Automaticity and invariant measures of linear cellular automata
AAUTOMATICITY AND INVARIANT MEASURESOF LINEAR CELLULAR AUTOMATA
ERIC ROWLAND AND REEM YASSAWI
Abstract.
We show that spacetime diagrams of linear cellular automataΦ : F Z p → F Z p with ( − p )-automatic initial conditions are automatic. This ex-tends existing results on initial conditions which are eventually constant. Eachautomatic spacetime diagram defines a ( σ, Φ)-invariant subset of F Z p , where σ is the left shift map, and if the initial condition is not eventually periodic thenthis invariant set is nontrivial. For the Ledrappier cellular automaton we con-struct a family of nontrivial ( σ, Φ)-invariant measures on F Z . Finally, given alinear cellular automaton Φ, we construct a nontrivial ( σ, Φ)-invariant measureon F Z p for all but finitely many p . Introduction
In this article, we study the relationship between p -automatic sequences andspacetime diagrams of linear cellular automata over the finite field F p , where p isprime. For definitions, see Section 2.There are many characterisations of p -automatic sequences. For readers familiarwith substitutions, Cobham’s theorem [18] tells us that they are codings of fixedpoints of length- p substitutions. In an algebraic setting, Christol’s theorem tellsus that they are precisely those sequences whose generating functions are algebraicover F p ( x ). In [35], we characterise p -automatic sequences as those sequences thatoccur as columns of two-dimensional spacetime diagrams of linear cellular automataΦ : F Z p → F Z p , starting with an eventually periodic initial condition.We investigate the nature of a spacetime diagram as a function of its initialcondition, when the initial condition is p -automatic. In the special case when theinitial condition is eventually 0 in both directions and the cellular automaton hasright radius 0, this question has been thoroughly studied in a series of articles byAllouche, von Haeseler, Lange, Petersen, Peitgen, and Skordev [5, 6, 7]. Amongstother things, the authors show that an N × N -configuration which is generated bya linear cellular automaton, whose right radius is 0, and an eventually 0 initialcondition, is [ p, p ] -automatic . In [31], Pivato and the second author have alsostudied the substitutional nature of spacetime diagrams of more general cellularautomata with eventually periodic initial conditions.In Sections 3 and 4 we extend these previous results by relaxing the constraintsimposed on the initial conditions and the cellular automata. We allow initial condi-tions to be bi-infinite ( − p )-automatic sequences or, equivalently, concatenations of Date : February 19, 2020.2010
Mathematics Subject Classification. a r X i v : . [ c s . F L ] F e b ERIC ROWLAND AND REEM YASSAWI two p -automatic sequences. Iterating Φ produces a Z × N -configuration, and we showin Theorem 3.10, Theorem 3.14, and Corollary 3.15, that such spacetime diagramsare automatic, with two possible definitions of automaticity: either by shearing a configuration supported on a cone or by considering [ − p, p ]-automaticity. Ourresults are constructive, in that given an automaton that generates an automaticinitial condition, we can compute an automaton that generates the spacetime dia-gram. We perform such a computation in Example 3.11, which we use as a runningexample throughout the article. While the spacetime diagram has a substitutionalnature, the alphabet size makes the computation of this substitution by hand in-feasible, and indeed difficult even using software.We can also extend a spacetime diagram backward in time to obtain a Z × Z -configuration where each row is the image of the previous row under the actionof the cellular automaton. In Lemma 4.2 we show that the initial conditions thatgenerate a Z × Z -configuration are supported on a finite collection of lines. InTheorem 4.5, we show that if the initial conditions are chosen to be p -automatic,then the resulting spacetime diagram is a concatenation of four [ p, p ]-automaticconfigurations.Apart from the intrinsic interest of studying automaticity of spacetime diagrams,one motivation for our study is a search for closed nontrivial sets in F Z p which areinvariant under the action of both the left shift map σ and a fixed linear cellularautomaton Φ. Analogously, we also search for measures µ on one-dimensionalsubshifts ( X, σ ) that are invariant under the action of both σ and Φ.We give a brief background. Furstenberg [22] showed that any closed subset of theunit interval I which is invariant under both maps x (cid:55)→ x mod 1 and x (cid:55)→ x mod 1must be either I or finite. This is an example of topological rigidity . Furstenbergasked if there also exists a measure rigidity , i.e. if there exists a nontrivial measure µ on I which is invariant under these same two maps. By “nontrivial” we meanthat µ is neither the Lebesgue measure nor finitely supported. This question isknown as the ( × , ×
3) problem.The ( × , ×
3) problem has a symbolic interpretation, which is to find a measureon F N which is invariant under both σ , which corresponds to multiplication by 2,and the map u (cid:55)→ u + σ ( u ), which corresponds to multiplication by 3 and where+ represents addition with carry. One can ask a similar question for σ and the Ledrappier cellular automaton u (cid:55)→ u + σ ( u ), where + represents coordinate-wiseaddition modulo 2. One way to produce such measures is to average iterates, underthe cellular automaton, of a shift-invariant measure, and to take a limit measure.Pivato and the second author [30] show that starting with a Markov measure, thisprocedure only yields the Haar measure λ . Host, Maass, and Martinez [23] showthat if a ( σ, Φ)-invariant measure has positive entropy for Φ and is ergodic for theshift or the Z -action then µ = λ . The problem of identifying which measuresare ( σ, Φ)-invariant is an open problem; see for example Boyle’s survey article [14,Section 14] on open problems in symbolic dynamics or Pivato’s article [29, Section3] on the ergodic theory of cellular automata.In Sections 5 and 6 we apply results of Sections 3 and 4 to find ( σ, Φ)-invariantsets and measures. Spacetime diagrams generate subshifts (
X, σ , σ ), where σ and σ are the left and down shifts, and these subshifts project to closed sets in F Z p that are ( σ, Φ)-invariant. Similarly, we show in Proposition 6.1 that ( σ , σ )-invariant measures on X project to ( σ, Φ)-invariant measures supported on a subset
UTOMATICITY AND INVARIANT MEASURES OF LINEAR CELLULAR AUTOMATA 3 of F Z p . Einsiedler [21] constructs, for each s in the interval 0 ≤ s ≤
1, a ( σ , σ )-invariant set and a ( σ , σ )-invariant measure whose entropy in any direction is s times the maximal entropy in that direction. He builds invariant sets using intersection sets as described in Section 5.2 and asks if every ( σ , σ )-invariant setis an intersection set. He also asks for the nature of the invariant measures. Weshow in Theorem 5.8 that each automatic spacetime diagram generates a ( σ, Φ)-invariant set of small (one-dimensional) word complexity. If we assume that theinitial condition is not spatially periodic and the cellular automaton is not a shift,we show in Proposition 5.3 that these sets are nontrivial. The invariant sets wefind are not obviously intersection sets.The quest for nontrivial ( σ, Φ)-invariant measures appears to be more delicate.Let ( X U , σ , σ ) be a subshift generated by a [ − p, − p ]-automatic configuration U .Theorem 6.11 states that the measures supported on such subshifts are convex com-binations of measures supported on codings of substitution shifts. We show in The-orem 5.2 that U has at most polynomial complexity. Therefore the ( σ, Φ)-invariantmeasures guaranteed by Proposition 6.1 are not the Haar measure. However theymay be finitely supported: the shift X U generated by a nonperiodic spacetime dia-gram U can contain periodic points on which a shift-invariant measure is supported.In Theorems 6.13 and 6.15 we identify cellular automata and nonperiodic initialconditions that yield two-dimensional shifts containing constant configurations.We show in Corollary 6.3 that spacetime diagrams that do not contain largeone-dimensional repetitions support nontrivial ( σ, Φ)-invariant measures, and inTheorem 6.2 we show that this condition is decidable. In Theorem 6.4 we showthat for the Ledrappier cellular automaton there exists a family of substitutions allof whose spacetime diagrams, including our running example, support nontrivialmeasures. In Theorem 6.9, we generalise this last proof, showing that for any linearcellular automaton Φ, nontrivial ( σ, Φ)-invariant measures exist for all but finitelymany primes p . Given Φ : F Z p → F Z p , to what extent it is the case that a random p -automatic initial condition generates a nontrivial ( σ, Φ)-invariant measure? Thisremains open.We are indebted to Allouche and Shallit’s classical text [4], referring to proofstherein on many occasions, which carry through in our extended setting. In Sec-tion 2, we provide a brief background on linear cellular automata, larger ringsof generating functions in two variables, and p - and ( − p )-automaticity. In Sec-tion 3 we prove that Z × N -indexed spacetime diagrams are automatic if we startwith automatic initial conditions. In Section 4 we extend these results to include Z × Z -indexed spacetime diagrams. In Section 5 we show that automatic spacetimediagrams for Φ yield nontrivial ( σ, Φ)-invariant sets and discuss their relation tothe intersection sets defined by Kitchens and Schmidt [25]. Finally in Section 6, westudy ( σ, Φ)-invariant measures supported on automatic spacetime diagrams.2.
Preliminaries
Linear cellular automata.
Let A be a finite alphabet. An element in A Z iscalled a configuration and is written u = ( u m ) m ∈ Z . The (left) shift map σ : A Z →A Z is the map defined as ( σ ( u )) m := u m +1 . Let A be endowed with the discretetopology and A Z with the product topology; then A Z is a metrisable Cantor space.A (one-dimensional) cellular automaton is a continuous, σ -commuting map Φ : A Z → A Z . The Curtis–Hedlund–Lyndon theorem tells us that a cellular automaton ERIC ROWLAND AND REEM YASSAWI is determined by a local rule f : there exist integers (cid:96) and r with − (cid:96) ≤ r and f : A r + (cid:96) +1 → A such that, for all m ∈ Z , (Φ( u )) m = f ( u m − (cid:96) , . . . , u m + r ). Let N denote the set of non-negative integers. Definition 2.1.
Let Φ : A Z → A Z be a cellular automaton and let u ∈ A Z . If U ∈ A Z × N satisfies U | Z ×{ } = u and Φ( U | Z ×{ n } ) = U | Z ×{ n +1 } for each n ∈ N , wecall U = ST Φ ( u ) the spacetime diagram generated by Φ with initial condition u .For the cellular automata in this article, A = F p . The configuration space F Z p forms a group under componentwise addition; it is also an F p -vector space. Definition 2.2.
A cellular automaton Φ : F Z p → F Z p is linear if Φ is an F p -linearmap, i.e. (Φ( u )) m = α − (cid:96) u m − (cid:96) + · · · + α u m + · · · + α r u m + r for some nonnegativeintegers (cid:96) and r , called the left and right radius of Φ. The generating polynomial [6]of Φ, denoted φ , is the Laurent polynomial φ ( x ) := α − (cid:96) x (cid:96) + · · · + α + · · · + α r x − r . We remark that our use of φ for the generating polynomial differs from usagein the literature of φ as Φ’s local rule, which is the map ( u m − (cid:96) , . . . , u m + r ) (cid:55)→ α − (cid:96) u m − (cid:96) + · · · + α r u m + r .The generating polynomial has the property that φ ( x ) (cid:80) m ∈ Z u m x m = (cid:80) m ∈ Z (Φ( u )) m x m .We will identify sequences ( u m ) m ∈ Z with their generating function f ( x ) = (cid:80) m ∈ Z u m x m .Recall that F p [ x ] and F p (cid:74) x (cid:75) are the rings of polynomials and power series in thevariable x with coefficients in F p respectively. Let F p ( x ) and F p (( x )) be theirrespective fields of fractions: F p ( x ) is the field of rational functions and F p (( x ))is that of formal Laurent series; elements of F p (( x )) are expressions of the form f ( x ) = (cid:80) m ≥ m u m x m , where u m ∈ F p and m ∈ Z .2.2. Cones. A cone is a subset of Z × Z of the form { v + s v + t v : s ≥ , t ≥ } for some v , v , v ∈ Z × Z such that v and v are linearly independent. The conegenerated by v and v is the cone { s v + t v : s ≥ , t ≥ } .If a cellular automaton is begun from an initial condition u satisfying u m = 0 forall m ≤ −
1, then the spacetime diagram ST Φ ( u ) is supported on the cone generatedby (1 ,
0) and ( − r, r ≥ Φ ( u ) as a formalpower series in some ring. We follow the geometric exposition given by AparicioMonforte and Kauers [8].By definition, a cone C is line-free , that is, for every n ∈ C \ { (0 , } , we have − n (cid:54)∈ C . This places us within the scope of [8].For each cone C , let F p, C (cid:74) x, y (cid:75) be the set of all formal power series in x and y ,with coefficients in F p , whose support is in C . Then (ordinary) multiplication oftwo elements in F p, C (cid:74) x, y (cid:75) is well defined, and the product belongs to F p, C (cid:74) x, y (cid:75) ; infact F p, C (cid:74) x, y (cid:75) is an integral domain [8, Theorems 10 and 11].Let (cid:22) be the reverse lexicographic order on Z × Z , i.e. ( m , n ) (cid:22) ( m , n ) if n
0) : m ≥ } is compatible with (cid:22) . Let F p, (cid:22) (cid:74) x, y (cid:75) := (cid:91) C compatible with (cid:22) F p, C (cid:74) x, y (cid:75) . UTOMATICITY AND INVARIANT MEASURES OF LINEAR CELLULAR AUTOMATA 5
Figure 1.
