Antonin Callard

Léo Paviet

Pascal Vanier

Computability of extender sets in multidimensional subshifts

Keywords: 2012 ACM Subject Classification Mathematics of computing → Discrete mathematics; Theory of computation extender sets, extender entropy, computability, sofic shifts, subshifts, tilings, symbolic dynamics

Subshifts are colorings of Z d defined by families of forbidden patterns. Given a subshift and a finite pattern, its extender set is the set of admissible completions of this pattern. It has been conjectured that the behavior of extender sets, and in particular their growth called extender entropy ([9]), could provide a way to separate the classes of sofic and effective subshifts. We prove here that both classes have the same possible extender entropies: exactly the Π3 real numbers of [0, +∞). We also consider computational properties of extender entropies for subshifts with some language or dynamical properties: computable language, minimal and some mixing properties.

Introduction

Multidimensional subshifts are sets of colorings of Z d where a family of patterns, i.e. colorings of finite portions of Z d , have been forbidden. They have been introduced originally to discretize continuous dynamical systems [START_REF] Marston | Symbolic Dynamics[END_REF]. One of the main families of subshifts that has been studied is the class of subshifts of finite type (SFTs), which can be defined with a finite family of forbidden patterns. This class has independently been introduced under the formalism of Wang tiles [START_REF] Hao | Proving theorems by Pattern Recognition II[END_REF] in dimension 2 in order to study fragments of second order logic.

In dimension 1 the study of SFTs is done mainly through the study of its defining graph. They share most of their properties with the class of sofic shifts [START_REF] Weiss | Subshifts of finite type and sofic systems[END_REF], which can be obtained as letter-to-letter projections of SFTs. These can also be characterized as the biinfinite walks on some finite automaton. In dimension 2 and higher the main tool in the study of subshifts becomes computability theory. This has led to the introduction of a new class of subshifts, the effective subshifts, which can be defined by computably enumerable families of forbidden patterns [START_REF] Hochman | On the dynamics and recursive properties of multidimensional symbolic systems[END_REF]. Many examples of effective subshifts with interesting properties have been proven to be sofic, such as substitutive subshifts [START_REF] Mozes | Tilings, substitution systems and dynamical systems generated by them[END_REF], or even effective subshifts on {0, 1} whose densities of symbols 1 are sublinear [START_REF] Destombes | Algorithmic complexity and soficness of shifts in dimension two[END_REF]. It turns out that sofic subshifts of dimension d + 1 capture all the behaviors of effective shifts of dimension d [START_REF] Hochman | On the dynamics and recursive properties of multidimensional symbolic systems[END_REF][START_REF] Durand | Fixed-point tile sets and their applications[END_REF][START_REF] Aubrun | Simulation of effective subshifts by two-dimensional subshifts of finite type[END_REF], which makes it hard to distinguish the two classes in dimension d ≥ 2.

An important question in symbolic dynamics it thus to find criteria separating the two classes [START_REF] Kass | A sufficient condition for non-soficness of higher-dimensional subshifts[END_REF]. All the arguments used to prove some cases of non-soficity that are known by the authors all revolve around a counting argument: only a linear amount of information may cross the border of an n × n square pattern. The most recent argument in this vein uses resource-bounded Kolmogorov complexity [START_REF] Destombes | Resource-bounded Kolmogorov complexity provides an obstacle to soficness of multidimensional shifts[END_REF]. These arguments all depend on the structure of Z d and fail on groups which are not amenable, as in such cases the border of a pattern may carry as much information as its interior. There even exist groups where the class of sofic shifts and effective shifts coincide [START_REF] Barbieri | Groups with self-simulable zero-dimensional dynamics[END_REF].

One way to extend these counting arguments is through the notion of extender sets of patterns [START_REF] Kass | A sufficient condition for non-soficness of higher-dimensional subshifts[END_REF]: the extender set of a pattern p is the set of all configurations with a p-shaped hole that may extend p. In dimension 1, being sofic means that there is a constant bounding the number of extender sets for any pattern shape. In higher dimensions, this no longer holds, but one can look at the growth of the number of extender sets. For SFTs, a large enough boundary (depending only on the size of the forbidden patterns) entirely determines the extender set of a given pattern, which implies that the number of extender sets of an SFT cannot grow too quickly. In [START_REF] Ormes | Extender sets and multidimensional subshifts[END_REF] it was proved that a subshift whose number of extender sets for patterns of size n d is bounded by n must be sofic.

The study of the growth rate of the number of extender sets can be done asymptotically through the notion of the extender entropy, which is defined in a similar way to the classical notion of topological entropy [START_REF] Lind | An Introduction to Symbolic Dynamics and Coding[END_REF]. Extender entropies in fact relate to the to notion of follower entropies [START_REF] Buzzi | Subshifts of quasi-finite type[END_REF], but are more robust in the sense that the extender entropy of a subshift is a conjugacy invariant.

In this paper, we aim at a better understanding of this quantity, and we achieve characterizations of the possible extender entropies in terms of computability, in the same vein as recent results on conjugacy invariants [START_REF] Hochman | A characterization of the entropies of multidimensional shifts of finite type[END_REF][START_REF] Meyerovitch | Growth-type invariants for Z d subshifts of finite type and arithmetical classes of real numbers[END_REF].

▶ Theorem A. The set of extender entropies of Z effective subshifts is exactly Π 3 ∩ [0, +∞).

▶ Theorem B. The set of extender entropies of Z 2 sofic subshifts is exactly Π 3 ∩ [0, +∞). This result also disproves a conjecture made in [START_REF] Kass | A sufficient condition for non-soficness of higher-dimensional subshifts[END_REF] stating that being sofic implies having extender entropy zero: Π 3 numbers are dense in [0, +∞). It also shows that the value of extender entropy does not allow one to separate sofic and effective shifts, since both have the same possible values. Even in the case of subshifts with computable languages, extender entropies span a large number of possible values: ▶ Theorem C. The set of extender entropies of Z 2 (sofic) subshifts with computable language is exactly Π 2 ∩ [0, +∞).

Finally, we also study extender entropies of subshifts constrained by some dynamical assumptions, such as minimality or mixingness. What is known by the authors at this stage can be summed up by the following table: Results in the paper are proved for d = 2, and generalizations to d > 2 follow from Claim 7, as the free lift of a sofic (resp. effective, block-gluing, computable) subshift is still sofic (resp. effective, block-gluing, computable).

