EFFECT OF QUANTIFIED IRREDUCIBILITY ON THE COMPUTABILITY OF SUBSHIFT ENTROPY

. We study the algorithmic computability of topological entropy of subshifts subjected to a quantiﬁed version of a strong condition of mixing, called irreducibility. For subshifts of ﬁnite type, it is known that this problem goes from uncomputable to computable as the rate of irreducibility decreases. Furthermore, the set of possible values for the entropy goes from all right-recursively computable numbers to some subset of the computable numbers. However, the exact nature of the transition is not understood. In this text, we characterize a computability threshold for subshifts with decidable language (in any dimension), expressed as a summability condition on the rate function. This class includes subshifts of ﬁnite type under the threshold, and oﬀers more ﬂexibility for the constructions involved in the proof of uncomputability above the threshold. These constructions involve bounded density subshifts that control the density of particular symbols in all subwords.


1.
Introduction. Topological entropy is a real parameter widely used in the study of dynamical systems as a conjugacy invariant and a measure of dynamical complexity. The problem of effectively computing topological entropy -that is to say, given a description of a dynamical system and ε > 0, computing its topological entropy with a maximum error of ε -has been considered for many systems. These efforts, leading to positive as well as negative answers, have been documented by Milnor in 2002 [15].

SILVÈRE GANGLOFF AND BENJAMIN HELLOUIN DE MENIBUS
circumstances [16]. More examples can be found in [15]. In many cases, these works characterized the class of real numbers that can appear as entropy of a system in the studied class (see e.g. [8,7,5]).
The case of one dimensional subshifts of finite type (SFT) is well understood. Given a one-dimensional SFT, its entropy is known to be computable through a simple method based on computing the largest eigenvalue of an associated graph. Furthermore this method characterizes the numbers which are entropies of such a system by an algebraic condition: namely, they are exactly the non-negative rational multiples of logarithms of Perron numbers [14].
The case of higher-dimensional SFT remained open for a long time, with many specific examples being studied and solved approximately or exactly using ad hoc methods, especially by the statistical physics community; see [12,3,18] among many others. A negative answer came much later in the seminal work of [8], where the authors proved the following. Given a multidimensional SFT, its entropy is not computable in general; and real numbers that can appear as the entropy of a multidimensional SFT are characterized by a computability condition: the set of Π 1 -computable numbers (i.e. upper-semi-computable).
In both settings, various authors studied the effect of dynamical restrictions, particularly mixing properties, on the difficulty of computing entropy. While the situation for mixing one-dimensional SFT is unchanged [14], entropy becomes computable for higher-dimensional SFT with strong mixing properties [8]. For the particular case of two-dimensional SFT, Pavlov and Schraudner proved that entropy is even exptime-computable under a different type of mixing condition (block-gluing) [19], with a partial characterization. It seems natural in this context to introduce a notion of irreducibility rate that corresponds to the strength of the mixing restriction. We prove (Theorem 3.5) that entropy is computable when the irreducibility rate is below a certain level, but we are unable to locate the threshold marking the difficulty jump between computable and uncomputable cases.
In this article, we consider subshifts that are not necessarily of finite type but that can be described by an algorithm in some sense (decidable subshifts), hoping that results on this class will provide insights for the finite type case. In general entropy of these subshifts is not computable [20] and all Π 1 -computable numbers can be realised as entropy [6], but entropy becomes computable under strong mixing conditions [21]. This is very similar to the situation for multidimensional SFT.
In this more general context, we are able to characterize precisely the location of the threshold in the irreducibility rate marking the difficulty jump between the computable and uncomputable cases. More precisely, our new results are the following: Main Theorem (Theorem 3.5, Theorem 3.6 and Theorem 3.7). Let f : N → N be a nondecreasing function such that f (n) = o(n).
1. If the series n f (n) n 2 converges at a computable rate, there exists an algorithm that computes the entropy of f -irreducible decidable subshifts and f -irreducible SFT. 2. If this series diverges, the set of numbers that appear as the entropy of a decidable f -irreducible subshift are exactly the Π 1 -computable numbers, and there exists no algorithm that computes the entropy of these systems.
Missing definitions, in particular the notion of convergence of a series with computable rate, appear futher in this text. The first result applies, e.g., to any (Second line) Set of possible entropies. "Weak" and "Strong" mixing stand for irreducibility rates above or below the threshold, respectively; "Very strong" stands for constant irreducibility rates, or similar properties. "Π 1 -comp." means that the problem is Π 1 -computable, but not computable; "Π 1 reals" stands for the set of Π 1 -computable reals; † symbols indicate the contribution of the present article.
Results in this text are of two kinds. On the one hand, we describe explicit algorithms to approximate the entropy in some cases, providing a computational upper bound on the difficulty of the problem. On the other hand, given a class of real numbers defined by their computational complexity, we build a family of subshifts whose entropy take all values in this class. This proves that computing the entropy is at least as hard as computing all numbers from this class, giving a computational lower bound.

