Universality in Multidimensional Symbolic Dynamics

We show that in the category of effective $Z$ dynamical systems there is a universal system, i.e. one that factors onto every other effective system. In particular, for d $\geq 3$ there exist d-dimensional shifts of finite type which are universal for 1-dimensional subactions of SFTs. On the other hand, we show that there is no universal effective $Z^d$-system for $d>1$, and in particular SFTs cannot be universal for subactions of rank $d>1$. As a consequence, a decrease in entropy and Medvedev degree and periodic data are not sufficient for a factor map to exists between SFTs. We also discuss dynamics of cellular automata on their limit sets and show that (except for the unavoidable presence of a periodic point) they can model a large class of physical systems.

1. Introduction 1.1. Universality for shifts of finite type. A basic problem about any class of dynamical systems is to understand the factoring relation between its members.
Much of ergodic theory and topological dynamics, and particularly the theory of one-dimensional shifts of finite type (SFTs), has been motivated by the hope, which for some classes is satisfied, that the factoring relation reduce to some simple parameter, such as entropy, periodic point data or spectrum. For higher dimensional SFTs, which are the main subject of this note, partial results are known under certain mixing conditions [7], but it has become progressively clearer that the invariants which dictate the factoring relation in dimension 1 are only a part of the picture in dimensions d > 1.
We begin by reviewing some definitions; see also section 2. A Z d shift of finite type (SFT) is a subshift X of the full d-dimensional shift Σ Z d over Σ, defined by excluding all configurations containing patterns from some fixed finite set. More precisely, by a (d-dimensional) pattern we mean a coloring of a finite subset of Z d . For F ⊆ Z d and a ∈ Σ F , we say that the pattern a appears in a configuration x ∈ Σ Z d if (T u x)| F = a for some u ∈ Z d ; here, T u is the shift by u. For L a set of 2000 Mathematics Subject Classification. 37B15, 37B40, 37B50, 94A17, 03D45.
Thus, the questions of whether (d, k)-universal SFTs exist is closely related to the existence of universal systems in the class of effective systems; 1 and the class of effective systems, though countable, includes essentially every type of dynamics we can "describe". This would seem to indicate that we should not expect universal dynamics to exist, since they do not for general systems (at least if we stay in the context of separable spaces). Another relevant piece of information was recently provided by S. Simpson [13], who introduced Medvedev degrees as an invariant of SFTs. Simpson associates to each SFT X the Medvedev degree m(X) of its phase space, which is a measure of the recursive-theoretic complexity of the phase space as a subset of the full shift without reference to the dynamics (see section 3). Since factor maps between SFTs are sliding block codes they are computable, and therefore do not increase Medvedev degree. Simpson also showed that every Medvedev degree is realized as a 2-dimensional SFT. It follows that the factoring relation between SFTs is at least as complicated as the order relation between Medvedev degrees, and the latter is still not well understood.
What is relevant to our question, however, is that there exists a maximal Medvedev degree. Thus, from the point of view of recursion theory, there is no obstruction to the existence of SFTs whose (sub)actions factor onto a very broad class of systems; indeed, any set with maximal Medvedev degree at least maps (via a computable function) into, every effective set, and so into every SFT (this map has nothing to do with dynamics, but even so its existence is non-trivial).
It turns out that the existence of universal effective systems depends on the rank of the action. For Z-actions, such a system exists: Theorem 1. There exists a universal effective Z-system, that is, an effective Zsystem that factors onto every other effective Z-system.
In particular, for every d ≥ 3 there exist Z d -SFTs whose Z-subaction factors onto the Z-subaction of every other Z d -SFT; i.e. there are (d, 1)-universal SFTs for every d ≥ 3.