Definitions

Subshifts

Let A denote a finite set of symbols and d ∈ N the dimension. A configuration is a coloring x ∈ A Z d , and the color of x at position p ∈ Z d is denoted by x p . A (d-dimensional) pattern over A is a coloring w ∈ A P for some set P ⊆ Z d called its support 1 . For any pattern w over A of support P , we say that w appears in a configuration x (and we denote w ⊑ x) if there exists p 0 ∈ Z d such that w p = x p+p0 for all p ∈ P . We sometimes consider patterns or configuration by their restriction: for S ⊆ Z d either finite or infinite, and x ∈ A Z d a configuration (resp. w a pattern), we denote by x| S (resp. w| S ) the coloring of A S it induces on S.

▶ Definition 1 (Subshift). For any family of finite patterns F, we define

X F = x ∈ A Z d | ∀w ∈ F, w ̸ ⊑ x A set X ⊆ A Z d is called a subshift if it is equal to some X F .
Subshifts can also be defined via their topological properties. Let us endow A Z d with the product discrete topology. Given the shift functions (σ t ) t∈Z d defined as (σ t (x)) p = x p+t , subshifts are the closed and shift-invariant (i.e. for every

t ∈ Z d , σ t (X) = X) subsets of A Z d .
Given a subshift X and a finite support P ⊆ Z d , we define L P (X) as the set of patterns w of support P that appear in some configuration x ∈ X. We define the language of X as L(X) = P ⊆Z d finite L P (X). For a pattern w, we also say that it is globally admissible in X if it appears in some configuration of X. Slightly abusing notations, we denote

L n (X) = L 0,n-1 d (X) for n ∈ N.
We say that a subshift is a subshift of finite type (SFT) if it is equal to some X F for some finite family F of forbidden finite patterns. We say that a subshift X is effective if it is equal to some X F for F a computably enumerable family of finite forbidden patterns F. We say that a subshift X is computable if L(X) is computable. We say that a subshift Y is sofic if it is equal to φ(X) for X an SFT and φ : X → Y a factor map (i.e. a continuous and shift-commuting function).

SFTs are of course sofic subshifts, sofic subshifts are effective, and computable subshifts are effective too. On the other directions, let us define the two following lifts:

given a Z subshift X ⊆ A Z d , consider the periodic lift X ↑ = {y ∈ A Z 2 | ∃x ∈ X, ∀i ∈ Z, y Z×{i} = x}; the free lift X ⇑ = {y ∈ A Z 2 | ∀i ∈ Z, y Z×{i} ∈ X}.
(The periodic lift is easily defined on configurations) Then the following result holds:

▶ Theorem 2 ([12], [2, Theorem 3.1], [7, Theorem 10]). If X is an effective Z subshift, then X ↑ is a sofic Z 2 subshift.
To help with later proofs, we introduce the following notation: for I, J ⊆ Z two nonnecessarily finite intervals, and w ∈ L I×J (X ↑ ), we denote any row of w by w ↓ = w |I×{j} for any j ∈ J. As w ⊑ X ↑ , this does not depend on the chosen j ∈ J and so w ↓ is well-defined. For any z ∈ A Z , we have (z ↑ ) ↓ = z. Finally, for (L i ) i∈I a family of sets and J ⊆ I, we denote by π Lj 1 ×Lj 2 ×... : i∈I L i → j∈J L j the cartesian projection, which will in particular be used on subshifts on different layers.

Extender sets and entropy

▶ Definition 3 (Extender set). For X ⊆ A Z d a d-dimensional subshift, P ⊆ Z d and w ∈ A P a pattern of support P , the extender set of w is the set

E X (w) = {x ∈ A Z d \P | x ⊔ w ∈ X},
where (x ⊔ w) p = w p if p ∈ P and (x ⊔ w) p = x p otherwise.

In other words, E X (w) is the set of all possible valid "completions" of the pattern w in X. For example, for two patterns with the same support w, w ′ , we have E X (w) ⊆ E X (w ′ ) if and only if the pattern w can be replaced by w ′ every time it appears in any configuration of X.

In the case of Z subshifts, extender sets are similar to the more classical notions of follower (resp. predecessor) sets, which are the set of right-infinite (resp. left-infinite) words that complete a finite given pattern. We introduce these notions more formally in Section 8.

The traditional notion of complexity in subshifts is called pattern complexity, and is defined by

N X (n) = L n (X). The asymptotic growth of |N X (n)| is called the topological entropy, and is properly defined because |N X (•)| is a submultiplicative function.
Adopting a similar approach, for X a Z d subshift, we denote E X (n) = {E X (w) | w ∈ L n (X)} its set of extender sets. Using this, [START_REF] French | Characterizing follower and extender set sequences[END_REF] defines the extender set sequence of X as (|E X (n)|) n∈N . From this sequence, one can derive2 a notion of entropy : ▶ Definition 4 (Extender entropy, [9, Definition 2.17]). For a Z d subshift X, its extender entropy is

h E (X) = lim n→+∞ log |E X (n)| n d
The limit actually exists because the map |E X (•)| is sub-multiplicative: more generally, it verifies the conditions of the Ornstein-Weiss lemma from [START_REF] Krieger | The Ornstein-Weiss lemma for discrete amenable groups[END_REF].

Examples

1. Let us consider X = A Z d some full-shift in dimension d. Then X has maximal topological entropy, but h E (X) = 0: indeed, for any two patterns w, w ′ ∈ L n (X), we have

E X (w) = E X (w ′ ) = {A Z d \ 0,n-1 d }; which implies that |E X (n)| = 1 for every n ∈ N.
In particular, this shows that contrary to the usual topological entropy, extender entropy is not increasing with (subshift) inclusion. 2. On the other hand, let us consider X a (strongly) periodic subshift. In other words, we assume there exists p 1 , . . . , p d ∈ N such that, for every x ∈ X and i ≤ d, we have σ pi•ei (x) = x. Then X has zero topological entropy, and we also have h E (X) = 0. Indeed, for n ≥ max p i and w ∈ L n (X), we have E X (w) ≃ {w}. Then, for any w ′ ̸ = w, we have E X (w) ̸ = E X (w ′ ), and so

|E X (n)| ≤ |L n (X)| ≤ pA p , so |E X (•)| is eventually constant.

