Admissible reversing and extended symmetries for bijective substitutions
AADMISSIBLE REVERSING AND EXTENDED SYMMETRIES FORBIJECTIVE SUBSTITUTIONS ´ALVARO BUSTOS, DANIEL LUZ, AND NEIL MA ˜NIBO
Abstract.
In this paper, we deal with reversing and extended symmetries of shifts generatedby bijective substitutions. We provide equivalent conditions for a permutation on the alphabetto generate a reversing/extended symmetry, and algorithms how to check them. Moreover, weshow that, for any finite group G and any subgroup P of the d -dimensional hyperoctahedralgroup, there is a bijective substitution which generates an aperiodic hull with symmetry group Z d × G and extended symmetry group ( Z d (cid:111) P ) × G . Introduction
The study of symmetry groups, often also known as automorphism groups, is an importantpart of the analysis of a dynamical system, as it can offer insight on the behaviour of the system,as well as allowing classifications of distinct families of dynamical systems (acting as a conjugacyinvariant). In particular, symmetry groups of shift spaces have been thoroughly studied (see e.g.the analysis of the symmetry group of the full shift [BLR88], the series of works on symmetriesin low-complexity subshifts [CK16, CQY16, DDMP16], and recent works on shifts of algebraicand number-theoretic origin [BBH+19, FY18]).Symmetries of subshifts can be algebraically defined as elements of the topological centraliserof the group (cid:104) σ (cid:105) generated by the shift, seen as a subgroup of the space Aut( X ) of all self-homeomorphisms of X onto itself. Thus, a natural question at this point is whether the cor-responding normaliser has an interesting dynamical interpretation as well. This leads to theconcept of reversing symmetries (for d = 1); see [BR06, Goo99, BRY18], the monograph [OS15]for a group-theoretic exposition, and [LR98] for a more physical background. These are specialtypes of flip conjugacies; see [BM08]. In higher dimensions, one talks of extended symmetries ;see [Baa18, BRY18], which are examples of GL( d, Z )-conjugacies; compare [Lab19, BBH+19].These kinds of maps are related to phenomena such as palindromicity and several proper-ties of geometric and topological nature, which is more evident in the higher-dimensional set-ting [BRY18, Bus20].High complexity is often (but not always, see for instance the square-free subshift [BBH+19])linked to a complicated symmetry group. For instance, determining whether the symmetrygroups of the full shifts in two and three symbols are isomorphic has consistently proven tobe a difficult question [BLR88]. The low-complexity situation, thus, often allows for a morein-depth analysis and more complete descriptions, up to and including explicit computation ofthese groups in many cases.The particular case of substitutive subshifts has gathered significant attention and here alot of progress has been made; see [MY21, KY19]. Unsurprisingly, the presence of non-trivialsymmetries is also tied to the spectral structure of the underlying dynamical system; see [Que10, Mathematics Subject Classification.
Key words and phrases.
Extended symmetries, automorphism groups, substitution subshifts, aperiodic tilings. a r X i v : . [ m a t h . D S ] J a n ra05]. In this work, we restrict to systems generated by bijective substitutions, both in oneand in higher dimensions. These substitutions are typically n -to-1 extensions of odometers andgenerate coloured tilings of Z d by unit cubes, where one usually identifies a letter with a uniquecolour; see [Fra05]. We compile and extend known properties about this family of substitutivesubshifts regarding symmetries. Some natural questions in this direction are: (1) What kinds of groups can appear as symmetry groups/extended symmetry groups ofspecific substitutive subshifts? (2)
Given a specific group G , can we construct a substitution whose associated subshift has G as its symmetry group/extended symmetry group?Both questions are accessible for bijective substitutions. For symmetry groups, the secondquestion is answered in full in [DDMP16], which extends to higher dimensions with no addi-tional assumptions because the result does not depend on the geometry of the substitution;see [CP20] for realisation results for more general group actions. We add to such known resultsin Theorem 14. Aperiodicity also plays a key role here, which can easily be confirmed in thebijective setting; see Propositions 5 and 33.On the other hand, the existence of non-trivial reversing or extended symmetries dependsheavily on the geometry and requires more in terms of the relative positions of the permu-tations in the corresponding supertiles, the expansive maps, and the shape of the supertilesthemselves. In Theorem 22, we provide equivalent conditions for the existence of non-trivial re-versing symmetries, which we generalise to higher dimensions in Theorem 30 to cover extendedsymmetries.As a corollary, in any dimension d , given a finite group G and a subgroup P of the hy-peroctahedral group P , we provide a construction in Theorem 35 of a bijective substitutionwhose underlying shift space has symmetry group and extended symmetry group Z d × G and( Z d (cid:111) P ) × G , respectively. A similar construction with a different structure of the extendedsymmetry group is done in Theorem 41. We also provide algorithms on how one can checkwhether there exist non-trivial symmetries and extended symmetries for a given substitution (cid:37) ;see Sections 2.2 and 3.1.2. Bijective constant-length substitutions
Setting and basic properties.
Let A be a finite alphabet and A + = (cid:83) L (cid:62) A L be the setof finite non-empty words over A ; we shall write A ∗ = A + ∪ { ε } , where the latter is the emptyword. A substitution is a map (cid:37) : A → A + . If there exists an L ∈ N such that (cid:37) ( a ) ∈ A L forall a ∈ A , (cid:37) is called a constant-length substitution. If there exists a power k such that (cid:37) k ( a )contains all letters in A , for every a ∈ A , we call (cid:37) primitive .The full shift is the set A Z of all functions (configurations) x : Z → A . More generally, wedefine the d -dimensional full shift as the set A Z d . To this space, we assign the product topology,giving A the discrete topology. This is a particular version of the local topology used in tilingspaces and discrete point sets, in which two tilings (or point sets) x and y are said to be ε - close if a small translation of x (of magnitude less than ε ) matches y on a large ball (of radius at least1 /ε ) around the origin; this can be used to define a metric d( x, y ). In the particular case of shiftspaces seen as tiling spaces, since tiles are aligned with Z d , we can disregard the translation andget, e.g., the following as an equivalent metric:d( x, y ) = 2 − inf (cid:110) n : x | [ − n,n ] d (cid:54) = y | [ − n,n ] d (cid:111) . his space is endowed with the shift action of Z d on A Z d , which is the action of Z d overconfigurations by translation, and can be defined via the equality ( σ n ( x )) m = x n + m for all x ∈ A Z d , m , n ∈ Z d (in particular, in one dimension we have the shift map σ = σ , whichcompletely determines the group action).A subshift is a topologically closed subset X ⊆ A Z d which is also invariant under the shiftaction. Thus, a subshift combined with the restriction of this group action to X defines atopological dynamical system, which can be endowed with one or more measures to obtain ameasurable dynamical system. In the one-dimensional case, the language (or dictionary ) of asubshift X is the set of all words that may appear contained in some x ∈ X , that is: L ( X ) = { x | [0 ,n ] : x ∈ X , n (cid:62) } ∪ { ε } . We may verify that any nonempty set of words L which is extensible (that is, any w ∈ L is a subword of a longer word w (cid:48) ∈ L ) and closed under taking subwords is the language of asubshift, and two subshifts are equal if and only if they share the same language.Higher-dimensional subshifts have a similar combinatorial characterisation, where the roleof words is taken by patterns , finite configurations of the form P : U ⊂ Z d → A , | U | < ∞ ; weidentify a pattern with any of its translations. In most cases (and, in particular, in the rest ofthis work), it makes no difference to allow arbitrary “shapes” U or to restrict ourselves to onlyrectangular patterns, i.e. products of intervals of the form U = (cid:81) di =1 [0 , n i − x | U that appear in some x ∈ X into a set L ( X ) as above, which we once again call the language of X . As in the one-dimensional case, alanguage closed under taking subpatterns and where every pattern of shape U is contained ina pattern of shape V ⊃ U for any larger (finite) V defines a unique subshift, and vice versa.Thus, given that iterating a primitive substitution (cid:37) : A → A L of constant length L > a ∈ A produces words of increasing length, the set L (cid:37) of all words that aresubwords of some (cid:37) k ( a ) for some k (cid:62) a ∈ A is the language of a unique subshift thatdepends only on (cid:37) , which we shall call the substitutive subshift defined by (cid:37) and denote by X (cid:37) .This definition extends to d -dimensional rectangular substitutions (cid:37) : A → A R (where R is aproduct of intervals), which are higher-dimensional analogues of constant-length substitutions;see [Fra05, Que10, Bar18]. It is well known that the primitivity of (cid:37) implies that X (cid:37) is strictlyergodic (uniquely ergodic and minimal); see [Que10, BG13]. We refer the reader to [MR18] fora treatment of substitutions which are non-primitive. Definition 1.
A constant-length substitution (cid:37) : A → A L is called bijective if the map whichis given by (cid:37) j : a (cid:55)→ (cid:37) ( a ) j is a bijection on A , for all indices 0 (cid:54) j (cid:54) L −
1. Equivalently, (cid:37) is bijective if there exist L (not necessarily distinct) bijections (cid:37) , . . . , (cid:37) L − : A → A such that (cid:37) ( a ) = (cid:37) ( a ) . . . (cid:37) L − ( a ) for every a ∈ A . We shall refer to the mapping (cid:37) j as the j -th column of the substitution (cid:37) .Consider (cid:8) (cid:37) j (cid:9) L − j =0 ⊂ S |A| . Let Φ : S |A| → GL( |A| , Z ) be the representation via permutationmatrices. One then has the following; compare [Fra05, Cor. 1.2]. Pattern shapes do matter when studying certain generalisations of topological mixing in the d -dimensionalsetting, where either restricting ourselves to specific shapes (rectangles, L-shapes, hollow rectangles, etc.) orallowing arbitrary ones may be preferrable depending on context. However, we are not concerned with thesekinds of properties here. act 2. Let (cid:37) be a primitive, bijective substitution, whose columns are given by (cid:8) (cid:37) , . . . , (cid:37) L − (cid:9) .Then the substitution matrix M is given by M = (cid:80) L − j =0 Φ( (cid:37) − j ) . Moreover, (1 , , . . . , T isa right Perron–Frobenius eigenvector of M , so each letter has the same frequency for everyelement in the hull X (cid:37) , i.e., ν a = |A| for all a ∈ A and all x ∈ X (cid:37) . (cid:3) Define the n -th column group G ( n ) to be the following subgroup of the symmetric group ofbijections A → A : G ( n ) := (cid:104){ (cid:37) j ◦ · · · ◦ (cid:37) j n | (cid:54) j , . . . , j n (cid:54) L − }(cid:105) . As it turns out, the groups G ( n ) generated by the columns give a good description of thesubstitution (cid:37) in the bijective case; see [KY19] for its relation to the corresponding Ellis semi-group of X (cid:37) . The primitivity of (cid:37) may be characterised entirely by this family of groups, asseen below. Recall that a subgroup G (cid:54) S n of the symmetric group on { , . . . , n } is transitive if for all 1 (cid:54) j, k (cid:54) n there exists τ ∈ G such that τ ( j ) = k . Here, we let N ∈ N be the minimalpower such that (cid:37) Nj = id for some 0 (cid:54) j (cid:54) L N −
1; compare [Que10, Lem. 8.1]. In [KY19], G ( N ) is called the structure group of (cid:37) . Proposition 3.