Spacetime diagram ST Φ ( u ) for a cellular automatonwith generating polynomial φ ( x ) = x − + x − + x − ∈ F [ x ]. Thedimensions are 511 × u m ) m ≥ of the initial condition is the Thue–Morse sequence (thefixed point beginning with 0 of 0 → , → u m = 0 forall m ≤ −
1. By Theorem 3.5, this spacetime diagram, restrictedto the cone generated by the vectors (1 ,
0) and ( − , F p, (cid:22) (cid:74) x, y (cid:75) is a ring contained in the field (cid:83) ( m,n ) ∈ Z × Z x m y n F p, (cid:22) (cid:74) x, y (cid:75) [8, Theo-rem 15]. This field also contains the field F p ( x, y ) of rational functions. Researchersworking with automatic sequences have previously worked with F p, (cid:22) (cid:74) x, y (cid:75) [1, 3].2.3. Automatic initial conditions.
Next we define automatic sequences, whichwe will use as initial conditions for spacetime diagrams.
Definition 2.3. A deterministic finite automaton with output (DFAO) is a 6-tuple( S , Σ , δ, s , A , ω ), where S is a finite set (of states), s ∈ S (the initial state), Σ is afinite alphabet (the input alphabet), A is a finite alphabet (the output alphabet), ω : S → A (the output function), and δ : S × Σ → S (the transition function).In this article, our output alphabet is A = F p .The function δ extends in a natural way to the domain S × Σ ∗ , where Σ ∗ isthe set of all finite words on the alphabet Σ. Namely, define δ ( s, m (cid:96) · · · m m ) := δ ( δ ( s, m ) , m (cid:96) · · · m ) recursively. If Σ = { , . . . , p − } , this allows us to feed thestandard base- p representation m (cid:96) · · · m m of an integer m into an automaton,beginning with the least significant digit. (Recall that the standard base- p repre-sentation of 0 is the empty word.) All automata in this article process integers byreading their least significant digit first.A sequence ( u m ) m ≥ of elements in F p is p -automatic if there is a DFAO ( S , { , . . . , p − } , δ, s , F p , ω ) such that u m = ω ( δ ( s , m (cid:96) · · · m m )) for all m ≥
0, where m (cid:96) · · · m m is the standard base- p representation of m .Similarly, we say that a sequence ( U m,n ) ( m,n ) ∈ N × N is [ p, p ] -automatic if there isa DFAO ( S , { , . . . , p − } , δ, s , F p , ω ) such that U m,n = ω ( δ ( s , ( m (cid:96) , n (cid:96) ) · · · ( m , n )( m , n ))) ERIC ROWLAND AND REEM YASSAWI
Figure 2.
Spacetime diagram for a cellular automaton with gen-erating polynomial φ ( x ) = x +1+ x − ∈ F [ x ]. The dimensions are511 × u m ) m ≥ of the initial condition is theThue–Morse sequence, and the left half ( u − m ) m ≥ is the Toeplitzsequence (the fixed point of 0 → , → − , m, n ) ∈ N × N , where m (cid:96) · · · m m is a base- p representation of m and n (cid:96) · · · n n is a base- p representation of n . Here, if m and n have standard base- p representations of different lengths, then we pad, on the left, the shorter represen-tation with leading zeros.As defined, p -automatic sequences are one-sided. To specify a bi-infinite se-quence, we use base − p . Every integer has a unique representation in base − p withthe digit set { , , . . . , p − } [4, Theorem 3.7.2]. For example, 10 is written in base − − − · ( − + 1 · ( − + 1 · ( − + 1 · ( − + 0 · ( − , so its base-( −
2) representation is 11110, and − − − · ( − + 0 · ( − + 1 · ( − + 1 · ( − , so the base-( −
2) representation of − u m ) m ∈ Z is( − p ) -automatic if there is a DFAO ( S , { , . . . , p − } , δ, s , F p , ω ) such that u m = ω ( δ ( s , m (cid:96) · · · m m )) for all m ∈ Z , where m (cid:96) · · · m m is the standard base-( − p )representation of m . A sequence ( u m ) m ∈ Z is ( − p )-automatic if and only if thesequences ( u m ) m ≥ and ( u − m ) m ≥ are p -automatic [4, Theorem 5.3.2].In this article, we use ( − p )-automatic sequences in F Z p as initial conditions forcellular automata. For example, the spacetime diagram in Figure 2 is of a linearcellular automaton begun from a ( − Algebraicity and automaticity of spacetime diagrams
In this section we show that a spacetime diagram obtained by evolving a lin-ear cellular automaton from a ( − p )-automatic initial condition u is automatic in UTOMATICITY AND INVARIANT MEASURES OF LINEAR CELLULAR AUTOMATA 7 several senses. There is a natural notion of the [ p, p ]-kernel of a two-dimensionalconfiguration extending the usual definition. First, if we consider bi-infinite initialconditions that satisfy u m = 0 for all m ≤ −
1, we show in Theorem 3.5 that thegenerating functions of these cone-indexed configurations are algebraic and thatthey have finite [ p, p ]-kernels. Then in Section 3.2 we show that the shear of analgebraic cone-indexed configuration is [ p, p ]-automatic. Finally, in Section 3.3 westudy the [ − p, p ]-automaticity of spacetime diagrams, where the coordinates ( m, n )are processed by reading m in base − p . Specifically, we prove in Corollary 3.15that a spacetime diagram obtained by evolving a linear cellular automaton from ageneral ( − p )-automatic initial condition is [ − p, p ]-automatic.3.1. Algebraicity and finiteness of the [ p, p ] -kernel. Define the [ p, p ] -kernel of U = ( U m,n ) ( m,n ) ∈ Z × N to be the set (cid:110) ( U p e m + i,p e n + j ) ( m,n ) ∈ Z × N : e ≥ , ≤ i ≤ p e − , ≤ j ≤ p e − (cid:111) . The [ p, p ]-kernel of a cone-indexed sequence ( U m,n ) ( m,n ) ∈C is defined by extending U m,n = 0 for all ( m, n ) ∈ ( Z × N ) \ C .Given i, j ∈ { , , . . . , p − } , the Cartier operator Λ i,j : F p, (cid:22) (cid:74) x, y (cid:75) → F p, (cid:22) (cid:74) x, y (cid:75) is defined asΛ i,j (cid:88) ( m,n ) ∈C U m,n x m y n := (cid:88) ( m,n ):( mp + i,np + j ) ∈C U mp + i,np + j x m y n . Let C be a cone. The [ p, p ] -kernel of a power series F ( x, y ) = (cid:80) ( m,n ) ∈C U m,n x m y n ∈ F p, C (cid:74) x, y (cid:75) is the set { Λ i (cid:96) ,j (cid:96) · · · Λ i ,j ( F ( x, y )) : (cid:96) ≥ ≤ i k , j k ≤ p − ≤ k ≤ (cid:96) } . If the sequence ( U m,n ) ( m,n ) ∈C is indexed by a cone, then its [ p, p ]-kernel is theset of all sequences ( V m,n ) ( m,n ) ∈C ∗ where (cid:80) ( m,n ) ∈C ∗ V m,n x m y n belongs to the [ p, p ]-kernel of (cid:80) ( m,n ) ∈C U m,n x m y n . We show in Lemma 3.2 that such C ∗ are compatiblewith (cid:22) .We can define analogously the one-dimensional Cartier operator Λ i : F p (cid:74) x (cid:75) → F p (cid:74) x (cid:75) and also the p -kernel of a one-dimensional power series. Eilenberg’s the-orem [4, Theorem 6.6.2] states that a sequence ( u m ) m ≥ is p -automatic preciselywhen its p -kernel is finite; the same is true for a [ p, p ]-automatic sequence ( U m,n ) ( m,n ) ∈ N × N [4,Theorem 14.4.1]. For a recent extension of Eilenberg’s theorem to automatic se-quences based on some alternative numeration systems, see [26].A power series f ( x ) ∈ F p (cid:74) x (cid:75) is algebraic over F p ( x ) if there exists a nonzeropolynomial P ( x, z ) ∈ F p [ x, z ] such that P ( x, f ( x )) = 0. Similarly, the cone-indexedseries f ( x, y ) ∈ F p, (cid:22) (cid:74) x, y (cid:75) is algebraic over F p ( x, y ) if there exists a nonzero poly-nomial P ( x, y, z ) ∈ F p [ x, y, z ] such that P ( x, y, f ( x )) = 0. We recall Christol’stheorem for one-dimensional power series [16, 17], generalised to two-dimensionalpower series by Salon [36]. Theorem 3.1. (1)
A sequence ( u m ) m ≥ of elements in F p is p -automatic if and only if (cid:80) m ≥ u m x m is algebraic over F p ( x ) . (2) A sequence of elements ( U m,n ) ( m,n ) ∈ N × N in F p is [ p, p ] -automatic if andonly if (cid:80) ( m,n ) ∈ N × N U m,n x m y n is algebraic over F p ( x, y ) . ERIC ROWLAND AND REEM YASSAWI
We refer to [4, Theorems 12.2.5 and 14.4.1] for the proof of Theorem 3.1, where itis shown that the algebraicity of a power series over a finite field is equivalent to theautomaticity of its sequence of coefficients, which is equivalent to the finiteness of its p - or [ p, p ]-kernel. In related work, Allouche, Deshouillers, Kamae, and Koyanagi [3,Theorem 6] show that the coefficients of an algebraic power series in F p (( x )) (cid:74) y (cid:75) is p -automatic.In the next lemma we show that the image of F p, (cid:22) (cid:74) x, y (cid:75) under Λ i,j is indeedcontained in F p, (cid:22) (cid:74) x, y (cid:75) . We show more: although elements of the [ p, p ]-kernel of F ( x, y ) ∈ F p, C (cid:74) x, y (cid:75) do not necessarily belong to F p, C (cid:74) x, y (cid:75) , their indexing sets areone of a finite set of translates of C . Lemma 3.2.
Let r ≥ be an integer, let C be the cone generated by (1 , and ( − r, , and let F ( x, y ) ∈ F p, C (cid:74) x, y (cid:75) . Then every element of the [ p, p ] -kernel of F ( x, y ) is supported on C − ( t, for some ≤ t ≤ r .Proof. Let 0 ≤ i ≤ p −
1, and 0 ≤ j ≤ p −
1. We abuse notation and defineΛ i,j ( C ) := (cid:110)(cid:16) m − ip , n − jp (cid:17) : ( m, n ) ∈ C , m ≡ i mod p, n ≡ j mod p (cid:111) . Let 0 ≤ s ≤ r . Then we claim that Λ i,j ( C − ( s, C − ( t,
0) for some 0 ≤ t ≤ r . Thestatement of the lemma follows from the claim. Let ( m, n ) ∈ Z × Z be a pointsatisfying n ≥ − m − rn ≤ s , m ≡ i mod p , and n ≡ j mod p . Then Λ i,j maps( m, n ) to (cid:16) m − ip , n − jp (cid:17) , which satisfies n − jp ≥ − m − ip − r · n − jp ≤ i + s + rjp ≤ ( p − r + r ( p − p = r + 1 − p . Since − m − ip − r · n − jp is an integer, this implies − m − ip − r · n − jp ≤ t := (cid:106) i + s + rjp (cid:107) and t ≤ r . (cid:3) Example 3.3. If p = 2 and C is generated by (1 ,
0) and ( − , , andΛ , map C to itself. The other Cartier operators map Λ , ( C ) = C − (1 ,
0) andΛ , ( C ) = C − (2 , C − (3 ,
0) arises from Λ , Λ , ( C ) = C − (3 , F p, (cid:22) (cid:74) x, y (cid:75) . The case r = 0 is Salon’stheorem (Part (2) of Theorem 3.1). We omit the proof, since it is a straightforwardgeneralisation of the proofs in [4, Theorems 12.2.5 and 14.4.1]. Theorem 3.4.
Let F ( x, y ) ∈ F p, (cid:22) (cid:74) x, y (cid:75) . Then F ( x, y ) is algebraic over F p ( x, y ) ifand only if F ( x, y ) has a finite [ p, p ] -kernel. Next we prove that a linear cellular automaton begun from a p -automatic initialcondition produces an algebraic spacetime diagram. A special case appears inAllouche et al. [6, Lemma 2], when the initial condition is eventually 0 in bothdirections. The proof in the general case is similar. Theorem 3.5.
