=\tau(c)=y$, and we find that $Y=X_{\bar{\theta}}$ where

[TABLE]

One finds easily that there exists a nontrivial $\Phi\in\operatorname{Aut}(Y,\sigma)$ with $\kappa$-value zero, namely $(xy)$.

We remark here that $\Phi\in\operatorname{Aut}(Y,\sigma)$ does define a measurable automorphism $F$ of $(X_{\theta},\sigma)$. In particular the only points where $F$ is not defined are on the orbits of the fixed points, and also the orbit of the points in $\pi^{-1}(\ldots 111)=\pi^{-1}(-\frac{1}{2})$, with $\pi:X_{\theta}\rightarrow\mathbb{Z}_{3}$ the maximal equicontinuous factor map. The map $F$ is not defined almost everywhere by a radius one sliding block code (or by any finite sliding block code, for that matter), which highlights that the result of Host and Parreau [14] is not true for not-reduced substitutions.

3.2. Automatic shifts as substitution shifts

In this section we prove that any shift factor of a primitive constant length substitution shift is topologically conjugate, via a radius zero code, to a constant length substitution shift. Note that by Lemma 3, we can assume that the factor is automatic to begin with.

Definition 14**.**

We call $a\neq b\in\mathcal{A}$ a periodic pair if there exists $p=p(a,b)$ and $j<r^{p}$ such that $\theta^{p}(a)_{j}=a$ and $\theta^{p}(b)_{j}=b$. Note that this is equivalent to the requirement that for all $k>0$ there exists $j<r^{kp}$ such that $\theta^{kp}(a)_{j}=a$ and $\theta^{kp}(b)_{j}=b$. We define

[TABLE]

and call a substitution $\theta$ with $p(\theta)=1$ pair-aperiodic.

Readers who are familiar with techniques used in substitutions will be aware that one often replaces a substitution $\theta$ by an appropriate power so that, without loss of generality, one can assume that all $\theta$-periodic points are $\theta$-fixed. In a similar spirit, we show in the following lemma that without loss of generality we can work with pair-aperiodic substitutions.

Lemma 15**.**

Let $\theta$ be a substitution of constant length. Then $\theta^{p(\theta)}$ is pair-aperiodic.

Proof.

A simple computation shows that if $a,b$ are a periodic pair for $\theta$ with period $p_{1}$, then they are also a periodic pair for $\theta^{n}$ with period $p_{2}=p_{1}/\gcd(p_{1},n)$. Furthermore, one finds directly that any periodic pair for $\theta^{n}$ is also a periodic pair for $\theta$. The result follows. ∎

Definition 16**.**

We call $a,b\in\mathcal{A}$ an asymptotic disjoint pair if for all $k>0$ there exists $j_{k}<r^{k}$ such that $\theta^{k}(a)_{j_{k}}\neq\theta^{k}(b)_{j_{k}}$.

Definition 17**.**

We call a pair $(c,d)\in\mathcal{A}^{2}$ $k$-reachable from $(a,b)\in\mathcal{A}^{2}$ if there exists $j<r^{k}$ such that $\theta^{k}(a)_{j}=c$ and $\theta^{k}(b)_{j}=d$.

It is straightforward to check that reachability is transitive, i.e. if $(b_{1},b_{2})$ is $k_{1}$-reachable from $(a_{1},a_{2})$ and $(c_{1},c_{2})$ is $k_{2}$-reachable from $(b_{1},b_{2})$, then $(c_{1},c_{2})$ is $(k_{1}+k_{2})$-reachable from $(a_{1},a_{2})$.

Lemma 18**.**

Let $\theta$ be a pair-aperiodic substitution of constant length and $(a,b)$ an asymptotic disjoint pair. Then there exists a periodic pair $(a^{\prime},b^{\prime})$ such that for any $k\geq 2|\mathcal{A}|^{2}$, and any $(c,d)$ that is reachable from $(a^{\prime},b^{\prime})$, it is also $k$-reachable from $(a,b)$.

Proof.

As $(a,b)$ is an asymptotic disjoint pair, there exists for any $k\in\mathbb{N}$ some $j_{k}$ such that $\theta^{k}(a)_{j_{k}}\neq\theta^{k}(b)_{j_{k}}$. We consider the finite sequence $(\theta^{k}(a)_{j_{k}},\theta^{k}(b)_{j_{k}})_{k=0,\ldots,|\mathcal{A}|^{2}}$. We note that we can choose the $j_{k}$ such that $j_{k}=\lfloor j_{k+1}/r\rfloor$. (For example one first chooses $j_{|\mathcal{A}|^{2}}$.)

As every element in this sequence belongs to $\mathcal{A}^{2}$ and the sequence has $|\mathcal{A}|^{2}+1$ elements, there exists $(a^{\prime},b^{\prime})$ and $0\leq k_{1}<k_{2}\leq|\mathcal{A}|^{2}$ such that $\theta^{k_{i}}(a)_{j_{k_{i}}}=a^{\prime}$ and $\theta^{k_{i}}(b)_{j_{k_{i}}}=b^{\prime}$ for $i=1,2$. We claim that $(a^{\prime},b^{\prime})$ are a periodic pair. For, we know that $j_{k_{2}}=j_{k_{1}}r^{k_{2}-k_{1}}+j^{\prime}$, which gives

[TABLE]

and the analogous statement for $b,b^{\prime}$, and the claim follows. Since $\theta$ is pair-aperiodic, $(a^{\prime},b^{\prime})$ is $1$-reachable from $(a^{\prime},b^{\prime})$. By the transitivity of reachability it follows that $(a^{\prime},b^{\prime})$ is reachable from $(a,b)$ by $k$ steps for any $k\geq|\mathcal{A}|^{2}$. Finally, if $(c,d)$ is reachable from $(a^{\prime},b^{\prime})$, then by the pigeonhole principle, it must be $k^{\prime}$- reachable for some $k^{\prime}\leq|\mathcal{A}|^{2}$. The result follows by transitivity. ∎

In this last proof, we used pair-aperioidicty strongly. We use it in a similar fashion in the next lemma.

Lemma 19**.**

Let $\theta$ be a primitive, pair-aperiodic substitution. If $(a,b)$ is not an asymptotic disjoint pair, then $\theta^{|\mathcal{A}|^{2}}(a)=\theta^{|\mathcal{A}|^{2}}(b)$.

Proof.

Since $(a,b)$ is not asymptotic disjoint, there exists a minimal $K$ such that $\theta^{K}(a)=\theta^{K}(b)$. Suppose that $K>|\mathcal{A}|^{2}$. Then for each $0\leq i\leq K-1$ there is a $j_{i}$ such that $\alpha_{i}:=(\theta^{i}(a))_{j_{i}}\neq\beta_{i}:=(\theta^{i}(a))_{j_{i}}$ and where if $i<\bar{i}$ then $(\alpha_{\bar{i}},\beta_{\bar{i}})$ is reachable from $(\alpha_{i},\beta_{i})$. Since $K-1\geq|\mathcal{A}|^{2}$, we have $(\alpha_{i},\beta_{i})=(\alpha_{\bar{i}},\beta_{\bar{i}})$ for some $i<\bar{i}$. Thus $(\alpha_{i},\beta_{i})$ is a periodic pair. But now since $\theta$ is pair aperiodic, for each $n$ $(\alpha_{i},\beta_{i})$ appears as $(\theta^{n}(a))_{k_{n}}=(\theta^{n}(b))_{k_{n}}$ for some $k_{n}$. This contradicts the fact that $\theta^{K}(a)=\theta^{K}(b)$. ∎

Definition 20**.**

Let $(\theta,\tau)$ be an $r$-automatic pair. We call two letters $a\neq b$ indistinguishable (by $(\theta,\tau)$ ) if for any $k\geq 0$ and $0\leq j<r^{k}$ we have $\tau(\theta^{k}(a)_{j})=\tau(\theta^{k}(b)_{j})$. We call the pair $(\theta,\tau)$ minimal if every pair of letters is distinguishable by $(\theta,\tau)$.