Let (cid:37) : A → A L be a bijective substitution. Then, the following are equivalent: (1) The substitution (cid:37) is primitive. (2)
All groups G ( n ) , n ∈ N , are transitive. (3) The group G ( N ) is transitive.Proof. Evidently, (2) = ⇒ (3), so we only need to prove (3) = ⇒ (1) = ⇒ (2). To see the firstimplication, note first that the columns of the iterated substitution (cid:37) N are compositions of theform (cid:37) j ,...,j N := (cid:37) j ◦ · · · ◦ (cid:37) j N , (cid:54) j , . . . , j N (cid:54) L −
1, that is, for any a ∈ A the following holds: (cid:37) N ( a ) = (cid:37) ,..., , ( a ) (cid:37) ,..., , ( a ) . . . (cid:37) ,..., ,L − ( a ) (cid:37) ,..., , ( a ) . . . (cid:37) L − ,...,L − ,L − ( a ) . Since, by (3), the group G ( N ) is transitive, the substitution matrix M (cid:37) N is irreducible, i.e. it isthe adjacency matrix of a strongly connected digraph. In other words, for all a, b ∈ A , thereexists a composition of columns q, q (cid:48) , . . . , q (cid:48)(cid:48) of (cid:37) N such that q ◦ q (cid:48) ◦ · · · ◦ q (cid:48)(cid:48) ( a ) = b , which may beidentified with a path in the graph whose vertices are the letters of A and with one edge from c to r ( c ) for any c ∈ A and column r . The choice of N also shows that M (cid:37) N has a non-zerodiagonal, since one of the columns of (cid:37) N is the identity. These two conditions immediatelyimply that M (cid:37) N is a primitive matrix (see [LM95, Ch. 2]) which in turn implies primitivity of (cid:37) , as desired.To prove (1) = ⇒ (2), note that primitivity of (cid:37) implies that, for some k > a ∈ A , the word (cid:37) k ( a ) contains all symbols of the alphabet A , including a itself. Sincethe columns of (cid:37) k generate G ( k ) , this implies that for all a, b ∈ A there is some generator ofthis group that maps a to b , i.e. G ( k ) is transitive. Since (cid:37) k ( a ) contains a as a subword, thisimplies that (cid:37) k ( a ) contains (cid:37) k ( a ) as a subword, and, by induction, that (cid:37) mk ( a ) contains (cid:37) k ( a )as a subword for all m (cid:62)
1; thus, all groups G ( mk ) are transitive. Now, it is easy to see that G ( n ) (cid:54) G ( d ) if d | n . Then, for all n ∈ N , G ( n ) has G ( nk ) as a transitive subgroup and hence itis transitive. (cid:3) The bijective structure of (cid:37) can also be exploited to conclude the aperiodicity of X (cid:37) by justlooking at simple features of (cid:37) . Below, we provide several criteria for aperiodicity in terms of |A| , L , and the existence of certain legal words. roposition 4. Let X (cid:37) be the hull of a primitive, bijective substitution (cid:37) of length L on a finitealphabet A . If X (cid:37) is periodic with least period p , then it has to satisfy the following conditions: (1) |A| divides p (2) L does not divide |A| .Proof. From Fact 2 we know that every letter has the same frequency. An element of a periodicshift is just a concatenation of its periods and thus every letter has the same frequency in everyperiod. This is only possible if every letter appears equally often in the period and thus theperiod length has to be a multiple of the alphabet size, which settles the first claim.Let us assume that w ∞ is a periodic word with p as least period. Without loss of generality,choose a power (cid:37) k such that its first column is the identity, and so w ∞ is fixed by (cid:37) k . We choose c and d minimal such that: cL = dp ⇐⇒ cb |A| = da |A| . (1)We apply (cid:37) − to w ∞ | [0 ,cL − of length cL , which has a unique pre-image in L since (cid:37) is injectiveon letters, i.e., (cid:37) ( a ) = (cid:37) ( b ) if and only if a = b . Then this segment must be of the form x , · · · x c .Applied to w ∞ , it yields · · · x c x , · · · x c x · · · , which means that w ∞ is c -periodic. Since p is theleast period c = ep but then c = ea |A| , making the factor |A| in Eq. (1) redundant, and thuscontradicting the minimality of c . (cid:3) Another way to get aperiodicity is through the existence of proximal pairs; see [DDMP16,Sec. 3.2.1] and [BG13, Cor. 4.2 and Thm. 5.1]. Two elements x (cid:54) = y ∈ ( X , σ ) are said tobe proximal if there exists a subsequence { n k } of N or − N such that d( σ n k x, σ n k y ) → k → ∞ . A stronger notion is that of asymptoticity, which requires d( σ n x, σ n y ) → n → ∞ or −∞ . For bijective substitutions, these two notions are equivalent, and asymptotic pairs arecompletely characterised by fixed points of (cid:37) ; see [KY19].Consider a one-dimensional substitution (cid:37) and a fixed point w arising from a legal seed a | b ,i.e., w = (cid:37) ∞ ( a | b ). Here, the vertical bar represents the location of the origin, and the letter a generates all the letters at the negative positions, while b does the same for all non-negativeones. Two fixed points w , w ∈ X (cid:37) generated by a | b and a | b are left-asymptotic if they agreeat all negative positions and disagree for all non-negative positions. Right-asymptotic pairs aredefined in a similar manner. We have the following equivalent condition for aperiodicity interms of existence of certain legal words; compare [KY19, Prop. 4.1] Proposition 5.
Let (cid:37) be a primitive, bijective substitution on a finite alphabet A in one di-mension. Then the hull X (cid:37) is aperiodic if and only if there exist distinct legal words of length which either share the same starting or ending letter.Proof. Let (cid:37) := (cid:37) · · · (cid:37) L − , with (cid:37) i ∈ G . Choosing k = lcm( | (cid:37) | , | (cid:37) L − | ), we get that the firstand the last columns of (cid:37) k are both the identity, i.e., (cid:37) k ( a ) = (cid:37) kL k − ( a ) = a for all a ∈ A . Ifthere exist ab, ac ∈ L (cid:37) with b (cid:54) = c , the bi-infinite fixed points (cid:37) ∞ ( a | b ) and (cid:37) ∞ ( a | c ) they generateunder (cid:37) k coincide in all negative positions and differ in at least one non-negative position, andhence are left-asymptotic and proximal. Since X (cid:37) is minimal and admits a proximal pair, all ofits elements must then be aperiodic. Now suppose that every letter has a unique predecessorand successor in A . This means that every element x ∈ X (cid:37) is uniquely determined by the letterat the origin. From the finiteness of A , one gets x = w ∞ and hence is periodic, from which theperiodicity of the hull follows. (cid:3) xample 6. The substitution (cid:37) : a (cid:55)→ aba, b (cid:55)→ bab is primitive, bijective and admits a periodichull. Here, the only legal words of length 2 are ab and ba . Note that (cid:37) is of height 2 andgenerates the same hull as the substitution (cid:37) (cid:48) : a, b (cid:55)→ ab . ♦ Symmetries.
In the following sections, we deal with the symmetry groups of our subshiftsof interest, which are certain homeomorphisms of the shift space which preserve the dynamicsof the shift action in a specific sense.
Definition 7.
Let X be a Z d -subshift. The symmetry group (often called automorphism group )is the set S ( X ) of all homeomorphisms X → X which commute with the shift action, i.e.,(2) ( ∀ n ∈ Z d ) : σ n ◦ f = f ◦ σ n . That is, S ( X ) is the centraliser of the set of shift maps in the group of all self-homeomorphismsof the space X . In this context, every symmetry f ∈ S ( X ) is entirely determined by its localfunction , which is a mapping F : A U → A , with U ⊂ Z d finite, such that for every n ∈ Z d , f ( x ) n = F ( x | n + U ). This fact is known as the Curtis–Hedlund–Lyndon (or CHL) theorem ;see [LM95]. We say that f has radius r (cid:62) U ⊆ [ − r, r ] d .Symmetry groups of one-dimensional bijective substitutions are a thoroughly studied sub-ject, both in the topological and ergodic-theoretical contexts. Complete characterisations ofthese groups are known, as seen in e.g. [Cov71] for a two-symbol alphabet, or [LM88] for acharacterisation in the measurable case; see also [Fra05, CQY16] for further elaboration in thedescription of the symmetries in this category of subshifts. The following theorem summarizesthis classification: Theorem 8.
Let X (cid:37) be the hull generated by an aperiodic, primitive, bijective substitution (cid:37) on Z d . Then, the symmetry group S ( X (cid:37) ) is isomorphic to the direct product of Z d , generatedby the shift action, with a finite group of radius- sliding block codes τ ∞ : X (cid:37) → X (cid:37) given by τ ∞ (( x j ) j ∈ Z d ) = ( τ ( x j )) j ∈ Z d for some bijection τ : A → A .Furthermore, let N be any integer such that (cid:37) N j is the identity for some j (note that such an N always exists). Then, τ : A → A induces a symmetry if and only if τ ∈ cent S |A| G ( N ) . (cid:3) As a consequence, every symmetry on X (cid:37) is a composition of a shift map and a radius-zerosliding block code as above. These conditions arise as a consequence of such a symmetry havingto preserve the supertile structure of any x ∈ X (cid:37) at every scale, which in particular impliesthat a level- k supertile (cid:37) k ( a ) , a ∈ A has to be mapped to some (cid:37) k ( b ) for some other b ∈ A bythe “letter exchange map” τ . The choice of N above ensures that, when k is a multiple of N ,the equality a = b holds, which implies that τ commutes with the columns of (cid:37) N , and thus (cid:37) N ◦ τ ∞ = τ ∞ ◦ (cid:37) N . This in turn implies Eq. (2). For further elaboration on the proof of theabove result, the reader may consult [Fra05, CQY16], among others. Example 9.
Consider the following substitution (cid:37) on the three-letter alphabet A = { a, b, c } : (cid:37) : a (cid:55)→ abc,b (cid:55)→ bca,c (cid:55)→ cab. In this work, we follow the notational conventions of [BRY18], and thus we avoid the term “automorphismgroup” as it may be understood as the set of all homeomorphisms f : X → X . he columns correspond to the three elements of the cyclic group generated by τ = ( a b c ). Itis not hard to verify that the only elements of S = D that commute with τ are the powersof τ themselves, and thus S ( X (cid:37) ) (cid:39) Z × C , with the finite subgroup C being generated by thesymmetries induced by the powers of τ . ♦ As it turns out, Theorem 8 provides an algorithm to compute S ( X (cid:37) ) explicitly. To introducethis algorithm, let us recall some easily verifiable facts from group theory [Hall76, Ch. 1 and 5]: Fact 10.
Let G be any group and H = (cid:104) S (cid:105) (cid:54) G a subgroup generated by S ⊂ G . Then, cent G ( H ) = { c ∈ G | ( ∀ h ∈ H ) : ch = hc } = (cid:92) s ∈ S cent G ( s ) . (cid:3) Fact 11.
