In this article, we study the relationship between -automatic sequences and spacetime diagrams of linear cellular automata over the finite field , where is prime. For definitions, see Section 2.
There are many characterisations of -automatic sequences. For readers familiar with substitutions, Cobham’s theorem tells us that they are codings of fixed points of length- substitutions. In an algebraic setting, Christol’s theorem tells us that they are precisely those sequences whose generating functions are algebraic over . In , we characterise -automatic sequences as those sequences that occur as columns of two-dimensional spacetime diagrams of linear cellular automata , starting with an eventually periodic initial condition.
We investigate the nature of a spacetime diagram as a function of its initial condition, when the initial condition is -automatic. In the special case when the initial condition is eventually in both directions and the cellular automaton has right radius , this question has been thoroughly studied in a series of articles by Allouche, von Haeseler, Lange, Petersen, Peitgen, and Skordev [3, 4, 5]. Amongst other things, the authors show that an -configuration which is generated by a linear cellular automaton, whose right radius is , and an eventually initial condition, is -automatic. In , Pivato and the second author have also studied the substitutional nature of spacetime diagrams of more general cellular automata with eventually periodic initial conditions.
In Sections 3 and 4 we extend these previous results by relaxing the constraints imposed on the initial conditions and the cellular automata. We allow initial conditions to be bi-infinite -automatic sequences or, equivalently, concatenations of two -automatic sequences. Iterating produces a -configuration, and we show in Theorem 3.10, Theorem 3.14, and Corollary 3.15, that such spacetime diagrams are automatic, with two possible definitions of automaticity: either by shearing a configuration supported on a cone or by considering -automaticity. Our results are constructive, in that given an automaton that generates an automatic initial condition, we can compute an automaton that generates the spacetime diagram. We perform such a computation in Example 3.11, which we use as a running example throughout the article. While the spacetime diagram has a substitutional nature, the alphabet size makes the computation of this substitution by hand infeasible, and indeed difficult even using software.
We can also extend a spacetime diagram backward in time to obtain a -configuration where each row is the image of the previous row under the action of the cellular automaton. In Lemma 4.2 we show that the initial conditions that generate a -configuration are supported on a finite collection of lines. In Theorem 4.5, we show that if the initial conditions are chosen to be -automatic, then the resulting spacetime diagram is a concatenation of four -automatic configurations.
Apart from the intrinsic interest of studying automaticity of spacetime diagrams, one motivation for our study is a search for closed nontrivial sets in which are invariant under the action of the both left shift map and a fixed linear cellular automaton . Analogously, we also search for measures on one-dimensional subshifts that are invariant under the action of both and .
We give a brief background. Furstenberg  showed that any closed subset of the unit interval which is invariant under both maps and must be either or finite. This is an example of topological rigidity. Furstenberg asked if there also exists a measure rigidity, i.e. if there exists a nontrivial measure on which is invariant under these same two maps. By “nontrivial” we mean that is neither the Lebesgue measure nor finitely supported. This question is known as the problem.
The problem has a symbolic interpretation, which is to find a measure on which is invariant under both , which corresponds to multiplication by , and the map , which corresponds to multiplication by and where represents addition with carry. One can ask a similar question for and the Ledrappier cellular automaton , where represents coordinate-wise addition modulo . One way to produce such measures is to average iterates, under the cellular automaton, of a shift-invariant measure, and to take a limit measure. Pivato and the second author  show that starting with a Markov measure, this procedure only yields the Haar measure . Host, Maass, and Martinez  show that if a -invariant measure that has positive entropy for and is ergodic for the shift or the -action then . The problem of identifying which measures are -invariant is an open problem; see for example Boyle’s survey article [11, Section 14] on open problems in symbolic dynamics or Pivato’s article [24, Section 3] on the ergodic theory of cellular automata.