Background.
2.1. Subshifts. Let A be a finite set called alphabet. We call symbols the elements of the alphabet.
Let U ⊂ Z be a finite subset of Z. A pattern on the alphabet A and support U is some element of A U . Denote A × the set of patterns on A, that is, where the union is over the finite subsets. A pattern of support 0, n − 1 d is an n-block. When d = 1, n-blocks are called words and we denote A * the set of all words.
The set A Z d is the d-dimensional full shift. We often omit the dimension when a result is valid for all d. An element of A Z d is also called a configuration.
The cylinder associated to a pattern u ∈ A U and position i ∈ Z d is defined as We say that a pattern u ∈ A U appears in a configuration x ∈ A Z d when x ∈ [u] i for some i ∈ Z d . Similarly, a pattern v ∈ A V appears in (or is a subpattern of) another pattern u ∈ A U when V ⊂ U and u| V = v. For a pattern u ∈ A U , and a symbol a ∈ A, we denote the number of symbols a appearing in the pattern u as follows: # a w = #{ i ∈ U : w i = a}.
Denote e 1 , . . . , e d the canonical generators of Z d . For i ∈ {1, . . . , d}, we call the ith shift function the function σ i : Shifts define an action of Z d on the full shift denoted σ (the shift action). The d-dimensional full shift A Z d endowed with the product of the discrete topology is a topological space. A subshift is a closed subset of the full shift which is stable under the action of the shift functions defined above.
The language of a subshift Σ, denoted L(Σ), is the set of patterns that appear in some configuration of Σ. Formally, Any subshift Σ can be defined by a set of forbidden patterns, meaning there exists a set F ⊂ A × such that For example, one can choose F Σ = L(Σ) c , but this choice is not unique. A subshift is of finite type (denoted SFT) when it can be defined by a finite set of forbidden patterns.
In a context where some set of patterns F such that Σ = Σ F is fixed, a pattern w is said to be locally admissible for Σ if no pattern of F appears in w. By opposition, we say w is globally admissible if w ∈ L(Σ). Being locally admissible depends on the choice of F, but we omit the set when some result applies to any choice of F.
We denote C n = −n, n d . The following lemma is a classical result that expresses the fact of belonging in the language of a subshift in terms of locally admissible patterns. We include the proof for completeness. Lemma 2.1 (Folklore). Let u ∈ A Cn be a pattern. This pattern is globally admissible (u ∈ L(Σ)) if and only if for all N > n there exists some pattern p N ∈ A C N locally admissible for Σ such that p N | Cn = u.
In other words, a pattern u is not globally admissible if and only if there is a larger window such that the pattern cannot be extended to this window without creating a forbidden pattern.
Proof. Let u ∈ A Cn .
(⇒): If u appears in some configuration x ∈ Σ, the patterns p N = x| C N for all N ≥ n verify the second member of the equivalence.
(⇐): The reciprocal uses a compacity argument. Assume that u verifies the second assertion. Fix γ 0 = u. Since A Cn+1 is a finite set, we can extract a subsequence (p N1(k) ) k from (p N ) N such that p N1(k) | Cn+1 is constant; denote this value γ 1 . Then iterate this process, extracting for all i a subsequence (p Ni+1(k) ) k from (p Ni(k) ) k such that p Ni+1(k) | Cn+i+1 is constant, denoted γ i+1 . We constructed a sequence of locally admissible patterns γ k ∈ A C n+k such that γ 0 = u and γ k+1 | C n+k = γ k for all k. Now define the configuration x such that for all i, x i is the limit of the sequence (γ k i ) k . Since all the patterns γ k are locally admissible, it follows that x ∈ Σ. Because u appears in x, it follows that u ∈ L(Σ).
We denote d the distance on Z d defined for all i, j by d( i, j) = max 1≤k≤d | i k − j k |.

2.2.
Entropy. Let d ≥ 1, and Σ a d-dimensional subshift. The couple (Σ, σ) is a dynamical system. The entropy of Σ is the following number: The second equality is a well-known result. See [13], Chapter 4, for a proof and more information about this notion. In this formula, and in the remainder of the paper, the logarithm is in base two.
For 0 < ε < 1, the binary entropy of ε is given by Despite its name, it is not related to subshift entropy, but to the (informationtheoretical) entropy of a Bernoulli process and happens to be useful in our proofs.
there exists a configuration x ∈ Σ such that x U = u and x V = v.
If the previous definition is true only when U and V are n-blocks, Σ is said to be f -block gluing.
Remark 2. This notion is related to and extends various mixing properties that appear in other texts.
• Σ is strongly irreducible (also called the specification property) iff it is O(1)irreducible; • Σ is topologically mixing iff there exists some f such that Σ is f -irreducible; • Σ is block-gluing iff it is O(1)-block-gluing.
For a subset U ⊂ Z, we denote γ(U) the smallest connected subset containing U (in other words, its convex hull).
Proof. It is clear that f -irreducibility implies f -block gluing. We prove the converse. First, let us prove some property of couples of subsets of Z.
Let U, V be two non-empty subsets of Z such that d(U, V) > 0. Let (U i ) 1≤i≤k be the maximal subsets of U that satisfy γ(U i ) ∩ V = ∅, and define (V i ) 1≤i≤l similarly. Fixing U i = γ(U i ), U = i U i , and V i and V similarly, one can check that: 3. |l − k| ≤ 1 and one of the following statements is true: and u ∈ L U (Σ), v ∈ L V (Σ). Since u and v are globally admissible there exist Iterating the same argument, we have n ≥ max(δ(γ(U 1 ∪ V 1 )), δ(U 2 )) and Through an easy induction, we see that there exists w ∈ L γ(U ∪V ) such that w| U = u and w| V = v .