Any permutation decomposes uniquely (up to reordering) as a product of disjointcycles. Conjugation by some τ ∈ S n can be computed from this decomposition using the identity: τ ( a a . . . a n ) τ − = ( τ ( a ) τ ( a ) . . . τ ( a n )) . A permutation τ ∈ S n belongs to cent S n ( π ) if and only if τ πτ − = π , and thus: π = ( a a . . . a k )( b b . . . b k ) · · · ( c c . . . c k r )= ( τ ( a ) τ ( a ) . . . τ ( a k ))( τ ( b ) τ ( b ) . . . τ ( b k )) · · · ( τ ( c ) τ ( c ) . . . τ ( c k r )) . Hence, the uniqueness of this decomposition implies that every cycle in the second decompositionis equal to a cycle of the same length in the first one. (cid:3)
Thus, to compute the letter exchange maps that determine S ( X (cid:37) ), we need to find all per-mutations τ that preserve certain cycle decompositions. We obtain the following procedure: Algorithm.
Assuming that (cid:37) is a primitive, bijective, aperiodic substitution, the following algorithmcomputes S ( X (cid:37) ) explicitly. • Input: (cid:37) is a length- L bijective substitution, which may be represented as a function (dictio-nary) (cid:37) : A → A L or a set of L permutations (cid:37) , (cid:37) , . . . , (cid:37) L − : A → A , corresponding to eachcolumn. • Output:
A (finite) set of permutations C forming a group, so that S ( X (cid:37) ) = Z d × C .(1) Compute the least positive integer N such that (cid:37) N j is the identity on A for some column of thesubstitution (cid:37) . N equals the least common multiple of all cycle lengths in the decomposition ofthe columns (cid:37) j into disjoint cycles (and is thus finite).(2) Determine all columns (cid:37) j ◦ · · · ◦ (cid:37) j N of the iterated substitution (cid:37) N . This is a generating setfor the group G ( N ) .(3) For every column computed in (2), compute G j ,..., j N = cent S n ( (cid:37) j ◦ · · · ◦ (cid:37) j N ) by taking thecycle decomposition of this permutation (in where we identify A with the set { , , . . . , |A|} )and employing the characterisation above.(4) Let C = (cid:84) j ,..., j n G j ,..., j N . As C can be biunivocally identified with the set of valid letterexchange maps modulo a shift, return S ( X (cid:37) ) = Z d × C as output. Example 9 above corresponds to a simple case in which G ( N ) = G (1) is a cyclic group, andwe derive an abelian subgroup of S corresponding to the valid letter exchange maps. We canuse the above procedure to construct examples with more complicated symmetry groups, seeExample 12. xample 12. We take as alphabet the quaternion group Q = { e, i, j, k, ¯ e, ¯ ı, ¯ , ¯ k } (see [Hall76]for the multiplication table and basic properties of this group, which is generated by the twoelements i and j ). With this, we construct a length-3 bijective substitution defined by rightmultiplication, x (cid:55)→ ( x · i )( x · j )( x · k ), given in full by: e (cid:55)→ ijk, ¯ e (cid:55)→ ¯ ı ¯ ¯ k,i (cid:55)→ ¯ ek ¯ , ¯ ı (cid:55)→ e ¯ kj,j (cid:55)→ ¯ k ¯ ei, ¯ (cid:55)→ ke ¯ ı,k (cid:55)→ j ¯ ı ¯ e, ¯ k (cid:55)→ ¯ ie. By direct computation, G ( n ) = G (1) (cid:39) Q for all n , making the substitution primitive (as Q actstransitively on itself in an obvious way). The three columns which generate G (1) are: R i := ( e i ¯ e ¯ ı )( j ¯ k ¯ k ) ,R j := ( e j ¯ e ¯ )( i k ¯ ı ¯ k ) ,R k := ( e k ¯ e ¯ k )( j i ¯ ¯ ı ) . Thus, the substitution (cid:37) has as columns R xyz ( g ) = g · xyz with x, y, z ∈ { i, j, k } ; in particular,since jik = e , (cid:37) must have an identity column. Also, since G (3) = G (1) , this group is the rightCayley embedding of Q into S . By applying the above algorithm, we obtain that the group ofletter exchange maps is generated by the following two permutations: π := ( e i ¯ e ¯ ı )( j k ¯ j ¯ k ) ,π := ( e j ¯ e ¯ )( i ¯ k ¯ i k ) . We can verify that these permutations generate the left
Cayley embedding of Q into S . Al-ternatively, if we consider the transposition ν = ( k ¯ k ), we can use Fact 11 above to see that π = νR i ν − and π = νR j ν − , which in turn implies that the group generated by π and π isconjugate to the group generated by R i and R j , the latter being isomorphic to Q . This showsthat S ( X (cid:37) ) (cid:39) Z × Q . ♦ It is well known that symmetry groups of aperiodic minimal one-dimensional subshifts arevirtually Z . The following result gives a full converse for shifts generated by bijective substitu-tions. Theorem 13 ( [DDMP16, Thm. 3.6]) . For any finite group G , there exists an explicit primitive,bijective substitution (cid:37) , on an alphabet on | G | letters, such that S ( X (cid:37) ) (cid:39) Z × G . (cid:3) The proof, which may be consulted in [DDMP16], follows a similar schema to the analysisdone in Example 12 above. In [Fra05, Sec. 4.1], it was shown that the number of letters neededin Theorem 13 is actually a tight lower bound. Below, we actually prove something stronger.
Theorem 14.
Let (cid:37) be an aperiodic, primitive, bijective substitution on the alphabet A . If S ( X (cid:37) ) (cid:39) Z × G , then G must act freely on A , and the order of G has to divide |A| .Proof. As seen in [Fra05, Sec. 4.1], if we replace (cid:37) with a suitable power, we may ensure thatthe word (cid:37) q ( a ) starts with a and contains every other symbol, for all a ∈ A . Thus, for any π ∈ S n , the equality π ( a ) = b implies π ( (cid:37) q ( a )) = (cid:37) q ( b ), which in turn determines the images ofevery symbol in the alphabet; the bound | G | (cid:54) |A| follows from here. ote as well that, since (cid:37) is bijective, if π ( a ) (cid:54) = a , then π ( c ) (cid:54) = c for every c ∈ A as the words (cid:37) q ( a ) and (cid:37) q ( b ) are either equal or differ at every position. This implies that if π has any fixedpoint then it must be the identity, i.e. that, if we identify G with the corresponding groupof permutations over A , the action of G on the alphabet is free. Equivalently, the stabiliserStab( c ) of any c ∈ A is the trivial subgroup.The elements of G commute with every column of (cid:37) q . Due to primitivity, there always existsa column (cid:37) ∗ = (cid:37) j ◦ · · · ◦ (cid:37) j q which maps this a to any desired c ∈ A . Since (cid:37) ∗ commuteswith every π ∈ G (i.e., it is an equivariant bijection for the action of G on A ), we have thatOrb( c ) = (cid:37) ∗ [Orb( a )], i.e. the orbit of c under G is necessarily the image of the orbit of a under (cid:37) ∗ .Thus, every orbit is a set of the same cardinality. This means that G induces a partition of A into disjoint orbits of the same cardinality (cid:96) , which then must divide |A| . By the freeness ofthe group action and the orbit-stabiliser theorem, we have | G | = | Orb( a ) | · | Stab( a ) | = (cid:96) , andthus | G | divides |A| . (cid:3) Remark 15.
It follows from Theorem 14 that the substitution in Example 12 is a minimal onein the sense that for one to get a Q -extension in S ( X (cid:37) ), one needs at least eight letters. ♦ Remark 16.
At no point in the proof of Theorem 13 found in [DDMP16] nor in Theorem 14above the fact that the substitution was one-dimensional is actually used. Thus, since Theorem8 is known to be valid for general rectangular substitutions, the two theorems above must bevalid in this more general setting as well, provided that the substitution is aperiodic in Z d ,which one can always guarantee; see Propositions 5 and 33. ♦ Corollary 17.
For any finite group G , there exists an explicit primitive, bijective d -dimensionalrectangular substitution (cid:37) , on an alphabet of | G | letters, such that S ( X (cid:37) ) (cid:39) Z d × G . Furthermore,this is the least possible alphabet size: for any bijective, primitive and aperiodic d -dimensionalrectangular substitution (cid:37) on the alphabet A , if S ( X (cid:37) ) (cid:39) Z d × G , then G acts freely on A , and | G | divides |A| . (cid:3) Extended and reversing symmetries of substitution shifts
One-dimensional shifts.
Since the term symmetry group does not cover everything thatcan be thought of as a symmetry (in the geometric sense of the word) we introduce the notion ofthe reversing symmetry group; see [BRY18] for a detailed exposition. We will exclusively lookat shift spaces X (cid:37) which are given by a bijective, primitive substitution (cid:37) and we will exploitthis additional structure in determining the reversing symmetry group for this class. Definition 18.
The extended symmetry group of a shift space X is given as R ( X ) := norm Aut( X ) ( G ) = { H ∈ Aut( X ) | H G = G H } where G is the group generated by the shift. In the case where the shift space is one-dimensional,we call R ( X ) the reversing symmetry group given by R ( X ) = { H ∈ Aut( X ) | H ◦ σ ◦ H − = σ ± } . A homeomorphism H ∈ Aut( X ) which satisfies H ◦ σ ◦ H − = σ − is called a reversor or a reversing symmetry . A Curtis–Hedlund–Lyndon-type characterisation of reversing symmetries,which incorporates the mirroring component (GL( d, Z )-component in higher dimensions) canbe found in [BRY18]. n what follows, we investigate the effect of a reversor f on inflated words. Given a bijectivesubstitution (cid:37) : A → A L , (cid:37) := (cid:37) (cid:37) · · · (cid:37) L − , the mirroring operation m acts on the columns of (cid:37) via m ( (cid:37) ( a )) = (cid:37) L − ( a ) · · · (cid:37) ( a ) (cid:37) ( a ). We may extend this to infinite configurations over Z intwo non-equivalent ways, given by m ( x ) k = x − k and m (cid:48) ( x ) k = x − k , respectively; we shall referto both as basic mirroring maps. Proposition 19.
Let (cid:37) be an aperiodic, primitive, bijective substitution. Then, any reversor isa composition of a letter exchange map π ∈ S n , where n = |A| , a shift map σ k and one of thetwo basic mirroring maps m or m (cid:48) (depending only on whether the substitution has odd or evenlength, respectively). See [BRY18, Prop. 1] and Theorem 8. This result, while desirable, is not immediately obvious(and can indeed be false for non-bijective substitutions, which may have reversors whose localfunctions have positive radius), and thus we show this result as a consequence of bijectivity.
Proof.
Suppose f : X (cid:37) → X (cid:37) is a reversor of positive radius r (cid:62)
1, i.e., x | [ − r,r ] = y | [ − r,r ] impliesthat one has f ( x ) = f ( y ) . There is some power k (cid:62) (cid:37) k ( a ) of length L k are longer than the local window of f , which has length 2 r + 1 (say, k = (cid:100) log(2 r + 1) / log( L ) (cid:101) ).Any point of X (cid:37) is a concatenation of words of the form (cid:37) k ( a ) , a ∈ A , which is unique up toa shift because of aperiodicity; see [Sol98]. In particular, if we choose a fixed x ∈ X (cid:37) and let y = f ( x ), both points have such a decomposition.Now, suppose that the value L k = 2 (cid:96) + 1 is odd (the case where L is even is dealt withsimilarly). By composing f with an appropriate shift map (say ˜ f = f ◦ σ h ), we can ensure thatthe central word (cid:37) k ( a ) in the aforementioned decomposition has support [ − (cid:96), (cid:96) ] for both x and y (note that we employ the uniqueness of the decomposition here, to avoid ambiguity in thechosen h ). Since L k = 2 (cid:96) + 1 (cid:62) r + 1, we must have (cid:96) (cid:62) r , and thus y is entirely determinedby x | [ − (cid:96),(cid:96) ] , which is a substitutive word (cid:37) k ( a ). But, since (cid:37) is bijective, this word is in turncompletely determined by its central symbol x . Figure 1.