In Sections 5 and 6 we apply results of Sections 3 and 4 to find -invariant sets and measures. Spacetime diagrams generate subshifts , where and are the left and down shifts, and these subshifts project to closed sets in that are -invariant. Similarly, we show in Proposition 6.1 that -invariant measures on project to -invariant measures supported on a subset of . Einsiedler  constructs, for each , a -invariant set and a -invariant measure whose entropy in any direction is times the maximal entropy in that direction. He builds invariant sets using intersection sets as described in Section 5.2 and asks if every -invariant set is an intersection set. He also asks for the nature of the invariant measures. We show in Theorem 5.7 that each automatic spacetime diagram generates a -invariant set of small complexity. If we assume that the initial 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 we find are not obviously intersection sets.
The quest for nontrivial -invariant measures appears to be more delicate. Let be a subshift generated by a -automatic configuration . Theorem 6.8 states that the measures supported on such subshifts are convex combinations of measures supported on codings of substitution shifts. We show in Theorem 5.2 that has at most polynomial complexity. Therefore the )-invariant measures guaranteed by Proposition 6.1 are not the Haar measure. However they may be finitely supported: the shift generated by a nonperiodic spacetime diagram can contain periodic points on which a shift-invariant measure is supported. In Theorems 6.10 and 6.12 we identify cellular automata and nonperiodic initial conditions that yield two-dimensional shifts containing constant configurations.
We show in Corollary 6.3 that spacetime diagrams that do not contain large one-dimensional repetitions support nontrivial -invariant measures, and in Theorem 6.2 we show that this condition is decidable. In Theorem 6.4 we show that for the Ledrappier cellular automaton there exists a family of substitutions all of whose spacetime diagrams, including our running example, support nontrivial measures. A natural question that remains is whether such a family can be found for every nontrivial linear cellular automaton.
We are indebted to Allouche and Shallit’s classical text , referring to proofs therein on many occasions, which carry through in our extended setting. In Section 2, we provide a brief background on linear cellular automata, larger rings of generating functions in two variables, and - and -automaticity. In Section 3 we prove that -indexed spacetime diagrams are automatic if we start with automatic initial conditions. In Section 4 we extend these results to include -indexed spacetime diagrams. In Section 5 we show that automatic spacetime diagrams for yield nontrivial -invariant sets and discuss their relation to the intersections sets defined by Kitchens and Schmidt . Finally in Section 6, we study -invariant measures supported on automatic spacetime diagrams.
2.1. Linear cellular automata
Let be a finite alphabet. An element in is called a configuration and is written . The (left) shift map is the map defined as . Let be endowed with the discrete topology and with the product topology; then is a metrisable Cantor space. A (one-dimensional) cellular automaton is a continuous, -commuting map . The Curtis–Hedlund–Lyndon theorem tells us that a cellular automaton is determined by a local rule : there exist integers and with and such that, for all , . Let denote the set of non-negative integers.
Let be a cellular automaton and let . If satisfies and for each , we call the spacetime diagram generated by with initial condition .
For the cellular automata in this article, . The configuration space forms a group under componentwise addition; it is also an
A cellular automaton is linear if is an -linear map, i.e. for some nonnegative integers and , called the left and right radius of . The generating polynomial of , denoted , is the Laurent polynomial
We remark that our use of for the generating polynomial differs from usage in the literature of as ’s local rule, which is the map .
The generating polynomial has the property that . We will identify sequences with their generating function . Recall that and are the rings of polynomials and power series in the variable with coefficients in respectively. Let and be their respective fields of fractions: is the field of rational functions and is that of formal Laurent series; elements of are expressions of the form , where and .
A cone is a subset of of the form for some such that and are linearly independent. The cone generated by and is the cone .
If a cellular automaton is begun from an initial condition satisfying for all , then the spacetime diagram is supported on the cone generated by and . For example, see Figure 1. If then this cone contains points with negative entries, but we would still like to represent as a formal power series in some ring. We follow the geometric exposition given by Aparicio Monforte and Kauers . For a more algebraic exposition, see .
By definition, a cone is line-free, that is, for every , we have . This places us within the scope of .