Some properties

The following results were proved in [START_REF] French | Follower, predecessor, and extender entropies[END_REF] in the context of Z subshifts, but easily generalize to Z d subshifts:

▶ Theorem 5. The following facts hold for one-dimensional subshifts: h E is a conjugacy invariant. h E is not necessarily decreasing under factor map. h E is additive under product (i.e. for X, Y two subshifts,

h E (X × Y ) = h E (X) + h E (Y )).
This paper studies the potential values of h E (X) for various classes of multidimensional subshifts, depending on their computational/topological properties. We can already formulate the following property, which is folklore:

▶ Proposition 6 ([14, Section 2]). Let X a d-dimensional SFT. Then h E (X) = 0.

Sketch of proof.

In an SFT of neighborhood 1, the extender set of a pattern w ∈ A 0,n-1 d is entirely determined by its border, and there are at most 2 O(n d-1 ) such borders. ◀

Computability notions

Arithmetical hierarchy

The arithmetical hierarchy [START_REF] Robert I Soare | Turing computability: Theory and applications[END_REF]Chapter 4] stratifies formulas of first-order logic by the number of their alternating unbounded quantifiers: for n ∈ N, define

Π 0 n = {∀k 1 , ∃k 2 , ∀k 3 , . . . ϕ(k 1 , . . . , k n ) | ϕ only contains bounded quantifiers } Σ 0 n = {∃k 1 , ∀k 2 , ∃k 3 , . . . ϕ(k 1 , . . . , k n ) | ϕ only contains bounded quantifiers }
We say that a decision problem is in Π 0 n (resp. Σ 0 n ) if its set of solutions is described by a Π 0 n (resp. Σ 0 n ) formula: in other words, Π 0 0 = Σ 0 0 corresponds to the set of computable decision problems; Σ 1 0 is the set of computably enumerable decision problems, etc. . .

Arithmetical hierarchy of real numbers

The arithmetical hierarchy of real numbers [25] stratifies real numbers depending on the number of alternating limit operations required to define them. In what follows, we say that a sequence of rationals (β i ) i∈I ∈ Q I (for I any product

I 1 × • • • × I n of intervals of integers) is computable if the function i → β i is a total computable function.
We denote Π 0 = Σ 0 the set of computable real numbers, i.e. the set of real numbers α for which there exists some computable sequence (

β i ) i∈N ∈ Q N such that α = lim β i and ∀i ∈ N, |α -β i | ≤ 2 -i . In
∪ Σ n ) ⊊ Π n+1 ∩ Σ n+1 .

Elementary constructions on extender sets

The free lift

We use this construction to generalize results on Z or Z 2 subshifts to higher dimensions:

▷ Claim 7. For a subshift X ⊆ A Z d any subshift, h E (X) = h E (X ⇑ ). Proof. Let X ⊆ A Z d . Since the d-dimensional hyperplane of Z d+1 contain independent configurations in X ⇑ ⊆ A Z d+1 , we have |E X ⇑ (n)| = |E X (n)| n , so that h E (X ⇑ ) = h E (X). ◀
The (semi)-mirror construction ▷ Claim. Let Y be any Z 2 sofic subshift over an alphabet A. There exists a Z 2 sofic subshift

Y mirror such that h E (Y mirror ) = h(Y ).
A natural idea is the mirror construction: a line of some special symbol * separates two half-planes; the upper half-plane contains a half-configuration of Y , while the lower half-plane contains its reflection of by the line of * . This mirror construction generates a subshift To solve this, the the semi-mirror with large discrepancy from [6, Example 5 ′′ ] reflects a single symbol instead of the whole upper-plane: Sketch of proof. For A ′ = A ∪ {␣, * }, define Y mirror over the alphabet A ′ as follows:

Y ′ such that h E (Y ′ ) = h(Y ), but
Symbols * must be aligned in a row, and there is at most one such row per configuration. If a row of * appears in a configuration x, then the lower half-place is entirely ␣, with the exception of at most a single position colored by a symbol of A; and the upper half-plane must appear in a configuration of Y .

If x i,j = * and x i,j-k ∈ A for some i ∈ Z, j ∈ Z, k ∈ N, then x i,j+k = x i,j-k .
In other words, the only symbol of A in the lower half-plane must be the mirror of the same symbol in the upper half-plane, as reflected by the horizontal row of * symbols.

Then we claim that h E (Y mirror ) = h(Y ). Indeed, any two distinct patterns of Y must appear in Y mirror and have distinct extender sets. ◀

This construction shows that there exists subshifts whose extender entropy is arbitrary; and since by [START_REF] Hochman | A characterization of the entropies of multidimensional shifts of finite type[END_REF] every Π 1 real number is the topological entropy of some (SFT, thus) sofic subshift, every Π 1 real number can be realized as the extender entropy of some sofic subshift. In particular, this already disproves the conjecture from [START_REF] Kass | A sufficient condition for non-soficness of higher-dimensional subshifts[END_REF] mentioned in the introduction.

4

Decision problems on extender sets

Inclusion of extender sets

Let us consider the following decision problem:

Extender-inclusion Input: An effective subshift X ⊆ A Z d , and u, v ∈ L(X), Output: Whether EX (u) ⊆ EX (v)
▷ Claim 8. Extender-inclusion is a Π 0 2 problem. Furthermore, restricted on the class of subshifts with computable language, Extender-inclusion is a Π 0 1 problem.

Proof.

As:

E X (u) ⊆ E X (v) ⇐⇒ ∀B ∈ A * , u ⊑ B =⇒ B ̸ ∈ L(X) ∨ ((B \ u) ⊔ v) ∈ L(X).
we obtain the claim: indeed, deciding whether a pattern w belongs in L(X) is a Π 0 1 problem, and it becomes decidable if X has computable language. ◀ On the other hand, we can prove completeness.

▶ Proposition 9. Extender-inclusion is Π 0 2 -complete. On the class of subshifts with a computable language, it is Π 0 1 -complete.