A reversor f establishes a 1-1 correspondence between words (cid:37) k ( a )in a point x and its image f ( x ).A similar argument shows that, for any n ∈ Z , if n ∈ mL k + [ − (cid:96), (cid:96) ], then y n depends onlyon the word x | − mL k +[ − (cid:96),(cid:96) ] , which contains (and is thus entirely determined by) x − n . Since anypoint in X (cid:37) is transitive, ˜ f is entirely determined by the points x and y , and thus, ˜ f is a mapof radius 0. Equivalently, for some bijection π : A → A , we have ˜ f ( x ) − n = π ( x n ), that is,˜ f = f ◦ σ h = π ◦ m (identifying π with the letter exchange map A Z → A Z ). We conclude that f is a composition of a letter exchange map, a mirroring map and a shift map. (cid:3) Remark 20.
With some care, it can be shown that the same argument applies in the higher-dimensional case, where an element of the normaliser is a composition of a letter exchange map,a map of the form f ( x ) n = x A n , with A a linear map from the hyperoctahedral group (seeTheorem 26, below), and a shift map; see [BRY18, Prop. 3] for a more general formulation. ♦ his result leads to the following criterion for the existence of a reversor in terms of thecolumns (cid:37) i . Proposition 21.
Let (cid:37) be an aperiodic, primitive, bijective substitution (cid:37) of length L on a finitealphabet A of n letters. Suppose that there exists a letter-exchange map π ∈ S n , π : A → A which gives rise to a reversing symmetry. Then one has (3) π − ◦ (cid:37) i ◦ (cid:37) − j ◦ π = (cid:37) L − ( i +1) ◦ (cid:37) − L − ( j +1) for all (cid:54) i, j (cid:54) L − , where (cid:37) i is the i -th column of (cid:37) seen as an element of S n .Proof. Let a ∈ A . Let m be the mirroring operation and suppose that there exists π ∈ S n suchthat m ◦ π extends to a reversor f ∈ R ( X (cid:37) ). One then has (cid:37) ( a ) = (cid:37) ( a ) · · · (cid:37) L − ( a ) m (cid:55)−→ (cid:37) L − ( a ) · · · (cid:37) ( a ) π (cid:55)−→ π ◦ (cid:37) L − ( a ) · · · π ◦ (cid:37) ( a ) . Since Proposition 19 guarantees that this must result to mapping substituted words to substi-tuted words, one gets(4) π ◦ (cid:37) L − ( a ) · · · π ◦ (cid:37) ( a ) = (cid:37) ( b ) · · · (cid:37) L − ( b ) = (cid:37) ◦ τ ( a ) · · · (cid:37) L − ◦ τ ( a ) , where the permutation τ describes precisely this induced shuffling of inflation words. This yields τ = (cid:37) − j ◦ π ◦ (cid:37) L − ( j +1) for all 0 (cid:54) j (cid:54) L −
1. Equating the corresponding right hand-sides for some pair i, j yieldsEq. (3). The claim follows since this must hold for all 0 (cid:54) i, j (cid:54) L − (cid:3) Theorem 22.
Let (cid:37) be as in Proposition . Suppose further that (cid:37) i = (cid:37) L − ( i +1) = id for some (cid:54) i (cid:54) L − . Then, given a permutation (letter exchange map) π ∈ S n , π : A → A , thefollowing are equivalent: (i)
The letter exchange map π gives rise to a reversing symmetry f ∈ R ( X (cid:37) ) \ S ( X (cid:37) ) givenby either f ( x ) n = π ( x − n ) or f ( x ) n = π ( x − n ) . (ii) The permutation π satisfies the system of equations (5) π − ◦ (cid:37) i ◦ π = (cid:37) L − ( i +1) for all (cid:54) i (cid:54) L − . (iii) There exist κ , κ , . . . , κ L − ∈ S n , where each κ i satisfies κ − i ◦ (cid:37) i ◦ κ i = (cid:37) L − ( i +1) , suchthat the following intersection of cosets is non-empty: (6) K = L − (cid:92) i =0 cent S n ( (cid:37) i ) κ i , and π ∈ K .Proof. It is clear that Eq. (5) implies Eq. (3). Note that it is sufficient to satisfy Eq. (3) for j = i + 1 mod L as any term can be obtained by multiplying sufficient numbers of succeedingterms. Under the extra assumption that there exist a column pair which is the identity, Eq. (3)simplifies to Eq. (5). This shows that (i) = ⇒ (ii).For the other direction, we show that if Eq. (5) is satisfied at by the level-1 inflation words,then these sets of equations must also be fulfilled by any power (cid:37) k of (cid:37) . Remember that,from any arbitrary bijective substitution (cid:37) , we may derive another bijective substitution (cid:37) (cid:48) that satisfies the additional condition of having two identity columns in opposing positions bychoosing k = lcm( | (cid:37) | , | (cid:37) L − | ) and replacing (cid:37) by its k -th power, (cid:37) (cid:48) := (cid:37) k . This makes no ifference when studying R ( X (cid:37) ), because (cid:37) and (cid:37) k define the same subshift and the group ofreversing symmetries is a property of the hull.First, we prove an important property of the columns of powers. Fix a power k ∈ N and picka column (cid:37) i of (cid:37) k , where 0 (cid:54) i (cid:54) L k −
1. One then has (cid:37) i = (cid:37) i · · · (cid:37) i k − where i i · · · i k − isthe L -adic expansion of i and (cid:37) i (cid:96) are columns of the level-1 substitution (cid:37) .The corresponding L -adic expansion of L k − ( i + 1) is then given by L k − ( i + 1) = ( L − ( i + 1)) · · · ( L − ( i k − + 1)) . This can easily be shown via the following direct computation k − (cid:88) j =0 ( L − ( i j + 1)) L j = k − (cid:88) j =0 ( L j +1 − L j ) − k − (cid:88) j =0 i j L j = L k − ( i + 1) . This implies that if one considers the corresponding column (cid:37) L k − ( i +1) one gets that(7) (cid:37) L k − ( i +1) = (cid:37) L − ( i +1) · · · (cid:37) L − ( i k − +1) . This has two consequences. First, if (cid:37) has an identity column pair, then all powers of (cid:37) admit atleast one identity column pair. For each power k one just needs to choose (cid:37) j with j = iii · · · i ,which implies (cid:37) j = (cid:37) ki = id. By Eq. (7), we also get that (cid:37) L k − ( j +1) = ( (cid:37) L − ( i +1) ) k = id. In fact, (cid:37) k contains at least 2 k − pairs of identity columns.Second, this property allows one to prove that if (cid:37) satisfies the system of equations in Eq. (5),then it is satisfied at all levels, i.e., by all powers of (cid:37) . To this end, choose 0 (cid:54) i (cid:54) L k − L -adic expansion i i · · · i k − . From Eq. (5) one then obtains π − ◦ (cid:37) i ◦ π = π − ◦ (cid:37) i · · · (cid:37) i k − ◦ π = π − ◦ (cid:37) i ππ − · · · ππ − (cid:37) i k − π = (cid:37) L − ( i +1) · · · (cid:37) L − ( i k − +1) = (cid:37) L k − ( i +1) . Since i is chosen arbitrarily and π induces a permutation of the substituted words at all levels,this means it extends to a map f = σ n ◦ m ◦ π : X (cid:37) → X (cid:37) , which by Proposition 19 is a reversor.This shows (ii) = ⇒ (i).To prove the remaining equivalences, note that if π , π ∈ S n are two permutations satisfyingthe equality π − ◦ (cid:37) i ◦ π = (cid:37) L − ( i +1) , then we have: π ◦ (cid:37) L − ( i +1) ◦ π − = (cid:37) i = ⇒ ( π ◦ π − ) − ◦ (cid:37) i ◦ ( π ◦ π − ) = (cid:37) i , that is, ( π ◦ π − ) ∈ cent S n ( (cid:37) i ). As a consequence, π belongs to the right coset cent S n ( (cid:37) i ) π for any choice of π , π , and, since right cosets are either equal or disjoint, this means that allsolutions of Eq. (5), for a fixed i , lie in the same right coset of cent S n ( (cid:37) i ). Reciprocally, if π satisfies Eq. (5) and γ ∈ cent S n ( (cid:37) i ), it is easy to verify that γ ◦ π satisfies Eq. (5) as well. Thus,the set of solutions of this equation is either empty or the aforementioned uniquely defined rightcoset.Thus, suppose that π satisfies Eq. (5) for all 0 (cid:54) i (cid:54) L −
1. The set of solutions for each i equals the unique coset cent S n ( (cid:37) i ) π , and thus the set of all permutations that satisfy Eq. (5)for all i is exactly the intersection of all these cosets, i.e. (cid:84) L − i =0 cent S n ( (cid:37) i ) π . Taking κ i = π forall i , we see that this is exactly the set K from (6). Evidently, π belongs to this intersection,and so we conclude that (ii) = ⇒ (iii).As stated before, our choice of κ i ensures that the set cent S n ( (cid:37) i ) κ i is exactly the set ofsolutions of Eq. (5) for a given i ; thus, any permutation π that satisfies all of these equalities ust be in all of these cosets and thus in the intersection (6), which is therefore non-empty.This shows that (iii) = ⇒ (ii), concluding the proof. (cid:3) The following general criterion on when a letter-exchange map generates a reversor is givenin [BRY18].
Lemma 23 ( [BRY18, Lem. 2]) . Let (cid:37) be a primitive constant-length substitution of height and column number c (cid:37) . Suppose that (cid:37) is strongly injective. Then, a permutation π : A → A generates a reversor f ∈ R ( X (cid:37) ) if and only if (1) ab ∈ L (cid:37) = ⇒ π ( ba ) ∈ L (cid:37) (2) ( π ◦ (cid:37) c (cid:37) ! )( ab ) = ( (cid:37) c (cid:37) ! ◦ π )( ba ) for each ab ∈ L (cid:37) . (cid:3) For a primitive, aperiodic and bijective (cid:37) , one has c (cid:37) = |A| . Moreover, (cid:37) is always stronglyinjective. Note that Theorem 22 implies conditions (1) and (2) in Lemma 23. The first oneimmediately follows from primitivity, and the fact that any legal word ab appears in some level- n superword, which is sent to another level- n superword by π ◦ m , which guarantees the legalityof π ( ba ). The second follows from the fact that π is compatible with superwords of all levels.In fact, one has (cid:0) π ◦ (cid:37) n (cid:1) ( ab ) = (cid:0) (cid:37) n ◦ π (cid:1) ( ba ) for all n ∈ N . Remark 24.