Let
Φ : F Z p → F Z p be a linear cellular automaton. If u ∈ F Z p issuch that ( u m ) m ≥ is p -automatic and u m = 0 for all m ≤ − , then the generatingfunction of ST Φ ( u ) is algebraic and so has a finite [ p, p ] -kernel.Proof. Let the generating polynomial of Φ be φ ( x ) := α − (cid:96) x (cid:96) + · · · + α + · · · + α r x − r .Let f u ( x ) ∈ F p (cid:74) x (cid:75) be the generating function of u . The n -th row of ST Φ ( u ) is thesequence whose generating function is the Laurent series φ ( x ) n f u ( x ). Let C be thecone generated by (1 ,
0) and ( − r, U := ST Φ ( u ) is identically 0 on UTOMATICITY AND INVARIANT MEASURES OF LINEAR CELLULAR AUTOMATA 9 ( Z × N ) \ C , so its generating function satisfies F U ( x, y ) ∈ F p, C (cid:74) x, y (cid:75) ⊆ F p, (cid:22) (cid:74) x, y (cid:75) .Also, F U ( x, y ) = ∞ (cid:88) n =0 φ ( x ) n f u ( x ) y n = f u ( x )1 − φ ( x ) y . Since ( u m ) m ≥ is p -automatic, Part (1) of Theorem 3.1 guarantees the existenceof a polynomial P ( x, z ) ∈ F p [ x, z ] such that P ( x, f u ( x )) = 0. Let Q ( x, y, z ) := P ( x, (1 − φ ( x ) y ) z ). Then Q ( x, y, F U ( x, y )) = P ( x, (1 − φ ( x ) y ) F U ( x, y )) = P ( x, f u ( x )) = 0 , so F U ( x, y ) is algebraic. By Theorem 3.4, U = ( U m,n ) ( m,n ) ∈C has a finite [ p, p ]-kernel. (cid:3) In Figure 1 we have an illustration of a spacetime diagram satisfying the condi-tions of Theorem 3.5.Let C be the cone generated by (1 ,
0) and ( − r, Q ( x, y, z ) = 0 satisfied by z = F ( x, y ) ∈ F p, C (cid:74) x, y (cid:75) , is it decidable whether F ( x, y ) is the generating function of ST Φ ( u ) forsome linear cellular automaton Φ? The initial condition u is determined by F ( x, (cid:96) .3.2. Automaticity by shearing. If r ≥
1, then the cone generated by (1 ,
0) and( − r,
1) contains points ( m, n ) where m ≤ −
1. In this section, we feed these indicesinto an automaton by shearing the sequence so that it is supported on N × N . Definition 3.6.
Let C be the cone generated by (1 ,
0) and ( − r, s ≥ shear of a sequence ( U m,n ) ( m,n ) ∈C− ( s, is the sequence ( V m,n ) ( m,n ) ∈ N × N definedby V m,n = U m − s − rn,n for each ( m, n ) ∈ N × N .The next lemma enables us to move between the [ p, p ]-kernel of the generat-ing function (cid:80) ( m,n ) ∈C U m,n x m y n of a cone-indexed sequence and the generatingfunction (cid:80) ( m,n ) ∈ N × N V m,n x m y n of its shear. Lemma 3.7.
Let F ( x, y ) ∈ F p, (cid:22) (cid:74) x, y (cid:75) . Let ≤ i ≤ p − , ≤ j ≤ p − . Then Λ i,j (cid:0) x (cid:96) F ( x, y ) (cid:1) = x − (cid:98) i − (cid:96)p (cid:99) Λ ( i − (cid:96) ) mod p,j ( F ( x, y )) . Proof.
Let (cid:96) (cid:48) = − (cid:106) i − (cid:96)p (cid:107) . Let m, n ∈ Z . We prove the result for the monomial F ( x, y ) = x m y n ; the general result then follows from the linearity of Λ i,j . If n (cid:54)≡ j mod p , then both sides are 0. If n ≡ j mod p , we haveΛ i,j (cid:0) x (cid:96) · x m y n (cid:1) = Λ i,j (cid:0) x (cid:96) + m y n (cid:1) = (cid:40) x (cid:96) + m − ip y n − jp if (cid:96) + m ≡ i mod p (cid:40) x (cid:96) (cid:48) + m − ( i − (cid:96) + p(cid:96) (cid:48) ) p y n − jp if m ≡ i − (cid:96) + p(cid:96) (cid:48) mod p x (cid:96) (cid:48) Λ i − (cid:96) + p(cid:96) (cid:48) ,j ( x m y n )= x (cid:96) (cid:48) Λ ( i − (cid:96) ) mod p,j ( x m y n ) . (cid:3) Note here that for each fixed (cid:96) , the map i (cid:55)→ ( i − (cid:96) ) mod p is a bijection. Example 3.8.
Let p = 3, and let F ( x, y ) ∈ F , (cid:22) (cid:74) x, y (cid:75) . For each j , we haveΛ ,j ( x − F ( x, y )) = Λ ,j ( F ( x, y )) , Λ ,j ( x − F ( x, y )) = Λ ,j ( F ( x, y )) , Λ ,j ( x − F ( x, y )) = x − Λ ,j ( F ( x, y )) . We prove a version of Eilenberg’s theorem for cone-indexed automatic sequences.We show there exists an explicit automaton representation of the shear of a cone-indexed p -automatic sequence using its [ p, p ]-kernel. Theorem 3.9.
Let C be generated by (1 , and ( − r, for some r ≥ . A C -indexedsequence ( U m,n ) ( m,n ) ∈C of elements in F p has a finite [ p, p ] -kernel if and only if itsshear is [ p, p ] -automatic.Proof. Let ( V m,n ) ( m,n ) ∈ N × N be the shear of ( U m,n ) ( m,n ) ∈C . By [4, Theorem 14.2.2], V is [ p, p ]-automatic if and only if its [ p, p ]-kernel is finite. Hence we show that U has a finite [ p, p ]-kernel if and only if V has a finite [ p, p ]-kernel.By Lemma 3.2, every element of the [ p, p ]-kernel of U is supported on C − ( s,
0) for some 0 ≤ s ≤ r . Let W be an element of the [ p, p ]-kernel of U , sup-ported on C − ( s, F ( x, y ) = (cid:80) ( m,n ) ∈C− ( s, W m,n x m y n . Let G n ( x, y ) = x s + rn (cid:80) m ≥− s − rn W m,n x m y n , so that G ( x, y ) = (cid:80) n ≥ G n ( x, y ) is the generatingfunction of the shear of W . Similarly write F n ( x, y ) = (cid:80) m ≥− s − rn W m,n x m y n ;then G n ( x, y ) = x s + rn F n ( x, y ). Fix n ≡ j mod p , and write n = j + kp where k ≥
0. By Lemma 3.7, we haveΛ i,j ( F n ( x, y )) = Λ i,j (cid:0) x − s − rn G n ( x, y ) (cid:1) = x − (cid:98) i + s + rnp (cid:99) Λ ( i + s + rn ) mod p,j ( G n ( x, y ))= x − (cid:98) i + s + rjp (cid:99) − rk Λ ( i + s + rj ) mod p,j ( G j + kp ( x, y )) . Summing over k ≥ i,j ( F ( x, y )) = x − (cid:98) i + s + rjp (cid:99) (cid:88) k ≥ x − rk Λ ( i + s + rj ) mod p,j ( G j + kp ( x, y )) . Therefore the shear of x (cid:98) i + s + rjp (cid:99) Λ i,j ( F ( x, y )) is (cid:88) k ≥ Λ ( i + s + rj ) mod p,j ( G j + kp ( x, y )) = Λ ( i + s + rj ) mod p,j (cid:88) n ≥ G n ( x, y ) = Λ ( i + s + rj ) mod p,j ( G ( x, y )) . Inductively, suppose G ( x, y ) is the generating function of an element of the kernelof V . Then the shear of x (cid:98) i + s + rjp (cid:99) Λ i,j ( F ( x, y )) is an element of the [ p, p ]-kernel of V . Note that Λ i,j ( F ( x, y )) is supported on C − (cid:16)(cid:106) i + s + rjp (cid:107) , (cid:17) .We set up a map κ from the [ p, p ]-kernel of U to the [ p, p ]-kernel of V . Let κ ( U ) = V , and define κ recursively as follows. For each W in the [ p, p ]-kernel of U , let κ (Λ i,j ( W )) = Λ ( i + s + rj ) mod p,j ( κ ( W )) where W is supported on C − ( s, i (cid:55)→ ( i + s + rj ) mod p is a bijection on F p , κ maps { Λ i,j ( W ) : 0 ≤ i, j ≤ p − } surjectively onto { Λ i,j ( κ ( W )) : 0 ≤ i, j ≤ p − } . It follows inductivelythat κ is a surjection from the [ p, p ]-kernel of U to the [ p, p ]-kernel of V . UTOMATICITY AND INVARIANT MEASURES OF LINEAR CELLULAR AUTOMATA 11
Figure 3.
Automata from Example 3.11. The automaton on theleft generates the initial condition u , and the automaton on theright generates the spacetime diagram ST Φ ( u ).If the [ p, p ]-kernel of U is finite, then the surjectivity of κ implies that the [ p, p ]-kernel of V is finite. By Lemma 3.2, κ is at most ( r + 1)-to-one. Therefore if the[ p, p ]-kernel of V is finite then the [ p, p ]-kernel of U has at most r + 1 times as manyelements and is also finite. (cid:3) We can now extend Theorem 3.5.
Theorem 3.10.
Let
Φ : F Z p → F Z p be a linear cellular automaton. If u ∈ F Z p is suchthat ( u m ) m ≥ is p -automatic and u m = 0 for all m ≤ − , then the shear of ST Φ ( u ) is [ p, p ] -automatic.Proof. By Theorem 3.5, ST Φ ( u ) has a finite [ p, p ]-kernel. By Theorem 3.9, weconclude that the shear of ST Φ ( u ) is [ p, p ]-automatic. (cid:3) Example 3.11.
Let p = 3, and let φ ( x ) = x + 1 ∈ F [ x ]. Let ( u m ) m ≥ bethe 3-automatic sequence generated by the automaton on the left in Figure 3,whose first few terms are 001001112 · · · . The size of this automaton makes latercomputations feasible. Let u m = 0 for all m ≤ −
1; then the spacetime diagram U = ST Φ ( u ) is supported on N × N . See Figure 4. We compute an automaton forthe [3 , U | N × N . By Part (1) of Theorem 3.1, we can computea polynomial P ( x, y ) such that P ( x, f u ( x )) = 0. We compute P ( x, y ) = x y + 2 (cid:0) x + x + x + x + x + x (cid:1) y + (cid:0) x + x + x + x + x + 2 x + 2 x + x (cid:1) y + 2 (cid:0) x + x (cid:1) y . Note that this is not the minimal polynomial for f u ( x ), but it is in a convenientform for the subsequent computation. As in the proof of Theorem 3.5, the gener-ating function F U ( x, y ) of U satisfies P ( x, (1 − φ ( x ) y ) F U ( x, y )) = 0. By Part (2)of Theorem 3.1, we can use this polynomial equation to compute an automaton for U | N × N . The resulting automaton has 486 states; minimizing produces an equiva-lent automaton with 54 states. This automaton is shown without labels or edgedirections on the right in Figure 3. These computations were performed with the Mathematica package
IntegerSequences [34].
Figure 4.
Spacetime diagram for a cellular automaton with gen-erating polynomial φ ( x ) = x + 1 ∈ F [ x ]. The initial conditionis generated by the automaton in Example 3.11. The line n = m separates the diagram into two regions; the upper region containsarbitrarily large white patches, and the lower region does not. Thisis because the left half of the initial condition is identically 0. Thedimensions are 511 × Automaticity in base [ − p, p ] . Instead of shearing, we may evaluate an au-tomaton at negative integers by using base − p . This approach gives a variant ofTheorem 3.9 and a notion of automaticity of ST Φ ( u ) for a general ( − p )-automaticinitial condition u . Definition 3.12.
A sequence ( U m,n ) ( m,n ) ∈ Z × N is [ − p, p ] -automatic if there is aDFAO ( S , { , . . . , p − } , δ, s , F p , ω ) such that U m,n = ω ( δ ( s , ( m (cid:96) , n (cid:96) ) · · · ( m , n )( m , n )))for all ( m, n ) ∈ N × N , where m (cid:96) · · · m m is the standard base-( − p ) representationof m and n (cid:96) · · · n n is the standard base- p representation of n , padded with zerosif necessary, as in Section 2.3. Theorem 3.13.