It remains an open problem whether there exist (2, 1)-universal SFTs, i.e. Z 2systems whose 1-dimensional subactions is universal. This would follow if every Z-EDS could be realized as the subaction of a Z 2 -SFT; this is problem open [4]. The universal effective Z-system in the theorem is constructed by taking the product of all non-empty Z-EDS (this is a countable product so no topological difficulties arise). The fact that the non-empty Z-EDS can be effectively enumerated rests on the fact that one can decide whether a Z-SFT is empty based on the list 1 One must be careful what notion of morphism one chooses for effective systems, since not every factor map is an effective factor map. However in this paper both definitions lead to the same results, since a factor map from an EDS to a symbolic system is automatically effective. of forbidden patterns which defines it. In contrast, it is a classical result of Berger that this problem is formally undecidable for Z 2 -SFTs [12,1], so this construction cannot be imitated in higher dimensions. This is not a shortcoming of the method because Theorem 2. If d ≥ 2 then there is no universal effective Z d system, and there are no (d, k)-universal SFTs for k ≥ 2.
To prove this, we show that the existence of a universal system could be used as a component in an algorithm that would decide whether an arbitrary finite set of patterns L defines a non-empty SFT, which would contradict Berger's theorem.
1.2. Cellular automata. Another class of dynamical systems defined by local rules are cellular automata. Recall that a d-dimensional cellular automaton (CA) is a function f : Σ Z d → Σ Z d which commutes with the shift action. This is well known to be equivalent to being defined by a local, finite transition rule. See [8] for definitions and a recent survey of the subject.
We next present two applications of the results from [4] to the dynamics of CA.
The first is analogous to the universality question for SFTs, i.e.: are there universal CA? This questions seems to be more difficult than for SFTs, except for the case d = 1 where again entropy considerations show that no universal object can exist.
However, using the relation between CA and SFTs (see e.g. [4], section 5.1), we can show that for d ≥ 3 there exist CA whose dynamics is very close to the universal effective Z-system of theorem 1.
In discussing dynamics of CA one must first overcome the fact that their action is in general neither invertible nor surjective (we are interested in invertible dynamics, though the question makes sense also in other categories). The limit set Λ f of a CA f is the largest set on which f acts surjectively: If f act injectively on Λ f then the action is effective; in any case, the natural extension of this system is effective.
Note that the limit action always contains a periodic point; this imposes certain restrictions on the dynamics which can occur on limit actions. However this is the only limitation. Combining theorem 1 with the results of [4] we have: Corollary 3. There exists a 3-dimensional cellular automaton such that, after removing from Λ f a fixed point and its basin of attraction, is a universal Z-EDS, and in particular factors onto the natural extension of every CA.
It is known that there are CA f such that for any other CA g, one can encode the configurations of g into configurations of f in a spatially homogeneous way and so that the action of f simulates the action of g (for a precise definition see [11]). This notion is not directly related to universality in our sense.
Our second application concerns one of the motivations for studying CA in the first place, namely that they provide a simple model for evolution of physical systems. We would like to show that they live up to this expectation in the sense that, for a reasonably large class of such systems, we can find a CA which models their dynamics very closely. We note that although much has been made of the fact that CA can perform universal computation, this in itself does not say much about their dynamics. The dynamics of a computer simulating a dynamical system is quite distinct from the dynamics of the system it is simulating. Since effective systems can be modeled as (4) f | X is invertible.
Then we say that X is an effective attractor of f .
As before, the presence of periodic points in the limit action of a CA prevents CA from modeling arbitrary dynamics, but this is in a sense the only obstruction.
Theorem 5. If X ⊆ R d is an effective attractor for f then there is a 3-dimensional cellular automaton g such that, after removing from Λ g a fixed point and its basin of attraction, the action of g factors onto (X, f ).