It is a known fact from group theory that, if g , . . . , g r are elements of a group G and H , . . . , H r are subgroups of this group, the intersection of cosets (cid:84) ri =1 g i H i is eitherempty or a coset of (cid:84) ri =1 H i . In this case, the latter intersection is exactly the group of non-trivial standard symmetries modulo a shift (letter exchanges), and thus, if there exist non-trivial reversing symmetries, these must all belong to a single coset of the group of valid letterexchanges. This is consistent with the fact that R ( X (cid:37) ) is at most an index 2 group extensionof S ( X (cid:37) ). ♦ Item (3) in Theorem 22 provides an explicit algorithm to compute the group of permutations π which define extended symmetries, which is a counterpart to that in Section 2.2 for standardsymmetries. As stated previously, the centralisers cent S n ( (cid:37) j ) can be computed for each columnusing Fact 11, and thus the problem reduces to obtaining a suitable candidate for each κ i , whichonce again can be done by an application of Fact 11. The algorithm is as follows: Algorithm.
Assuming that (cid:37) is a primitive, bijective, aperiodic substitution, the following algorithmcomputes the set K of permutations that induce reversors, which determines R ( X (cid:37) ). • Input: (cid:37) is a length- L bijective substitution, represented either as a function or a set of columns. • Output:
A (finite) set of permutations K , either empty or a coset of the group C computedby the previous algorithm, so that R ( X (cid:37) ) / (cid:104) σ (cid:105) (cid:39) C ∪ K (i.e. R ( X (cid:37) ) (cid:39) Z (cid:111) ϕ ( C ∪ K ), with ϕ ( g, n ) = n if g ∈ C , and − n if g ∈ K ).(1) Let N be the least positive integer which ensures that two opposite columns of (cid:37) N are theidentity map. This can be computed as: N = min (cid:110) lcm(ord( (cid:37) i ) , ord( (cid:37) L − ( i +1) )) : 0 (cid:54) i (cid:54) N/ (cid:111) . (2) For each 0 (cid:54) i (cid:54) N/
2, compute κ i via the following subroutine:(2.i) If (cid:37) i and (cid:37) L − ( i +1) are non-conjugate (i.e., their cycle decomposition has a different numberof cycles of some length), stop the algorithm, as reversors do not exist (see Theorem 22). (cid:37) i by increasing order of length.Using this as a basis, by appropriately sorting the elements of each cycle in this decom-position, define a total order relation < on A , given by, say, a < · · · < a n , such that allof the elements of a given cycle come before the elements of the following cycle, in thesorting by left. Do the same for (cid:37) L − ( i +1) , defining a corresponding total order < (cid:48) givenby b < (cid:48) · · · < (cid:48) b n . This ensures that there are cycle decompositions of both permutationssuch that the corresponding cycles, ordered from left to right, have the same length, asfollows: (cid:37) i = ( a . . . a j )( a j +1 . . . a j (cid:48) ) · · · ( a j (cid:48)(cid:48) +1 . . . a n ) ,(cid:37) L − ( j +1) = ( b . . . b j )( b j +1 . . . b j (cid:48) ) · · · ( b j (cid:48)(cid:48) +1 . . . b n ) , with 1 (cid:54) j (cid:54) j (cid:48) (cid:54) . . . (cid:54) j (cid:48)(cid:48) (cid:54) n .(2.iii) Define: κ i = (cid:32) a a · · · a n b b · · · b n (cid:33) , κ L − ( i +1) = κ − i . (3) Compute each centraliser C ( i ) = cent S n ( (cid:37) i ), using the same procedure as in the computationof S ( X (cid:37) ).(4) Return K = (cid:84) Ni =1 C ( i ) κ i . Any element of K induces a reversor; if K is empty, reversors do notexist. Any programming environment with suitable data structures (e.g. computer algebra systemssuch as
Sagemath ® or Mathematica ® ) is amenable to the implementation of this algorithm,providing effective procedures to entirely characterise the groups S ( X (cid:37) ) and R ( X (cid:37) ) from asuitable description of the substitution (cid:37) , e.g. using a dictionary.3.2. Higher-dimensional subshifts.
Now, we turn our attention to the situation in higherdimensions. The extended symmetry group of a Z d -shift is defined as R ( X ) = norm Aut( X ) ( G ),where now G = (cid:104) σ e , . . . , σ e d (cid:105) (cid:39) Z d ; see [BRY18,Bus20,BBH+19]. In this more general context,an extended symmetry is a element f ∈ R ( X ) \ S ( X ).Similar to standard symmetries, there is a direct generalisation of the characterisation ofextended symmetries from Proposition 21 and the subsequent theorem to the higher-dimensionalsetting, which is given by the following. Proposition 25.
Let (cid:37) be an aperiodic, primitive, bijective, block substitution in Z d . Thenany extended symmetry f ∈ R ( X (cid:37) ) \ S ( X (cid:37) ) must be (up to a shift) a composition of a letterexchange map and a rearrangement function f A given by f A ( x ) n = x A n , where A ∈ GL( d, Z ) ,with A (cid:54) = I . (cid:3) For shifts generated by bijective rectangular substitutions one has the following restrictionon the linear component A of an extended symmetry f . Theorem 26 ( [Bus20, Thm. 18]) . Let (cid:37) an aperiodic, primitive, bijective rectangular substi-tution in Z d . One then has R ( X (cid:37) ) / S ( X (cid:37) ) (cid:39) P (cid:54) W d , where W d (cid:39) C d (cid:111) S d is the d -dimensional hyperoctahedral group, which represents the symmetriesof the d -dimensional cube. (cid:3) With this, one can show that all extended symmetries of such subshifts are of finite order.The proof of the following result is patterned from [BR06, Prop. 2], which deals with the rder of reversors of an automorphism h of a general dynamical system with ord( h ) = ∞ ;compare [Goo99]. Proposition 27.
Let X (cid:37) be the same as above with symmetry group S ( X (cid:37) ) = Z d × G . Let f ∈ R ( X (cid:37) ) \ S ( X (cid:37) ) be an extended symmetry, whose associated matrix is A ∈ W d . Then ord( f ) divides ord( A ) · | G | . Moreover, ord( f ) (cid:54) | G | · max { ord( τ ) | τ ∈ S d } .Proof. Under the given assumptions, f ◦ σ m ◦ f − = σ A m holds for all m ∈ Z d , which yields f (cid:96) ◦ σ m ◦ f − (cid:96) = σ A (cid:96) m (8) f ◦ σ n m ◦ f − = σ nA m (9)for all (cid:96), n ∈ N . Choosing (cid:96) = ord( A ), Eq. (8) gives f ord( A ) ∈ S ( X (cid:37) ). From Theorem 8, f ord( A ) = σ p ◦ π , for some p ∈ Z d and letter-exchange map π . From the direct product structureof the symmetry group, one has σ p ◦ π = π ◦ σ p , which implies f ord( A ) ·| G | = σ | G | p ◦ π | G | = σ | G | p .Using the two equations above, one gets f ord( A ) ·| G | = σ | G | A (cid:96) ( p ) for all (cid:96) ∈ N . Since f is anextended symmetry, A (cid:54) = I . Next we show that p cannot be an eigenvector of A .Suppose A p = p with p (cid:54) = . Note that f − ord( A ) | G | = σ −| G | p . From Eqs. (8) and (9), one alsohas f − ◦ σ | G | A − p ◦ f = σ −| G | p , which implies A − p = − p , contradicting the assumption on p .Since ord( σ p ) = ∞ , this forces p = and hence f ord( A ) ·| G | = id from which the first claim isimmediate. The upper bound for the order follows from the upper bound for the order of theelements of the hyperoctahedral group W d ; see [Baa84]. (cid:3) Due to the fact that R ( X (cid:37) ) is (possibly) a larger extension of S ( X (cid:37) ) (that is, the correspondingquotient can have up to 2 d d ! − W d except the identity. This leads us to another problem of differentnature: if the rectangle R , which is the support of the level-1 supertiles of (cid:37) , is not a cube in Z d ,some symmetries from W d may not be compatible with R , i.e., they may map R to a differentrectangle that is not a translation of R , so the corresponding equation does not have a propermeaning (as it may compare an existing column with a non-existent one). (cid:55)→ (cid:55)→ Figure 2.
A non-square substitution that generates the two-dimensional Thue-Morse hull.This could be taken as a suggestion that such symmetries cannot actually happen, imposingfurther limitations on the quotient R ( X (cid:37) ) / S ( X (cid:37) ). Interestingly, this is not actually the case. Forinstance, consider the two-dimensional rectangular substitution from Figure 2. As the supportfor this substitution is a 4 × × W = D . Thus, only geometricalconsiderations are not enough to exclude candidates for extended symmetries.Fortunately, there is a subcase of particular interest in which this geometrical intuition isactually correct, which involves an arithmetic restriction on the side lengths of the support ectangle R . It turns out that coprimality of the side lengths is a sufficient condition (althoughit can be weakened even further) to rule out such symmetries, e.g. there are no extendedsymmetries compatible with rotations when R is a, say, 2 × Theorem 28.
Let (cid:37) : A → A R be a bijective rectangular substitution with faithful associatedshift action. Suppose that R = [ , L − ] with L = ( L , . . . , L d ) (that is, R is a d -dimensionalrectangle with side lengths L , L , . . . , L d ) and that for some indices i, j there is a prime p suchthat p | L j but p (cid:45) L i , i.e. L i and L j have different sets of prime factors. Let A ∈ W d (cid:54) GL( d, Z ) and suppose that A is the underlying matrix associated to an extended symmetry f ∈ R ( X (cid:37) ) .Then A ij = A ji = 0 . The underlying idea is that, if A ∈ W d induces a valid extended symmetry for some sub-stitution (cid:37) with support U , we can find another substitution η with support A · U (up to anappropriate translation) such that X (cid:37) = X η , and then we use the known factor map from anaperiodic substitutive subshift onto an associated odometer to rule out certain matrices A .Similar exclusion results have been studied by Cortez and Durand [CD08]. Proof.