Proof of Π 0 2 -hardness for Z subshifts. We reduce the following known Π 0 2 problem3 :

Det-Rec-state Input:

A deterministic Turing Machine M , and a state q Output:

Whether q is visited infinitely often by M on the empty input Let (M, q) an instance of this Det-Rec-state. We construct an effective subshift X on two layers over the alphabet {0, 1} × { * , □} as follows:

The symbol 1 can only appear above a symbol * ; and if there are at least two symbols * on the second layer, no symbol 1 can appear on the first layer;

If there are at least two symbols * on the second layer, at distance, say, n > 0, then the configuration is n-periodic; and if M enters q at least n ′ times, then we impose n > n ′ . As the rules above forbid an enumerable set of patterns, X is an effective subshift. We then claim that E X ((0, * )) ⊆ E X ((1, * )) if and only if M enters q infinitely many times.

Indeed, the symbol (0, * ) can be extended either by semi-infinite lines of symbols (0, □), which are configurations that also extend the symbol (1, * ) ; or by configurations containing n-periodic symbols (0, * ), which do not extend the symbol (1, * ) because of the first rule. However, the second rule implies that for every n > 0, this n-periodic configuration exists if and only if M visits q less than n times. This concludes the proof. ◀

Proof of Π 0 1 -hardness for computable Z subshifts. We slightly modify the previous construction to reduce co-Halt = {M | M is a Turing Machine that does not stop on the empty input} by forbidding periods < n when the machine is still running after n steps. The same argument goes through. ◀

Computing the number of extender sets

Let us determine the computational complexity of the formula |E X (n)| ≥ k, given a subshift X and some size n and some k. It is equivalent to the following:

v1,...,v k ∈Ln(X) 1≤i<j≤k E X (v i ) ̸ = E X (v j )
As |E X (n)| ≥ k is expressible as finite disjunctions/conjunctions of a Σ 0 2 (even Σ 0 1 if X has computable language) problem over the set L n (X), we conclude:

▶ Lemma 10. For an effective subshift X, |E X (n)| ≥ k is a Σ 0
2 formula, and even Σ 0 1 on subshifts with computable language.

Upper computational bounds on extender entropies

As

h E (X) = inf n log |E X (n)| n 2
, the following corollary follow from Lemma 10:

▶ Corollary 11. For an effective subshift X, h E (X) ∈ Π 3 . For a computable subshift X, h E (X) ∈ Π 2 . Proof. For fixed X, n, Lemma 10 shows that the set {k ≤ |E X (n)|} is a Σ 0 2 set if X is effective (resp. Σ 0 1 set if X has computable language). This means that the |E X (n)|, and therefore log |E X (n)| n d , are Σ 2 (resp. Σ 1 ) and so h E (X) ∈ Π 3 (resp. h E (X) ∈ Π 2 ) as the infimum of Σ 2 (resp. Σ 1 ) real numbers. ◀ 5
Realizing extender entropies

Summary

The fact that sofic and effective subshifts have the same bounds should not come out as a surprise at this point: most of our proofs rely on finding some one-dimensional effective subshift realizing some property, and using Theorem 2 as a black-box: its periodic lift is a Z 2 sofic shift. This section follows the same scheme: Section 5.2: we prove in Theorem A that effective Z subshifts have exactly Π 3 entropies; Section 5.3: we lift this construction to Z 2 subshifts and add slight modifications to obtain the same result in Theorem B.

One-dimensional effective subshifts

Let us focus on one-dimensional subshifts for the time being.

▶ Theorem A. The set of extender entropies of Z effective subshifts is exactly

Π 3 ∩ [0, +∞).
In order to construct a subshift Z α with h E (Z α ) = α, we would like to have |E Zα (n)| ≃ 2 αn . A way to have this would be, as in Section 3, to create subshifts with one extender set per pattern, and with 2 αn patterns of size n; however, since effective subshifts have Π 1 entropies, this would not realize the whole class of Π 3 numbers.

Yet, realizing the right number of patterns is the main idea behind the proof that follows: we just do not blindly create one extender set per pattern, but only separate extender sets when some conditions are met.

Proof. Let α ∈ Π 3 a positive real number, α = inf i sup j inf k β i,j,k for some computable sequence (β i,j,k ). We can assume α ≤ 1, and using [25, Lemma 3.1] we can assume that (β i,j,k ) satisfies some monotonicity properties: for all i, j, (β i,j,k ) k∈N is (non-strictly) decreasing and converges towards some α i,j ; for all i, (α i,j ) j∈N is (non-strictly) increasing and converges towards some α i ; and the sequence (α i ) i∈N is (non-strictly) decreasing and converges towards α.

Sketch of the construction

We construct a subshift Z α with several layers:

The first layer encodes an integer i. Intuitively, it tells us which of the α i we are trying to approximate in this configuration.

The second layer encodes some other integer j. Its purpose is to control the "precision" at which we try to approximate α i .

The density layer contains a binary word w i,j of density roughly α i,j . Note that α i,j is not computable, as it is a Π 1 real number.

The free layer is a way to increase the entropy, by adding "free bits" on top of the 1s of the density layer.

More formally, we first define an auxiliary subshift Z ′ α with an extra layer. From Z ′ α , we will later define Z α and prove that h E (Z α ) = α. Auxiliary subshift Consider this set of five layers:

1. First layer L 1 : For A * = { * , ␣}, we set L 1 = X * where X * = i∈N Orb( * ␣ i-1
) is the subshift composed of (the closure of) periodic configurations of * symbols separated by ␣ symbols. In other words, denoting ⟨i⟩ k1 ∈ A Z * the i-periodic configuration defined by (⟨i⟩ k1 ) p = * ⇐⇒ p = k 1 mod i, and ⟨∞⟩ = {x ∈ A Z * | |x| * ≤ 1} the set of configurations having at most a single symbol * , we have:

X * = ⟨∞⟩ ∪ i∈N {x ∈ A Z * | ∃k 1 ∈ N, x = ⟨i⟩ k1 }.
T (β) = β T0 β T1 . . . β Ti • • • ∈ {0, 1} N and denote T (β, i) k1 ∈ A Z d the i-periodic configuration defined as (T (β, i) k1 ) p = T (β) k1+p mod i .
We will use configurations T (β, i) k1 to realize specific densities of symbols 1. ▶ Remark. Several densities β can realize the same configuration T (β, i) k1 . In the rest of the article, when considering a configuration x = T (β, i) k1 , β will implicitly be the minimal density realizing x. 4. Free layer L f : Let us now add free bits above symbols 1 of A d . More precisely, denoting

A f = {0, b, b ′ }, define the free layer as L f = A Z f . Given the synchronization function π sync : A f → A d by π sync (b) = π sync (b ′ ) = 1 (and π sync (0) = 0), we say that a pair of configurations (z (d) , z (f ) ) ∈ A Z d × A Z f are synchronized if π sync (z (f ) ) = z (d) .