For the proof one shows that effective attractors can be extended to EDSs, and applies the machinery from [4]. As we have indicated, the statement above is in a sense the best possible for invertible dynamics without fixed points. It is true that in some ways, the dynamics of the CA given by the theorem do not "look like" the original system: tracing back to the construction in [4] one sees that the basin of attraction that we have thrown out is dense, and for product measures on the configuration space typical points will converge under the CA action to the fixed point. However, all the invariant measures on the system, except for the point measure on the fixed point, are pullbacks of measures from the original system. Furthermore, under the technical assumption that the effective attractor to be modeled satisfies the small boundary property, the EDS extending it can be made to be injective on the complement of a universally null set. Thus from the point of view of stationary dynamics the CA looks very much like an extension of the original system.
Organization. In section 3 we recall some of the recursive-theoretic machinery we will need and define effective dynamics. In section 4 we prove theorem 1, and in section 5 prove theorem 2. Section 6 discusses realization of effective attractors.

Section 7 contains some open problems.
Acknowledgement. I would like to thank Mike Boyle for some interesting discussions.

Definitions and notation
A Z d -dynamical system is the action of Z d by homeomorphisms on a compact metric space X; the action of u ∈ Z d is denoted usually by T u : X → X. A factor map between systems X, Y acted on by the same group is a continuous, onto map π : X → Y which commutes with the action in the sense that πT u = T u π for every u.
Let Σ be a finite set of symbols. The space Σ Z d of colorings of Z d by Σ is called the full d-dimensional shift over Σ, or just the full shift, and its points are called configurations. Topologically the full shift is a Cantor set, and it comes equipped with a natural Z d action, called the shift action, in which u ∈ Z d acts via the A subset X ⊆ Σ Z d which is closed and invariant to the shift (i.e. T u X = X for u ∈ Z d ) is called a subshift, or a symbolic system. By the Curtis-Hedlund-Lyndon theorem [3], factor maps between subshifts of the same dimension (but possibly distinct alphabets) are given by a block code: if Y ⊆ ∆ Z d , X ⊆ Σ Z d and π : Y → X is a factor map, then there is a finite F ⊆ Z d and a function π 0 : ∆ F → Σ, so that π acts on each site of x ∈ ∆ Z d by applying π 0 to the local neighborhood of the site: (πx)(u) = π 0 ((T u x)| F ). The diameter of F is called the window size of π. Conversely, any such map π 0 : ∆ F → Σ gives rise to a factor map π in this way (the image is automatically a subshift).
3. Recursive sets and effective dynamical systems 3.1. Some recursion theory. We require some basic facts from recursion theory; see [6] for a formal introduction. Recursion theory provides a classification of certain subsets of N according to the extent to which the set may be described algorithmically. By an algorithm we mean a finite set of instructions which can be carried out automatically, i.e. by computer program or, more formally, a Turing machine.
A subset A ⊆ N is recursive (R) if there is an algorithm which, given n ∈ N, outputs "yes" if n ∈ A and "no" otherwise. A function f : N → N is recursive if there is an algorithm which, given n, outputs f (n).
A set A is recursively enumerable (RE) if there is an algorithm which, on input n, returns "yes" if n ∈ A and otherwise runs forever. Alternatively, a non-empty set A ⊆ N is RE if there is an algorithm which, given n ∈ N, outputs a n ∈ N so that A = {a n : n ∈ N}; in other words, it is the image of a recursive function.
By fixing a bijection between N and another countable set U , we can speak of R and RE subsets of U . Thus we will speak of R and RE subsets of pairs of integers, finite sequences or patterns over a finite set Σ, etc. We will always assume that the objects have been placed in correspondence with N in some effective way (for the purpose of classifying subsets as R or RE, two identifications which can be algorithmically reduced to each other are equivalent).
Since there are countably many algorithms there are countably many R and RE sets. We note that every recursive set is RE, but not vice-versa. However, the examples of this tend to be rather artificial, e.g. the set of provable theorems in number theory (Gödel's theorem), or the set of halting Turing machines (Turing's theorem).
The following standard facts will be useful: Then M is RE.