A sequence ( U m,n ) ( m,n ) ∈ Z × N has a finite [ p, p ] -kernel if and onlyif it is [ − p, p ] -automatic.Proof. Define the [ − p, p ]-Cartier operator ¯Λ i,j by¯Λ i,j (cid:0) ( W m,n ) ( m,n ) ∈ Z × N (cid:1) := ( W − pm + i,pn + j ) ( m,n ) ∈ Z × N . Define the [ − p, p ] -kernel of U = ( U m,n ) ( m,n ) ∈ Z × N to be the smallest set containing U that is closed under ¯Λ i,j for all i, j ∈ { , , . . . , p − } . We show that the [ p, p ]-kernel of U is finite if and only if the [ − p, p ]-kernel of U is finite.For a sequence ( W m,n ) ( m,n ) ∈ Z × N , define ρ ( W ) := ( W − m,n ) ( m,n ) ∈ Z × N and σ − ( W ) :=( W m − ,n ) ( m,n ) ∈ Z × N . Let K be the union, over all elements W in the [ p, p ]-kernel of U , of the set (cid:8) W, ρ ( W ) , σ − ( W ) , ρ ( σ − ( W )) (cid:9) . UTOMATICITY AND INVARIANT MEASURES OF LINEAR CELLULAR AUTOMATA 13
We claim that the [ − p, p ]-kernel of U is a subset of K . One verifies that ¯Λ i,j ( K ) ⊆ K : ¯Λ i,j ( W ) = ρ (Λ i,j ( W ))¯Λ i,j ( ρ ( W )) = (cid:40) Λ ,j ( W ) if i = 0 σ − (Λ p − i,j ( W )) if i (cid:54) = 0¯Λ i,j ( σ − ( W )) = (cid:40) ρ ( σ − (Λ p − ,j ( W ))) if i = 0 ρ (Λ i − ,j ( W )) if i (cid:54) = 0¯Λ i,j ( ρ ( σ − ( W ))) = σ − (Λ p − − i,j ( W )) . For example, if i (cid:54) = 0 we have¯Λ i,j ( ρ ( W )) = ¯Λ i,j (cid:0) ( W − m,n ) ( m,n ) ∈ Z × N (cid:1) = ( W − ( − pm + i ) ,pn + j ) ( m,n ) ∈ Z × N = ( W p ( m − p − i,pn + j ) ( m,n ) ∈ Z × N = σ − (cid:0) ( W pm + p − i,pn + j ) ( m,n ) ∈ Z × N (cid:1) = σ − (Λ p − i,j ( W ));the other identities follow similarly. Since U ∈ K , it follows that the [ − p, p ]-kernelof U is a subset of K . Therefore there are at most four times as many elementsin the [ − p, p ]-kernel as in the [ p, p ]-kernel, so if the [ p, p ]-kernel is finite then the[ − p, p ]-kernel is also finite.Similarly, we can emulate Λ i,j by taking the four states W, ρ ( W ) , σ ( W ) , σ ( ρ ( W ))for each element W in the [ − p, p ]-kernel of U , where σ ( W ) := ( W m +1 ,n ) ( m,n ) ∈ Z × N :Λ i,j ( W ) = ρ (¯Λ i,j ( W ))Λ i,j ( ρ ( W )) = (cid:40) ¯Λ ,j ( W ) if i = 0 σ (¯Λ p − i,j ( W )) if i (cid:54) = 0Λ i,j ( σ ( W )) = (cid:40) σ ( ρ (¯Λ ,j ( W ))) if i = p − ρ (¯Λ i +1 ,j ( W )) if i (cid:54) = p − i,j ( σ ( ρ ( W ))) = σ (¯Λ p − − i,j ( W )) . It follows that there are at most four times as many elements in the [ p, p ]-kernelas in the [ − p, p ]-kernel, so if the [ − p, p ]-kernel is finite then the [ p, p ]-kernel is alsofinite.Now we show that the [ − p, p ]-kernel of U is finite if and only if U is [ − p, p ]-automatic. The proof is similar to the usual proof of Eilenberg’s characterisation, asin [4, Theorem 6.6.2]. If the [ − p, p ]-kernel of U is finite, then the automaton whosestates are the elements of the [ − p, p ]-kernel and whose transitions are determinedby the action of ¯Λ i,j is finite; moreover, this automaton outputs U m,n when fed thebase-[ − p, p ] representation of ( m, n ). Conversely, if there is such an automaton,then the [ − p, p ]-kernel is finite since it can be embedded into the set of states ofthe automaton. (cid:3) Theorem 3.14.
Let
Φ : F Z p → F Z p be a linear cellular automaton. If u ∈ F Z p issuch that ( u m ) m ≥ is p -automatic and u m = 0 for all m ≤ − , then ST Φ ( u ) is [ − p, p ] -automatic. Figure 5.
Spacetime diagram for the linear cellular automatonwith generating polynomial φ ( x ) = x + 1 ∈ F [ x ] begun fromthe 3-automatic initial condition described in Example 3.16. Thedimensions are 511 × Proof.
By Theorem 3.5, ST Φ ( u ) has a finite [ p, p ]-kernel. By Theorem 3.13, ST Φ ( u )is [ − p, p ]-automatic. (cid:3) Corollary 3.15.
Let
Φ : F Z p → F Z p be a linear cellular automaton. If u ∈ F Z p is ( − p ) -automatic, then ST Φ ( u ) is [ − p, p ] -automatic.Proof. Consider the two initial conditions · · · u − u − · · · · and · · · · u u · · · .By Theorem 3.14, ST Φ ( · · · · u u · · · ) is [ − p, p ]-automatic. A straightforwardmodification of Theorem 3.14 shows that ST Φ ( · · · u − u − · · · · ) is also [ − p, p ]-automatic. Since Φ is linear, ST Φ ( u ) is the termwise sum of these two spacetimediagrams. The sum of two [ − p, p ]-automatic sequences is automatic; thereforeST Φ ( u ) is [ − p, p ]-automatic. (cid:3) Example 3.16.
As in Example 3.11, let p = 3, let φ ( x ) = x + 1 ∈ F [ x ], andlet ( u m ) m ≥ be 3-automatic sequence generated by the automaton on the left inFigure 3. We extend ( u m ) m ≥ to a ( − u m ) m ∈ Z by setting u m = u − m for all m ≤ −
1. The resulting spacetime diagram is shown in Figure 5.By Corollary 3.15, ST Φ ( u ) is [ − , Φ ( u ), we start with the 54-state automatoncomputed in Example 3.11 for the right half ( U m,n ) ( m,n ) ∈ N × N of the spacetimediagram in Figure 4. We convert this [3 , − , U m,n ) ( m,n ) ∈ Z × N in Figure 4 whoseleft half is identically 0; minimizing produces an automaton M with 204 states.We also need an automaton for the Z × N -indexed spacetime diagram with initialcondition · · · u − u − · · · , shown in Figure 6. The symmetry x − φ ( x ) = φ ( x − )implies that a shear of this diagram is the left–right reflection ( U − m,n ) ( m,n ) ∈ Z × N of the diagram in Figure 4. Since ( U − m,n ) ( m,n ) ∈ Z × N is an element of the [ − , U , we obtain an automaton for ( U − m,n ) ( m,n ) ∈ Z × N simply by changingthe initial state in M to be the state corresponding to this kernel sequence; hence( U − m,n ) ( m,n ) ∈ Z × N is generated by an automaton M (cid:48) with 204 states. Shearing UTOMATICITY AND INVARIANT MEASURES OF LINEAR CELLULAR AUTOMATA 15
Figure 6.
Spacetime diagram whose sum with the diagram inFigure 4 is the diagram in Figure 5.( U − m,n ) ( m,n ) ∈ Z × N produces ( U − m + n,n ) ( m,n ) ∈ Z × N , the spacetime diagram in Fig-ure 6. Using a variant of Theorem 3.9 for the [ − p, p ]-kernel of a Z × N -indexedsequence, we compute an automaton with 204 states for this spacetime diagram.Finally, since u = 0, the product of the automata for ( U m,n ) ( m,n ) ∈ Z × N and( U − m + n,n ) ( m,n ) ∈ Z × N is an automaton for the sum ST Φ ( u ) of the spacetime diagramsin Figures 4 and 6, which is the diagram in Figure 5. The product automaton has204 states, but minimizing reduces this to 1908 states.4. Automaticity of Z × Z -indexed spacetime diagrams In Corollary 3.15, we showed that if u is ( − p )-automatic then the Z × N -configuration ST Φ ( u ) is [ − p, p ]-automatic. Our aim in this section is to extendCorollary 3.15 to Z × Z -configurations. We remark that the results of this sectioncan be further extended to statements about two-dimensional linear recurrenceswith constant coefficients. We also note that Bousquet-M´elou and Petkoˇvsek [13]prove similar results, with different proofs, for linear recurrences on N × N overfields of characteristic 0. Definition 4.1. If U ∈ F Z × Z p satisfies Φ( U | Z ×{ n } ) = U | Z ×{ n +1 } for each n ∈ Z , wecall U a spacetime diagram for Φ.Note that if Φ : F Z p → F Z p is a linear cellular automaton with left and rightradii (cid:96) and r respectively, then it is surjective, and every sequence in F Z p has p (cid:96) + r preimages. Hence if (cid:96) + r ≥ Z × Z -indexed spacetimediagrams U such that U | Z ×{ } = u .Let Φ have generating polynomial φ ( x ) = α − (cid:96) x (cid:96) + · · · + α + · · · + α r x − r . Aconfiguration U = ( U m,n ) ( m,n ) ∈ Z × Z is a spacetime diagram for Φ if and only if(1 − φ ( x ) y ) (cid:88) ( m,n ) ∈ Z × Z U m,n x m y n = 0 . In the following lemma we identify which initial conditions determine a spacetimediagram for Φ.
Figure 7. A Z × Z -indexed spacetime diagram for the Ledrappiercellular automaton. The initial conditions are U m, = T ( m ) for m ≥ U m, = T ( − m ) for m ≤ −
1, and U ,n = T ( − n ) for n ≤ − T ( m ) m ≥ is the Thue–Morse sequence. The dimensions are511 × Lemma 4.2.
Let
Φ : F Z p → F Z p be a linear cellular automaton with generatingpolynomial φ ( x ) = α − (cid:96) x (cid:96) + · · · + α + · · · + α r x − r . Let I = ( Z × { } ) ∪ (cid:96) + r − (cid:91) i =0 ( { i } × − N ) . Then every U ∈ F I p can be uniquely extended to a spacetime diagram U ∈ F Z × Z p for Φ .Proof. Note that U | Z ×{ } uniquely determines a Z × N -indexed spacetime diagramfor Φ. Next we observe that U | ( Z ×{ } ) ∪{ (0 , − ,..., ( (cid:96) + r − , − } determines U | Z ×{− } for Φ. For, given a word w ∈ F (cid:96) + rp , there is a unique sequence v ∈ F Z p such that v · · · v (cid:96) + r − = w and Φ( v ) = U | Z ×{ } . Similarly, U | ( Z ×{− n } ) ∪{ (0 , − n − ,..., ( (cid:96) + r − , − n − } determines U | Z ×{− n − } . We can repeat this, determining one row at a time, oncewe have specified a word of length (cid:96) + r in that row. (cid:3) Example 4.3.
Consider the
Ledrappier cellular automaton Φ, whose generatingpolynomial is φ ( x ) = 1 + x − . By Lemma 4.2, U is determined by its values on( Z × { } ) ∪ ( { } × − N ). See Figure 7 for an example of a spacetime diagram for Φ. UTOMATICITY AND INVARIANT MEASURES OF LINEAR CELLULAR AUTOMATA 17
Definition 3.12 naturally generalises to [ p, q ]-automaticity for any integers p, q with | p | ≥ | q | ≥
2. Therefore we may consider [ − p, − p ]-automaticity. Onecan also define [ p, p ]-automaticity for any of the four quadrants ( ± N ) × ( ± N ). Proposition 4.4.
A sequence U ∈ F Z × Z p is [ − p, − p ] -automatic if and only if eachof U | ( ± N ) × ( ± N ) is [ p, p ] -automatic. The proof of Proposition 4.4 follows the same lines as that of [4, Theorem 5.3.2].
Theorem 4.5.
Let
Φ : F Z p → F Z p be a linear cellular automaton with left and rightradii (cid:96) and r . Let U ∈ F Z × Z p be a spacetime diagram for Φ . If U | { i }×− N is p -automatic for each i in the interval − (cid:96) ≤ i ≤ r − and U | Z ×{ } is ( − p ) -automatic,then U is [ − p, − p ] -automatic.Proof. By Lemma 4.2, U is uniquely determined by its values on ( Z × { } ) ∪ (cid:83) ri = − (cid:96) ( { i } × − N ). By Proposition 4.4 it is sufficient to show that each of thefour quadrants U | ( ± N ) × ( ± N ) is [ p, p ]-automatic.By Corollary 3.15, U | Z × N is [ − p, p ]-automatic. By Theorem 3.13, U | Z × N has afinite [ p, p ]-kernel. Thus each of U | ± N × N has a finite [ p, p ]-kernel. By Theorem 3.9with r = 0, each of U | ± N × N is [ p, p ]-automatic.We show that U | N ×− N is [ p, p ]-automatic; the automaticity of U | − N ×− N followsby a similar argument. Let φ ( x ) = α − (cid:96) x (cid:96) + · · · + α + · · · + α r x − r be the generatingpolynomial of Φ. For S ⊆ Z × Z , let F | S denote the generating function of U | S .Since U is a spacetime diagram for Φ, we have U m,n +1 − (cid:80) ri = − (cid:96) α i U m + i,n = 0 foreach ( m, n ) ∈ Z × Z . Multiplying by x m y n +1 and summing over m ≥ n ≤ − (cid:88) m ≥ n ≤− U m,n +1 x m y n +1 − (cid:88) m ≥ n ≤− r (cid:88) i = − (cid:96) α i U m + i,n x m y n +1 = F | N ×− N − r (cid:88) i = − (cid:96) α i x − i y (cid:88) m ≥ n ≤− U m + i,n x m + i y n = F | N ×− N − − (cid:88) i = − (cid:96) α i x − i y (cid:32) i (cid:88) k = − (cid:96) F | { k }×− N + F | N ×− N − F | N ×{ } − P i ( x ) (cid:33) − α y (cid:0) F | N ×− N − F | N ×{ } (cid:1) − r − (cid:88) i =1 α i x − i y (cid:32) F | N ×− N − i − (cid:88) k =0 F | { k }×− N − F | N ×{ } + P i ( x ) (cid:33) = (1 − φ ( x ) y ) F | N ×− N + φ ( x ) yF | N ×{ } − − (cid:88) i = − (cid:96) α i x − i y (cid:32) i (cid:88) k = − (cid:96) F | { k }×− N (cid:33) + r − (cid:88) i =1 α i x − i y (cid:32) i − (cid:88) k =0 F | { k }×− N + P i ( x ) (cid:33) , where P i ( x ) are Laurent polynomials to account for over- and under-counting. Sinceeach U | { k }×− N and U | N ×{ } is automatic, each F | { k }×− N and F | N ×{ } are algebraicby Part (1) of Theorem 3.1. Hence F | N ×− N = G ( x, y )1 − φ ( x ) y where G ( x, y ) is algebraic. Therefore F | N ×− N is algebraic, and U | N ×− N is [ p, p ]-automatic by Part (2) of Theorem 3.1. (cid:3) Example 4.6.