Note that the letters $a,b$ are indistinguishable if and only if for any $(c,d)$ that is reachable from $(a,b)$ (which includes $(a,b)$ by [math] steps) we have $\tau(c)=\tau(d)$.

In the next lemma, a fundamental result from automata theory enables us to replace an $r$-automatic pair by a minimal $r$-automatic pair.

Lemma 21**.**

Let $(Y,\sigma)$ be an infinite $r$-automatic shift generated by $(\theta^{\prime},\tau^{\prime})$ where $\theta^{\prime}$ is primitive and pair-aperiodic. Then there exists a minimal $r$-automatic pair $(\theta,\tau)$ which generates $(Y,\sigma)$, where $\theta$ is primitive and pair-aperiodic.

Proof.

We define an equivalence relation $a\sim b$ if $a,b$ are indistinguishable. This equivalence relation is obviously compatible with $\theta^{\prime}$, so we can define

[TABLE]

and it is also compatible with $\tau$, so we set

[TABLE]

We need to check that $\theta$ is primitive and pair-aperiodic. As $\theta^{\prime}$ is primitive, we know that there exists some $n$ such that for all $a,b\in\mathcal{A}^{\prime}$ we have that $(\theta^{\prime})^{n}(a)$ contains $b$ as a subword. Let us take $[c],[d]\in\mathcal{A}$. We know that $(\theta^{\prime})^{n}(c)$ contains $d$ as a subword and therefore $\theta^{n}([c])$ contains $[d]$ as a subword. In other words $\theta$ is primitive.

Let us now assume that $\theta^{k}([a])_{j}=[a]$ and $\theta^{k}([b])_{j}=[b]$, i.e. $([a],[b])$ is a periodic pair. This implies that $(\theta^{\prime})^{k}_{j}$ is a map from $[a]$ to itself and also from $[b]$ to itself. Thus a power must have a fixed-point:

[TABLE]

for some $a_{0}\in[a],b_{0}\in[b]$. Therefore we have

[TABLE]

which means that $(a_{0},b_{0})$ is a periodic pair for $\theta^{\prime}$. As $\theta^{\prime}$ is pair-aperiodic we know that there exists $j_{0}$ such that

[TABLE]

and, therefore,

[TABLE]

This shows that $\theta$ is pair-aperiodic.

Furthermore, by definition

[TABLE]

which shows that the language of $(Y\sigma)$ is the same as the language of $(\tau(X_{\theta}),\sigma)$. Since shifts are uniquely determined by their language, the result follows. ∎

We can now prove the main result of this section.

Theorem 22**.**

Let $(Y,\sigma)$ be an infinite $r$-automatic shift, generated by $(\theta^{\prime},\tau^{\prime})$ where $\theta^{\prime}$ is primitive. Then there exists an $r$-automatic pair $(\theta,\tau)$ that generates $(Y,\sigma)$ and such that $\tau:X_{\theta}\rightarrow Y$ is a bijection.

Proof.

By Lemmas 15 and 21, we can assume that $(Y,\sigma)$ is generated by the minimal $r$-automatic pair $(\theta,\tau)$ where $\theta$ is pair-aperiodic.

First we suppose that there exist $x\neq\tilde{x}\in X_{\theta}$ such that $\tau(x)=\tau(\tilde{x})$. As $X_{\theta}$ is recognizable [19] we have $x^{\prime},\tilde{x}^{\prime}$ such that $x=\sigma^{\ell}\theta^{2|\mathcal{A}|^{2}}(x^{\prime})$ and $\tilde{x}=\sigma^{\tilde{\ell}}\theta^{2|\mathcal{A}|^{2}}(\tilde{x}^{\prime})$. Now Theorem 5 tells us that $\ell=\tilde{\ell}$. Thus, we have $x^{\prime},\tilde{x}^{\prime}$ such that $\theta^{2|\mathcal{A}|^{2}}(x^{\prime})\neq\theta^{2|\mathcal{A}|^{2}}(\tilde{x}^{\prime})$ and $\tau(\theta^{2|\mathcal{A}|^{2}}(x^{\prime}))=\tau(\theta^{2|\mathcal{A}|^{2}}(\tilde{x}^{\prime}))$.

We consider the pair $(x^{\prime}_{n},\tilde{x}^{\prime}_{n})\in\mathcal{A}^{2}$. Suppose that for some $n\in\mathbb{Z}$, $(x^{\prime}_{n},\tilde{x}^{\prime}_{n})$ is an asymptotic disjoint pair. By Lemma 18 we know that there exists a periodic pair $(a^{\prime},b^{\prime})$ such that for any $(c,d)$ that is reachable from $(a^{\prime},b^{\prime})$, we have that $(c,d)$ is $(2\left|\mathcal{A}\right|^{2})$-reachable from $(a,b)$. As $\tau(\theta^{2|\mathcal{A}|^{2}}(x^{\prime}_{n}))=\tau(\theta^{2|\mathcal{A}|^{2}}(\tilde{x}^{\prime}_{n}))$, this gives that $\tau(c)=\tau(d)$ and therefore that $a^{\prime}$ and $b^{\prime}$ are indistinguishable. This contradicts our assumption that $(\theta,\tau)$ is minimal.

Thus $(x^{\prime}_{n},\tilde{x}^{\prime}_{n})$ is not an asymptotic disjoint pair for any $n\in\mathbb{Z}$. By Lemma 19 we know that $\theta^{2|\mathcal{A}|^{2}}(x^{\prime}_{n})=\theta^{2|\mathcal{A}|^{2}}(\tilde{x}^{\prime}_{n})$ for each $n\in\mathbb{Z}$, i.e. $x=\tilde{x}$, another contradiction. The result follows.

∎

The following example tells us that starting with a pair-aperiodic substitution before we minimise is necessary in Theorem 22.

Example 23**.**

We consider the substitution $\theta^{\prime}:\{a,b,c,d\}\to\{a,b,c,d\}^{3}$ and $\tau^{\prime}:\{a,b,c,d\}\to\{x,y,z\}$ where

[TABLE]

and

[TABLE]

We find that $(\theta^{\prime},\tau^{\prime})$ is minimal, as the only possible indistinguishable pair is $(a,c)$, but $\tau^{\prime}(\theta^{\prime}(a)_{1})=y\neq\tau^{\prime}(\theta^{\prime}(c)_{1})=z$.

However $\tau^{\prime}:X_{\theta^{\prime}}\to Y$ is not injective. For, let $x=\lim_{n\to\infty}\sigma^{(3^{2n}-1)/2}\theta^{\prime 2n}(a)$ and let $\tilde{x}=\lim_{n\to\infty}\sigma^{(3^{2n}-1)/2}\theta^{\prime 2n}(c)$. One can verify that $x_{n}=\tilde{x}_{n}$ for all $n\neq 0$ and $x_{0}=a,\tilde{x}_{0}=c$. Thus $\tau^{\prime}(x)=\tau^{\prime}(\tilde{x})$.

The “problem” here is that $(a,c)$ is a periodic pair of period $2$ and we need to actually consider $\theta^{\prime 2}$ instead of $\theta^{\prime}$.

[TABLE]

Here we see that $(a,c)$ is indeed indistinguishable, so that $(\theta^{\prime 2},\tau)$ is not minimal. In this case we minimise and we find the new substitution

[TABLE]

along with $\tau(a)=x,\tau(b)=y,\tau(d)=z$ gives us the correct representation.

3.3. Automorphism groups and uniform-to-one extensions.

In this section we investigate topological factors of minimal dynamical systems $(X,T)$ where $X$ is compact and metric, and where the factor map is uniform-to-one.

Let $(X,T)$ be a such a dynamical system. If there exists an automorphism $\Phi$ of $(X,T)$ with $\Phi^{k}=\operatorname{Id}$ for some minimal $k$, then we claim that we can define an everywhere $k$-to-one factor of $(X,T)$. First note that if $k$ is minimal, then for no $\ell<k$ do we have $\Phi^{\ell}(x)=x$ for some $x\in X$. For, if that were the case, then $\Phi^{\ell}$ equals the identity on a dense subset of $X$. By minimality, $\Phi^{\ell}=\operatorname{Id}$ and this contradicts minimality of $k$. Now we can identify points $\{x,\Phi(x),\ldots,\Phi^{k-1}(x)\}$ that are in the same orbit and this is a well defined $T$-commuting map on $X$.