Periodicity layer L p : Denote

A p = {p, ∞} and X p the finite subshift L p = {p Z , ∞ Z }.
We now use these five layers to define an auxiliary effective subshift Z ′ α :

Z ′ α = (z (1) , z (2) , z (d) , z (1) , z (2) ∈ ⟨∞⟩ or z (2) ∈ ⟨∞⟩) and π sync (z

(f ) , z (p) ) ∈ L 1 × L 2 × L d × L f × L p | (z
(f ) ) = z (d) ∪ i,j∈N
(z (1) , z (2) , z (d) , z

(f ) , z (p) ) ∈ L 1 × L 2 × L d × L f × L p | ∃i, j, k 1 , k 2 ∈ N, j ≥ i, z (1) = ⟨i⟩ k1 , z (2) = ⟨j⟩ k2 , π d (z (f ) ) = z (d) , z (p) = p Z , ∃β ≤ α i,j , z (d) = T (β, i) k1 and z (f ) is i-periodic Z ′
α is an effective subshift. Indeed, the structure of the configurations is straighforward to enforce, and we need to forbid configurations in which z (1) = ⟨i⟩ k1 , z (2) = ⟨j⟩ k2 and z (d) = T (β, i) k1 with β > α i,j . As α i,j is a Π 1 real number, we can enumerate the densities larger than α i,j and forbid the corresponding patterns. * * * * * * *

1 0 1 1 1 0 1 0 1 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 1 0 1 b'
z (f ) = (b ′ 0b0b0b ′ bb0bb ′ 0b ′ b) ∞ ).
The red line indicates the origin.

The periodicity layer L p is actually not needed in this proof, but is later used for Theorem B.

The subshift Z α From Z ′ α , we define the effective subshift Z α by forgetting the L p component of Z ′ α . As the projection of an effective subshift, Z α is also an effective subshift:

Z α = π L1×L2×L d ×L f (Z ′ α )
. Denoting the configurations of Z α by z = (z (1) , z (2) , z (d) , z (f ) ), we separate the configurations of Z α into two types:

If the first two layers actually encode integers, i.e. if z = (⟨i⟩ k1 , ⟨j⟩ k2 , T (β, i) k1 , z (f ) ) with z (f ) being both i-periodic and synchronized with T (β, i) k1 (which forces one extender set per pattern) ; we call z a Type 1 configuration. In that case, notice that the density β verifies the following inequalities:

β ≤ α i,j ≤ α i . Otherwise, z is either z = (⟨∞⟩ , ⟨∞⟩ , z (d) , z (f ) ), z = (⟨i⟩ k , ⟨∞⟩ , z (d) , z (f ) ) or z = (⟨∞⟩ , ⟨j⟩ k , z (d) , z (f )
). In that case, z (d) can be any configuration and z (f ) 's only constraint is to have b, b ′ exactly on top of z (d) 's symbols 1 (which means that there are very few extender sets). Intuitively, these configurations have no meaning in our construction, and we call them Type 2 configurations.

We formally prove in Appendix A.1 that h E (Z α ) = lim i α i = α, which concludes the proof. Intuitively, consider two patterns w, w ′ that can appear in some configurations of Type 1:

If they differ on their L 1 , L 2 or L d layers, then they have different extender sets. Otherwise, they differ on their free bit layer, and they therefore have different extender sets. Indeed, say without loss of generality that w (f ) 0 = b, and w

′(f ) 0 = b ′ .
Consider z ∈ Z α extending w, with z (1) = ⟨i⟩ k1 for some i, k 1 . By definition of Z α , z (f ) is i-periodic, and in particular z (f ) | iZ is constant equal to b. This implies that z cannot extend w ′ , as w ′f 0 = b ′ . This implies that there is one extender set per pattern appearing in Type 1 configurations. For given n and i ≤ n, there are roughly 2 i•αi patterns of size n in configurations z of Type 1 where z (1) = ⟨i⟩. Summing over i ≤ n and taking the limit, we obtain h E (Z α ) = lim n α n = α. ◀

Two-dimensional sofic subshifts

In order to extend Theorem A to multidimensional sofic shifts, an idea could be to generalize the previous construction. Indeed, in the construction of the Z effective subshift Z α , Type 1 configurations that encode an integer i on their first layer have a density (less than or equal to) α i of free bits which are furthermore i-periodic, making one extender set per such pattern; on Z 2 , we could make these free-bits (i, i)-periodic and obtain the same result. Unfortunately, a short argument shows that such a subshift would not be sofic 4 .

To solve this issue, we use the same trick that the semi-mirror shift solves in Section 3: instead of "reflecting" the whole half-plane (in our case, the whole i × i square), reflecting a single bit "non-deterministically" (in our case, a single bit per i × i square) is enough to separate patterns into distinct extender sets; and having to synchronize a single bit is an amount of information low enough as to make the subshift sofic.

▶ Theorem B. The set of extender entropies of Z 2 sofic subshifts is exactly Π 3 ∩ [0, +∞).

Proof. We use the notations introduced in the proof of Theorem A.

Auxiliary subshift

We first define an auxiliary subshift Y ′ α as follows. Its layers are: Lifted layers: We define the first four layers of

Y ′ α as L ↑ 1 × L ↑ 2 × L ↑ d × L ↑ p , where L 1 , L 2 , L d and L p are defined in the proof of Theorem A. Free layer L f : Still denoting A f = {0, b, b ′ }, we define the free layer as L f = A Z 2
f . Once again, we define a synchronization function π sync :

A f → A d by π sync (b) = π sync (b ′ ) = 1
(and π sync (0) = 0) to synchronize this free layer with the density layer from the lift Z ↑ aux : we say that a pair of configurations (z (d) . ▶ Remark. Note that L f is not obtained by lifting the L f layer of the previous proof, as this would lead to a linear instead of a quadratic growth.