Let ϕ : X (cid:37) (cid:16) Z L × · · · × Z L d = Z L be the standard factor map from the substitutivesubshift to the corresponding product of odometers. It is known [BRY18, Thm. 5] that, for anyextended symmetry f : X (cid:37) → X (cid:37) with associated matrix A , there exists k f = ( k , . . . , k d ) ∈ Z L and a group automorphism α f : Z L → Z L satisfying the following equation:(10) ϕ ( f ( x )) = k f + α f ( ϕ ( x )) , where α f is the unique extension of the map n (cid:55)→ A n , defined in the dense subset Z d , to Z L .In particular, for any n ∈ Z d , if f = σ n is a shift map, then k σ n = n and α σ n = id Z L .Now, consider the sequence h m = L mi e i , and suppose A ji = ±
1. Equivalently, A e i = ± e j ,since A is a signed permutation matrix. Without loss of generality, we may assume the sign to be+. One has L mi m →∞ −−−−→ L i -adic topology, and thus ϕ ( σ h m ( x )) = h m + ϕ ( x ) m →∞ −−−−→ ϕ ( x ),as it does so componentwise. By compactness, we may take a subsequence h β ( m ) such that σ h β ( m ) ( x ) converges to some x ∗ ; then, as the factor map ϕ is continuous, we have ϕ ( x ∗ ) = ϕ ( x ).Eq. (10) and this last equality imply that ϕ ( f ( x )) = ϕ ( f ( x ∗ )) as well. Writing x ∗ as a limit,we obtain from continuity that ϕ ( x ∗ ) = lim m →∞ ϕ ( f ( σ h β ( m ) ( x ))) = lim m →∞ ϕ ( σ A h β ( m ) ( f ( x )))= ϕ ( x ) + lim m →∞ A h β ( m ) = ϕ ( x ) + lim m →∞ L β ( m ) i A e i = ⇒ lim m →∞ L β ( m ) i e j = ϕ ( x ∗ ) − ϕ ( x ) = . The last equality implies that, in the topology of Z L j , the sequence L β ( m ) i converges to 0.However, since there is a prime p that divides L j but not L i , due to transitivity we must have L j (cid:45) L ni for all n , as otherwise p | L ni and thus p | L i . Thus, in base L j , the last digit of L β ( m ) i is never zero, and thus L β ( m ) i remains at fixed distance 1 from (in the L j -adic metric),contradicting this convergence. Thus, A ji cannot be 1 and must necessarily equal 0. For A ij ,the same reasoning applies to f − . Since A is a signed permutation matrix, A ij = ± A − ) ji = ±
1, again a contradiction. (cid:3)
We now proceed to the generalisation of Theorem 22 in higher dimensions. As before, fora block substitution (cid:37) , we have R = (cid:81) di =1 [0 , L i − L i (cid:62) = diag( L , L , . . . , L d ). Let A ∈ W d (cid:54) GL( d, Z ) be a signed permutation matrix. First, weassume that the location of a tile in any supertile is given by the location of its centre. Definethe affine map A (1) : R → R via A (1) ( i ) = A ( i − x ) + | A | x where i ∈ R and x = Q v − v with v = (1 , , . . . , T . Here, ( | A | ) ij = | A ij | . The vector | A | x is the translation needed to shift thecentre of the supertile to the origin, which we will need before applying the map A and shiftingit back again. We extend A (1) to any level- k supertile by defining the map A ( k ) : R ( k ) → R ( k ) given by(11) A ( k ) ( i ) = A ( i − x k ) + | A | x k , with i ∈ R ( k ) and x k = Q k v − v . Here R ( k ) := (cid:81) di =1 [0 , L ki −
1] is the set of locations of tiles ina level- k supertile. Example 29.
Let (cid:37) be a two-dimensional block substitution with Q = ( ) and A be thecounterclockwise rotation by 90 degrees, with corresponding matrix A = (cid:0) −
11 0 (cid:1) . Consider thelevel-3 supertile and let i = (7 , T ∈ R (3) , with Q -adic expansion i (cid:98) = i i i . Here one has i = i = e + e and i = e . One then gets A (3) ( i ) = (4 , T ; see Figure 3. One can checkthat (cid:80) j =0 Q j ( A (1) ( i j )) = A (3) ( i ). ♦ Figure 3.
The transformation of a marked level-3 location set R (3) under themap A (3) .The following result is the analogue of Theorem 22 in Z d . Theorem 30.
Let (cid:37) be an aperiodic, primitive, bijective block substitution (cid:37) : A → A R . Let W d be the d -dimensional hyperoctahedral group and let A ∈ W d . Suppose there exists (cid:96) ∈ R suchthat (cid:37) (cid:96) (cid:48) = id for all (cid:96) (cid:48) ∈ Orb A ( (cid:96) ) . Assume further that [ A, Q ] = 0 and | A | x = x . Then π ,together with A , gives rise to an extended symmetry f ∈ R ( X (cid:37) ) if and only if (12) π − ◦ (cid:37) i ◦ π = (cid:37) A (1) ( i ) for all i ∈ R . roof. Most parts of the proof mimics those of the proof of Theorem 22, where one replaces themirroring operation m with a more general map A ∈ W d . One then gets an analogous systemof equations, as in those coming from Eq. (4). Using this, one can show the necessity direction.To prove sufficiency, we show that if Eq. (12) is satisfied for all i ∈ R , then it also holds forall positions in any level- k supertile. Let i ∈ R ( k ) , which admits the unique Q -adic expansiongiven by i (cid:98) = i k − i k − · · · i i , i.e., i = (cid:80) k − j =0 Q j ( i j ). We now show that the Q -adic expansionof A ( k ) ( i ) is given by A ( k ) ( i ) (cid:98) = A (1) ( i k − ) A (1) ( i k − ) · · · A (1) ( i ). Plugging in the expansion of i into Eq. (11), one gets A ( k ) ( i ) = (cid:16)(cid:80) k − j =0 AQ j ( i j ) (cid:17) − A x k + x k . On the other hand, one also has k − (cid:88) j =0 Q j ( A (1) ( i j )) = k − (cid:88) j =0 Q j (cid:0) A ( i − Q v + v ) + Q v − v (cid:1) = k − (cid:88) j =0 Q j A ( i j ) + k − (cid:88) j =0 (cid:0) − AQ j +1 v + AQ j v (cid:1) + k − (cid:88) j =0 (cid:0) Q j v − Q j v (cid:1) = k − (cid:88) j =0 AQ j ( i j ) − AQ k v + A v (cid:124) (cid:123)(cid:122) (cid:125) − A x k + Q k v − v (cid:124) (cid:123)(cid:122) (cid:125) x k = A ( k ) ( i ) , where the penultimate equality follows from [ A, Q ] = 0 and the evaluation of the two telescopingsums. As in Theorem 22, one then obtains π − ◦ (cid:37) i ◦ π = π − ◦ (cid:37) i k − ◦ (cid:37) i k − ◦ · · · (cid:37) i ◦ π = (cid:37) A ( k ) ( i ) , whenever i (cid:98) = i k − i k − · · · i and π − ◦ (cid:37) i s ◦ π = (cid:37) A (1) ( i s ) for all i s ∈ R , which finishes theproof. (cid:3) Remark 31.
The conditions [
A, Q ] = 0 and | A | x = x in Theorem 30 are automaticallysatisfied if (cid:37) is a cubic substitution, i.e., L i = L for all 1 (cid:54) i (cid:54) d , which means one can useEq. (12) to check whether a given letter-exchange map works for any A ∈ W d . For general (cid:37) , these relations are only satisfied for certain A ∈ W d , e.g. reflections along coordinate axes,which means one needs a different tool to ascertain whether it is possible for other rigid motionsto generate extended symmetries. For example, one can use Theorem 28 to exclude somesymmetries. ♦ Before we proceed, we need a higher-dimensional generalisation of Proposition 5 regardingaperiodicity. For this, we use the following result, which is formulated in terms of Delone sets.Here, S d − is the unit sphere in R d . Theorem 32 ( [BG13, Thm. 5.1]) . Let X ( Λ ) be the continuous hull of a repetitive Delone set Λ ⊂ R d . Let (cid:8) b i ∈ S d − | (cid:54) i (cid:54) d (cid:9) be a basis of R d such that for each i , there are two distinctelements of X ( Λ ) which agree on the half-space { x | (cid:104) b i | x (cid:105) > α i } for some α i ∈ R d . Then onehas that X ( Λ ) is aperiodic. (cid:3) The proof of the previous theorem relies on the generalisation of the notion of proximalityfor tilings and Delone sets in R d , which is proximality along s ∈ S d − ; see [BG13, Sec. 5.5] forfurther details. Note that from a Z d -tiling generated by a rectangular substitution, one canderive a (coloured) Delone set Λ by choosing a consistent control point for each cube (usuallyone of the corners or the centre). Primitivity guarantees that Λ is repetitive and the notion f proximality extends trivially to coloured Delone sets using the same metric. The two hulls X ( Λ ) and X (cid:37) are then mutually locally derivable, and the aperiodicity of one implies that of theother. We then have a sufficient criterion for the aperiodicity of X (cid:37) in higher dimensions. Proposition 33.
Let (cid:37) : A → A R be a d -dimensional rectangular substitution which is bijectiveand primitive. If there exist two legal blocks u, v ∈ L of side-length in each direction such that u and v disagrees at exactly one position and coincides at all other positions, then the hull X (cid:37) is aperiodic.Proof. The proof proceeds in analogy to Proposition 5. Here we choose the appropriate powerto be k = lcm (cid:110) | (cid:37) r | : r = (cid:80) di =1 r i e i , r i ∈ { , L i − } (cid:111) . If we then place u and v at the origin, theresulting fixed points x = (cid:37) ∞ ( u ) and x (cid:48) = (cid:37) ∞ ( v ) which cover Z d will coincide at every sectorexcept at the one where u j (cid:54) = v j . One can then choose b i = e i and α i = 0 in Theorem 32, andfor each i , x and x (cid:48) to be the two elements which agree on a half-space, which guarantees theaperiodicity of X (cid:37) . More concretely, x and x (cid:48) are asymptotic, and hence proximal, along e i forall 1 (cid:54) i (cid:54) d . (cid:3) Remark 34.
Obviously, one can have a lattice of periods of rank less than d in higher dimen-sions. An example would be when (cid:37) = (cid:37) × (cid:37) , where (cid:37) is the trivial substitution a (cid:55)→ aa, b (cid:55)→ bb and (cid:37) is Thue–Morse. Although (cid:37) is itself not primitive, the product (cid:37) is and admits the legalblocks given in Figure 4, which generate fixed points that are Z e -periodic. If one requires thatthe shift component in S ( X (cid:37) ) is Z d , one needs all elements of X (cid:37) to be aperiodic in all cardinaldirections, hence the stronger criterion in Proposition 33. ♦ Figure 4.
The image of two distinct blocks under (cid:37) coincide in the upperhalf-plane and are distinct in the lower half-plane. In the limit, these legal seedsgenerate two fixed points which are neither left nor right asymptotic with respectto σ e .The next result is the analogue of Theorem 13 for extended symmetries, which holds in anydimension. Theorem 35.