This assertion extends to the measurable setting. Namely, if there exists a measurbale ergodic bijection $\Phi$ of $(X,T,\mu)$, then we can use ergodicity to assert that there is a minimal $n$ such that for $\mu$-almost all $x$, $\Phi^{n}(x)=x$ and $n$ is minimal for that $x$. Now we can proceed with that $n$ as above.

In particular, if $\theta$ has column number $c$ and $\Phi^{c}=\operatorname{Id}$ with $c$ minimal, then the factor defined above is an almost everywhere one-to-one extension of its maximal equicontinuous factor, and this is a convenient topological factor for a dynamical system to have, see for example [1], [17], or [22] for uses. It is not always the case that such a factor exists [12]. It is natural to investigate how closely related automorphisms and everywhere $k$-to-$1$ factors are. The following result shows that for the case $k=2$ we have a $1$-to-$1$ correspondence. We are grateful to T. Downarowicz and M. Lemanczyk who pointed this general result out to us via private communication [7].

Theorem 24**.**

Let $(X,T)$ and $(Y,S)$ be minimal topological dynamical systems on compact metric spaces $X$ and $Y$. If $\pi:(X,T)\rightarrow(Y,S)$ is a uniformly 2-to-1 topological factor map, then there exists $\Phi\in\operatorname{Aut}(X,T)$ such that $\Phi^{2}=\operatorname{Id}$.

Proof.

Several times in this proof, compactness allows us to drop to a subsequence; we will do this without re-indexing.

For $y\in Y$, write $\{x,\tilde{x}\}=\pi^{-1}(y)$; define $\Phi(x)=\tilde{x}$ and $\Phi(\tilde{x})=x$. It is clear that $\Phi$ is bijective, and the fact that $\pi\circ T=S\circ\pi$ implies that $\Phi\circ T=T\circ\Phi$. It remains to show that $\Phi$ is continuous.

Define $D:Y\rightarrow[0,\infty)$ by $D(y)=d(x,\tilde{x})$. If $y_{n}\rightarrow y$ then it can be verified that $D(y)\geq\limsup D(y_{n})$, i.e. $D$ is upper semi-continuous. Suppose that $D$ is bounded away from 0. Then $\Phi$ is continuous. Let $x_{n}\to x$ and $\{x_{n},\tilde{x}_{n}\}=\pi^{-1}(y_{n})$. By compactness $\tilde{x}_{n}\to\tilde{x}$ along one subsequence and let us assume that $\tilde{x}_{n}\to x^{\prime}$ along another subsequence, where $\tilde{x}\neq x^{\prime}$. Since $D$ is bounded away from 0, we also have $x^{\prime}\neq x$. But since $\pi$ is continuous, we have $\pi(x)=\pi(\tilde{x})=\pi(x^{\prime})=y$, a contradiction.

It remains to show that $D$ is bounded away from 0. Suppose that $D(y_{n})\rightarrow 0$. The continuity of $T$ implies that for each $j$, $D(S^{j}y_{n}))\rightarrow 0$ as $n\rightarrow\infty$. If $y_{n}\rightarrow y$, the minimality of $S$ implies that in any open subset of $Y$ we can find points whose $D$ value is arbitrarily small. By upper semi-continuity, $\{y:D(y)<r\}$ is open and by the above it is also dense. Hence $D=0$ on a dense $G_{\delta}$ set, which is a contradiction.

∎

Unfortunately Theorem 24 is no longer true for a uniform $k$-to-$1$ factor map, where $k>2$.

Example 25**.**

Let $\theta(a)=abb$, $\theta(b)=bac$, $\theta(c)=cca$, and let $\eta(x)=yxz$, $\eta(y)=yxx$, $\eta(z)=yxy$. Then $\eta$ has column number $1$, and $(X_{\theta},\sigma)$ is an almost everywhere three-to-one extension of $(\mathbb{Z}_{3},+1)$. One verifies that the left radius one, right radius zero rule in Table 2 defines an exactly three-to-one shift commuting map $\pi:X_{\theta}\rightarrow X_{\eta}$. However $(X_{\theta},\sigma)$ has a trivial automorphism group.

4. Strongly injective substitutions and language automorphisms

Definition 26**.**

Let $\mathcal{L}\subset\mathcal{A}^{*}$ be a language. We say that the bijection $\Phi:\mathcal{L}\rightarrow\mathcal{L}$ is an $\mathcal{L}$-automorphism if $\Phi(w_{1}w_{2})=\Phi(w_{1})\Phi(w_{2})$ whenever $w_{1}$, $w_{2}$ and $w_{1}w_{2}$ belong to $\mathcal{L}$.

We have the following proposition which links language automorphisms to automorphisms of the corresponding dynamical system. Recall that any shift space $X$ is defined by its language, which is closed under the taking of subwords, and where every word is left and right extendable to a word in $\mathcal{L}$. Conversely, any language $\mathcal{L}$ satisfying these two properties defines a shift $X_{\mathcal{L}}$.

Proposition 27**.**

Let $\mathcal{L}$ be the language of a shift $(X,\sigma)$. Then $\Phi$ is an $\mathcal{L}$-automorphism if and only if $\Phi\in\operatorname{Aut}(X,\sigma)$ has zero radius.

Proof.

Suppose that $\Phi$ is an $\mathcal{L}$-automorphism. Since $\Phi$ is bijective, it must map letters to letters, i.e. $\Phi|_{\mathcal{A}}$ is a bijection $\phi:\mathcal{A}\rightarrow\mathcal{A}$. Since it is a language morphism, $\phi$ defines $\Phi$. We can extend $\Phi$, via concatenation, to a map on bi-infinite sequences $u$ in $X$, namely $\Phi((u_{n})_{n\in\mathbb{Z}}=(\phi(u_{n}))_{n\in\mathbb{Z}}$. Let $u\in X$ and let $v:=\Phi(u)$. Since $\Phi$ maps $\mathcal{L}$ to $\mathcal{L}$, $v\in X$ and so the image of $\Phi(X)$ is contained in $X$. Since $\Phi$ is defined by a local rule it is shift-commuting, and it is injective on $X$ since it is injective on $\mathcal{L}$. By minimality, $\Phi$ is surjective.

The converse is straightforward.∎

Remark 28**.**

In the substitutional case, we note that if $\Phi$ has radius zero then $\kappa(\Phi)=0$. For if $\Phi$ has radius zero, then for some $n$, $\Phi^{n}=\operatorname{Id}$. But then $n\kappa(\Phi)=0$, and since $\mathbb{Z}_{r}$ is torsion free, our assertion follows.

In an ideal world any automorphism $\Phi$ with $\kappa(\Phi)=0$ would have zero radius. Unfortunately this is not the case.

Example 29**.**

Let $\theta(a)=ab$, $\theta(b)=ca$ and $\theta(c)=ba$ and define the local rule $\phi$ with left radius one, as $\phi(b)=a$, $\phi(c)=a$, $\phi(aa)=c$, $\phi(ba)=b$ and $\phi(ca)=b$. Then $\phi$ is the local rule of an involution in $\operatorname{Aut}(X_{\theta},\sigma)$.

Definition 30**.**

We say that $\theta$ is strongly right- (left-) injective if $\theta$ is injective and does not have any right- (left-) infinite fixed points which differ only in their 0-th entry, and we say that $\theta$ is strongly injective if it is both strongly right- and left-injective.

Definition 31**.**

Let $\theta:\mathcal{A}\rightarrow\mathcal{A}^{r}$ be a length $r$ substitution with column number $c$. A minimal set (for $\theta$) is a set of letters $\{\alpha_{1},\ldots,\alpha_{c}\}\subset\mathcal{A}$ of cardinality $c$ such that this set appears as $(\theta^{n}(\mathcal{A}))_{i}$ for some $n\in\mathbb{N}$ and some $0\leq i<r^{n}$. Let $\mathcal{X}$ denote the collection of all minimal sets:

[TABLE]

Theorem 32**.**

Let $\theta$ be a primitive length $r$ substitution such that $X_{\theta}$ is infinite. Then $(X_{\theta},\sigma)$ is topologically conjugate to $(X_{\eta},\sigma)$ where $\eta$ is a strongly injective length $r$ substitution.