Consider the Ledrappier cellular automaton with φ ( x ) = 1 + x − ,and let L = N × { } L = { } × − N so that U | L ∪ L determines U | N ×− N for Φ.We have U m,n + U m +1 ,n − U m,n +1 = 0 for each ( m, n ) ∈ Z × Z , so, following theproof and notation of Theorem 4.5, we have0 = F | N ×− N − y ( F | N ×− N − F | L ) − x − y ( F | N ×− N − F | L − F | L + U , )and therefore F | N ×− N = x − yU , − (1 + x − ) yF | L − x − yF | L − (1 + x − ) y . If F | L and F | L are both algebraic, then F | N ×− N is also.As we converted the [ p, p ]-kernel to the [ − p, p ]-kernel in Theorem 3.13, onecan also convert the [ − p, p ]-kernel of a spacetime diagram in Theorem 4.5 to the[ − p, − p ]-kernel. For example, this enables one to compute a [ − p, − p ]-automatonfor the spacetime diagram in Figure 7.5. Invariant sets for linear cellular automata
In this section and the next we apply the automaticity of spacetime diagrams,as shown in Corollary 3.15 and Theorem 4.5, to two related questions in symbolicdynamics. We consider the Z × Z -dynamical system ( F Z p , σ, Φ) generated by the leftshift map σ and a linear cellular automaton Φ, and we find closed subsets of F Z p which are invariant under both σ and Φ. In Section 6 we find nontrivial measures µ on F Z p that are invariant under the action of σ and Φ.By a simple transfer principle, these questions can be approached by consideringdynamical systems generated by spacetime diagrams U for Φ. Given a spacetimediagram U , one considers the subshift ( X U , σ , σ ), a Z × Z -dynamical systemgenerated by U ; this is defined in Section 5.1. If U is automatic, then X U is smallin the sense of Theorem 5.2.The maps σ and Φ do not exhibit the topological rigidity that Furstenberg’ssetting yields, as mentioned in the Introduction. An example of a ( σ, Φ)-invariantset was first pointed out by Kitchens and Schmidt [25, Construction 5.2] and elabo-rated by Einsiedler [21]. In Theorem 5.8 we identify a large family of ( σ, Φ)-invariantsets, and we discuss the relationship between our invariant sets and those that areobtained by the method in [25].5.1.
Subshifts generated by [ − p, − p ] -automatic spacetime diagrams. Inthis section we set up the necessary background, define subshifts generated bya spacetime diagram, and show that the subshift generated by an automatic space-time diagram is small but infinite. We also define substitutions, linking them toautomaticity.We equip F p with the discrete topology and the sets F Z p and F Z × Z p with the metris-able product topology, noting that with this topology they are compact. Let σ : UTOMATICITY AND INVARIANT MEASURES OF LINEAR CELLULAR AUTOMATA 19 F Z × Z p → F Z × Z p denote the left shift map ( U m,n ) ( m,n ) ∈ Z × Z (cid:55)→ ( U m +1 ,n ) ( m,n ) ∈ Z × Z , andlet σ : F Z × Z p → F Z × Z p denote the down shift map ( U m,n ) ( m,n ) ∈ Z × Z (cid:55)→ ( U m,n +1 ) ( m,n ) ∈ Z × Z .With the notation of Section 2.1, applying the left shift (down shift) to a sequenceis equivalent to multiplying its generating function by x − ( y − ). Definition 5.1.
Let S and T be transformations on X . A set Z ⊂ X is T -invariant if T ( Z ) ⊂ Z , and Z is ( S, T ) -invariant if it is both S - and T -invariant. A (two-dimensional) subshift ( X, σ , σ ) is a dynamical system with X a closed, σ - and σ -invariant subset of F Z × Z p .We can similarly define a one-dimensional subshift ( X, σ ): here X is a closed, σ -invariant subset of F Z p and σ is the left shift map. We call X the shift space .Let S ⊆ Z × Z be a rectangle [ m , m ] × [ n , n ]. A word on S is a map w : S → F p .These words are higher-dimensional analogues of words in one dimension, i.e. thoseindexed by a finite interval in Z . If U ∈ F Z × Z p , then U | S is the word ( U m,n ) ( m,n ) ∈ S ,and we say that the word U | S occurs in U . Given a configuration U ∈ F Z × Z p , the language L U of U is the set of all words that occur in U . The language L X of ashift space X is the set of all words that occur in some configuration U ∈ X . A subword of the word w : S → F p is a restriction of w to some rectangular S (cid:48) ⊆ S .The language L X is closed under the taking of subwords, and every word in thelanguage is extendable to a configuration in X . Conversely, a language L on F p which is closed under the taking of subwords defines a (possibly empty) subshift( X L , σ , σ ), where X L is the set of configurations all of whose subwords belongto L .Note that we can also define the language of an N × N - or Z × N -configuration U and, in an analogous manner, of the Z × Z -subshift ( X U , σ , σ ).Let U be a two-dimensional configuration. Recall the complexity function c U : N × N → N , where c U ( m, n ) is the number of distinct m × n words that occur in U . We remark that the second statement of the following theorem can be improvedbut is sufficient for our purposes. Theorem 5.2. (1)
If the sequence U ∈ F N × N p is [ p, p ] -automatic, then for some K , its complex-ity function satisfies c U ( m, n ) ≤ K max { m, n } . (2) If the sequence U ∈ F Z × Z p is [ − p, − p ] -automatic, then for some K , itscomplexity function satisfies c U ( m, n ) ≤ K max { m, n } .Proof. The proof of Part (1) is in [4, Corollary 14.3.2]. See also [2] and [11].To see Part (2), we recall first that, by Proposition 4.4, each of U | ± N ×± N is[ p, p ]-automatic, so by Part (1), for each of them there exists a constant K ± N ×± N such that c U | ± N ×± N ( m, n ) ≤ K ± N ×± N max { m, n } . Let K ∗ be the maximum ofthe four constants K ± N ×± N and let K := ( K ∗ ) . Let w be a rectangular m × n word that occurs in U . If each occurrence of w is entirely contained in one of thequadrants ± N × ± N , then w is counted by the complexity of U restricted to thatquadrant, and this count is bounded above by K max { m, n } . Otherwise, either S is partitioned into two rectangles, each of which lies in a distinct quadrant, or S is partitioned into four rectangles lying in distinct quadrants. The worst case iswhen S is a concatenation of four subrectangles, so we assume this. There are atmost K (cid:80) mi =1 (cid:80) nj =1 max { i, j } max { i, n − j } max { m − i, j } max { m − i, m − j }
20 ERIC ROWLAND AND REEM YASSAWI of these subrectangles, and a crude upper estimate tells us that there are at most K max { m, n } such words. (cid:3) Theorem 5.2 tells us the languages generated by [ − p, − p ]-automatic configura-tions are small. On the other hand, provided that the initial conditions generating U are not periodic, we now also show that they are not too small.Let f u ( x ) = (cid:80) m ∈ Z u m x m be the generating function of u ∈ F Z p and let F U ( x ) = (cid:80) m ∈ Z ,n ∈ Z U m,n x m y n be the generating function of U ∈ F Z × Z p . Recall that theconfiguration u is periodic if x − i f u ( x ) = f u ( x ) for some i ≥ nonperiodic otherwise. Similarly the configuration U is periodic if there exists ( i, j ) (cid:54) = (0 , x − i y − j F U ( x, y ) = F U ( x, y ) and nonperiodic otherwise. We say that( u m ) m ≥ is eventually periodic if ( x − i f u ( x )) | N is periodic for some i ≥ Proposition 5.3.
Let u ∈ F Z p be ( − p ) -automatic, let Φ : F Z p → F Z p be a linearcellular automaton whose generating polynomial is neither nor a monomial, andlet U ∈ F Z × Z p be a spacetime diagram for Φ with U | Z ×{ } = u . If ( u m ) m ≥ is noteventually periodic, then U is nonperiodic.Proof. Suppose that U is periodic. Then there is ( i, j ) (cid:54) = (0 ,
0) such that x − i y − j F U ( x, y ) = F U ( x, y ). We can assume without loss of generality that − j ≥
0. We have x i F U ( x, y ) = y − j F U ( x, y ). Restricting to Z × { } , we get x i f u ( x ) = φ ( x ) − j f u ( x ),where φ ( x ) is the generating polynomial of Φ. In other words (cid:0) φ ( x ) − j − x i (cid:1) f u ( x ) =0, where by assumption φ ( x ) − j − x i (cid:54) = 0. Thus ( u m ) m ≥ min { i,rj } satisfies a linearrecurrence and hence is eventually periodic. (cid:3) Corollary 5.4.
Under the conditions of Proposition 5.3, if ( u m ) m ≥ is not even-tually periodic, then c U ( m, n ) > mn for each m and n ∈ N .Proof. This follows directly from [24, Corollary 9 and the remark following it], whereKari and Moutot show that Nivat’s conjecture holds for Z × Z -indexed spacetimediagrams U of a linear cellular automaton: If c U ( m, n ) ≤ mn for some m and n ,then U is periodic. (cid:3) We remark that in [32] and [19] there are more general but less sharp resultsconcerning Nivat’s conjecture.Let Φ : F Z p → F Z p be a linear cellular automaton, and let U in F Z × Z p or F Z × N p bea spacetime diagram for Φ. Define X U := { V ∈ F Z × Z p : L V ⊆ L U } . We call ( X U , σ , σ ) the Z × Z -subshift defined by U . We consider spacetime di-agrams U ∈ F Z × Z p which are [ − p, − p ]-automatic. By Theorem 4.5, we obtainthese once we choose automatic sequences as initial conditions, in U | { i }×− N , for − (cid:96) ≤ i ≤ r −
1, in U | − N ×{ } , and in U | N ×{ } . Lemma 5.5.
Let
Φ : F Z p → F Z p be a linear cellular automaton, let U ∈ F Z × Z p be aspacetime diagram for Φ , and let ( X U , σ , σ ) be the Z × Z -subshift defined by U .Then every element of X U is a spacetime diagram for Φ .Proof. Let φ ( x ) = α − (cid:96) x (cid:96) + · · · + α + · · · + α r x − r be the generating polyno-mial of Φ. If some element V ∈ X U is not a spacetime diagram for Φ, thenΦ’s local rule is violated somewhere, i.e. for some m, n we have α − (cid:96) V m,n + · · · + α V m + (cid:96),n + · · · + α r V m + (cid:96) + r − ,n (cid:54) = V m + (cid:96),n +1 . By definition the rectangular word UTOMATICITY AND INVARIANT MEASURES OF LINEAR CELLULAR AUTOMATA 21 w := ( V i,j ) m ≤ i ≤ m + (cid:96) + r − ,n ≤ j ≤ n +1 belongs to the language of U ; that is, w occursin U and agrees with Φ’s local rule, a contradiction. (cid:3) We collect some facts about constant-length substitutive sequences, referring thereader to [4] for a thorough exposition. A substitution of length p is a map θ : A →A p . We use concatenation to extend θ to a map on finite and infinite words from A .By iterating θ on any fixed letter a ∈ A , we obtain infinite configurations u ∈ A N such that θ j ( u ) = u for some natural number j ; we call such configurations θ -periodic , or θ -fixed if j = 1. We write θ ∞ ( a ) to denote a fixed point. The pigeonholeprinciple implies that θ has a θ -periodic configuration. We can also define bi-infinitefixed points of θ . Given a bi-infinite sequence u = · · · u − u − · u u · · · ∈ A Z andsubstitution θ on A , define θ ( u ) = · · · θ ( u − ) θ ( u − ) · θ ( u ) θ ( u ) · · · . If a, b are letterssuch that θ ( a ) starts with a , θ ( b ) ends with b , and the word ba occurs in θ n ( c ) forsome letter c , then we call the unique sequence u = · · · b · a · · · that satisfies θ ( u ) = u a bi-infinite fixed point of θ . Bi-infinite fixed points of a length- p substitution θ are( − p )-automatic, since p -automatic sequences are closed under shifting to the rightand the addition of finitely many new entries; see [4, Theorem 6.8.4].We can similarly define two-dimensional substitutions θ : A → A p × p and two-dimensional θ -fixed points.We recall Cobham’s theorem [18]. We refer to [4, Theorems 6.3.2 and 14.2.3] forthe proof. Theorem 5.6. (1)
The sequence ( u m ) m ≥ ∈ F N p is p -automatic if and only if it is the image,under a coding, of a fixed point of a length- p substitution θ . (2) The sequence ( U m,n ) m ≥ ,n ≥ ∈ F N × N p is [ p, p ] -automatic if and only if it isthe image, under a coding, of a fixed point of a substitution θ : A → A p × p . Example 5.7.