For each cone , let be the set of all formal power series in and , with coefficients in , whose support is in . Then (ordinary) multiplication of two elements in is well defined, and the product belongs to ; in fact is an integral domain [6, Theorems 10 and 11].
Let be the reverse lexicographic order on , i.e. if or if and . A cone is compatible with if for all . Every cone contained in the set is compatible with . Let
Then is a ring contained in the field [6, Theorem 15]. This field also contains the field of rational functions.
2.3. Automatic initial conditions
Next we define automatic sequences, which we will use as initial conditions for spacetime diagrams.
A deterministic finite automaton with output (DFAO) is a -tuple , where is a finite set (of states), (the initial state), is a finite alphabet (the input alphabet), is a finite alphabet (the output alphabet), (the output function), and (the transition function).
In this article, our output alphabet is .
The function extends in a natural way to the domain , where is the set of all finite words on the alphabet . Namely, define recursively. If , this allows us to feed the standard base- representation of an integer into an automaton, beginning with the least significant digit. (Recall that the standard base- representation of is the empty word.) All automata in this article process integers by reading their least significant digit first.
A sequence of elements in is -automatic if there is a DFAO such that for all , where is the standard base- representation of .
Similarly, we say that a sequence is -automatic if there is a DFAO such that
for all , where is a base- representation of and is a base- representation of . Here, if and have standard base-
representations of different lengths, then the shorter representation is padded with zeros.
As defined, -automatic sequences are one-sided. To specify a bi-infinite sequence, we use base . Every integer has a unique representation in base with the digit set [2, Theorem 3.7.2]. For example, is written in base as
so its base- representation is . We say that a sequence is -automatic if there is a DFAO such that for all , where is the standard base- representation of . A sequence is -automatic if and only if the sequences and are -automatic [2, Theorem 5.3.2].
In this article, we use -automatic sequences in as initial conditions for cellular automata. For example, the spacetime diagram in Figure 2 is of a linear cellular automaton begun from a -automatic initial condition.
3. Algebraicity and automaticity of spacetime diagrams
In this section we show that a spacetime diagram obtained by evolving a linear cellular automaton from a -automatic initial condition is automatic in several senses. There is a natural notion of the -kernel of a two-dimensional configuration extending the usual definition. First, if we consider bi-infinite initial conditions that satisfy for all , we show in Theorem 3.5 that the generating functions of these cone-indexed configurations are algebraic and that they have finite -kernels. Then in Section 3.2 we show that the shear of an algebraic cone-indexed configuration is -automatic. Finally, in Section 3.3 we study the -automaticity of spacetime diagrams, where the coordinates are processed by reading in base . Specifically, we prove in Corollary 3.15 that a spacetime diagram obtained by evolving a linear cellular automaton from a general -automatic initial condition is -automatic.
3.1. Algebraicity and finiteness of the -kernel
Define the -kernel of to be the set
The -kernel of a cone-indexed sequence is defined by extending for all .
Given , the Cartier operator is defined as
Let be a cone. The -kernel of a power series is the set
If the sequence is indexed by a cone, then its -kernel is the set of all sequences where belongs to the -kernel of . We show in Lemma 3.2 that such are compatible with .
We can define analogously the one-dimensional Cartier operator and also the -kernel of a one-dimensional power series. Eilenberg’s theorem [2, Theorem 6.6.2] states that a sequence is -automatic precisely when its -kernel is finite; the same is true for a -automatic sequence [2, Theorem 14.4.1].
A power series is algebraic over if there exists a nonzero polynomial such that . Similarly, the cone-indexed series is algebraic over if there exists a nonzero polynomial such that . We recall Christol’s theorem for one-dimensional power series [13, 14], generalised to two-dimensional power series by Salon .
A sequence of elements in is -automatic if and only if is algebraic over .
A sequence of elements in is -automatic if and only if is algebraic over .
We refer to [2, Theorems 12.2.5 and 14.4.1] for the proof of Theorem 3.1, where it is shown that the algebraicity of a power series over a finite field is equivalent to the automaticity of its sequence of coefficients, which is equivalent to the finiteness of its - or -kernel.