Proof.

Note that if a substitution $\theta$ has column number $c$, then it has at least $c$ right-infinite and $c$ left-infinite fixed points. We claim that if $\theta$ has exactly $c$ right- (left-) infinite fixed points, then it must be right- (left-) strongly injective. For, in this case there is a minimal set $\mathcal{A}_{r}$ ($\mathcal{A}_{l}$) of $c$ letters that generates the $c$ right- (left-) infinite fixed points. The fact that $\theta$ has column number $c$ implies that these fixed points are pairwise distinct at each index $n\geq 0$, and our claim follows.

Now suppose that $\theta$ is not strongly injective. By replacing $\theta$ by a power if necessary, we can assume that there are three consecutive indices $k-1,k,k+1$, with $k\in\{1,\ldots r-1\}$, where $\theta$’s column number is achieved jointly for these indices, i.e. $|\{\theta(a)[k-1,k+1]:a\in\mathcal{A}\}|=c$. Now let $\bar{\eta}:=\theta^{(2,k)}$ be the $k$-shifted 2-sliding block representation of $\theta$. By construction, $\bar{\eta}$ has exactly $c$ right-infinite fixed points and $c$ left-infinite fixed points. If $\bar{\eta}$ is injective on letters, then by our previous claim, it is strongly injective and in this case let $\eta=\bar{\eta}$. If $\bar{\eta}$ is not injective on letters, define an equivalence relation on its alphabet where $\alpha\sim\beta$ if $\bar{\eta}(\alpha)=\bar{\eta}(\beta)$. Let $\mathcal{B}$ be the set of equivalence classes for $\sim$ and define $\eta([\beta]):=\bar{\eta}(\beta)$. Then $(X_{\eta},\sigma)$ is topologically conjugate to $(X_{\bar{\eta}},\sigma)$ via the map $\Phi((x_{n})_{n\in\mathbb{Z}})=([x_{n}])_{n\in\mathbb{Z}}$. (See for example the proof of Theorem 7 in [10].) By construction $\eta$ is injective on letters. Since $\kappa(\phi)=0$, the fixed points of $\bar{\eta}$ are in one-to-one correspondence with those of $\eta$. The result follows by our claim.

∎

Theorem 33**.**

Let $\theta$ be a strongly injective primitive length $r$ substitution such that $X_{\theta}$ is infinite. If $\kappa(\Phi)=0$ then $\Phi$ has radius 0 and so in particular it is a language automorphism.

Proof.

We suppose that $\Phi$ is nontrivial; then it must permute fixed points, and it cannot fix either the right ray or the left ray of a fixed point. We assume that $\Phi$ sends the fixed point $u$, whose right-infinite ray is $u_{[0,\infty)}=\lim_{k}\theta^{k}(a)$, to the fixed point $v\neq u$, whose right-infinite ray is $v_{[0,\infty]}=\lim_{k}\theta^{k}(b)$.

If $\Phi$ does not have radius 0, then there exist letters $c,d$ and $e$, with $d\neq e$, and indices $i\neq j$ such that $u_{i}=c=u_{j}$, $v_{i}=d$ and $v_{j}=e$. This implies that for all $n$, $\theta^{n}(c)$ appears starting at the indices $r^{n}i$ and $r^{n}j$ in $u$, $\theta^{n}(d)$ appears starting at index $r^{n}i$ in $v$, and $\theta^{n}(e)$ appears starting at index $r^{n}j$ in $v$. Suppose that the left radius of $\Phi$ is $\ell>0$. Then for all large $n$, $\theta^{n}(d)_{[\ell,r^{n}-1]}=\theta^{n}(e)_{[\ell,r^{n}-1]}$ and thus we have two fixed points $u^{\prime}=\ldots\cdot d^{\prime}D^{\prime}x$ and $v^{\prime}=\ldots\cdot e^{\prime}E^{\prime}x$ which disagree on their initial entries: $d^{\prime}\neq e^{\prime}$, where the length of $D^{\prime}$ equals that of $E^{\prime}$, and where the suffix of $D^{\prime}$ does not equal that of $E^{\prime}$, and finally where $u^{\prime}$ and $v^{\prime}$ agree on their right rays $x$ starting at index at most $\ell$. If $D^{\prime}\neq\emptyset$, so that also $E^{\prime}\neq\emptyset$, then the image of their suffixes under $\theta$ must agree, and this contradicts our assumption that $\theta$ is injective. (Here we are using the fact that $r$ must be at least two, of course.) Thus $D^{\prime}=\emptyset=E^{\prime}$ and thus we have two fixed points $u^{\prime}=\ldots\cdot d^{\prime}x$ and $v^{\prime}=\ldots\cdot e^{\prime}x$ which disagree on their initial entry: $d^{\prime}\neq e^{\prime}$, and which agree on their right rays starting at index 1. This contradicts our assumption that $\theta$ is strongly right-injective. If $\Phi$ has right radius $r>0$, we would obtain a contradiction to our assumption that $\theta$ is strongly left-injective. ∎

Our next goal is to characterize the automorphisms of $(X_{\theta},\sigma)$ which have radius [math]. This is particularly interesting as, by Theorem 32 we can assume without loss of generality that $\theta$ is a strongly injective substitution. Furthermore, by Theorem 33 we know that in this situation any $\Phi$ with $\kappa(\Phi)=0$ has radius [math]. In particular the local rule is just a map from $\mathcal{A}$ to itself.

The characterisation that we will give is inspired by the special case of bijective substitutions, and it is instructive to recall this special case first. We note that bijective substitutions are strongly injective, so any element of $\ker\kappa$ is a letter exchanging map. Lemanczyk and Mentzen [16, Theorem 5] gave a closed form for the automorphism group of bijective substitutions in the measurable setting. In particular, they consider the group $G$ of permutations generated by the bijections $\{\theta_{i}:1\leq i\leq r\}$ of $\mathcal{A}$, and they show that a permutation $\tau$ defines a shift commuting measurable bijection of $(X_{\theta},\sigma)$ if and only if it belongs to the centralizer of $G$ in the symmetric group $S_{|\mathcal{A}|}$. This statement is still true if we pass to the topological setting. For, first the fact that $\ldots 00$ is the only singular fibre of the equicontinuous factor map tells us that the automorphisms of $X_{\theta}$ must have $\kappa$-value zero. Our claim can now be seen using Proposition 10, which is effectively telling us that $\tau$ must commute with $G$. In fact we only need a simpler version of Proposition 10, namely one that takes into account the fact that $\tau$ must have radius zero.

We recall the definition of the column number in (1), which denotes the minimal number of elements of $\mathcal{A}$ one can see in one column when writing the words $\theta^{k}(a)$ for $a\in\mathcal{A}$ underneath each other. Recall that we use $\mathcal{X}\subseteq\mathcal{P}(\mathcal{A}^{c(\theta)})$ to denote the collection of all possible minimal sets for $\theta$. The minimality of $M\in\mathcal{X}$ implies some useful properties: for any $a,b\in M$ we have that $\theta^{k}(a)_{j}\neq\theta^{k}(b)_{j}$ for all $k\geq 1$ and $j<r^{k}$. Otherwise, we would have a contradiction to the minimality of the column number $c$. This implies that $\theta$ is well-defined on $\mathcal{X}$ and we denote it by $\tilde{\theta}$. It follows quite easily that $\tilde{\theta}$ is primitive (see [17, Proposition 6.3]).

Lemma 34**.**

$\mathcal{X}$* covers $\mathcal{A}$. In other words, every letter of $\mathcal{A}$ belongs to a set in $\mathcal{X}$.*

Proof.