Proof. Let A be an algorithm that on input a ∈ U halts if a ∈ L and runs forever otherwise. Let B be the algorithm which, upon input b ∈ V , iterates over all pairs (n, a) ∈ N × U , and for each pair runs the algorithm A for n steps (or until it halts) on the input a. If A halts before n steps are up, it checks whether (a, b) ∈ R, and if so it halts; otherwise it continues to the next pair (n ′ , a ′ ). It is easily seen that this algorithm halts on input b if and only if b ∈ M , so M is RE.
Proof. Let A and B be algorithms which, given x ∈ U , halt if x ∈ L or x ∈ U \ L, respectively, and otherwise run forever. Consider the algorithm which accepts x as input and iterates over n ∈ N. For each n it simulates n steps of the computation of A on input x, and if that computation halted it announces that a ∈ L and halts.
Otherwise it simulates n steps of the computation of B on x and if that computation halts, it announces x / ∈ L and halts. If neither simulations terminates, it proceeds to the next n. Clearly, our algorithm always halts and gives the correct answer. Let Since there are countably many algorithms, there are countably many EDS, representing only countable many isomorphism types of systems. In spite of this we do not know of any "natural" invariant of dynamical systems which cannot be realized as an EDS.
In [4] we showed: Theorem 9. The subaction of an SFT is an EDS.
More important for our present discussion is the partial converse obtained there:

Universal effective Z-systems
In this section we prove the following result: Theorem 11. There exists a Z-EDS which factors onto every other Z-EDS.
Together with theorem 10 this proves theorem 1.
A sequence L n ⊆ N of sets is uniformly recursively enumerable if ∪ ∞ n=1 L n ×{n} ⊆ N 2 is RE, or equivalently, if there is an algorithm A which, given input i, j ∈ N, halts if j ∈ L i and otherwise runs forever.
Lemma 12. Let L n ⊆ P * d be a uniformly RE sequence of sets. Write X n = Ω Ln . Then the product system ∞ n=1 X n is an EDS.
Proof. The proof is based on the fact that K ∼ = ∞ n=1 K, and this isomorphism can be made effective. Fix some recursive bijection ϕ : N 2 → N, and let which is a partition of N into disjoint infinite sets. We may identify I n with N via ϕ(·, n) : N → I n . Let π n : K → K be the projection x → x| In where identify x| In with a point in K using this bijection of I n and N. Extend π n to patterns over K pointwise, so if a ∈ K E for a finite set E ⊆ Z d then (π n (a))(u) = π n (a(u)), u ∈ Z d . Then x → (π 1 (x), π 2 (x), . . .) is a homeomorphism from Ω to We also need the following: Lemma 13. Given a finite L ⊆ P * 1 , it is decidable whether Ω L = ∅ or not.
Proof. Suppose L is given. Let I ⊆ N be large enough that if a ∈ L, a : E → P, then every pattern a(i), i ∈ E is supported in I. It is not hard to check that Ω L = ∅ if and only if there is an infinite sequence x = (x(n)) n∈Z over the alphabet {0, 1} I so that, for every every k ∈ N there is no a ∈ L, a : E → P, which satisfies x(k + i)(n) = (a(i))(n) for i ∈ E and all n at which a(i) is defined. Thus, deciding whether Ω L is empty is equivalent to deciding whether a certain Z-SFT over the alphabet {0, 1} I is empty (the last condition, though cumbersome, is a finite, local restriction symbols in x and is equivalent to excluding a finite number of patterns). Clearly the reduction from the first problem to the second is computable, and the second problem is decidable, since it is equivalent to deciding whether there are any cycles in a finite graph associated to the given SFT in an effective way (see e.g. [9]). Thus the original problem is decidable as well.
Proof. (of theorem 11) We show that there is a uniformly recursive sequence L n ⊆ P * 1 so that X n = Ω Ln = ∅ for every n and every Z-EDS appears as one of the X n 's. Given such a sequence, the product ∞ n=1 X n is universal for EDS and is itself an EDS by the previous lemma.