As in Examples 3.11 and 3.16, let p = 3, and let φ ( x ) = x +1 ∈ F [ x ]. We perform a search to find substitutions θ : F → F with fixedpoints θ ∞ ( a ) generated by small automata under Part (1) of Theorem 5.6, sincea small automaton makes subsequent computations feasible. We also require that θ is primitive, that the fixed point ( u m ) m ≥ is not eventually periodic, and that( u m ) m ≥ , ( u m +1 ) m ≥ , and ( u m +2 ) m ≥ are not eventually periodic. Among thesubstitutions satisfying these criteria, the substitution θ defined by θ (0) = 001, θ (1) = 112, and θ (2) = 220 minimizes the number of states in the correspondingautomaton, producing the automaton on the left in Figure 3 for the fixed point θ ∞ (0). Indeed this is how we chose that automaton. From the 54-state automatonfor U | N × N , we compute by Part (2) of Theorem 5.6 a substitution Θ : A → A × and coding τ : A → F such that τ (Θ ∞ ( a )) = U | N × N for a particular letter a ∈ A .The size of the alphabet is |A| = 75.Note that while the spacetime diagram has a substitutional nature, the alphabetsize makes the computation of this substitution by hand infeasible. This is pre-sumably why such substitutions have not been studied in the symbolic dynamicsliterature.5.2. Automatic invariant sets and intersection sets.
For a linear cellularautomaton Φ : F Z p → F Z p , let X Φ = { V ∈ F Z × Z p : V is a spacetime diagram for Φ } . Then X Φ is closed in F Z × Z p and ( X Φ , σ , σ ) is a Z × Z -subshift, an example of a Markov subgroup or algebraic shift [37].We define π : X Φ → F Z p by π ( V ) = V | Z ×{ } . Let Z ⊂ X Φ be a closed and ( σ , σ )-invariant subset. Note that by construction Φ maps π ( Z ) onto π ( Z ), though Φ isnot necessarily invertible on π ( Z ); i.e. we have two commuting transformations σ and Φ defined on π ( Z ) that define a monoid action of Z × N . The reader whoprefers to work with a Z × Z action can take the natural extension of ( π ( Z ) , σ, Φ);see for example the exposition in [20]. We have(1) π ◦ σ = σ ◦ π and π ◦ σ = Φ ◦ π. Theorem 5.8.
Let
Φ : F Z p → F Z p be a linear cellular automaton whose generat-ing polynomial is neither nor a monomial, and let u ∈ F Z p be a ( − p ) -automaticsequence which is not eventually periodic. Then π ( X ST Φ ( u ) ) is a closed ( σ, Φ) -invariant subset of F Z p which is neither finite nor equal to F Z p .Proof. By the identities in (1), any closed ( σ , σ )-invariant set in X Φ projects toa closed ( σ, Φ)-invariant subset of F Z p . Thus π ( X ST Φ ( u ) ) is ( σ, Φ)-invariant, andcompactness implies that it is closed in F Z p . By Proposition 5.3, π ( X ST Φ ( u ) ) is notfinite. By Theorem 5.2, π ( X ST Φ ( u ) ) (cid:54) = F Z p . (cid:3) There are other examples of invariant sets for linear cellular automata. This wasfirst touched on by Kitchens and Schmidt [25, Construction 5.2] [37, Example 29.8]and by Silberger [38, Example 3.4], where the following construction is described.One starts with a finite set H ⊂ F jp and considers H Z . There is a natural injection i : H Z → F Z p obtained by concatenating. Note that i ( H Z ) is not necessarily invariantunder the left shift σ , but ¯ Y := ∪ j − m =0 σ m ( i ( H Z )) is. It is clear that ¯ Y is a propersubset of F Z p . However, to extend ¯ Y to a “small” set which is invariant under Φ,Kitchens and Schmidt [25, Construction 5.2] assume in addition that H is a groupand that j has a simple base- p representation. For example, they take j = p k ,and then the assumption that H = H k is a group and the “freshman’s dream”(which is that if Φ has generating polynomial φ ( x ) = α − (cid:96) x (cid:96) + · · · + α + · · · + α r x − r then Φ p k has generating polynomial φ ( x ) p k = α − (cid:96) x (cid:96)p k + · · · + α + · · · + α r x − rp k )imply that Φ p k ( ¯ Y k ) ⊆ ¯ Y k . Therefore Y k := ∪ p k − n =0 Φ n ( ¯ Y k ) is ( σ, Φ)-invariant and isalso a proper subset of F Z p . One can also obtain more complex subshifts by takingan infinite intersection ∩ k Y k of nested shift spaces where Y k is built from a group H k ⊂ F p k p and k → ∞ . Example 5.9.
Let p = 2, let Φ be the Ledrappier cellular automaton, and let H k = { k , θ k (0) , θ k (1) , k } where θ is the Thue–Morse substitution. Then, usingthe freshman’s dream, ∩ k Y k contains π ( X ST Φ ( u ) ), where u ∈ F Z p is any bi-infinitefixed point of the Thue–Morse substitution. Note that in fact here π ( X ST Φ ( u ) ) isalmost all of ∩ k Y k , as ∩ k Y k \ π ( X ST Φ ( u ) ) consists of bi-infinite sequences which areidentically 0 to the left of some index and which are a θ -fixed point to the right ofthat index, or vice versa. We can rectify this discrepancy by changing our initialcondition. If one starts with the ( − u whose right halfis a fixed point of θ and whose left half is identically 0, then π ( X ST Φ ( u ) ) = ∩ k Y k .This construction is explored in greater detail by Einsiedler [21], who showsthat one can find ( σ , σ )-invariant sets of any possible entropy. His construction UTOMATICITY AND INVARIANT MEASURES OF LINEAR CELLULAR AUTOMATA 23 is based on the construction of Kitchens and Schmidt, although he expresses itdifferently. Precisely, recall that X Φ is the set of all spacetime diagrams for Φ.Einsiedler works with a group Z ⊂ X Φ which is invariant under the action of some σ m σ n . For example, if one considers the group Z := { V ∈ X Φ : V m, n = 0 for each m, n ∈ Z } , then this group is invariant under σ σ . Using the Kitchens–Schmidt construc-tion, it can be generated by taking spacetime diagrams of sequences on H = { (0 , , (1 , } ∈ F with the Ledrappier cellular automaton Φ. For, the imageof a sequence in H Z under Φ contains a 0 in every even index, and the image of asequence in H Z under Φ is a sequence in H Z . Einsiedler also allows addition of Z by a finite set F . He calls sets Z = ∩ k ( Z k + F k ) intersection sets , and he askswhether there is a description of every ( σ , σ )-invariant set in terms of intersectionsets. Theorem 5.10.
Let
Φ : F Z p → F Z p be a linear cellular automaton, and let u ∈ F Z p bea ( − p ) -automatic sequence which is not eventually periodic. Then π ( X ST Φ ( u ) ) is a ( σ, Φ) -invariant proper subset of F Z p which is a subset of an intersection set.Proof. By assumption, u is a concatenation of two p -automatic sequences. ByCobham’s theorem, there are substitutions θ : A → A p and θ : A → A p ,and codings τ : A → F p and τ : A → F p such that u | N is the τ -coding ofa right-infinite fixed point of θ , and u | − N is the τ -coding of a left-infinite fixedpoint of θ . For each k let H k be the group in F p k p generated by { τ ( θ k ( a )) : a ∈ A } ∪ { τ ( θ k ( a )) : a ∈ A } . Let Y k be the ( σ, Φ)-invariant subset of F Z p as defined above using the group H k . Then for each k , π ( X ST Φ ( u ) ) ⊂ Y k , so π ( X ST Φ ( u ) ) ⊂ ∩ k Y k . (cid:3) In Example 5.9, we can find u such that the set π ( X ST Φ ( u ) ) is equal to an inter-section set ∩ k Y k . This is because for each k the group generated by { θ k (0) , θ k (1) } is very close to the set { θ k (0) , θ k (1) } . Example 5.11.
We continue with our running example, last seen in Example 5.7,where p = 3, Φ is the cellular automaton with generating function x + 1, and theinitial condition is generated by the substitution θ (0) = 001, θ (1) = 112, θ (2) = 220.Every word of length 2 occurs in every fixed point of θ . One shows by inductionthat(2) θ k (0) + θ k (1) + θ k (2) = 0 k for each k . We also have(3) 2 θ k (0) + θ k (1) = 2 θ k (1) + θ k (2) = 2 θ k (2) + θ k (0) = 1 k , so that the group generated by { θ k (0) , θ k (1) , θ k (2) } is H k = { k , k , k , θ k (0) , θ k (0) , θ k (1) , θ k (1) , θ k (2) , θ k (2) } . Let ( u m ) m ≥ be the fixed point θ ∞ (0) and let ( u − m ) m ≥ be the constant 0 sequence.Its spacetime diagram ST Φ ( u ) is shown in Figure 4. We claim that all words in H k occur horizontally in ST Φ ( u ). The words 0 k , θ k (0), θ k (1), and θ k (2) occur in the0-th row of ST Φ ( u ). Since all possible words of length 2 occur in u , each element of S k = { θ k ( a ) + θ k ( b ) : ab ∈ F × F } = { θ k (0) , θ k (1) , θ k (2) } occurs in the 3 k -th row of ST Φ ( u ). Also, since ( x + 1) · k = x · k + x · k + x k + 1,Equation (3) impliesΦ · k ( u ) | [3 · k , · k − = u | [3 · k , · k − + u | [2 · k , · k − + u | [0 , k − + u | [ − k , − = θ k (0) + θ k (1) + θ k (0) + 0 k = 1 k . It follows that 2 k − occurs in row 4 · k + 1; this is true for all k , so 2 k also occurs.Therefore all words in H k occur in ST Φ ( u ), and by approximation arguments onesees that π ( X ST Φ ( u ) ) = ∩ k Y k .In contrast, for the initial condition u in Figure 5, it is not so clear that π ( X ST Φ ( u ) )is an intersection set. In Example 6.6, for a different initial condition u , which isalso not eventually periodic in either direction, we describe π ( X ST Φ ( u ) ) as a modi-fied intersection set ∩ k Y k , where Y k is defined with sets of words H k which are notgroups, but which nevertheless capture the words we see at levels p k . Question 5.12.
Can all of the invariant sets in Theorem 5.8 be written as inter-section sets? Invariant measures for linear cellular automata
In this section we study the ( σ, Φ)-invariant measures that are supported on theinvariant sets found in Theorem 5.8. By the same transfer principle mentioned inSection 5, a measure supported on X U that is invariant under σ and σ trans-fers to a measure on F Z p which is invariant under σ and Φ. By Proposition 6.1,these measures are never the Haar measure. In Theorem 6.2 we identify a de-cidable condition which guarantees that the measure µ in question is not finitelysupported, and in Theorem 6.4 we identify a family of nontrivial ( σ, Φ)-invariantmeasures when Φ is the Ledrappier cellular automaton. In Theorem 6.11 we iden-tify ( σ, Φ)-invariant measures as belonging to simplices whose extreme points areergodic measures supported on codings of substitutional shifts. This statement im-plicitly contains another method by which to determine whether µ is trivial, as thereexist algorithms to compute the frequency of a word for such a measure. Finally,in Theorems 6.13 and 6.15, we give conditions that guarantee that the shifts westudy contain constant configurations and hence possibly lead to finitely supported( σ, Φ)-invariant measures.Throughout this section, we make use of the substitutional characterisation ofautomatic sequences to state and prove our results.6.1.
Invariant measures on [ − p, − p ] -automatic spacetime diagrams. Recallthat a subshift (
X, σ ) is aperiodic if each x ∈ X is aperiodic. We consider measureson the Borel σ -algebra of X . Let S, T : X → X be transformations on X . Ameasure µ on X is T -invariant if µ ( Z ) = µ ( T − ( Z )) for every measurable Z , and itis ( S, T ) -invariant if it is both S - and T -invariant. A measure µ has finite support { x , . . . , x n } if it is a finite weighted sum of Dirac measures µ = (cid:80) ni =1 w i δ x i . Ifthe finitely-supported Borel measure µ on a shift space X ⊆ F Z p is also σ -invariant,then each configuration in the support of µ is periodic. The same is true if µ isfinitely supported on a two-dimensional shift space and is ( σ , σ )-invariant. In thenext proposition we list some elementary observations about the measures on Y U that are projections of measures on X U . By the Krylov–Bogolyubov theorem [39, UTOMATICITY AND INVARIANT MEASURES OF LINEAR CELLULAR AUTOMATA 25
Theorem 6.9], there exist ( σ , σ )-invariant measures supported on X U . Recall thatthe map π : X U → F Z p is defined by π ( V ) = V | Z ×{ } . Proposition 6.1.