(d) , z (f ) ) ∈ A Z 2 d × A Z 2 f are synchronized if π sync (z (f ) ) = z

Marker layer

G = [∞] ∪ i∈N {x ∈ A Z 2 m | ∃m 1 , m 2 ∈ Z, x = [2i] m1,m2 }
From these layers, we define Y ′ α as follows. Let

Z aux = π L1×L2×L d ×Lp (Z ′ α )
where Z ′ α is the auxiliary subshift defined in the proof of Theorem A. Then:

Y ′ α = (y (1)↑ , y (2)↑ , y (d)↑ , y (p)↑ , y (f ) , y (m) ) ∈ L ↑ 1 × L ↑ 2 × L ↑ d × L ↑ p × L f × L m ∃i, k 1 , m 1 , m 2 ,
(y (1) , y (2) , y (d) , y (p) ) ∈ Z aux , π sync (y

(f ) ) = y (d)↑ , y (1) ∈ ⟨∞⟩ =⇒ y (m) ∈ [∞] ,
y (1) = ⟨i⟩ k1 =⇒ (y (m) = [2i] m1,m2 and

y (p)↑ = p Z 2 =⇒ y (f ) | (m1+iZ)×(m2+iZ) is constant)
See Figure 3 for an illustration. In other words, Y ′ α is the lift Z ↑ aux with two additional layers L f (containing free bits) and L m (containing markers ■). We say that y ∈ Y ′ α is of Type 1 (resp. Type 2) if its first four layers (y (1) , y (2) , y (d) , y (p) ) are the lift of a Type 1 (resp. Type 2) configuration of Z aux .

In a configuration of Type 1 with first and second layers y (1) = ⟨i⟩ ↑ k1 and y (2) = ⟨j⟩ ↑ k2 , the density of the layer y (d) is lower than α i,j , and in the layer y (f ) there is a single periodic free bit of period (i, i) whose position modulo (i, i) is determined by the layer y (m) = [2i] m1,m2 . The actual set of configurations of Y ′ α is:

Y ′ α = (y (1)↑ , y (2)↑ , y (d)↑ , y (p)↑ , y (f ) , y (m) ) ∈ L ↑ 1 × L ↑ 2 × L ↑ d × L ↑ p × L f × L m
y (1) , y (2) ∈ ⟨∞⟩ and y (m) ∈ [∞] and π sync (y

(f ) ) = y (d)↑ ∪ i∈N (y (1)↑ , y (2)↑ , y (d)↑ , y (p)↑ , y (f ) , y (m) ) ∈ L ↑ 1 × L ↑ 2 × L ↑ d × L ↑ p × L f × L m y (1) = ⟨i⟩ k1 , y (m) = [2i] m1,m2 , y (2) ∈ ⟨∞⟩ , π sync (y (f ) ) = y (d)↑ y (p)↑ = p Z 2 =⇒ y (f ) | (m1+iZ)×(m2+iZ) is constant ∪ i,j∈N y (1)↑ , y (2)↑ , y (d)↑ , y (p)↑ , y (f ) , y (m) ∈ L ↑ 1 × L ↑ 2 × L ↑ d × L ↑ p × L f × L m j ≥ i, y (1) = ⟨i⟩ k1 , y (2) = ⟨j⟩ k2 , y (m) = [2i] m1,m2 , π sync (y (f ) ) = y (d)↑ , y (p)↑ = p Z 2 ∃β ≤ α i,j , y (d) = T (β, i) k1 and y (f ) | (m1+iZ)×(m2+iZ) is constant
By Theorem 2, Z ↑ aux is sofic. We prove in Appendix A.3 that the subshift Y ′ α is a sofic subshift, that is, the extra conditions on the layers L p , L f , L m are sofic.

The subshift Y α Finally, we define the subshift Y α by forgetting the L p component of Y ′ α . Contrary to the proof of Theorem A, the layer L p is actually important here, as it ensures that only Type 1 configurations are forced to have some periodic free bit (more details below). As the projection of a sofic subshift, Y α is also a sofic subshift:

Y α = π L ↑ 1 ×L ↑ 2 ×L ↑ d ×L f ×Lm (Y ′ α )
4 Namely, this is the argument that shows that the full-mirror subshift cannot be sofic: there would be 2 O(i 2 ) distinct i × i patterns, but only 2 (O(i)) borders.

We finally claim that h E (Y α ) = α. The proof is very similar to the proof of Theorem A, and the following paragraphs highlight the main modifications:

Fix w, w ′ two different i × i patterns that do not contain a marker ■ and that can both appear in Type 1 configurations of the form z = (⟨i⟩ ↑ k1 , . . . ). If w, w ′ differ on their first, second or density layers, they have different extender sets. Otherwise, if they only differ on their free bits layer, the forced periodic free bit of Type 1 configurations ensures that they have different extender sets. Indeed, assume without loss of generality that w, w ′ differ in position (0, 0). Since w appears in some Type 1 configuration y = (⟨i⟩ ↑ k1 , ⟨j⟩ ↑ k2 , T (β, i) k1 , y (f ) , y (m) ) ∈ Y α , and that w does not contain a marker ■, we can pick y (m) = [2i] 0+i,0+i . Having markers in position (0 + i mod 2i, 0 + i mod 2i) forces y (f ) | (iZ×iZ) to be constant 5 , i.e. to have a periodic free bit; therefore y is not a valid completion of w ′ , as w ′ (0,0) ̸ = y