Computability.
Turing machines are a model of computation that is believed to capture the informal notion of computation (Church-Turing thesis). Although it is not needed in this text, we give the formal definition of a Turing machine for completeness.
A Turing machine consists in a head attached with a symbol representing its internal state, which is able to move over and read/write on an infinite tape filled with symbols, and to change its internal state each time it goes over a symbol according to a transition function.
Formally, a Turing machine is a tuple (A, Q, q 0 , q h , #, δ) where A is a finite set (the tape alphabet), Q is a finite set (the states alphabet), q 0 ∈ Q is the initial state, and q h ∈ Q the halting state, # / ∈ Q is a special blank tape symbol and is called the transition function. When a machine is in state q and reads a symbol a at its current position, δ(a, q) specifies the new tape symbol being written, the new state of the head and the direction in which the head moves at this step.
Initially, the tape contains a finite word from A * (the input) surrounded by blank symbols, the head is on position 0 with state q 0 . The evolution of the machine is then determined by the rules described above. If the machine enters the halting state q h (the machine halts), the output is the finite non-blank portion of the tape at this point.
A function A * → A * is said to be computable if there exists a Turing machine that, given as input u ∈ A * , eventually halts and outputs f (u). This notion extends to functions N → N through a binary encoding, and to other countable input and output sets by appropriate encodings. We present a notion of computability for real numbers and subshifts as needed in this text.
Computability of real numbers: Remark 3. In the definition for a Π 1 -computable number given above, we also have where ϕ α (n) = min k≤n ϕ α (k) for all n. ϕ α is also computable and is nonincreasing, which means that the function ϕ α in the definition can be chosen nonincreasing.

Remark 4.
A computable real number is in particular Π 1 -computable. However, there exist Π 1 -computable numbers that are not computable. For instance, take the real number where ε k = 0 if the kth Turing machine stops on the empty input and ε k = 1 otherwise. This number is Π 1 -computable but not computable: otherwise, the halting problem would be decidable.

Computability of subshifts:
General subshifts do not admit a description by a finite sequence of symbols, so we need to restrict to a countable subclass: decidable subshifts, defined below.
is a computable function.