This is Equation (37) in [17], but we include the verification here for completeness. We need to show that for any $b\in\mathcal{A}$ there exists $M^{\prime}\in\mathcal{X}$ with $b\in M^{\prime}$. Obviously $\mathcal{X}\neq\emptyset$. Thus, there exists some $M$ and $a$ such that $a\in M\in\mathcal{X}$. As $\theta$ is primitive, there exist $k,j<r^{k}$ with $\theta^{k}(a)_{j}=b$. Now obviously $b\in\tilde{\theta}^{k}(M)_{j}=:M^{\prime}$. ∎

This allows us to find our first restrictions for automorphisms with radius [math].

Lemma 35**.**

Suppose that $\tau$ is an automorphism of $(X_{\theta},\sigma)$ with radius [math]. Then, for any $M\in\mathcal{X}$ we have that $\tau|_{M}$ is a bijection.

Proof.

We know by a modified version of Proposition 10 (see for example [2, Lemma 2]) that

[TABLE]

Thus by changing to $\theta^{c!}$ we can assume without loss of generality that

[TABLE]

By the surjectivity of $\tau$ it follows directly that $\tau$ is a bijection on $\mathcal{A}$. We know that for any $M\in\mathcal{X}$ there exists $k\in\mathbb{N},j<r^{k}$ with $\theta^{k}(\mathcal{A})_{j}=M$. Thus, for any $a\in M$ there exists $b\in\mathcal{A}$ such that $\theta^{k}(b)_{j}=a$. This gives by (4)

[TABLE]

So $\tau$ maps $M$ to itself and as $\tau$ is injective it is bijective on $M$. ∎

So we know that $\tau$ is a bijection on every minimal set and by (4) we would expect these bijections to be compatible. Next we aim to relate elements of minimal sets to each other. Therefore we fix some $M_{0}\in\mathcal{X}$ and $k_{0}$ with

[TABLE]

Furthermore, we can assume without loss of generality that $\theta^{k_{0}}(a)_{j_{0}}=a$ for every $a\in M_{0}$. This is due to the fact that $\theta^{k_{0}}_{j_{0}}$ defines a bijection on $M_{0}$ and taking a power yields the result. Now we fix an arbitrary bijection $f_{0}:M_{0}\to\{1,\ldots,c\}$ which allows us to define $f:\mathcal{A}\to\{1,\ldots,c\}$ by

[TABLE]

In particular, we have $f|_{M_{0}}=f_{0}$. Note also that the minimality of $c$ implies that for any $M\in\mathcal{X}$ we have that $f|_{M}$ is a bijection between $M$ and $\{1,\ldots,c\}$.

Lemma 36**.**

There exists a bijection $\tau^{\prime}$ on $\{1,\ldots,c\}$ such that $f\circ\tau=\tau^{\prime}\circ f$.

Proof.

Let us take $a_{1}\neq a_{2}$ such that $f(a_{1})=f(a_{2})$. We need to show that $f(\tau(a_{1}))=f(\tau(a_{2}))$. By the definition of $f$ we have

[TABLE]

By (4) this gives

[TABLE]

∎

However, there is still one more condition that $\tau$ has to satisfy. For this remaining condition we use a generalisation of Lemanczyk and Mentzen’s idea for bijective substitutions to assign to a general $\theta$ a group of permutations called $G$. The following was presented by the first author and Lemanczyk in [17] and we follow their notation. Let us fix an arbitrary $M\in\mathcal{X}$. Then $\theta_{j}$ is a bijection from $M$ to $\tilde{\theta}(M)_{j}$. Thus it corresponds via $f$ to a permutation of $\{1,\ldots,c\}$ which we denote by $\sigma_{M,j}$. More precisely we have $\sigma_{M,j}(m)=n$ if there exist $a\in M,b\in\tilde{\theta}(M)_{j}$ with $f(a)=n,f(b)=m$ such that $\theta(a)_{j}=b$. Another way to define it is via the relation

[TABLE]

Furthermore, we denote by $G$ the group generated by all the $\sigma_{M,j}$:

[TABLE]

We have gathered now all the information needed for the following Theorem. In what follows the function $f$ is that defined in Equation (6).

Theorem 37**.**

Let $\theta$ be a primitive and injective length $r$ substitution of height $1$ such that $X_{\theta}$ is infinite. Then $\tau$ is an automorphism with radius [math] if and only if

1. (i)

$\tau$* is a bijection on every minimal set $M\in\mathcal{X}(\theta)$.* 2. (ii)

$\tau$* is well-defined on $\{1,\ldots,c\}$ via $f$, which we denote by $\tau^{\prime}$, i.e. $f\circ\tau=\tau^{\prime}\circ f$.* 3. (iii)

$\tau^{\prime}$* belongs to the centralizer of $G(\theta)$.*

Proof.

First we still need to show that any automorphism $\tau$ with radius [math] fulfills (iii): Obviously it is sufficient to show that $\tau^{\prime}$ commutes with every $\sigma^{-1}_{M,j}$. We find for any $a\in M$,

[TABLE]

Here the first and fourth equation are due to (7), the second and fifth equation follow from (ii) and the third equation from (4).

It remains to show that any $\tau$ that satisfies (i) - (iii) indeed gives an automorphism. It is sufficient to check (4) as this implies $\tau(\theta^{k}(a)[j-1,j+1])=\theta^{k}(\tau(a))[j-1,j+1]\in\mathcal{L}_{\theta}$. Let $a\in\mathcal{A}$ be an arbitrary element and $j<r$. By Lemma 34 we know that $a\in M^{\prime}$ for some $M^{\prime}\in\mathcal{X}$. By (i) we have that $\tau(a)\in M^{\prime}$. It follows by the definition of $\tilde{\theta}$ that $\theta(a)_{j},\theta(\tau(a))_{j}\in\tilde{\theta}(M^{\prime})_{j}=:M$. This gives

[TABLE]

Here the second and fifth equation hold by (7), the third and sixth equation by (ii) and the fourth equation by (iii). ∎

Remark 38**.**

In the case of bijective substitution we have that for any element in the centralizer of $G$ there exists an automorphism. This is no longer the case in the general situation as the following example shows.

Example 39**.**

We consider the strongly injective substitution

[TABLE]

We find directly that $\mathcal{X}=\{\{a,b\},\{a,c\}\}$. We have $\theta(a)_{0}=a,\theta(b)_{0}=b,\theta(c)_{0}=b$. Thus we define $f(a)=1,f(b)=f(c)=2$. This gives

[TABLE]

Therefore, we have $G(\theta)=\{id,(12)\}$ and the centralizer of $G$ is $G$ itself. However, as $a\in\{a,b\}$ and $a\in\{a,c\}$ we know by (i) that $\tau(a)=a$ and therefore $\tau(b)=b,\tau(c)=c$. This gives an automorphism with $\tau=id$ which corresponds to $\tau^{\prime}=id$, but we don’t have any automorphism corresponding to $\tau^{\prime}=(12)$.

Because of the situation in Example 39, Theorem 37 is not entirely satisfactory. We recall that Lemanczyk and Mentzen were working in a measurable context, and this raises the question of whether we can hope for a better version of Theorem 37 if we move to the measurable setting. It turns out that this is indeed the case, as we shall next show in Theorem 44. To do this we recall the notion of a reduced substitution, due to Host and Parreau [14].

We denote by $d^{*}:(\mathcal{A}^{2})^{*}\to\mathbb{R}$,

[TABLE]

Definition 40**.**

We call a primitive length $r$ substitution reduced if for all $a,b\in\mathcal{A}$ we have

[TABLE]

One can reduce a substitution by identifying all $a,b\in\mathcal{A}$ for which

[TABLE]

When reducing a substitution, the related dynamical system does not change measure-theoretically.

Lemma 41**.**

Let $\theta$ be a primitive length $r$ substitution. Then we have for any $a,b\in\mathcal{A}$

[TABLE]

if and only if for each $k$, we have

[TABLE]

Proof.

Let $a\in M\in\mathcal{X}(\theta)$. Then $\theta^{k}(a)_{j}$ is uniquely determined by $f(\theta^{k}(a)_{j})$ and $\tilde{\theta}^{k}(M)_{j}$. Furthermore, as $c(\tilde{\theta})=1$ we know that

[TABLE]

for all $M_{1},M_{2}\in\mathcal{X}(\theta)$. This shows that (9) implies (8).