Let
Φ : F Z p → F Z p be a linear cellular automaton, and let U ∈ F Z × Z p be a [ − p, − p ] -automatic spacetime diagram for Φ . Let ( Y U , σ ) be the Z -subshiftdefined by U . Let µ be a ( σ , σ ) -invariant measure on X U , and let λ := µ ◦ π − . (1) Then λ is a ( σ, Φ) -invariant measure on Y U that is not the Haar measure. (2) Moreover, if µ is not finitely supported, then λ is not finitely supported.Proof. By Equations (1), any Borel measure µ on X U which is ( σ , σ )-invariantdefines a ( σ, Φ)-invariant Borel measure λ := µ ◦ π − on Y U . By Part (2) ofTheorem 5.2, there is a K such that there are at most Km words on an m × L U , so there are at most Km words of length m in the language of Y U . Thus for large m , there exists a word w of length m such that λ ( w ) = 0. Thisproves the first assertion.To see the second assertion, if λ is supported on a finite set { y , . . . , y n } , then, as λ is invariant under Φ − , for each i we have Φ − ( y i ) ∩ { y , . . . , y n } (cid:54) = ∅ . For each i ,this implies that Φ − ( y i ) ∩ { y , . . . , y n } consists of exactly one element. ThereforeΦ is a permutation on { y , . . . , y n } . For each cycle in this permutation, consider the Z × Z -configurations whose rows are elements of the cycle. Then µ is supported onthe union of the ( σ , σ )-orbits of these Z × Z -configurations. Since λ is invariantunder the left shift, each y i is periodic. Therefore µ is finitely supported. (cid:3) In the following theorem we give a condition that guarantees the existence ofmeasures on Y U which are ( σ, Φ)-invariant and which are not finitely supported.We say that a two-dimensional configuration U is horizontally M -power-free if no m × w M with m ≥ U . Theorem 6.2.
Let U ∈ F Z × Z p be a [ − p, − p ] -automatic sequence, specified by anautomaton. It is decidable whether there exists M ≥ such that U is horizontally M -power-free.Proof. We reduce the decidability of horizontal M -power-freeness of U to that ofeach quadrant.An occurrence of a horizontal M -power w M with | w | = (cid:96) in the sequence( U m,n ) ( m,n ) ∈ Z × Z is a word of the form U m,n · · · U m + M(cid:96) − ,n satisfying U i,n = U i + (cid:96),n for all i in the interval m ≤ i ≤ m + ( M − (cid:96) −
1. Therefore U is horizontally M -power-free if and only if the set S := { ( M, (cid:96) ) : ( ∃ m ≥ ∃ n ≥ ∀ i )((0 ≤ i ≤ M (cid:96) − → ( U m + i,n = U m + i + (cid:96),n )) } is empty. We follow Charlier, Rampersad, and Shallit [15, Theorem 4]. The con-figuration U is horizontally M -power-free for arbitrarily large M if and only if forall k ≥ S contains a pair ( M, (cid:96) ) with
M > (cid:96)p k . Padding the shorter wordwith zeros if necessary, we write the base- p representation of the pair ( M, (cid:96) ) as( M e , (cid:96) e ) , ( M e − , (cid:96) e − ) , . . . , ( M , (cid:96) ). Thus for every k ≥ S contains a pair ( M, (cid:96) )with M ≥ (cid:96)p k if and only if S contains a pair ( M, (cid:96) ) whose base- p representationstarts with ( d , , ( d , , . . . , ( d k , d (cid:54) = 0 and each other d i ∈ F p . Giventhe automaton M which generates χ S , S contains a pair ( M, (cid:96) ) with M ≥ (cid:96)p k forarbitrarily large k if and only if there are words u , w , and v on the alphabet F p × F p with the second entries of all letters in w and v all equal to 0, and where u is thelabel of a path from the initial state of M to a state s , w is the label of a cycle at s , and v is the label of a path from s to a state whose corresponding output is 1.Whether three such words exist is decidable. (cid:3) For fixed M , the set S in the proof is a p -definable set (see [33, Definition 6.34]),and horizontal M -power-freeness can be determined by constructing an automaton;see [33, Section 6.4] and [28]. Corollary 6.3.
Let
Φ : F Z p → F Z p be a linear cellular automaton, let U ∈ F Z × Z p be a [ − p, − p ] -automatic spacetime diagram for Φ , and let ( Y U , σ ) be the Z -subshiftdefined by U . If U is horizontally M -power-free for some M ≥ , then there existsa ( σ, Φ) -invariant measure λ on Y U which is neither the Haar measure, nor finitelysupported.Proof. Recall that a finitely-supported σ -invariant measure λ is supported on aset { y , y , . . . , y n } ⊆ Y U where each y i is periodic. If X U is horizontally M -power-free, then Y U is aperiodic. Thus for any ( σ , σ )-invariant measure µ on( X, σ , σ ), µ ◦ π − is a ( σ, Φ)-invariant measure which is not finitely supported.By Proposition 6.1, µ ◦ π − is not the Haar measure. (cid:3) Note that if we take the initial condition u to be an aperiodic fixed point of aprimitive substitution, then, by results of Moss´e [27], u is M -power-free for some M .Continuing with Example 5.9, Schmidt [37, Example 29.8] identifies a ( σ, Φ)-invariant measure which is supported on π ( X U ), where Φ is the Ledrappier cellularautomaton, U = ST Φ ( u ), u m = 0 for all m ≤ −
1, and ( u m ) m ≥ is a fixed point ofthe Thue–Morse substitution. He does not study whether this measure is finitelysupported; our experiments suggest that this measure is a point mass supportedon the constant zero configuration. However in the next theorem we identify afamily of substitutions which do yield nontrivial ( σ, Φ)-invariant measures for theLedrappier cellular automaton.Given a substitution θ : F p → F pp , we write θ ( a ) = θ ( a ) · · · θ p − ( a ). We say that θ is bijective if, for each i in the interval 0 ≤ i ≤ p − { θ i ( a ) : a ∈ F p } = F p . Theorem 6.4.
Let
Φ : F Z → F Z be the linear cellular automaton with generatingpolynomial φ ( x ) = x + 1 , let θ be a primitive bijective substitution on F , andsuppose that u ∈ F Z is a bi-infinite aperiodic fixed point of θ . Then there exists M such that ST Φ ( u ) is horizontally M -power-free.Proof. Since θ is bijective, θ satisfies Identity (2): θ k (0) + θ k (1) + θ k (2) = 0 k . We claim that, for each k ≥
1, for each n ≥
0, and for each m ∈ Z , we have(4) Φ n · k ( u ) | [ m k , ( m +1)3 k − ∈ (cid:40) { θ k (0) , θ k (1) , θ k (2) } if n is even { θ k (0) , θ k (1) , θ k (2) } if n is odd.Fix k ≥
1. Since u is a bi-infinite fixed point of θ , we have u | [ m k , ( m +1)3 k − ∈{ θ k (0) , θ k (1) , θ k (2) } . Let n = 1. Since ( x + 1) k = x k + 1, we haveΦ k ( u ) | [ m k , ( m +1)3 k − = u | [ m k , ( m +1)3 k − + u | [( m − k ,m k − = θ k ( u m ) + θ k ( u m − ) ∈ { θ k (0) , θ k (1) , θ k (2) } UTOMATICITY AND INVARIANT MEASURES OF LINEAR CELLULAR AUTOMATA 27 for each m ∈ Z . The claim follows by induction on n by replacing u with Φ k ( u ).For each k , let H k = { θ k (0) , θ k (0) , θ k (1) , θ k (1) , θ k (2) , θ k (2) } . (Note that H k is not a group, contrary to the definition of an intersection set.) Since u is an aperiodic fixed point of a primitive substitution, Moss´e’s theorem [27] tellsus that u is M -power-free for some M ≥
2. This implies that θ k ( a ) is M -power-freefor each a ∈ F , and hence 2 θ k ( a ) is also M -power-free. Thus all words in H k are M -power-free, so if a power w l occurs as a subword of a word in H k , then l < M .Next note that, again because words in H k are M -power-free, if a word in H k is tiled by a word w (that is, is a subword of w ∞ ), then | w | > k M . This impliesthat if | w | ≤ k M and w l occurs as a subword of W · · · W j ∈ H jk , then w l occurs asa subword of W i W i +1 for some 1 ≤ i ≤ j −
1, and so l ≤ M − w = w · · · w m of length m ≥
2, define Φ( w ) := ( w + w ) · · · ( w m − + w m ). Suppose w l occurs in the n -th row of ST Φ ( u ). We show that l < M . Let k be such that 3 k +1 ≤ | w l | = l | w | < k +2 . Then | w | < k +2 l . Let N be such that N · k ≤ n < ( N + 1) · k . Write Φ ( N +1) · k − n ( w l ) = ¯ w ¯ l ¯ v , where the words ¯ w and ¯ v are such that ¯ l ≥ v is a prefix of ¯ w with 0 ≤ | ¯ v | ≤ | ¯ w | −
1. Wehave | ¯ w | ≤ | w | since the period length of a word does not increase after applyingΦ. There are two cases.If | ¯ w | ≥ k M , then k M ≤ | ¯ w | ≤ | w | < k +2 l , so l < M .If | ¯ w | < k M , then, since ¯ w ¯ l occurs on row ( N + 1) · k , by (4) ¯ w ¯ l occurs as asubword of W · · · W j ∈ H jk for some j . By the argument above, ¯ w ¯ l occurs as asubword of W i W i +1 and therefore ¯ l ≤ M −
2. We also have | ¯ w ¯ l ¯ v | = | Φ ( N +1) · k − n ( w l ) | = | w l | − (cid:0) ( N + 1) · k − n (cid:1) ≥ k +1 − ( N + 1) · k + N · k = 2 · k , so 2 · k ≤ | ¯ w ¯ l ¯ v | < (¯ l + 1) | ¯ w | ≤ (2 M − | ¯ w | ≤ (2 M − | w | . Therefore · k M − < | w | < k +2 l , so l < (2 M − < M .It follows that ST Φ ( u ) is (9 M )-power-free. (cid:3) Remark 6.5.
Analogous to the construction preceding Example 5.9, we constructthe shift Y k using H k . We do not need H k to be a group since we have shown thatΦ n · k ( u ) is a concatenation of words that belong to H k . Since (4) holds for each k ≥
1, we have π ( X ST Φ ( u ) ) = ∩ k Y k . Example 6.6.
We continue with our running example, in particular from Exam-ple 5.11, where p = 3, Φ is the cellular automaton with generating function φ ( x ) = x + 1, and the initial condition is generated by the substitution θ (0) = 001, θ (1) =112, θ (2) = 220. We saw that H k , the group generated by { θ k (0) , θ k (1) , θ k (2) } , is H k = { k , k , k , θ k (0) , θ k (0) , θ k (1) , θ k (1) , θ k (2) , θ k (2) } . If we take u = · · · u − u − · u u · · · to be any bi-infinite fixed point of θ , thenST Φ ( u ) is horizontally M -power-free for some M by Theorem 6.4. In Theorem 6.4, we fixed the cellular automaton and prime p , and we let θ varyover a family of substitutions. Next, for each p we fix a substitution and vary thecellular automaton to obtain nontrivial ( σ, Φ)-invariant measures for a family ofcellular automata.
Definition 6.7.
For fixed p , let W := 01 · · · ( p −
1) and define θ : F p → F pp by θ ( a ) = W + a p , where a p denotes the word aa · · · a of length p . We call θ the(base- p ) parity substitution .If u ∈ F N p is the fixed point of the parity substitution starting with 0, then u m isthe sum, modulo p , of the digits in the base- p representation of m . Lemma 6.8.
The fixed point u ∈ F N p of the parity substitution θ : F p → F pp is noteventually periodic.Proof. For each candidate period length k , we show that there are arbitrarily large m such that u m (cid:54) = u m + k . Let k (cid:96) · · · k k be the base- p representation of k , with k (cid:96) (cid:54) = 0. If u k (cid:54) = 0, let m = p N for some N > (cid:96) ; then u m = 1 (cid:54)≡ u k ≡ u m + k mod p . If u k = 0, let m = p N +( p − k (cid:96) ) p (cid:96) for some N > (cid:96) +1; then u m ≡ p − k (cid:96) (cid:54)≡ − k (cid:96) ≡ u m + k mod p . (cid:3) Theorem 6.9.