Let (A n ) ∞ n=1 be a fixed recursive enumeration of all algorithms and let L ′ n = {b : A n halts on input b} The sequence L ′ n is uniformly RE, because given n and b, in order to determine if b ∈ L ′ n one first computes A n (we can because the sequence A n is recursive) and then simulates the computation of A n on input b, halting only if this computation halts. This achieves the first of our goals, since clearly the sequence Ω L ′ n will contain all effective subshifts. The problem is that some Ω L ′ n 's will be empty. We therefore will define an RE sequence L n with L n = L ′ n if Ω L ′ n = ∅, and with Ω Ln = ∅ in any case.
First, given n, k ∈ M and a ∈ P * d we say that a is (n, k)-recognized if the algorithm A n halts on input a within k steps. Note that this condition is recursive, since A n can be computed from n and then its computation on input a can be simulated for k steps to see if it halts. Choose an enumeration b 1 , b 2 , . . . of P * d . Given k let Clearly the L n,k can be computed given n, k, they are increasing in k, and their union is L ′ n . Finally, let L n = {a ∈ P * d : Ω L n,k = ∅ and a ∈ L n,k for some k ∈ N} By lemma 6 we see that L n is RE. Also, if Ω L ′ n = ∅ then Ω L n,k = ∅ for every k so a ∈ L n if and only if a ∈ L ′ n , or in other words, L n = L ′ n . On the other hand, if Ω L ′ n = ∅ then by compactness there is a k for which Ω L n,k = ∅; let k 0 be the minimal such k. One sees that , a ∈ L n if and only if a ∈ L n,k0−1 , so L n = L n,k0−1 and by definition Ω L n,k 0 = ∅. This completes the proof.

Nonexistence of universal SFTs
The proof from the last section cannot be adapted to Z d -EDS because, for d ≥ 2, one cannot decide in general if a d-dimensional SFT is empty; this is Berger's theorem. Although this in itself is not a proof that no (d, k)-universal SFTs exist for k > 1, the proof in fact involves showing that if one did exist then it could be used as a component in an algorithm for deciding the emptiness of SFTs, contradicting Berger's theorem.
Although the proof that universal Z d -EDS don't exist for d ≥ 2 is not conceptually difficult, it will be more transparent to first establish the weaker claim that there are no (d, d)-universal SFTs, i.e. no Z d -SFTs which factor onto every other Z d -SFT. This follows easily from entropy considerations, but the proof we give is of a recursive nature.
Fix d. It will be convenient to consider SFTs over alphabets which are subsets of N; this allows us to examine sets of SFTs without restricting the alphabet size, and is no restriction since any SFT is isomorphic to one over the alphabet N.
For a language L over Σ, we say that a pattern a ∈ Σ E is L-admissible if, whenever b ∈ Σ F is a pattern in L and F + u ⊆ E, the pattern b does not appear at u in a; i.e. a(u We return to the question of (d, d)-universal SFTs. It is well-known that the set L = {L : L is a finite set of finite patterns over N and X L = ∅} is RE. To see this consider the algorithm that is given as input a finite set L over a finite Σ ⊆ N, and iterates over n ∈ N; for each n it checks if there exist L-admissible a ∈ Σ [−n;n] . If none exist it announces that X L = ∅ and halts. Otherwise, is goes on to the next n. Clearly if X L = ∅ then the algorithm will not halt, and a compactness argument shows that if X L = ∅ it will.
Thus, in order to prove the claim about non-existence of (d, d)-universal SFTs, we show that, if there is some finite L * so that X = X L * ⊆ Σ Z d factors onto every Berger's theorem.
The following algorithm establishes that M is RE. As input it accepts a finite set L of patterns over N and decides if X L is empty. Recall that L * is assumed to be a finite set of patterns so that X L * factors onto every 0-entropy SFT; we shall use L * in constructing our algorithm. Let R ∈ N be such that each pattern in L * and L is supported in [−R, R] d .