Let us now assume that there exists $k^{\prime}$ and $j^{\prime}<r^{k^{\prime}}$ such that

[TABLE]

Recall the definition of $M_{0}$, $k_{0}$, $j_{0}$ and $f$ via $(\ref{definition-of-M_0})$ and $(\ref{definition-of-f})$. We know that

[TABLE]

As $f_{0}$ is a bijection between $\{1,\ldots,c\}$ and $M_{0}$ we know that

[TABLE]

and also $a^{\prime},b^{\prime}\in M_{0}$. We know by the definition of minimal sets that for all $k$ and $j<r^{k}$

[TABLE]

Thus we find that

[TABLE]

∎

Corollary 42**.**

Let $\theta$ be a reduced and primitive substitution of constant length $r$. Then every $\tau^{\prime}$ in the centraliser of $G(\theta)$ defines an automorphism $\tau\in\operatorname{Aut}(X_{\theta},\sigma).$

Proof.

As $\theta$ is reduced this implies that $\theta$ is strongly injective, so we know that any automorphism with $\kappa$-value [math] has radius [math]. We need to define $\tau:\mathcal{A}\to\mathcal{A}$ that satisfies the conditions of Theorem 37. Fix any $f$ defined as in (6). Let $a\in M\in\mathcal{X}$. Then we define

[TABLE]

and we need to make sure that this definition is independent of the choice of $M$:

Let $a\in M_{1}\cap M_{2}$. Then we set

[TABLE]

So we have $b_{i}\in M_{i}$ and

[TABLE]

We compute

[TABLE]

for $i=1,2$. Thus we know that $f(\theta^{k}(b_{1})_{j})=f(\theta^{k}(b_{2})_{j})$ for all $k,j<r^{k}$. By Lemma 41 we have

[TABLE]

As $\theta$ is reduced we conclude that $b_{1}=b_{2}$, so that (10) is well defined and one checks easily that $\tau$ fulfils the conditions of Theorem 37. ∎

Lemma 43**.**

Let $\theta$ be a primitive length $r$ substitution and let $\theta^{\prime}$ be the reduced substitution of $\theta$. Then $G(\theta)=G(\theta^{\prime})$.

Proof.

Let $\mathcal{A}^{\prime}$ be the alphabet on which $\theta^{\prime}$ is defined. By Lemma 41 we can define $f$ on $\mathcal{A}^{\prime}$ as in particular it says that if $a\sim b$ then $f(a)=f(b)$, so we can simply define $f([a]):=f(a)$. This shows that $|\theta^{\prime k}([M])_{j}|=c(\theta)$ for all $k$ and $j<r^{k}$ which means that $c(\theta^{\prime})\geq c(\theta)$. As $\theta^{\prime}$ is constructed by identifying letters of $\mathcal{A}$ we have $c(\theta)\geq c(\theta^{\prime})$ and so $c(\theta)=c(\theta^{\prime})$. It suffices to show that

[TABLE]

First we find directly that for all $M\in\mathcal{X}(\theta)$ we have $[M]\in\mathcal{X}(\theta^{\prime})$ and as $f([a])=f(a)$ we know that

[TABLE]

Furthermore, for any $M^{\prime}\in\mathcal{X}(\theta^{\prime})$ there exist $k^{\prime}$ and $j^{\prime}<r^{k^{\prime}}$ such that $\theta^{\prime k^{\prime}}(\mathcal{A}^{\prime})_{j^{\prime}}=M^{\prime}$. We recall that as in (5), we have $M_{0}=\theta^{k_{0}}(\mathcal{A})_{j_{0}}$ and, therefore, we find

[TABLE]

Thus, we can write $M^{\prime}=[M]$ for some $M\in\mathcal{X}(\theta)$, which shows that

[TABLE]

and in total (11), which finishes the proof. ∎

Recall that if $\theta$ is primitive, then there is a unique probability measure $\mu$ such that $(X_{\theta},\sigma,\mu)$ is a measure preserving dynamical system. The essential centraliser of $(X_{\theta},\sigma,\mu)$ is the set of all measure-preserving maps of $X_{\theta}$ which commute with $\sigma$, quotiented out by $\{\sigma^{n}:n\in\mathbb{Z}\}$. Combining Corollary 42 and Lemma 43, we deduce the following.

Theorem 44**.**

Let $\theta$ be a primitive length $r$ substitution. Then $G(\theta)$ corresponds to those elements of the essential centraliser of $(X_{\theta},\sigma,\mu)$ that are a root of the identity.

5. Rational orders in an automorphism group, twisted substitutions and $k$-compressions.

Theorem 32 tells us that we can assume that our substitution is strongly injective, and henceforth we do this. Theorem 33 tells us that all automorphisms whose $\kappa$-value is zero arise from alphabet permutations. In this section we investigate the structure of substitutions $\theta$ which possess other automorphisms, namely, automorphisms with a non-integer $\kappa$-value. We show in Theorem 46 that these automorphisms indicate that $\theta$ has a structure coming from some other constant length substitution.

Given a sequence $(x_{n})_{n\in\mathbb{Z}}\in\mathcal{A}^{\mathbb{Z}}$, we can consider what Cobham [3] calls its $k$-compression, which is the sequence $(y_{n})_{n\in\mathbb{Z}}\in({\mathcal{A}^{k}})^{\mathbb{Z}}$ where $y_{n}:=x_{nk}\ldots x_{(n+1)k-1}$. The operation of $k$-compression also applies to words whose length is a multiple of $k$. If $\theta:\mathcal{A}\rightarrow\mathcal{A}^{r}$, and $\alpha_{1}\ldots\alpha_{k}\in\mathcal{A}^{k}$, then the $k$-compression of the word $\theta(\alpha_{1}\ldots\alpha_{k})$ is a word in $\mathcal{A}^{k}$ of length $r$; this defines a substitution $\theta^{(k)}:\mathcal{B}\rightarrow\mathcal{B}^{r}$, where $\mathcal{B}\subset\mathcal{A}^{k}$ is the set of words $\mathcal{B}=\{u[mk,(m+1)k-1]:m\geq 0\}$ where $(u_{n})_{n\in\mathbb{N}}$ is a fixed point of $\theta$. Cobham studied the notion of $k$-compression and showed that if $x$ is $r$-automatic and primitive, then for any $k$, its $k$-compression is also $r$-automatic and primitive, generated by the substitution $\theta^{(k)}$. We call $\theta^{(k)}$ (resp. $(X_{\theta^{(k)}},\sigma)$) the $k$*-compression of $\theta$ * (resp. $(X_{\theta},\sigma)$). It is straightforward to see that $(X_{\theta},\sigma^{k})$ is topologically conjugate to $(X_{\theta^{(k)}},\sigma)$.

Definition 45**.**

Let $\theta:\mathcal{A}\rightarrow\mathcal{A}^{r}$ be a substitution and let $\tau\in\operatorname{Aut}(X_{\theta},\sigma)$ with radius [math] satisfying $\kappa(\tau)=0$ and $\tau^{r}=\tau$. Define the substitution $\theta_{\tau}:\mathcal{A}\rightarrow\mathcal{A}^{r}$ by

[TABLE]

We call $(X_{\theta_{\tau}},\sigma)$ the $\tau$-twist of $(X_{\theta},\sigma)$.

Recall $\kappa_{\theta}:\operatorname{Aut}(X_{\theta},\sigma)\rightarrow\mathbb{Z}_{r}$, the group homomorphism defined in Theorem 6. By [5, Theorem 3.16 and Lemma 3.2], $\kappa(\operatorname{Aut}(X_{\theta},\sigma))$ is always cyclic and generated by $1/k$ for some $k$.

Theorem 46**.**

Let $\theta$ be a strongly injective primitive length $r$ substitution of height one such that $X_{\theta}$ is infinite. If $\kappa_{\theta}(\operatorname{Aut}(X_{\theta},\sigma))=\langle\frac{1}{k}\rangle$ then $(X_{\theta},\sigma)$ is topologically conjugate to a twist of a $k$-compression of $(X_{\eta},\sigma)$, with $\eta$ a primitive length $r$ substitution.