Given a finite group G and a subgroup P of the d -dimensional hyperoctahedralgroup W d , there is an aperiodic, primitive, bijective d -dimensional substitution (cid:37) whose shiftspace satisfies S ( X (cid:37) ) (cid:39) Z d × G R ( X (cid:37) ) (cid:39) ( Z d (cid:111) P ) × G. roof. We start by taking a cursory look at the proof of Theorem 3.6 in [DDMP16]. For agiven finite group G , we choose a generating set S = { s , . . . , s r } that does not contain theidentity, and build a substitution whose columns correspond to the left multiplication maps L s j ( g ) = s j · g , seen as permutations of the alphabet A = G . These permutations generatethe left Cayley embedding of G in the symmetric group on | G | elements, whose correspondingcentraliser, which induces all of the letter exchanges in S ( X (cid:37) ), is the right Cayley embedding of G generated by the maps R s j ( g ) = g · s j .In what follows, we shall assume first that the group G is non-trivial, as the case in which G is trivial requires a slightly different construction. We also assume that the rectangularsubstitution we will construct engenders an aperiodic subshift, so that the group generated bythe shifts is isomorphic to Z d . We delay the proof of this until later on, to avoid cluttering ourconstruction with extraneous details.Since S ( X (cid:37) ) depends only on the columns of the underlying substitution and not their relativeposition, we shall construct a d -dimensional rectangular substitution (cid:37) with cubic support whosecolumns correspond to copies of the aforementioned L s j , placed in adequate positions along thecube. We start with a cube R = [0 , | S | + 2 d + 1] d of side length 2 | S | + 2 d + 2, where theadditional layer corresponding to the term 2 will be used below to ensure aperiodicity. Thiscube is comprised of N = | S | + d + 1 “shells” or “layers”, which are the boundaries of the innercubes [ j, | S | + 2 d + 2 − j ] d ; we shall denote each of them by Λ j , where j can vary from 0 to N − i -th inner shell Λ N − i with copies of the column L s i , for all 1 (cid:54) i (cid:54) r . This ensuresthat, as long as every other column is a copy of L s j for some j or an identity column, thesymmetry group S ( X (cid:37) ) of the corresponding subshift will be isomorphic to G , because in ourconstruction the 2 d corners of the point will always be identity columns.Now, note that N is chosen large enough so that the point p = (0 , , . . . , d −
1) lies in theouter N − r (cid:62) d shells and, moreover, the cube [0 , d − d is contained in these outer shellsas well. Thus, any permutation of the coordinates maps the cube [0 , d − d to itself and, inparticular, two different permutations map this point to two different points in this cube, thatis, the orbit of p has d ! different points. Combining this with the fact that the mirroring mapssend this cube to one of 2 d disjoint cubes (translations of [0 , d − d ) in the corners of the largercube [0 , N − d , it can be seen that W d acts freely on the orbit of the point p , that is, there isa bijection between the hyperoctahedral group W d and the set Orb( p ).Next, choose a fixed s j ∈ S that is not the identity element of G , so that L s j is not an identitycolumn. As P is a subgroup of W d , it is bijectively mapped to the set P · p = { g · p : g ∈ P } .Place a copy of L s j in each position from P · p , and an identity column in every other positionfrom Orb( p ). Fill every remaining position in the cube with identity columns. This ensuresthat the group of letter exchanges will remain isomorphic to G , and, for each matrix A ∈ W d associated with some element g ∈ P , the map f A given by the relation f A ( x ) n = x A n will be avalid extended symmetry, as a consequence of Theorem 30.Since every other extended symmetry is a product of such an f A with some letter-exchangemap that has to satisfy the conditions given by Eq. (12) due to our construction, and L s j cannotbe conjugate to the identity column, the only other extended symmetries are compositions ofthe already extant f A with elements from S ( X (cid:37) ), i.e. R ( X (cid:37) ) / S ( X (cid:37) ) has the equivalence classesof each f A as its only elements. As the set of all f A is an isomorphic copy of P contained in R ( X (cid:37) ), we conclude that R ( X (cid:37) ) is isomorphic to the semi-direct product S ( X (cid:37) ) (cid:111) P . However, igure 5. Examples of substitutions obtained by the above construction, for theKlein 4-group C × C , the cyclic group C and the whole W = D , respectively.The thicker lines mark the layer of identity columns separating the inner cubefrom the outer shell.since every letter exchanges from G commutes with every f A trivially, this semi-direct productmay be written as R ( X (cid:37) ) (cid:39) ( Z d (cid:111) P ) × G , as desired.In the case where G is trivial, we may choose an alphabet with at least three symbols (toensure that S |A| is non-Abelian) and repeat the construction above with a collection of columns (cid:37) , . . . , (cid:37) r − that generates some subgroup of S |A| with trivial centraliser (e.g. the two generatorsof S |A| itself). The rest of the proof proceeds in the same way.To properly conclude the proof, we need to verify that the constructed substitution generatesan aperiodic shift space. We focus on the case d >
1, as the one-dimensional case is a straight-forward modification of the construction from Theorem 13. Since our d -dimensional cube hasat least d + 1 (cid:62) × · · · × R contained in theouter layers that does not overlap any of the 2 d cubes of size d × · · · × d on the corners northe inner cube of size 2 | S | × · · · × | S | . As a consequence, this cube R contains only identitycolumns. Since we have a layer Λ d consists only of identity columns directly enveloping the innercube Λ d +1 ∪ · · · ∪ Λ d + | S | , the layer immediately following Λ d is comprised only of non-identitycolumns, which are copies of the same bijection π : A → A . . Thus, the 2 d corners of the hollowcube Λ d ∪ Λ d +1 are 2 × · · · × R , · · · , R d having exactly one non-identity column each,with this non-identity column τ being placed in every one of the 2 d possible positions on thesecubes.Since τ is not the identity, there must exist some a ∈ A such that τ ( a ) (cid:54) = a . The previousdiscussion thus implies that there is an admissible pattern P a of size 2 × · · · × a , and 2 d + 1 other admissible patterns P ( n ) a that differ from P a only inthe position n ∈ [0 , d . Using the proximality criterion from Proposition 33, we conclude thatthe subshift obtained is indeed aperiodic, as desired. (cid:3) Remark 36.
An alternative Cantor-type construction, which produces the prescribed symme-try and extended symmetry groups, involves putting the non-trivial columns on the faces of R and labelling all columns in the interior to be the identity. Let G and P be given. From Theo-rem 13, there exists a substitution on A with S ( X (cid:37) ) = Z × G . Let (cid:37) , . . . , (cid:37) r − be the non-trivialcolumns of (cid:37) . Pick L to be large enough such that W d acts freely on the faces of R = [0 , L − d .Choose j ∈ R and consider the orbit of j under P , i.e., O := P · j = { A · j | A ∈ P } where A · j = A (1) ( j ) as in Eq. (11) . Label all the columns in O with (cid:37) . We then expand R via Q = diag( L, . . . , L ) to get the d -dimensional cube Q ( R ) of side length L . Consider B := Q ( O ) + R , pick j ∈ B and let O = P · j . Relabel all columns in B \ O with (cid:37) and ll columns in O with (cid:37) . One can continue this process until all needed column labels appear;see Figure 6 for a two-dimensional example. (a) O in blue (b) B \ O inblue, O in red (c) B \ O ingreen Figure 6.
An example in Z with three non-trivial columns (cid:37) (blue), (cid:37) (red)and (cid:37) (green). Here, one has G = cent S |A| (cid:104) (cid:37) , (cid:37) , (cid:37) (cid:105) and P (cid:39) V , where V (cid:54) D = W is the Klein-4 group.Note that one has (cid:37) i = (cid:37) A (1) ( i ) for all A ∈ P and i ∈ R = [0 , L − d by construction, whichmeans π = id gives rise to an element of R ( X (cid:37) ) for all A ∈ P by Theorem 30. No other extendedsymmetries can occur because all the location sets B i only contain non-trivial labels and are P -invariant, whereas if A / ∈ P induces an extended symmetry, one must have (cid:37) (cid:96) = id for some (cid:96) ∈ B r .The resulting block substitution is primitive, since reordering the columns does not affectprimitivity. It is also aperiodic because one has enough identity columns, and hence one canfind the legal words required in Proposition 33. For example, in the constructed substitution inFigure 6, the legal seeds can be derived from the 2 × S ( X (cid:37) ) (cid:39) Z d × G and R ( X (cid:37) ) (cid:39) ( Z d (cid:111) P ) × G . ♦ We now turn our attention to examples where the letter-exchange map π that generates f ∈ R ( X (cid:37) ) is not given by the identity. In particular, in these examples, π does not commutewith the letter-exchanges which correspond to the standard symmetries in S ( X (cid:37) ). To avoidconfusion, we will use letters for our substitution and the action of the hyperoctahedral groupwill be given by numbers, seen as permutations of the coordinates. Mirroring along a hyperplanewill be denoted by m i , where i is the respective coordinate. Example 37.
We explicitly give a substitution whose symmetry group is S ( X ε ) = Z d × C and build another C component in R ( X ε ), which produces reversors of order 9. With therequirement on R ( X ε ) / S ( X ε ), the space has to be at least of dimension 3. ε = ( a d g )( b e h )( c f i ) ε = ( a b c )( d e f )( g h i ) ε = id ε = ( a g d )( b h e )( c i f ) ε = ( b c d )( e f g )( h i a ) ε = ( c d e )( f g h )( i a b ) ere one has S ( X ε ) = Z × C , which is generated by ( a d g )( b e h )( c f i ). Depending on thepositioning of the columns, R ( X ε ) can either be Z (cid:111) C , Z (cid:111) C × C or Z × C . The group Z (cid:111) C × C can be realised using the construction from Theorem 35. On the other hand, Z × C is obtained if one orbit of maximal size is labelled with just one non-identity ε i once,and the rest with ε .Note that π = ( a b c d e f g h i ) sends ε → ε → ε → ε and ε → ε , ε → ε . Takingthe cube of ( a b c d e f g h i ) gives ( a d g )( b e h )( c f i ) ∈ cent S ( G (1) ), where G (1) is the groupgenerated by the columns. This is consistent with the bounds calculated in Proposition 27.We will illustrate the positioning of a few elements following the construction in Theorem 35.We look at a position that has the maximum orbit size under W , for example (0 , , ∈ R .The orbit under C is (0 , , , (1 , , , (2 , , ε at position (0,1,2), ε at position (1,2,0) and ε at position (2,0,1).Since ε , ε ∈ cent S ( G ), we will position them each along a different orbit. All remainingpositions will be filled with the identity to ensure that we cannot have additional symmetries.We use Proposition 33 to ensure aperiodicity. It is easy to see that one gets the required patchesby choosing the 2 × × R ( X ε ) = Z (cid:111) C . ♦ Figure 7.
The gray cubes are filled with ε (the identity). Yellow and browncan be filled by either ε , ε , respectively. Lastly, ε is blue, ε is green and ε is red, where one has the obvious freedom in choosing the colours due to the C -symmetry. Remark 38.
As a generalisation of Example 37, for any given cyclic groups C n and C k , we canconstruct a substitution (cid:37) in Z n , such that X (cid:37) has the symmetry group Z n × C k and its extendedsymmetry group is given by ( Z n × C k ) (cid:111) C n . More precisely, since the extended symmetry groupcontains an element of order nk , R ( X (cid:37) ) = Z n (cid:111) C nk . The substitution can be realised by thefollowing columns ε = ( a a k +1 · · · a ( n − k +1 ) · · · ( a k · · · a nk ) ε i = ( a i a i +1 · · · a k − i )( a k + i a k + i +1 · · · a k + i − ) · · · ( a ( n − k + i a ( n − k + i +1 · · · a nk − i ) ε n +1 = idwhere i runs from 1 to n , where the values are seen modulo nk .From the columns ε i with i (cid:54) = 0 we can see that the centraliser can only be the permutationof the cycles limiting the centraliser to S k , while ε limits it further to be C k , since this copyof S k operates on the cycles independently and the centraliser of a cycle is just the cycle itself. he extended symmetry is realised by the permutation ( a · · · a kn ) which maps ε i to ε i +1 . Itsorbit is determined by the action of C n (cid:54) W n on the positioning of the columns. ♦ In the next example we illustrate how important it is to choose compatible structures for theletter-exchange map and the corresponding action in W d . Example 39.
We look at a four-letter alphabet with the following columns in Eq. (13) whichgenerate S as a subgroup of S , thus implying that the shift space to have a trivial centraliser.We plan to have S (cid:39) R ( X ε ) / S ( X ε ), so we place the columns in a three-dimensional cube. ε = id ε = ( a b c d ) ε = ( a c d b ) ε = ( a b d c ) ε = ( a d b c )(13) ε = ( a c b d ) ε = ( a d c b )The symmetry group is trivial since the columns generate S . Conjugation with τ = ( c d ) maps ε to ε , just as any τ κ , with κ ∈ cent S ( ε ). Figure 8.