In the next lemma we show that the image of under is indeed contained in . We show more: although elements of the -kernel of do not necessarily belong to , their indexing sets are one of a finite set of translates of .
Let be an integer, let be the cone generated by and , and let . Then every element of the -kernel of is supported on for some .
Let , and . We abuse notation and define . Let . Then we claim that for some . The statement of the lemma follows from the claim. Let be a point satisfying , , , and . Then maps to , which satisfies and
Since is an integer, this implies and . ∎
If and is generated by and , then and map to itself. The other Cartier operators map and . The cone arises from .
We now state Christol’s theorem for . The case is Salon’s theorem (Part (2) of Theorem 3.1). We omit the proof, since it is a straightforward generalisation of the proofs in [2, Theorems 12.2.5 and 14.4.1].
Let . Then is algebraic over if and only if has a finite -kernel.
Next we prove that a linear cellular automaton begun from a -automatic initial condition produces an algebraic spacetime diagram. A special case appears in Allouche et al. [4, Lemma 2], when the initial condition is eventually in both directions. The proof in the general case is similar.
Let be a linear cellular automaton. If is such that is -automatic and for all , then the generating function of is algebraic and so has a finite -kernel.
Let the generating polynomial of be . Let be the generating function of . The -th row of is the sequence whose generating function is the Laurent series . Let be the cone generated by and . Note that is identically on , so its generating function satisfies . Also,
Let be the cone generated by and . An interesting question is the following. Given a polynomial equation satisfied by , is it decidable whether is the generating function of for some linear cellular automaton ? The initial condition is determined by , so it would suffice to obtain an upper bound on the left radius .
3.2. Automaticity by shearing
If , then the cone generated by and contains points where . In this section, we feed these indices into an automaton by shearing the sequence so that it is supported on .
Let be the cone generated by and , and let . The shear of a sequence is the sequence defined by for each .
The next lemma enables us to move between the -kernel of the generating function of a cone-indexed sequence and the generating function of its shear.
Let . Let , . Then
Let . Let . We prove the result for the monomial ; the general result then follows from the linearity of . If , then both sides are . If , we have
Note here that for each fixed , the map is a bijection.
Let , and let . For each , we have
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 -automatic sequence using its -kernel.
Let be generated by and for some . A -indexed sequence of elements in has a finite -kernel if and only if its shear is -automatic.
Let be the shear of . By [2, Theorem 14.2.2], is -automatic if and only if its -kernel is finite. Hence we show that has a finite -kernel if and only if has a finite -kernel.
By Lemma 3.2, every element of the -kernel of is supported on for some . Let be an element of the -kernel of , supported on . Let . Let , so that is the generating function of the shear of . Similarly write ; then . Fix , and write where . By Lemma 3.7, we have
Summing over gives
Therefore the shear of is
Inductively, suppose is the generating function of an element of the kernel of . Then the shear of is an element of the -kernel of . Note that is supported on .
We set up a map from the -kernel of to the -kernel of . Let , and define recursively as follows. For each in the -kernel of , let where is supported on . Since the map is a bijection on , maps surjectively onto . It follows inductively that is a surjection from the -kernel of to the -kernel of .
If the -kernel of is finite, then the surjectivity of implies that the -kernel of is finite. By Lemma 3.2, is at most -to-one. Therefore if the -kernel of is finite then the -kernel of has at most times as many elements and is also finite. ∎
We can now extend Theorem 3.5.
Let be a linear cellular automaton. If is such that is -automatic and for all , then the shear of is -automatic.
Let , and let . Let be the -automatic sequence generated by the automaton on the left in Figure 3, whose first few terms are . The size of this automaton makes later computations feasible. Let for all ; then the spacetime diagram is supported on . See Figure 4. We compute an automaton for the -automatic sequence . By Part (1) of Theorem 3.1, we can compute a polynomial such that . We compute