2.5.
Irreducibility and decidability for subshifts of finite type. It is a wellknown fact that multidimensional SFT (d ≥ 2) are not decidable in general; this is sometimes referred to as the undecidability of the extension problem. However, some dynamical properties (minimality, block gluing, etc.) make subshifts of finite type decidable.
Remark 5. More generally, the following proofs and lemmas work for f (n)-irreducible SFT where f is such that f (n) − n 2 has no upper bound as n → ∞. On the other hand, one can deduce from the main construction of [4] the existence of a linearly irreducible subshift whose language is undecidable.
Remark 6. In the following of this text, all the considered functions f verify the condition f (n) = o(n). Under this constraint, Proposition 2 means that the class of decidable subshifts extends the class of subshifts of finite type.
The remainder of this section is dedicated to a proof of this statement. It is a slight generalization from the proof of Corollary 3.5 in [8], which proved the same result for O(1)-irreducible subshifts.
Let Σ be a non-empty d-dimensional SFT for some d ≥ 1 defined by a set of forbidden patterns F. We assume without loss of generality that elements of F are r-blocks for some r > 0 (by replacing every pattern u in F by the set of r-blocks that contain it, assuming r ≥ δ(u)).
Remember that C n = −n, n d , and denote D m = C m+r \C m for all integer m. Lemma 2.7. Let Σ be a o(n)-irreducible, non-empty SFT. A pattern u ∈ A Cn is globally admissible for Σ if and only if there exists some integers N 0 and m such that • N 0 ≥ m + r and m ≥ n; • for every v ∈ A Dm such that w| Dm = v for some locally admissible pattern w ∈ A C N 0 , there exists a locally admissible pattern w ∈ A C N 0 such that w | Cn = u and w | Dm = v. Proof.
(⇐): Assume u verifies the last condition. Consider x ∈ Σ. Taking v = x| Dm and w = x| C N 0 in the condition, we get a locally admissible pattern In the first case, it is a subpattern of w , and it can not be a forbidden pattern of Σ since w is locally admissible. In the second case, it is a subpattern of x, which is in Σ. As well, it can not be forbidden in Σ. Since forbidden patterns of Σ are r-blocks, this means that x ∈ Σ. Hence u is globally admissible. exists some k such that by contraposition. • Using the irreducibility property on C n and D k : Assume that Σ is f -irreducible, and assume that u ∈ L(Σ).
For all k ≥ n, C n and D k have both diameter at most 2k + 2r and d(C n , D k ) = k − n. Let m be the smallest of the integers k such that From the f -irreducibility property of Σ, we have that for any globally admissible v ∈ A D k , there exists a configuration x ∈ Σ such that x| D k = v and x| Cn = u. • Sufficiently large locally admissible patterns have globally admissible centers: It follows that all the locally admissible patterns w ∈ A C N 0 satisfy w| D k ∈ L(Σ). • Using the irreducibility on these centers and replacing the inside: Consider any locally admissible pattern w ∈ A C N 0 . As noted above, w| Dm ∈ L(Σ), so by f -irreducibility there exists x ∈ L Cm+r (Σ) such that x| Cn = u and x| Dm = w| Dm . Now define w by putting w | Cm+r = x and w | C N 0 \Cm+r = w| C N 0 \Cm+r . This pattern is locally admissible, with the same argument as in the proof of the indirect implication. This process is illustrated in Figure 1. Using the previous lemmas, we finish the proof of Proposition 2.
Take as input a pattern u and assume without loss of generality that u ∈ A Cn . Do the following in parallel: 1. For every integer N , check whether Lemma 2.1 applies; if for some N this is not the case, then output u / ∈ L(Σ). 2. For every integers m and N 0 , check whether Lemma 2.7 applies; when this is the case, output u ∈ L(Σ). By Lemma 2.1 and Lemma 2.7, exactly one of these processes will stop and output the correct answer. This means that Σ is decidable.
3. Computability of the entropy of subshifts. From now on we consider decidable subshifts. This section is organized as follows: • In Section 3.1 we present the state of the art for computing entropy in SFT and decidable subshifts. • In Section 3.2 we prove that when some sum related to f converges at a computable rate, then the entropy of f -irreducible decidable subshifts is computable. • In Section 3.3, we prove that when the same sum diverges, every Π 1 -computable number can be realised as the entropy of an f -irreducible decidable subshift, which implies that the entropy of this class of subshifts is uncomputable in general.
Hence this summability condition defines some kind of threshold that delimits the computable and uncomputable cases, leaving open only the case when the sum converges at a non-computable rate.
3.1. State of the art. In this section, we present known results that are either folklore or appeared in the litterature.
Definition 3.1. Let d ≥ 1 and C a class of d-dimensional decidable subshifts. We say there is an algorithm that computes the entropy of C when there is a Turing machine that takes as input some integers (n, m) and, if the nth Turing machine decides a subshift Σ ∈ C, outputs some rational number r n,m such that: We say that the algorithm upper semi-computes the entropy of C when instead h top (Σ) = inf m r n,m .
Remark 7. In Definition 3.1, if the number given as input to the algorithm does not correspond to a Turing machine that decides the language of a d-dimensional decidable subshift in C, the behaviour of the algorithm is unspecified.
Intuitively, there is a uniform way to compute the entropy for this class of subshifts, but the validity of the input cannot be checked.

Proposition 3.
There is an algorithm that upper-semi-computes the entropy of decidable d-dimensional subshifts. In particular, entropies of decidable d-dimensional subshifts are Π 1 -computable real numbers.
This result also holds for d-dimensional SFT. The following result states that this upper bound is tight. A similar result for d-dimensional SFT, d > 1, was obtained in [8]. This proposition can also be obtained as a corollary of Theorem 3.7.
The situation improves when considering subshifts with mixing properties: This last result suggests that irreducibility relates to the computability of the entropy in SFT. It is natural to ask for which rates of irreducibility the entropy of SFT is computable or uncomputable.
We consider in the rest of this paper decidable subshifts, a more flexible class that lets us determine a threshold delimiting the irreducibility rates for which the entropy is computable or uncomputable. Proposition 2 implies that SFT whose irreducibilty rate is near or under the threshold are included in this class.

3.2.
Under the threshold. In this section, we prove that when the irreducibility rate f satisfies some summability condition, the entropy of the class of f -irreducible decidable subshifts is computable. These results apply to f -irreducible SFT as well.
Definition 3.3. Let (a n ) n be a sequence of non-negative numbers. The series a n is said to converge at a computable rate when there is a computable function n : N → N (the rate) such that for all t ∈ N, Lemma 3.4. Let (a n ) n and (b n ) n two series of non-negative integers, such that for all n, a n ≤ b n . If the series b n converges at a computable rate, then the series a n also converges at computable rate.
Before stating the main theorems for this section, we prove two technical results related to the sum we consider in the hypothesis of the theorem. Proof.
1. Comparing the two series: Since f is non-decreasing, we have that for all k, The first inequality comes from the fact that f (2 k ) ≤ f (i) for i ≥ 2 k , and for the second inequality we use that i ≤ 2 k+1 . The last two inequalities are similar. Thus the two series have the same convergence. 2. Computable rate: As a consequence of Lemma 3.4, the series Proof. By the previous proposition, the condition f (n) n 2 < +∞ implies f (2 n ) 2 n < +∞. As a consequence, f (2 n ) 2 n → 0. Since f is nondecreasing, we thus have f (n) ≤ f (2 log 2 (n) +1 ) = o(2 log 2 (n) +1 ) = o(n).