It is often the case that $\tau=\operatorname{Id}$, and in this case one can conclude that $(X_{\theta},\sigma)$ is topologically conjugate to a $k$-compression. However, situations where $\tau$ is nontrivial do occur; see Example 50.

To prove Theorem 46 we prove some preliminary lemmas.

Lemma 47**.**

Let $\theta$ be a primitive length $r$ substitution of height one such that $X_{\theta}$ is infinite. If $\frac{1}{k}\in\kappa(\operatorname{Aut}(X_{\theta},\sigma))$ then, by taking a power of $\theta$ if necessary, there is an automorphism $\Phi$ with $\kappa(\Phi)=\frac{1}{k}$ satisfying

[TABLE]

and an automorphism $\tau\in\operatorname{Aut}(X_{\theta},\sigma)$ such that $\tau^{r}=\tau$ and

[TABLE]

Proof.

Let $\Phi$ be any automorphism with $\kappa(\Phi)=\frac{1}{k}$. Then we have

[TABLE]

and since $\frac{1}{k}\in\mathbb{Z}_{r}$, this implies that $\Phi^{-1}\circ\theta^{-1}\circ\Phi^{r}\circ\theta:X_{\theta}\rightarrow X_{\theta}$ is a well defined bijection which also commutes with $\sigma$, i.e. is an automorphism. We also have $\kappa(\Phi^{-1}\circ\theta^{-1}\circ\Phi^{r}\circ\theta)=0$. We claim, by taking a power of $\theta$ if necessary, that there exists an automorphism $\Phi$ such that $\kappa(\Phi)=\frac{1}{k}$ and Equation (12) holds.

If $\kappa$ is injective, then $\kappa(\Phi^{-1}\circ\theta^{-1}\circ\Phi^{r}\circ\theta)=0$ implies that $\Phi^{-1}\circ\theta^{-1}\circ\Phi^{r}\circ\theta=\operatorname{Id}$ and we are done. If $\kappa$ is not injective, consider the map $f:\kappa^{-1}(\frac{1}{k})\rightarrow\kappa^{-1}(\frac{1}{k})$ given by $\Phi\mapsto\theta^{-1}\Phi^{r}\theta$. For some $n\geq 1$, and for some $\Phi$, we have $f^{n}(\Phi)=\Phi$. One verifies that in this case $\Phi$ satisfies $\theta^{n}\Phi=\Phi^{r^{n}}\theta^{n}$. As $X_{\theta^{n}}=X_{\theta}$, we can replace $\theta$ by $\theta^{n}$ if necessary, to assume that we started with a $\theta$ for which there exists an automorphism $\Phi$ satisfying (12).

For this $\Phi$ satisfying $\eqref{eq:thetaPhicomm}$, we have that $\Phi^{k}=\sigma\circ\tau$ for some $\tau\in\ker{\kappa}$. It remains to show that $\tau^{r}=\tau$. Considering the power $\theta^{c!p}$ instead of $\theta$, we have by Equation (2) that $\sigma^{N}\circ\Phi\circ\theta=\theta\circ\Phi$ for some integer $N$. Using Equation (12), we have $\sigma^{N}\circ\Phi\circ\theta=\Phi^{r}\circ\theta$, so that $\sigma^{N}\Phi=\Phi^{r}$. Using Equation (13), we have

[TABLE]

Both sides being automorphisms with necessarily equal $\kappa$-values, this implies that $Nk+1=r$, and the result follows.

∎

Lemma 48**.**

Let $\theta$ be a strongly injective primitive length $r$ substitution of height one such that $X_{\theta}$ is infinite. If $\Phi\in(X_{\theta},\sigma)$ satisfies

[TABLE]

for some automorphism $\tau\in\operatorname{Aut}(X_{\theta},\sigma)$ such that $\tau^{r}=\tau$, then there exists a substitution $\eta$ such that $(X_{\theta},\Phi)$ is topologically conjugate to $(X_{\eta},\sigma)$.

Proof.

By Theorem 33, $\tau$ has radius zero. If $\theta$ is defined on the alphabet $\mathcal{A}$, define a map $\pi\colon X_{\theta}\to\mathcal{A}^{\mathbb{Z}}$ by $\pi(x)_{n}=(\Phi^{n}x)_{0}$. As we assume that $k>1$, then by an analogue of Proposition 8, $\Phi$ has left radius 0 and right radius at most 1. We notice that

[TABLE]

where we use $\sigma$ to denote the shift on either of our spaces. Let $\bar{X}:=\pi(X_{\theta})$; then $\bar{X}$ is closed and shift invariant. We have

[TABLE]

and since $\tau$ must be injective on letters, this implies that $\pi$ is injective.

Since $\pi$ conjugates $(X_{\theta},\Phi)$ to $(\bar{X},\sigma)$, and $\Phi^{k}=\sigma\circ\tau$, then $\pi$ conjugates $(X_{\theta},\sigma\circ\tau)$ to $(\bar{X},\sigma^{k})$. Since $(X_{\theta},\sigma)$ is minimal and $\tau$ is an automorphism, then $(X_{\theta},\sigma\circ\tau)$ is minimal and so $(\bar{X},\sigma^{k})$ is minimal; hence $(\bar{X},\sigma)$ is also minimal.

We define a map on $\bar{X}$ by $\eta=\pi\circ\theta\circ\pi^{-1}$. We have

[TABLE]

where the second, third and fourth equalities are by (14), (12) and (14) respectively.

We claim that $\eta$ is a length $r$ substitution. By (15), it suffices to check that if $z,z^{\prime}\in\bar{X}$ satisfy $z_{0}=z_{0}^{\prime}$, then $\eta(z)_{i}=\eta(z^{\prime})_{i}$ for $i=0,\ldots,r-1$. Suppose then that $z$ and $z^{\prime}\in\bar{X}$ satisfy $z_{0}=z^{\prime}_{0}$. Let $x=\pi^{-1}z$ and $x^{\prime}=\pi^{-1}z^{\prime}$. Notice that if $z=\pi(x)$, then $z_{0}=x_{0}$, so that $x_{0}=x^{\prime}_{0}$. Hence $\theta(x)_{i}=\theta(x^{\prime})_{i}$ for $0\leq i<r$. As $\Phi$ has right radius at most one and left radius 0, then $\eta(z)_{i}=(\pi\circ\theta(x))_{i}=(\pi\circ\theta(x^{\prime}))_{i}=\eta(z^{\prime})_{i}$ for $0\leq i\leq r-1$. It follows that $\eta$ is a length $r$ substitution map, as required.

If $x\in X_{\theta}$ is a fixed point of $\theta$, then $\pi(x)\in\bar{X}$ is a fixed point of $\eta$. Furthermore, by minimality its orbit is dense in $\bar{X}$, so that $\bar{X}$ is the substitution space of $\eta$. By (14), $\pi$ conjugates $\Phi\colon X_{\theta}\to X_{\theta}$ to $\sigma:X_{\eta}\to X_{\eta}$. The result follows.

∎

Proof of Theorem 46.

We continue with the notations of Lemmas 47 and 48. By Lemma 47, there exist automorphisms $\Phi$ and $\tau$, with $\kappa$ values $\frac{1}{k}$ and 0 respectively, and satisfying Equations (12) and (13). Also $\tau^{r}=\tau$. By Lemma 48, $(X_{\theta},\Phi^{k})=(X_{\theta},\sigma\circ\tau)$ is conjugate to the $k$-compression $(X_{\eta^{(k)}},\sigma)$ of the substitution shift $(X_{\eta},\sigma)$. Let $\bar{\pi}:(X_{\theta},\sigma\circ\tau)\rightarrow(X_{\eta^{(k)}},\sigma)$ be a conjugacy. Since $\tau^{-1}$ is an automorphism of $(X_{\theta},\sigma\circ\tau)$, this implies that $(X_{\theta},\sigma)$ is conjugate to $(X_{\eta^{(k)}},\sigma\circ\bar{\tau})$ for some automorphism $\bar{\tau}$ of $(X_{\eta^{(k)}},\sigma)$ with $\bar{\tau}\in\ker\kappa_{\eta^{(k)}}$.