The columns assigned to the colors are as follows: ε (blue), ε (yellow), ε (green) ε (purple), ε (black) and ε (red).Here C (cid:111) C (cid:39) S is realised by ( b c d )(0 1 2) and ( c d )(0 1). The transposition ( c d ) cannotbe realised in W d by mirroring along an axis in the cube since that is not consistent with theinteraction between ( b c d ) and ( c d ). This can be easily be seen by looking at mirroring alongall hyperplanes. ( a b c d )(2 , ,
0) ( a c d b )(0 , , a b d c )(1 , ,
3) ( a d c b ) / ( a c b d )(3 , , ( b c d )(0 1 2) m ( c d ) m ( c d )( b c d )(0 1 2) We see that the diagram does not commute, thus there is no way to assign a single column tothe vertex (3, 1, 2). One can do this for all axes, which rules out the C component in W , thusyielding R ( X (cid:37) ) = Z d (cid:111) S . ♦ emark 40. One can also ask whether, starting with a group G , one can build the centraliser S ( X (cid:37) ) and normaliser R ( X (cid:37) ) organically from G , under a suitable embedding of G . Considerthe Cayley embedding G (cid:44) → S | G | as in Example 12. We know that cent S | G | ( G ) (cid:39) G andnorm S | G | ( G ) (cid:39) G (cid:111) Aut( G ); see [Seh89]. Since the automorphisms of G are given by conjugationin S | G | , they define letter-exchange maps which are compatible with reversors in R ( X (cid:37) ). Bychoosing the dimension appropriately, one can construct a substitution (cid:37) on A = G such thatthe extended symmetry group is given by R ( X (cid:37) ) = (cid:0) Z d ( G ) × G (cid:1) (cid:111) Aut( G ) , where we choose d ( G ) such that Aut( G ) (cid:54) W d ( G ) . This can always be done for d ( G ) = | G | , butdepending on Aut( G ), a smaller dimension is possible. Let π ∈ Aut( G ) and let A π ∈ W d . Theconstruction from the proof of Theorem 35 can be applied. Here, the orbits of A π will not befilled with the same element, but with columns that are determined by π , i.e., (cid:37) A π ( i ) = π ◦ (cid:37) i ◦ π − ,where π is seen as an element of S | G | . ♦ These series of examples with more complicated structure can be generalised for arbitrarygroups G and P . Here, we have the following version of Theorem 35 where the letter exchangemap is no longer π = id, which we build from a specific set of columns. Theorem 41.
Let
H, P be arbitrary finite groups. Then for all (cid:96) (cid:62) c ( P ) , where c ( P ) is aconstant which depends only on the group P , there is a shift space X (cid:37) originating from anaperiodic, primitive and bijective substitution (cid:37) such that S ( X (cid:37) ) = Z (cid:96) × H R ( X (cid:37) ) = ( Z (cid:96) × H ) (cid:111) P. Proof.
The proof will be divided into two parts, beginning with a manual for the constructionof the substitution and a second part where we verify the claims made in the construction andcheck if the subshift has the desired properties. • We first turn our attention to the construction of P which later is supposed to beisomorphic to R ( X (cid:37) ) / S ( X (cid:37) ). For that purpose we embed P (cid:44) → S (cid:96) which is certainlypossible for some (cid:96) . It is clear that there is a minimal c ( P ) ∈ N for which this embeddingis possible, and that every (cid:96) (cid:62) c ( P ) gives a valid embedding as well. This means thechoice of (cid:96) has a lower bound, but can be increased arbitrarily. This chosen (cid:96) determinesthe dimension of the space Z (cid:96) where the subshift is constructed. Let us now fix a suitable (cid:96) , excluding (cid:96) = 2 , , S (cid:96) ( S (cid:96) ) = Inn S (cid:96) ( S (cid:96) ) (cid:39) S (cid:96) which doesnot hold for these values of (cid:96) ; see [Seg40]. • Next, we look for suitable columns for our substitution. Choose the set T = { (cid:15) , · · · (cid:15) k } of all transpositions in S (cid:96) , together with the identity column as the set of columns. T generates S (cid:96) and the action of S (cid:96) (viewed as the automorphism group) acts faithfullyon T . From this, we get that P ⊂ S (cid:96) (cid:39) Inn S (cid:96) ( S (cid:96) ) ⊂ norm S (cid:96) ( { (cid:15) , · · · , (cid:15) k } ). This isenough for now, since P ⊂ norm S (cid:96) ( { (cid:15) , · · · , (cid:15) k } ) and we can exclude the surplus later. • Now, we compute the centraliser of the column group. In our current constructionthe centraliser is trivial, which is why we need to modify our columns. We do this byextending our alphabet { a, · · · , (cid:96) } to { a , · · · , a | H | , b , · · · , b | H | , · · · , (cid:96) , · · · , (cid:96) | H | } . Wesimply duplicate the cycles in each column: The permutations of the columns aremapped by ρ → ρ (cid:48) sending (cid:15) i = ( x y ) (cid:55)→ ε i = ( x y ) · · · ( x | H | y | H | ). We embed
G (cid:44) → S | H | with the usual Cayley embedding. This group is only acting onthe indices of the letters in the new alphabet. The action on the indices is applied toevery { a, . . . , (cid:96) } , giving the final set of columns { η , . . . , η m } added to the substitution ρ (cid:48) giving a new substitution (cid:37) . • The Cayley embedding guarantees that cent S | H | ( G (cid:37) ) (cid:39) H , where G (cid:37) is the columngroup of (cid:37) . We can decrease the size of R ( X (cid:37) ) / S ( X (cid:37) ) with the same arguments as inTheorem 35. This way we achieve a group R ( X (cid:37) ) / S ( X (cid:37) ) (cid:39) P where the letter exchangecomponent π of the extended symmetries are not in cent S | H | ( G (cid:37) ).Aperiodicity of X (cid:37) can be easily obtained via proximal pairs. Regarding primitivity, it issufficient to check the transitivity of G (cid:37) and use Proposition 3. For any pair ( x j , y k ) of letterswith indices chosen from the alphabet we need to find a g ∈ G (cid:37) such that gx j = y k . Note thatthe permutation ( x y ) · · · ( x j y j ) · · · ( x | H | y | H | ) ∈ G (cid:37) and maps x j to y j . Now we need to map y j to y k , which is an action solely on the indices. The mapping on the indices can be realizedby the right embedding copy of H in S | H | and thus by an element composed of the columns { η , · · · , η m } .Let us prove that the centraliser is indeed isomorphic to G . The centraliser of G { ε , ··· ε k } canonly contain elements that are pure index permutations, since those columns generate S (cid:96) . Sincethe structure of the cycles in each column are independent of the index, any index permutationis an element of cent S (cid:96) ( G { ε , ··· ε k } ) = S (cid:96) .We continue by determining cent S (cid:96) | H | ( G { η , ··· ,η m } ) (cid:84) S (cid:96) . The group S (cid:96) are the pure indexswitches and since η , · · · , η m are the columns generated by the Cayley embedding of H into S (cid:96) their centraliser is isomorphic to H .The following rule lifts an automorphism h (cid:48) on G ρ to h on G (cid:37) . h ( (cid:15) i ) = h (cid:48) ( (cid:15) ) i Thus S | H | (cid:54) Aut S (cid:96) | H | ( G { ε , ··· ,ε k } ). It is sufficient to prove that the automorphism group did notdecrease in size by the addition of the columns ( η , · · · , η m ). Then we can use the geometricplacement of the columns in Theorem 35 in the substitution to exclude any unwanted W d -component. Any lifted automorphism h still only maps the letters and fixes the indices. Sincethe cycles in any η , · · · , η m contain only the same letter with different indices and the indexstructure is independent of the letter, every h is in cent S | H | (cid:96) ( G { η , ··· ,η m } ) and surely legal. Thusit is an automorphism on the whole of G (cid:37) . (cid:3) Remark 42.
Theorems 35 and 41 fall under realisation theorems for shift spaces. The mostgeneral current result along this vein known to the authors is that of Cortez and Petite, whichstates that every countable group G can be realised as a subgroup G (cid:54) R ( X , Γ ), where R ( X , Γ )is the normaliser of the action of a free abelian group Γ on an aperiodic minimal Cantor space X ; see [CP20]. ♦ Concluding remarks
While the higher-dimensional criteria in Theorems 30 and 28, which confirm or rule out theexistence of extended symmetries, are rather general, it remains unclear how to find a way toextend this to a larger (possibly all) class of systems, with no constraints on the geometry of thesupertiles. This is related to a question of determining whether, given a substitution in Z d (or R d ), one can come up with an algorithm which decides whether there is a simpler substitutionwhich generates the same or a topologically conjugate hull, which is easier to investigate. This s exactly the case for the two-dimensional Thue–Morse substitution in Figure 2. Such an issueis non-trivial both in the tiling and the subshift context; see [CD08, DL18, HRS05].Note that the letter-exchange map π ∈ S |A| in Theorem 30 always induces a conjugacybetween columns whenever it generates a valid reversor. It would be interesting to know whetherouter automorphisms in this case can yield valid reversors for a bijective substitution subshiftin Z d , for example for those whose geometries are not covered by Theorems 30 and 28. Forinstance, Aut( S ) contains elements which are not realised by conjugation.Another natural question would be to determine other possibilities for S ( X (cid:37) ) and R ( X (cid:37) )outside the class of bijective, constant-length substitutions. Here, the higher-dimensional gen-eralisations of the Rudin–Shapiro substitution would be good candidates; see [Fra03]. Thereare also substitutive planar tilings with |R ( X ) / S ( X ) | = D , which arises from the hexagonalsymmetry satisfied by the underlying tiling. For these classes, and in the examples treatedabove, the simple geometry of the tiles introduces a form of rigidity which leads to R ( X ) beinga finite extension of S ( X ); see [BRY18, Sec. 5] for the notion of hypercubic shifts. There aresubstitution tilings whose expansive maps Q are no longer diagonal matrices, and whose super-tiles have fractal boundaries; compare [Fra20, Ex. 12], which allows more freedom in terms ofadmissible elements of GL( d, Z ) which generate reversors. This raises the following question: Question 43.
What is the weakest condition on the shift space/tiling dynamical system X which guarantees [ R ( X ) : S ( X )] < ∞ ?This is always true in one dimension regardless of complexity, since either the subshift isreversible or not, but is non-trivial in higher dimensions because | GL( d, Z ) | = ∞ for d >
1, soinfinite extensions are possible; see [BBH+19]. We suspect that this is connected to the notionsof linear repetitivity, finite local complexity, and rotational complexity; compare [BRY18, Cor. 4]and [HRS05]. For inflation systems, the compatibility condition [
A, Q ] = 0 in Theorem 30 mightalso be necessary in general when the maximal equicontinuous factor (MEF) has an explicit form.5.
Acknowledgements
The authors would like to thank Michael Baake for fruitful discussions and for valuablecomments on the manuscript. AB is grateful to ANID (formerly CONICYT) for the finan-cial support received under the Doctoral Fellowship ANID-PFCHA/Doctorado Nacional/2017-21171061. NM would like to acknowledge the support of the German Research Foundation(DFG) through the CRC 1283.
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Email address : [email protected] Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld,Postfach 100131, 33501 Bielefeld, Germany
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