Algorithm. For each triple r, k, ϕ with r, k ∈ N, r > R + k + 1, and ϕ : To see that this works, note that if the algorithm halts in (2) for some triple (r, k, ϕ) then the image under ϕ of any point in X L * gives a point in X L , implying that X L = ∅. Conversely, if X L = ∅ then by universality of X L * there is some k and factor map X L * → X L which is defined locally by some ϕ : A compactness argument shows that for these k, ϕ, if the condition in (2) fails for every r then there is a point x ∈ X L * with ϕ(x) / ∈ X L , a contradiction.
We now deal with the general case.
Proof. (of theorem 2) The proof that there is no universal EDS follows the same argument as above, the only difference being that factor maps are no longer sliding block codes, which makes the notation more cumbersome. If (X, Z d ) is a totally disconnected system and π : X → Y ⊆ Σ Z d is a factor map to a subshift Y , then the factor map is determined by the partition of X into closed and open sets, and conversely if {U σ } σ∈Σ is such a partition then it defines a factor map π into Σ Z d , where (πx)(u) = σ if and only if T u x ∈ U σ . Thus in order to adapt the proof above to general EDS we iterate over partitions rather than sliding block codes; note that the finite partitions of Ω into closed and open sets can be effectively enumerated.
Fix d and suppose L ⊆ P * d is an RE set and that Ω L factors onto every Z d -EDS. As before, we obtain a contradiction by showing that the set L ′ above is RE.
Let L n be a recursive increasing sequence of sets with ∪L n = L, which exists since L is RE. Consider the following algorithm, which as input accepts a finite set M of finite patterns over N: (1) Let R be an upper bound on the diameter of the patterns in M .
(2) For each triple r, n and (U σ ) σ∈Σ , with r, n ∈ N, r > R and (U σ ) a closed and open partition of Ω, do: • If for every x ∈ Ω the condition implies that f (x)| [R;R] d is M -admissible, announce that Y = ∅ and halt.
Although the condition in (a) is not phrased as a finite condition and may appear hard to check, it is actually a question about the non-emptiness of the intersection of finitely many cylinder sets which are given in the data, and can therefore be effectively checked. The proof that this algorithm produces the correct output is the same as the one given above for SFTs, and we omit it.

Realization of effective attractors
Let U ⊆ R d be open and let f : U → U be an effective map with attractor X ⊆ U as in definition 4. We wish to show that there is an EDS which extends (X, f ); theorem 5 then follows.
Let us say that a dyadic rational is one of the form k 2 n ; a dyadic interval is a closed interval of the form [ k 2 n , k+1 Given a binary representation x of x ∈ R let D N (x) denote the set of all y ∈ R whose first N binary digits after the "decimal" point agree with x; this is a closed dyadic interval of length 2 −N . For x = (x 1 , . . . , x d ) ∈ R d and binary representations which is a dyadic cell with x ∈ D N (x). There are at most 2 d binary representations of (x 1 , . . . , x d ), each giving rise to a dyadic cell containing x.
Lemma 14. Let f : U → U ⊆ R d be an effective map. Then there is an algorithm which, given a closed dyadic cell D ⊆ U and an integer n, outputs a finite set of rational points in R d which are 1 n -dense in f (D). Proof. Let A be the algorithm given by the definition of an effective function; for a point x ∈ R d with binary representation x and for given n let A(x, n) denote the approximation of f (x) produced by A on inputs x, n, so A(x, n) − f (x) < 1 n . Let N = N (x, n) denote the number of digits from x that is used by A in computing A(x, n). If x ′ ∈ D N (x) and x ′ is a binary representation of x ′ ∈ R d agreeing with x for the first N digits, then A(x ′ , n) = A(x, n), since the algorithm halts before having a chance to detect that x = x ′ . Hence, for There are at most 2 d binary representations x of x, and for each we get an N as above. Let N * = N * (x, n) be the maximum of these numbers and let U n (x) be the interior of the union of dyadic cells of side 2 −N * containing x. This is an open set containing x, and from the discussion above we see that the diameter of f (U n (x)) is < 5 n , since the image of each cell has diameter < 2 n and is within distance 1 n of f (x). Note that for a dyadic point r ∈ Q d , both N * (r, n) and U n (r) are computable.