(f ) (0+i,0+i) since y (f ) (0+i,0+i) = w (0,0) .
The layer L p (that is forgotten after the final projection) is used to prevent Type 2 configurations from forcing (i, i)-periodicity on the free bit marked by L m . Indeed, there exists Type 2 configurations with marked free bits: y (m) is some [2i] m1,m2 as soon as y (1) = ⟨i⟩ ↑ k1 , even when y (2) actually belongs in ⟨∞⟩ ↑ . In that case, periodicity on the free bits should not be enforced, as it would lead to one extender set per value of the free bits layer, of which there are too many since the density of y (d) can be arbitrary. Layer L p solves this problem: indeed, if y (1) = ⟨i⟩ ↑ k1 and y (2) ∈ ⟨∞⟩ ↑ , both layers y (p) = p Z 2 (which forces (i, i)-periodicity of the free bit marked by [2i] m1,m2 ) and

y (p) = ∞ Z 2 (
which does not) can exist; after the projection, these two cases are merged and no free bit is forced to be (i, i)-periodic. On the other hands, we force y (p) = p Z 2 on Type 1 configurations, which (even after projection) enforces (i, i)-periodicity of the free bit marked by y (m) . The complete proof that h E (Y α ) = α can be found in Appendix A.2. ◀

Realizing extender entropies: computable subshifts

The upper-bound from Corollary 11 shows that if X is a Z 2 shift with decidable language, then its extender entropy is a Π 2 real number. We show that the converse holds and obtain:

▶ Theorem C. The set of extender entropies of Z 2 (sofic) subshifts with computable language is exactly Π 2 ∩ [0, +∞).

Sketch of proof.

We slightly alter the previous construction. Let α = inf i α i = inf i sup j β i,j ∈ Π 2 , for (β i,j ) a computable sequence, while verifying the monotonicity conditions from [25, Lemma 3.1]. We redefine the subshift Z ′ α constructed in Theorem A by taking β i,j,k = β i,j for every i, j, k ∈ N. The resulting subshift is computable: indeed, the non-computability of L(Z ′ α ) came from the fact that, given some i, j ∈ N and word T (β, i), it was undecidable to know whether β ≤ α i,j . In this new case, since α i,j = β i,j is now computable, this becomes decidable. In turn, the subshift now has computable language.

The rest of the proofs, both on Z and Z 2 , goes through in the exact same way.

◀ Extender sets of minimal subshifts

On minimal subshifts, the problem is much easier and does not even depend on the computability of the language.

▶ Proposition 12. Let X a minimal subshift over Z d . Then for any n > 0 and any patterns

u, v ∈ L n (X), E X (u) ⊆ E X (v) ⇐⇒ u = v.
Proof. Let u, v ∈ L n (X) and suppose that E X (u) ⊆ E X (v). Then in any configuration, any appearance of u can be replaced by v: if u ̸ = v, we obtain by compactness that forbidding u in X leads to a non-empty proper subshift of X, which contradicts its minimality. ◀

For X a minimal subshift, this implies in particular that h E (X) = h(X). Since minimal effective subshifts can have arbitrary Π 1 entropy (consider the proof of [START_REF] Kůrka | Topological and Symbolic Dynamics[END_REF]Theorem 4.77] on computable sequences (k n ) n∈N of integers), we obtain: ▶ Corollary 13. The possible extender entropies of minimal effective subshifts are exactly the Π 1 real numbers.

8

Extender sets of subshifts with mixing properties

Mixing Z subshifts

Being mixing is a general notion defined on all dynamical systems. We recall here the definition in the case of Z subshifts: intuitively, being mixing implies that for any pair of admissible words, there exists a configuration containing both of them at arbitrary positions, provided we set them sufficiently far apart.

▶ Definition 14 (Mixing subshift). A Z subshift X is mixing if

∀n > 0, ∃N > 0, ∀u, v ∈ L n (X), ∀k ≥ N, ∃w ∈ L k (X), uwv ∈ L(X)
We say that X is f (n)-mixing for some function f if N can be taken equal to f (n) in the previous definition. In the special case where f is constant f (n) = N , we simply write that X is N -mixing.

It seemed natural to expect that strong mixing conditions would imply restrictions on the extender entropy of a subshift. Indeed, the examples we mentioned so far either have strong mixing properties (the full shift, Z SFTs, . . . ) and zero extender entropy, or contain strong obstructions to mixing properties (periodicity, reflected positions, . . . ) to increase their number of extender sets. Furthermore, similar problems have been studied for topological entropies: e.g. [START_REF] Gangloff | Quantified block gluing for multidimensional subshifts of finite type: Aperiodicity and entropy[END_REF] shows that some mixing properties (namely, linear block-gluing) do not restrict the possible entropies of multidimensional SFTs, while [START_REF] Gangloff | Effect of quantified irreducibility on the computability of subshift entropy[END_REF] proves that the set of possible entropies depends on the growth rate of a mixing function of the subshift.

However, we show in this section that even very restrictive mixing properties do not imply anything on the extender entropies. ▶ Proposition 15. Let X a one-dimensional subshift. There exists X # a 1-mixing subshift with h E (X) = h E (X # ).

▶ Remark. If X is sofic (resp. effective), then X # can also be taken sofic (resp. effective).

Proof. Let X be a subshift defined by a family of forbidden patterns F over an alphabet A, and denote α = h E (X). We can assume that F = A * \ L(X).

Let us define a subshift X # over the alphabet A ⊔ {#} (assuming that # is a free symbol not in A) by the same family of forbidden patterns F. In other words, configurations of X # are composed of (possibly infinite) words of L(X) separated by the safe symbol #. It is clear that X # is 1-mixing, as for any u, v ∈ L(Y ), we have u#v ∈ L(Y ). On the other hand, we prove in Appendix B.1 that h E (X # ) = h E (X). ◀ This can be generalized to higher dimensional subshifts, as in the following proposition. There are various mixing notions one can study, and we choose to formulate our results for block-gluing subshifts; it is unclear how those could be adapted to other mixing properties.

▶ Proposition 16. For any d-dimensional effective subshift X, there exists a 1-block-gluing effective subshift

X # with h E (X) = h E (X # ).
▶ Remark. As opposed to the one-dimensional case, the hypothesis that X is effective is important in this proposition.

For the definition of block-gluingness and the proof of Proposition 16, see Appendix B.2.

Reformulation as growth rates of syntactic monoids

In this section, we briefly show a different point on view on the previous results, by relating extender sets with the classical syntactic monoid from the study of formal languages. For any finite alphabet A and any language L ⊆ A * , one can define an equivalence relation, called the syntactic congruence, as (see for example [1, Definition 3.6]):

∀u, v ∈ L, u ∼ L v ⇐⇒ (∀x, y ∈ L, xuy ∈ L ⇐⇒ xvy ∈ L)
▶ Definition 17 (Syntactic monoid). Let L a language. Then M (L) = L/ ∼ L with the concatenation operation is a monoid, called the syntactic monoid of L.