SILVÈRE GANGLOFF AND BENJAMIN HELLOUIN DE MENIBUS
The following theorem is stated for one-dimensional subshifts, but we extend it to higher dimensions afterwards.
Theorem 3.5. Let f be a non-decreasing computable function such that the series f (n) n 2 converges at a computable rate. Then there is an algorithm that computes the entropy of f -block gluing one-dimensional decidable subshifts.
Remark 9. It is actually enough that f is non-decreasing for n large enough.
Remark 10. In particular, values of the entropy for subshifts in this class are computable real numbers. In this theorem, f -block gluing can be replaced by firreducible.
Remark 11. In the statement of Theorem 3.5, the algorithm depends on f . However, a careful reading of the proof shows that there exists an algorithm which takes as input positive integers c and m and, assuming that m represents a Turing machine that decides the language of some c-block gluing Z-subshift, outputs the entropy of this subshift. In other words, a single algorithm computes the entropy of all O(1)-block-gluing decidable subshifts, which we will use in a later proof.
Proof. Let Σ ⊂ A Z be a one-dimensional f -block gluing subshift. By definition of the f -block gluing property, for any two words u, v ∈ L n (Σ), there exists a word w ∈ A f (n) such that uwv ∈ L 2n+f (n) (Σ). Because u, v can be chosen freely we have , the second inequality coming from the fact that any globally admissible word of length 2n + f (n) can be decomposed into a globally admissible word of length 2n and some word of length f (n) for a crude upper bound. By taking the logarithm of this inequality and dividing by 2n, we get: The previous inequality being true from any n, we apply it iteratively on the sequence (2 n+k ) k≥0 , and combining the l ≥ 1 first inequalities we get: converges at a computable rate (by Proposition 5 and the hypothesis of the theorem), there exists a computable function t → n(t) such that for all t, To conclude, given m some integer such that the mth Turing machine decides the language of one-dimensional f -block gluing subshift Σ and a precision t, the algorithm to approximate h top (Σ) runs as follows: 1. compute n(t), 2. then count all words of L 2 n (t) (Σ) (this is possible because Σ is decidable),

then compute a rational approximation of
, using an approximation of the logarithm function up to a precision 2 −t−1 , which is in turn a rational approximation of h top (Σ) up to precision 2 −t . We conclude that the entropy h top (Σ) is uniformly computable.
We now extend this proof to the case of d-dimensional subshifts, for d ≥ 2.
Theorem 3.6. Let d ≥ 1 and f be a non-decreasing computable function such the series f (n) n 2 converges at a computable rate. Then there is an algorithm that computes the entropy of f -block gluing d-dimensional decidable subshifts.

Corollary 1.
For a function f that verifies the same conditions as Theorem 3.6, there is an algorithm that computes the entropy of f -irreducible d-dimensional decidable subshifts. In particular, this algorithm also computes the entropy of f -block gluing or f -irreducible SFT.
Proof. Let d ≥ 1 and Σ some d-dimensional decidable subshift which is f -block gluing. Denote for n 1 , . . . , n k ∈ N, The diameter of this set is N = max k n k .
Fix U = C n1,...,n d and V = U + i where i = (n k + f (N )) e k for some k. From the f -block gluing property of Σ, for all u ∈ L U (Σ) and v ∈ L V (Σ), there exists a pattern w ∈ L W (Σ) where W = C[n 1 , n 2 ...n k−1 , 2n k + f (N ), n k+1 , ...n d ] such that w| U = u and w| V = v.
Therefore, taking k = 1: where the first inequality is by the above remark, the second inequality comes from decomposing w into two patterns on C[2n 1 , n 2 , .., n d ] and on C[f (N ), n 2 , .., n d ], respectively, and the last is by definition of N . We have δ(C[2n 1 , n 2 , ..., n d ]) ≤ 2N , so applying the same argument on k = 2: . Applying this argument for each k ≤ d, we have: where we used the fact that f is nondecreasing. Applying this equation to n 1 = ... = n d = n for some n, The end of the proof is similar to the proof of Theorem 3.5.
Remark 12. The constant c is only needed so that the function is nonincreasing, which can be checked by a straightforward computation.
Using Proposition 5, we consider the sum For all n, and this sum converges at a computable rate n : t → t + 1, hence the same result follows for f . We conclude using Theorem 3.5.
3.3. Above the threshold. In this section, we consider decidable subshifts whose irreducibility rate is above the threshold, and prove the following theorem: Then there exists a decidable f -irreducible one dimensional subshift with entropy α.
First we prove this theorem for d = 1 using bounded density subshifts, introduced by B. Stanley [22] (as bounded density shifts). We introduce these objects in Section 3.3.1 and study their properties in the rest of the section, the actual proof consisting in a construction presented in Section 3.3.10. The result is then extended to any d ≥ 2 in Section 3.3.11.
In all this section, f is a function which verifies the conditions of Theorem 3.7, and we denote F : n → 2n + f (n).