Let u ∈ F Z p be a fixed point of the parity substitution θ : F p → F pp ,let Φ : F Z p → F Z p be a linear cellular automaton, and let L be the number of nonzeromonomials in the generating polynomial of Φ . If p does not divide L , then thereexists M such that ST Φ ( u ) is horizontally M -power-free.Proof. The proof is similar to that of Theorem 6.4. We refine Equation (4) to claimthat(5) Φ n · p k ( u ) | [ mp k , ( m +1) p k − ∈ { ( L n mod p ) θ k (0) + a p : a ∈ F p } for each n ∈ N . The proof of this claim is by induction, as in Theorem 6.4. Notethat L n (cid:54)≡ p for every n , since p does not divide L . Next we let H k = { jθ k (0) + a p : a ∈ F p and j ≡ L n mod p for some n ∈ N } . As in the proof of Theorem 6.4, there exists M ≥ H k are M -power-free. Also, if | w | ≤ p k M and w l occurs as a subword of W · · · W j ∈ H jk ,then l ≤ M − (cid:96) and r be the left and right radii of Φ. Given a word w = w · · · w m oflength m ≥
2, define Φ( w ) to be the word of length m − (cid:96) − r obtained by applyingΦ’s local rule. Suppose w l occurs in the n -th row of ST Φ ( u ). We show that l ≤ max (cid:18) ( (cid:96) + r ) p M, (cid:24) p p − M − (cid:25)(cid:19) . If | w l | (cid:96) + r < p , then l ≤ l | w | < ( (cid:96) + r ) p < ( (cid:96) + r ) p M . If | w l | (cid:96) + r ≥ p , let k be such that( (cid:96) + r ) p k +1 ≤ | w l | = l | w | < ( (cid:96) + r ) p k +2 . Then | w | < ( (cid:96) + r ) p k +2 l . Let N be such that N · p k ≤ n < ( N + 1) · p k . Write Φ ( N +1) · p k − n ( w l ) = ¯ w ¯ l ¯ v , where the words ¯ w and ¯ v are such that ¯ l ≥ v is a prefix of ¯ w with 0 ≤ | ¯ v | ≤ | ¯ w | −
1. Wehave | ¯ w | ≤ | w | since the period length of a word does not increase after applyingΦ. There are two cases.If | ¯ w | ≥ p k M , then p k M ≤ | ¯ w | ≤ | w | < ( (cid:96) + r ) p k +2 l , so l < ( (cid:96) + r ) p M . UTOMATICITY AND INVARIANT MEASURES OF LINEAR CELLULAR AUTOMATA 29 If | ¯ w | < p k M , then, since ¯ w ¯ l occurs on row ( N + 1) · p k , by (5) ¯ w ¯ l occurs as asubword of W · · · W j ∈ H jk for some j . By the same argument in the proof ofTheorem 6.4, ¯ w ¯ l occurs as a subword of W i W i +1 and therefore ¯ l ≤ M −
2. Wealso have | ¯ w ¯ l ¯ v | = | Φ ( N +1) · p k − n ( w l ) | = | w l | − (cid:0) ( N + 1) · p k − n (cid:1) ( (cid:96) + r ) ≥ ( (cid:96) + r ) p k +1 − ( N + 1) · p k ( (cid:96) + r ) + N · p k ( (cid:96) + r )= ( (cid:96) + r ) p k +1 − p k ( (cid:96) + r )= ( (cid:96) + r )( p − p k so ( (cid:96) + r )( p − p k ≤ | ¯ w ¯ l ¯ v | < (¯ l + 1) | ¯ w | ≤ (2 M − | ¯ w | ≤ (2 M − | w | . Therefore ( (cid:96) + r )( p − p k M − < | w | < ( (cid:96) + r ) p k +2 l , so l < p p − (2 M − ≤ (cid:108) p p − (2 M − (cid:109) .It follows that ST Φ ( u ) is max (cid:16) ( (cid:96) + r ) p M, (cid:108) p p − (2 M − (cid:109)(cid:17) -power-free. (cid:3) Question 6.10.
Given a linear cellular automaton
Φ : F Z p → F Z p , what is theproportion of length- p substitutions θ : F p → F pp , with a bi-infinite θ -fixed point u ,for which there exists an M ≥ such that ST Φ ( u ) is horizontally M -power-free? Einsiedler [21], as well as finding the invariant sets that are discussed in Sec-tion 5.2, shows the existence of shift-invariant measures supported on a subset of X Φ (the set of spacetime diagrams for a linear cellular automaton Φ). He asks:What are the ergodic measures on X ? Our contribution is to identify simplices ofinvariant measures that are generated by ergodic measures supported on codings ofsubstitutional sets. The invariant measures of a substitutional dynamical systemcan be derived from its incidence matrix: see [12] for a thorough description ofhow to compute them from the relevant Perron vectors of the matrix. The the-ory for two-dimensional substitutions is very similar and is described for primitivesubstitutions in [10]. Theorem 6.11.
Let
Φ : F Z p → F Z p be a linear cellular automaton, and let U ∈ F Z × Z p be a [ − p, − p ] -automatic spacetime diagram for Φ . Then there exists a simplex of ( σ , σ ) -invariant measures generated by the relevant Perron vectors of the incidencematrices of the four substitutions defining U . Automatic spacetime diagrams with finitely supported invariant mea-sures.
Given a length- p substitution θ : A → A ∗ , recall that we write θ ( a ) = θ ( a ) · · · θ p − ( a ), i.e. for 0 ≤ i ≤ p − θ i : A → A where θ i ( a ) is the( i + 1)-st letter of θ ( a ). We say that θ has a coincidence if there exists k ≥ i , . . . , i k such that | θ i ◦ · · · ◦ θ i k ( A ) | = 1 . (The notion of a coincidence has dynamical significance, as a constant-length sub-stitution with a coincidence defines a subshift which has discrete spectrum andso is measure theoretically a group rotation. There are various generalisations ofthe notion of a coincidence, such as the strong coincidence condition [9] for non-constant-length substitutions; it is conjectured that a substitution satisfying thestrong coincidence condition also has discrete spectrum.) By considering a powerof θ if necessary, we assume that the coincidence is achieved by θ , i.e. | θ i ( A ) | = 1for some i . Analogously, we say that a p -automatic sequence u has a coincidence if u = τ ( θ ∞ ( a )) for some length- p substitution θ with a coincidence. Given a word w = w w · · · w n , let w [ i,j ) := w i w i +1 · · · w j − .Let Φ be a linear cellular automaton, let u ∈ F Z p , and let U = ST Φ ( u ). Noticethat X U contains the constant zero configuration if for all N and m there exists n > N and k such that 0 m occurs in the row Φ n ( u ) starting at index k , as thisimplies that ST Φ ( u ) contains arbitrarily large triangles of 0’s. We investigate when X U contains constant configurations. Remark 6.12.
In the following two theorems we assume that the cellular automa-ton Φ has left radius 0. This is not a serious restriction for the following reason.If Φ has generating polynomial φ ( x ) and has left radius (cid:96) , then the generatingpolynomial x − (cid:96) φ ( x ) is the generating polynomial of a linear cellular automaton Ψwith left radius 0. Further, the n -th row of ST Ψ ( u ) is the left shift, by (cid:96)n units,of the n -th row of ST Φ ( u ). In the case where u m = 0 for m ≤
0, this tells us thatthe shears of ST Ψ ( u ) and ST Φ ( u ) coincide. By Theorem 3.9, the unsheared space-time diagram ST Φ ( u ) has a finite [ p, p ]-kernel if and only if the sheared spacetimediagram ST Ψ ( u ) is [ p, p ]-automatic.Note that Theorems 6.13 and 6.15 do not apply to the generating polynomial φ ( x ) = x + 1 ∈ F [ x ] in Examples 3.11, 3.16, and 5.7 (even after shearing as inRemark 6.12), since (cid:80) ri = − (cid:96) α i (cid:54) = 0. Theorem 6.13.
Let u ∈ F Z p be such that ( u m ) m ≥ is p -automatic with a coinci-dence, and let U = ST Φ ( u ) . Let Φ : F Z p → F Z p be a linear cellular automaton of leftradius with generating polynomial φ ( x ) = (cid:80) ri =0 α i x − i ∈ F p [ x − ] . If (cid:80) ri =0 α i = 0 ,then the constant zero configuration is an element of X U .Proof. Let θ : A → A p and τ : A → F p be the underlying substitution andcoding defining ( u m ) m ≥ . Suppose first that |{ θ ( a ) : a ∈ A}| = 1, i.e. that thecoincidence is achieved in the leftmost column θ , and also that the coincidenceis attained by θ . Thus there exists a ∗ such that θ ( a ) = a ∗ for each a ∈ A and u np = τ ( a ∗ ) for each n ≥
0. Since u is the coding of a θ -fixed point, we have that u [ np j +1 ,np j +1 + p j ) = τ ( θ j ( a ∗ )) for each j ≥ n ≥ p (cid:96) has generating polynomial (cid:80) ri =0 α i x − ip (cid:96) , thenΦ p j +1 ( u ) [0 ,p j ) = r (cid:88) i =0 α i u [ ip j +1 ,ip j +1 + p j ) = r (cid:88) i =0 α i τ ( θ j ( a ∗ )) = 0 p j , and in fact for each m ≥ p j +1 ( u ) [ mp j +1 ,mp j +1 + p j ) = r (cid:88) i =0 α i u [ ip j +1 + mp j +1 ,ip j +1 + mp j +1 + p j ) = r (cid:88) i =0 α i τ ( θ j ( a ∗ )) = 0 p j . If the coincidence is achieved in the column θ L , we translate the above argument,starting with the modification that u np + L = τ ( a ∗ ) for each n ≥
0, and adjustingaccordingly. (cid:3)
Example 6.14.
Let θ be the substitution θ ( a ) = ab, θ ( b ) = cd, θ ( c ) = ac, θ ( d ) = da ,and let τ ( a ) = τ ( c ) = 0 , τ ( b ) = τ ( d ) = 1. Then θ has a coincidence in the 5-thcolumn. Let u := τ ( θ ∞ ( a )) and let φ ( x ) = 1 + x − . Theorem 6.13 tells us thatST Φ ( u ) contains arbitrarily large patches of 0; see Figure 8. The left half of theinitial condition is the image under τ of the left-infinite fixed point of θ endingwith a . UTOMATICITY AND INVARIANT MEASURES OF LINEAR CELLULAR AUTOMATA 31
Figure 8.
Spacetime diagram for the Ledrappier cellular automa-ton, whose generating polynomial is φ ( x ) = 1 + x − , with a 2-automatic initial condition generated by the substitution in Ex-ample 6.14. The dimensions are 511 × Theorem 6.15.
Let u ∈ F Z p be such that ( u m ) m ≥ is p -automatic, and let U =ST Φ ( u ) . Let θ : A → A p and τ : A → F p be such that ( u m ) m ≥ = τ ( θ ∞ ( a )) .Let Φ : F Z p → F Z p be a linear cellular automaton of left radius with generatingpolynomial φ ( x ) = (cid:80) ri =0 α i x − i ∈ F p [ x − ] such that (cid:80) ri =0 α i = 0 . If there exists afinite word w = w w · · · w r ∈ A r +1 such that w occurs in θ ∞ ( a ) and |{ w i : α i (cid:54) =0 }| = 1 , then X U contains the constant zero configuration.Proof. Let { b } = { w i : α i (cid:54) = 0 } . For each j ≥
0, since w occurs in θ ∞ ( a ), then θ j ( w ) also occurs in θ ∞ ( a ). Also, for each i in the interval 0 ≤ i ≤ r such that α i (cid:54) = 0, θ j ( b ) occurs at θ j ( w ) [ p j i,p j ( i +1)) . Since Φ p j has generating polynomial φ ( x ) p j = (cid:80) ri =0 α i x − p j i , we have, for each k in the interval 0 ≤ k < p j , (cid:16) Φ p j τ (cid:0) θ j ( w ) (cid:1)(cid:17) k = r (cid:88) i =0 α i τ (cid:0) θ j ( w ) p j i + k (cid:1) = r (cid:88) i =0 α i τ (cid:0) θ j ( b ) k (cid:1) = (cid:32) r (cid:88) i =0 α i (cid:33) τ (cid:0) θ j ( b ) k (cid:1) = 0 , so that the word 0 p j occurs in ST Φ ( u ). The result follows. (cid:3) We remark that in the previous proof, it is sufficient that the word w occurs oncein θ ∞ ( a ), since for each j we obtain a triangular region of 0’s. Also, appropriateversions of the previous two theorems could be stated without left radius 0; then wewould also need to specify the left side of the initial condition. Finally, given a p -automatic initial condition u , one can always find a linear cellular automaton Φ suchthat ST Φ ( u ) contains arbitrarily large words which are identically zero. Conversely,given a linear cellular automaton Φ whose generating polynomial satisfies φ (1) = 0,one can find an initial condition such that ST Φ ( u ) contains large words which areidentically zero. Theorems 6.13 and 6.15 are useful tools in Section 6.1, where wewished to avoid finitely supported invariant measures. Corollary 6.16.
Let u ∈ F Z be such that ( u m ) m ≥ is -automatic, and let U =ST Φ ( u ) . Let Φ : F Z → F Z be the Ledrappier cellular automaton with generatingpolynomial φ ( x ) = 1 + x − ∈ F [ x − ] . Then X U contains the constant zero config-uration.Proof. If 00 or 11 occurs in ( u m ) m ≥ , we are done by Theorem 6.15. Otherwise,( u m ) m ≥ is 0101 · · · or 1010 · · · . Since each of these sequences has a coincidence,we are done by Theorem 6.13. (cid:3) Example 6.17.
Let Φ be the Ledrappier cellular automaton Let θ be the Thue–Morse substitution, θ (0) = 01 and θ (1) = 10, and let p = 2. Then 00 and 11occur in both fixed points of θ and the conditions of Corollary 6.16 are satisfied;see Figure 7. Acknowledgement
We thank Benjamin Hellouin de Menibus and Marcus Pivato for helpful discus-sions, and the referee for a careful reading. Reem Yassawi thanks IRIF, Universit´eParis Diderot-Paris 7, for its hospitality and support.
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