We claim that $\bar{\tau}$ has radius zero. To see this it is sufficient to show that $\eta$ is strongly injective, as this will imply that $\eta^{(k)}$ is also strongly injective and we are done by Theorem 33.

Note that the column number of $\eta$ is at most the column number of $\theta$. If $\theta$ has column number $c$, we can assume, by considering the rotation in the proof of Theorem 32 if necessary, where we replace $\theta$ by the appropriate $k$-shifted 2-sliding block representation, that $\theta$ has exactly $c$ right-infinite fixed points and $c$ left-infinite fixed points. The map $\Phi$ has left radius zero and right radius one, and each right-infinite $\theta$-fixed point is mapped to a right-infinite $\eta$-fixed point. Furthermore any right-infinite $\eta$-fixed point is the image under $\pi$, of a right-infinite $\theta$-fixed point. Thus $\eta$ has $c$ right-fixed points and hence its column number must also be $c$. Now since $\kappa(\Phi^{-1})=-\kappa(\Phi^{-1})$, $\Phi^{-1}$ has left radius one and right radius 0. Working with $\Phi^{-1}$ (i.e. moving up this spacetime diagram, instead of down it), we see as above that $\eta$ has exactly $c$ left fixed points. By the discussion in the proof of Theorem 32, $\eta$ is strongly injective.

Thus $\bar{\tau}$ has radius zero. Suppose that $\eta^{(k)}$ is defined on an alphabet $\mathcal{B}$. Define a map $p:X_{\eta^{(k)}}\rightarrow\mathcal{B}^{\mathbb{Z}}$ by $p(x)_{n}=((\sigma\circ\bar{\tau})^{n}(x))_{0}$. One verifies that if $x\in X_{\eta^{(k)}}$ is a fixed point for $\eta^{(k)}$, then $p(x)_{n}={\bar{\tau}}^{n}(x_{n})$, i.e. $p(x)$ is a $\left({\eta^{(k)}}\right)_{\bar{\tau}}$-fixed point. Now $p$ is an injection, since $\bar{\tau}$ is an automorphism with radius zero. Furthermore $p\circ\sigma\circ\bar{\tau}=\sigma\circ p$, i.e. $p$ is a conjugacy between $(X_{\eta^{(k)}},\sigma\circ\bar{\tau})$ and $(X_{({\eta^{(k)}})_{\bar{\tau}}},\sigma)$. We have shown that $(X_{\theta},\sigma)$ is conjugate to $(X_{\eta^{(k)}},\sigma\circ\bar{\tau})$, which is conjugate to $(X_{({\eta^{(k)}})_{\bar{\tau}}},\sigma)$, i.e. that $(X_{\theta},\sigma)$ is conjugate to the twist of a $k$-compression. ∎

We specialise to the following case, which includes the case when $\operatorname{Aut}(X_{\theta},\sigma)$ is cyclic, generated by an automorphism whose $\kappa$-value is $\frac{1}{k}$. This happens, for instance, if $\theta$ has a coincidence, and more generally if $\kappa$ is injective and $\theta$ has height coprime to $k$ [5, Corollary 3.7]. Then we can restate our result as follows.

Corollary 49**.**

Let $\theta$ be a primitive length $r$ substitution such that $X_{\theta}$ is infinite. If $\Phi\in\operatorname{Aut}(X_{\theta},\sigma)$ and $\Phi^{k}=\sigma$ then $(X_{\theta},\sigma)$ is topologically conjugate to the $k$-compression of a substitution shift $(X_{\eta},\sigma)$.

The following example shows that the twist in Theorem 46 is indeed necessary.

Example 50**.**

Consider the substitution

[TABLE]

It has an automorphism $\Phi$ with $\kappa$-value $\frac{1}{3}$, and whose local rule $\phi$ is given in Table 3.

Let

[TABLE]

Now $\eta$ has an order two automorphism $\tau$ defined by the permutation $(ae)(bf)(cg)(dh)$, $(X_{\theta},\Phi)$ is conjugate to $(X_{\eta},\sigma)$, and $(X_{\theta},\sigma)$ is conjugate to a twist (by $\tau$) of the $3$-compression of $(X_{\eta},\sigma)$.

In general, what can one say about the $\kappa$-kernel of the substitution $\theta$ versus that of the substitution $\eta$ in the statement of Theorem 46? The following examples tell us that in general, there is no elegant statement. In this next example, we were still able to cling to a sense that we could set things up so that the $\kappa$-kernels of the relevant maps are isomorphic.

Example 51**.**

Consider the substitution $\theta$ on the alphabet $\mathcal{A}$:

[TABLE]

Then $\theta$ has one nontrivial automorphism $\tau$ belonging to $\ker\kappa_{\theta}$, defined by the permutation $(AK)(BG)(CD)(EH)(FI)(JL)$. It also has two square roots of the shift, one of which has local rule in Table 4 and which we call $\Phi$, the other of which is $\tau\circ\Phi$. Now using $\Phi$, and working through the proof of Lemma 48, we can show that $(X_{\theta},\Phi)$ is topologically conjugate to $(X_{\eta},\sigma)$ where

[TABLE]

So $\theta$ is the twist of a compression of $\eta$, but $\ker\kappa_{\eta}$ is isomorphic to $S_{3}$, so here it is not the case that the $\kappa$-kernels of the two maps $\theta$ and $\eta$ are the same. However, note that if we use $\tau\circ\Phi$ in Lemma 48, we obtain that $(X_{\theta},\tau\circ\Phi)$ is topologically conjugate to $(X_{\eta_{\tau}},\sigma)$, and that $\ker\kappa_{\eta_{\tau}}$ is isomorphic to $\ker\kappa_{\theta}$.

However, whatever hope we clung to with Example 51 was dashed with this next example.

Example 52**.**

Consider

[TABLE]

which is a two-compression of $(X_{\eta_{1}},\sigma)$ where

[TABLE]

Note that $\theta$ has a nontrivial kernel automorphism $\tau$ given by the permutation $(ADH)(BGC)(EIF)$, and so, since it is a two-compression, it also has three automorphisms with $\kappa$ value $1/2$, $\Phi_{1}$, $\Phi_{2}=\tau\circ\Phi_{1}$ and $\Phi_{3}=\tau^{2}\circ\Phi_{1}$. Note also that $\tau$ and $\Phi_{1}$ do not commute. Finally, $\Phi_{1}^{2}=\Phi_{2}^{2}=\Phi_{3}^{2}=\sigma$, and using $\Phi_{i}$ in Lemma 48, we get that $(X_{\theta},\Phi_{i})$ is conjugate to a two-compression of $(X_{\eta_{i}},\sigma)$ where

[TABLE]

all three of whose automorphism groups are trivial.

Remark 53**.**

What causes this breakdown? We have seen in Theorem 37 that the automorphisms of $\eta$ with $\kappa$-value [math] can be characterized in terms of the minimal sets and the group

[TABLE]

Let us now consider the $k$-compression $\eta^{(k)}$ of $\eta$ for some $k\mid r-1$. One can then show that

[TABLE]

Thus, we might be in the situation where $G(\eta^{(k)})$ is strictly smaller then $G(\eta)$, or one where the centralizer of $G(\eta^{(k)})$ is strictly larger then the centralizer of $G(\eta)$.

This is precisely what happens in this last example, where $G(\eta_{i})=S_{3}$, so that $\eta_{i}$ has a trivial automorphism group, but $G(\eta_{i}^{(2)})=A_{3}=\{\operatorname{Id},(123),(132)\}=C(G(\eta_{i}^{(2)}))$, so that we have non-trivial automorphisms with $\kappa$ value [math].

Acknowledgements.

We thank Anthony Quas for the key insights in Corollary 49, and Tomasz Downarowicz and Mariusz Lemanczyk for pointing out Theorem 24 to us.