Bounded density subshifts.
Definition 3.8 (Bounded density subshift). Let p = (p n ) n≥1 be a sequence of positive integers. The bounded density subshift associated to p, denoted Σ p ⊂ {0, 1} Z , is the one-dimensional subshift defined by the set of forbidden words F p = n≥1 F p n , where for all n: F p n = {u ∈ A n : # 1 u > p n } . Example.
• Reciprocally, take w ∈ L n (Σ p ). Since 11 is forbidden, every 1 in w is followed by a zero, except if it is the last symbol. It follows that 2# 1 w − 1 ≤ n, and therefore # 1 w ≤ n/2 = p n , which means that w ∈ L n (Σ p ). We conclude that Σ p ⊂ Σ p . 2. For k ∈ N, define Σ k the bounded density subshift associated to the sequence p k defined as follows: • if the k-th Turing machine starting on an empty input stops before computing n steps, then p k n = p n−1 ; • otherwise, p k n = p n−1 + 1. If the k-th Turing machine never stops on the empty input, then p k n = n for all n and Σ k = {0, 1} Z the full shift, with entropy 1. On the other hand, if the k-th Turing machine stops at some point, then p k is ultimately constant and Σ k has entropy zero by Lemma 3.11.

3.3.2.
Bounded density subshifts are decidable. To use bounded density subshifts in the proof of Theorem 3.7, we prove sufficient conditions for them to be decidable in Lemma 3.10. Lemma 3.9 is used in the proof of Lemma 3.10, and is used again throughout the section. Lemma 3.9. Let p be a non-decreasing sequence of integers, F p the set of forbidden patterns defined in Definition 3.8, and w a word.
If w / ∈ F p , then 0 m w0 n / ∈ F p for all integers m, n. If w is locally admissible for Σ p (it has no subword in F p ), then 0 m w0 n is globally admissible for Σ p for all integers n, m. In particular, any locally admissible word is globally admissible.
Proof. The first statement follows from the fact that # 1 0 n w0 m = # 1 w and that p is non-decreasing.
For the second statement, the element x of {0, 1} Z such that • x 0,|w|−1 = w • and for all i ∈ Z\ 0, |w| − 1 , x i = 0. belongs to Σ p . Indeed, any finite word appearing in x is amongst the following types: 1. 0 k u, where k ≥ 0 and u is some prefix of w, 2. v0 k , where k ≥ 0 and v is some suffix of w, 3. 0 m w0 n , where n, m ≥ 0. In each case, u, v and w cannot be forbidden patterns since w is locally admissible. By the first statement, none of these words can be a forbidden pattern. As a consequence, for all n, m ≥ 0, 0 m w0 n is globally admissible. Lemma 3.10. If p is non-decreasing and computable, then Σ p is decidable.
Proof. Since p is non-decreasing, Lemma 3.9 implies that any locally admissible pattern of Σ p is globally admissible. Hence the algorithm to decide if an input word w ∈ {0, 1} l is in L(Σ p ) runs as follows: • compute the value of p k for all k ≤ l; • compute the set of forbidden patterns of length ≤ l; • check if one of these patterns appears in w.