If D ⊆ U is a closed dyadic cell and n ∈ N, then there is a finite cover of D by sets of the form U n (r), and such a cover can be computed by iterating over all finite collections of the sets U n (r) for r a dyadic rational, until such a collection is found that covers D. If {U n (r i )} M i=1 is such a collection, then as we have seen, the set , where A(r, n) is the output of the algorithm on input n and the binary representation r of r. We do not claim that C can be found effectively but it exists and can be described by finite data, and we may use it in the algorithm that we now present. As input the algorithm takes a dyadic cell D. It then iterates over n and for each n it computes a finite set of points F which are 1 n -dense in f (C) (this can be done by the previous lemma). If every point in f has distance > 1 n from D the algorithm halts; otherwise it proceeds to the next n.
In the same way we have Lemma 16. Let X be the attractor of an effective map f : U → U ⊆ R. Then there is an algorithm which, given as input two (closed) dyadic cells D ′ , D ′′ , halts f (D ′ ) ∩ D ′′ = ∅ and otherwise runs forever.
We can now prove theorem 5: Proof. Let f : U → U be an effective map with attractor X ⊆ U . Since X is bounded and we can assume that U is bounded, and without loss of generality U ⊆ [0, 1] d . Let K = {0, 1} N be the Cantor set and π : K → [0, 1] d be given by x → (x 1 , . . . , x d ) where Let Y ⊆ Ω = K Z be those points ω so that π(ω(n)) ∈ X , n ∈ Z and π(ω(n + 1)) = f (π(ω(n))) Clearly Y is a subshift and π : Y → X is a factor map from Y to X. It remains to show that Y is an effective subshift. One may verify that Y = Ω L ; the proof will be complete if we show that L is RE. In order to so this it suffices to show that there is an algorithm which, given a : {n, n + 1} → {0, 1} {1,2...dk} , halts if π([a(n)]) ∩ X = ∅ or π([a(n + 1)]) ∩ X = ∅ or f (π([a(n)])) ∩ π([a(n + 1)]) = ∅, and otherwise runs forever. But this is a direct consequence of the lemmas preceding this proof.

Open Problems
We conclude with a few problems which arise in connection with this work.
We have shown that universality cannot occur for effective Z d systems. This can be rephrased as follows: If d > 2 then for every effective Z d -system X, there is a Z d -SFT Y so that X does not factor onto Y . If we relax the effectiveness requirement, a stronger result is true: under mild assumptions on the original system X, there is a system disjoint from X in the sense of Furstenberg.
Problem. Given a minimal Z d -EDS, does there exists a Z d -EDS disjoint from it? (one may of course ask this about other classes).
(minimality is required to avoid trivial counterexamples). The same question may be asked for minimal SFTs.
The non-existence of universal EDS was demonstrated using purely recursiontheoretic considerations. It is not clear how to handle some restricted interesting classes of EDS. Of particular interest are the minimal SFTs, nontrivial examples of which were constructed in [10] and more examples follow from [5]. We note that for SFTs, minimality implies zero entropy, so entropy cannot rule out a universal minimal SFT. We also note that, besides being an interesting class dynamically, minimal SFTs have the additional feature that the set of patterns appearing in a minimal SFT is recursive, and the extension problem is decidable for them, i.e.
given a locally admissible pattern one can decide if it can be extended to an infinite configuration [4]. Thus this class does not exhibit the recursive complexity of SFTs in general. Problem. If a recursive product SFTs has finite entropy, can it be extended to an SFT?