SILVÈRE GANGLOFF AND BENJAMIN HELLOUIN DE MENIBUS
• if this is the case, return 0 ; else, return 1.
3.3.3. Zero limit density implies zero entropy. In this section, we prove a technical lemma related to the entropy of bounded density subshifts. This lemma will be used in the proof of Theorem 3.7 to control the entropy of the constructed subshift.
For p a sequence of positive integers, the limit density of p is the number lim pn n when it exists.
Lemma 3.11. Let p be some sequence such that pn n → 0. Then h top (Σ p ) = 0. In other words, if the limit density is zero, then the entropy is also zero. This result holds under the weaker condition inf n pn n = 0. Proof. Consider a sequence p such that inf n pn n = 0. Since for all n, any word of L n (Σ p ) has less than p n symbols equal to 1, we have: Take some n such that pn n ≤ 1 2 . We have: pn k=0 n k ≤ (p n + 1) n p n , since k → n k is non-increasing on 0, n 2 . Therefore: Then, we define p(f, β) : N → N by If β = (β k ) k≤n is a finite sequence, we define a corresponding infinite sequence β as follows: By abuse of notation, we denote ϕ(f, β) = ϕ(f, β ) and p(f, β) = p(f, β ). The bounded density subshift obtained from a sequence p(f, β), where β can be finite or infinite, is denoted Σ f,β . Figure 2 illustrate schematically this definition. Lemma 3.13. Let β be a non-increasing sequence of positive rational numbers such that β n → 0. Then the entropy of Σ f,β is zero.
Proof. Since β is non-increasing, for all n ≥ 1 and t ≥ 0, and as a consequence, Hence This means that lim sup This is true for all n and β n → 0, so pn n → 0. By Lemma 3.11, we get h top (Σ f,β ) = 0.
Moreover, using the Lemma hypothesis, we have p 2n+f (n) ≥ 2p n + 4. Combining the above inequalities, we obtain # 1 w = p |u | + p |v | ≤ p |v |+|u |+f (n) = p |w| , and therefore w is not a forbidden pattern. This means that u0 f (n) v is locally admissible for Σ p , and globally admissible as a consequence of Lemma 3.9.
3.3.6. Computability of the entropy of bounded density subshifts associated to a finite sequence. In the construction, we will need to compute an approximation of the entropy of the subshift obtained after defining only the n first terms of the sequence. This is the object of the following Lemma.
Lemma 3.18. Let f be a computable integer function. There exists an algorithm that, given as input a non-increasing finite sequence (β n ) n≤N of positive rational numbers, computes the entropy of Σ f,β .
Proof. Let β be a non-increasing sequence of non-negative rational numbers. Since β is non-increasing, ϕ(f, β) is concave and therefore p(f, β) is concave.
For any n, c ∈ N, applying Lemma 3.16 with k = n + c and l = F N (1) − n, and the definition of p(f, β) for the rightmost inequality, we have: As a consequence, we get: With the choice we obtain p(f, β) 2n+c ≥ 2p(f, β) n + 4. By Proposition 3.17, Σ p is c-irreducible, and c is computable from the knowledge of β and f . The sequence p(f, β) is also computable since f and β are computable. It follows, by Lemma 3.10, that Σ f,β is decidable, and one can construct the Turing machine which decides its language from the knowledge of β.
The Lemma follows by application of Theorem 3.6.

3.3.7.
Controlling the change of entropy. In this section, we give an upper bound on the decrease of entropy when defining the (N + 1) th term of the sequence β when the N first terms are already defined. In the construction, these conditions will be verified asymptotically.
Lemma 3.19. Let (β n ) n≤N be a finite non-increasing sequence of positive rational numbers, and f a computable integer function. Denote (β n ) n≤N the sequence defined by β N = β N − 1 F N (1) and β n = β n for n < N . Then we have the following inequality: where H is the binary entropy.
Proof. The left-hand inequality comes from Σ ⊂ Σ, since β is non-increasing. Let us prove the right-hand inequality. We define a family of functions δ N : A * → A * such that δ N (L n (Σ)) ⊂ L n (Σ ) for all n, and evaluate the size of the pre-images of each word.
A remark on p(f, β) and p(f, β ): We prove that for all n, . , using the Lemma hypothesis, and where ϕ is defined in Definition 3.12. Definition of the function δ N : Take w ∈ A n and define inductively (when possible): , N j (w) = min{k > N j−1 : k > jF N (1) and w k = 1}.
N j (w) is left undefined when the corresponding set is empty (as a consequence, the following ones are also left undefined). Now define: Intuitively, the function marks every F N (1)-th letter of the word. Going from left to right, it turns the first symbol 1 it meets after each mark. Our goal is that most words of length n lose 1 F N (1) symbols 1 that are well-distributed along the word. This definition is illustrated in Figure 3.  The image of L n (Σ) is in L n (Σ ) for all n: Take w ∈ L n (Σ). We prove that δ N (w) ∈ L n (Σ ). Using Lemma 3.9, it is sufficient to prove that this word is locally admissible for Σ . By the same argument as Lemma 3.9, any forbidden pattern for Σ contains another forbidden pattern ending in a 1. Therefore we take two arbitrary coordinates i, i+m−1 such that 0 ≤ i < i+m−1 < n and δ N (w) [i,i+m−1] = 1 and we prove that δ N (w) [i,i+m−1] is not a forbidden pattern for Σ .
Take any j such that i ≤ jF N (1) ≤ i + m − 1. If N j (w) ≥ i + m − 1 or is undefined, than δ N (w) i+m−1 = 0, a contradiction. Therefore we can assume that N j (w) is well-defined and i ≤ N j (w) ≤ i + m − 1. Since this holds for Since we took w ∈ L n (Σ), we have # 1 (w [i,i+m−1] ) ≤ p(f, β) m , and therefore using the first part of the current proof. This means that δ N (w) [i,i+m−1] is not a forbidden pattern for Σ . Since this is true for all i and m, w is locally admissible, and therefore globally admissible, for Σ . Entropy inequality: Take any w ∈ L n (Σ ). From the definition of δ N , a preimage of w by δ N must be equal to w except for at most n F N (1) coordinates. Therefore w has at most n n/F N (1) different preimages. Since we proved δ N (L n (Σ)) ⊂ L n (Σ ) for all n, it follows that: log #L n (Σ) n ≤ log n n/F N (1) + log #L n (Σ )) n By Stirling's formula, log(n!) ∼ n log n. It follows: .
3.3.8. Sketch of the proof of Theorem 3.7. The idea of the proof for Theorem 3.7 is as follows. Given a Π 1 -computable number α > 0 and a computable non-decreasing function f : N → N, we build an algorithm that computes a non-increasing sequence of rational numbers β such that: • Σ f,β has entropy α; • p(f, β) satisfies the conditions of Lemma 3.17 (ensuring the f -irreducibility). The decidability is ensured by the computability of β.
Since we only know approximate values α n of α with no known rate of convergence, we build intermediate subshifts with entropies approximately α n , and use the summability condition on f to ensure that the final subshift has entropy α.
3.3.9. Description of the algorithm. Let α > 0 be a Π 1 -computable real number and (α n ) n a non-increasing computable sequence of rational numbers such that α n α. We define inductively a sequence (β n ) n∈N such that the associated bounded density subshift Σ f,β satisfies h top (Σ f,β ) = α; this induction can be seen as a recursive algorithm.
In the following, we use a few shorthands for notation: Remark 13. A consequence of the entropy condition is that for all ,  Figure 4. Illustration of the definition of the algorithm. The sequence is already defined up to F n+1 (1). The number m * n is the smallest one such that the mixing condition is verified.
Proof of Lemma 3.20.
Lower bound: Since L n (Σ α ) = L n (Σ ,α ), we have that Since for all , h top (Σ ,α ) ≥ α ≥ α by the entropy condition of the algorithm, Upper bound: For the sake of contradiction, assume that Behavior of the algorithm: Since h top (Σ α ) = inf n h top (Σ n,α ), it follows that h top (Σ n,α ) − α ≥ H( 1 n ) + 2 −n for all n large enough. By Lemma 3.19, this means that in the definition of Σ n,α , the choice m = m * n + 1 would not break the entropy condition for all n large enough. As a consequence, m = m * n + 1 breaks the mixing condition. In other words (using the notation from the algorithm): Since η (1) and η m * n +1 k = η m * n k for k ≤ n, it follows that: • Using these equalities in Equation 3, we obtain: By definition of p α , and since f (k) = o(k), this means that for n large enough, p α F n+1 (1) ≤ 2p α F n (1) + 6.

SILVÈRE GANGLOFF AND BENJAMIN HELLOUIN DE MENIBUS
Contradiction using the limit density: By the previous equation, there is a constant N such that for all n ≥ N , (1) Rewriting this equation, (1) Applying this equation inductively, we get for all m ≥ 0: Since the sequence (p n /n) n is bounded, the left-hand side of Equation (4) is bounded. Since F (n) ≥ 2n for all n, we have F k (1) ≥ 2 k for all k, and 6 F k+1 (1) ≤ 6 2 k+1 , so that ∞ k=0 6 F k+1 (1) < ∞, and therefore inf n p n n · +∞ k=n f (F k (1)) F k+1 (1) < ∞.
Since f is nondecreasing, we have for all i: Since ∞ k f (k) k 2 = +∞ by hypothesis, we have ∞ i f (F i (n)) F i+1 (n) = +∞ as well. By considering Equation 4 as m → ∞, we see that we must have inf n pn n = 0. By Lemma 3.11, this implies that h top (Σ α ) = 0. We have reached a contradiction.
As a consequence, h top (Σ p ) = α which is the desired statement.
The Theorem for d = 1 follows from Lemma 3.10 and Lemmas 3.17 and 3.20.
3.3.11. Proof for d ≥ 1. In order to obtain the same result in higher dimension, notice that for any one-dimensional subshift Σ, the subshift has the same entropy and decidability properties as Σ. We prove that if Σ is firreducible, then Σ d is also f -irreducible.
Indeed, let U, V two finite subsets of Z d such that d(U, V) ≥ f (max(δ(U), δ(V))) and u, v two patterns on U, V respectively. For all k ∈ Z d−1 , denote H k = {(i, k) : i ∈ Z}. Consider the sets U k = U ∩ H k and V k = V ∩ H k , and put u k = u| U k , v k = v| V k .
Since the function f is non-increasing, that d(U k , V k ) ≥ δ(U, V) and that furthermore max(δ(U), δ(V)) ≥ max(δ(U k ), δ(V k )), we have: By f -irreducibility of Σ, this implies that there exists some x k ∈ Σ whose restrictions on U k and V k are u k and v k , respectively. Let x be the configuration defined by x| H k = x k for all k ∈ Z d−1 . Then x ∈ Σ and x| U = u and x| V = v by construction.

Conclusion.
Our main result is the proof of a jump in the difficulty of computing entropy of decidable subshifts when a measure of mixing strength, the irreducibility rate, passes a certain threshold. We offer some perspectives for further research: • We do not have a characterisation of real numbers that can be reached as entropies of decidable subshifts whose irreducibility rate is under the threshold. We conjecture that all computable real numbers can be reached in this way. • The main question, and the initial motivation of this work, is whether the same threshold marks the jump between computable and uncomputable entropy for subshifts of finite type of higher dimension.