For L a language over A and u ∈ L, we call reduced length of u the non-negative integer ∥u∥ L = min v∼ L u |v|. The growth rate of M (L) is then

h(M (L)) = lim n→+∞ log |{[u] ∈ M (L) | u ∈ L, ∥u∥ L ≤ n}| n
For a Z-subshift X, we define its syntactic monoid M (X) as M (L(X)). In this setting, we can reformulate Theorem A as: ▶ Theorem 18. The growth rates of syntactic monoids of effective Z-subshift are exactly the non-negative Π 3 real numbers.

Sketch of proof.

Consider the subshift Z α from the proof of Theorem A. Then we claim that M (Z α ) has growth rate α. Indeed, for any two words u

, v ∈ L n (Z α ) such that u ̸ ∼ L(Zα) v, we have E Zα (u) ̸ = E Zα (v). So |{[u] ∈ M (X) | u ∈ L X (Z α ), ∥u∥ L(Zα) ≤ n}| ≤ |E Zα (n)|.
On the other hand, the argument computing a lower bound on the number of extender sets of Z α in Appendix A.1 exhibits a family of roughly 2 nα+o(n) words of L n (Z α ) such that any two words of this family cannot be syntactically congruent. This concludes the proof. ◀ Then, we obtain for T 1 the following upper-bound:

|T 1 | ≤ n i=1 i-1 k1=0 n j=i j k2=0 2 ⌈log i⌉ • 2 i•αi+o(i) + n k1=0 n k2=0 2 ⌈log i⌉ • 2 n•αn+o(n) ≤ poly(n) n i=1 2 i•αi+o(i)
Counting both T 1 and T 2 , we obtain:

|E Zα (n)| ≤ poly(n) • 1 + n i=1 2 i•αi+o(i) ≤ poly(n) • n i=1 2 i•αi+o(i)
and a brief computation shows that h

E (Z α ) = lim log |E Zα (n)| n ≤ lim α i = α. ◀ Proof of h E (Z α ) ≥ α.
To compute a lower bound, if suffices to observe that for each n ∈ N, we can find roughly 2 n•αn patterns with different extender sets for patterns of length n. Indeed, for n ∈ N, let β = α n -2 -⌊log n⌋ . Since α n = sup j α n,j , there exists some

j n ≥ n such that α n,jn ≥ α n -2 -⌊log n⌋ = β. Now, let W = T (β, j n ) 0,n-1 . Then, the number of symbols 1 in W is bounded from below by |W | 1 ≥ n • β -O(log n) = n • α n -O(log n).
Now, there are more than 2 n•αn-O(log n) patterns w in L n (Z α ) w (1) = ⟨n⟩ 0 = w (2) = ⟨j n ⟩ 0 and w (d) = W (because of the density layer (d) , that contains either b or b ′ under each symbol 1 of W ). Moreover, each of these patterns w has a different extender set. Indeed, as W is T (β, j n ) 0,n-1 with β ≤ α n,jn , each of these patterns can be extended by a configuration of Type 1 z = (⟨n⟩ 0 , ⟨j n ⟩ 0 , T (β, j n ) 0 , (w (f ) ) ∞ ) that is a valid extender for w only because of the i-periodicity of z (f ) .

In particular, we get that |E Zα (n)| ≥ 2 n•αn-o(n) , and we conclude that h E (Z α ) ≥ α. ◀

A.2 Computations for the proof of Theorem B

Proof of h E (Y α ) ≤ α. Fix n > 0. The proof follows the case analysis performed in Theorem A with the same notations. As in the computation of h E (Z α ) in the Z construction, we partition the set L n (Y α ) into T 1 ⊔ T 2 , where T 1 denotes the set of patterns that can appear in a configuration of Type 1 (and T 2 the set of patterns that only appear in configurations of Type 2). Furthermore, we refine the partition of T 1 ⊆ L n (Y α ) as follows: denote T 1,■ the patterns of T 1 containing a ■ on their L m layer, and T 1,□ = T 1 \ T 1,■ those who do not. We then have

L n (X) = T 1 ⊔ T 2 = (T 1,□ ⊔ T 1,■ ) ⊔ T 2 .
We will now bound the number of extender sets for patterns of each family.

The extender sets of patterns w ∈ T 2 are entirely determined by those of

π L ↑ 1 ×L ↑ 2 (w) × π Lm (w)
, that is, by its layers L 1 and L 2 on any of its rows, and by the L m layer. Indeed, for every w ∈ T 2 , (π

L d (E X (w))) ↓ is a full-shift, so (π L f (E X (w)))
↓ does not depend on w (d) ; in particular, E X (w) does not depend on w (d) at all. Similarly, since the layer L m enforces (i, i)-periodicity of some free bit only in Type 1 configurations (none of those extend w since w ∈ T 2 ), (π L f (E X (w)))

↓ does not depend on w (f ) either, so E X (w) also does not depend on w (f ) at all. As L m is periodic, it is itself completely determined by its period and by the position of the unique ■ in each of its period, so we have |π Lm (T (w) × π Lm (w) and by the value of the marked bit in L f . Indeed, even in Type 1 configurations, all the bits of L f that are not under a marker ■ are free. More precisely, for any w, w

′ ∈ T 1,■ with π L ↑ 1 ×L ↑ 2 ×L ↑ d ×Lm (w) = π L ↑ 1 ×L ↑ 2 ×L ↑ d ×Lm (w ′ ), denote by (m 1 , m 2 ) ∈ 0, n-1 2 a position having a marker, i.e. w (m) (m1,m2) = w ′(m) (m1,m2) = ■.
Then E X (w) = E X (w ′ ) if and only if (w (f ) ) m1,m2 = (w ) ) m1,m2 , that is, if their bit 6 Finally, we bound the number of extender sets for patterns of T 1,□ . Observe that a Type 1 configuration extending some pattern of w ∈ T 1,□ must have a layer L 1 equal to ⟨i⟩ ↑ k for some i, k, and a periodic (2i, 2i)-periodic layer L m . Therefore, it verifies i ≥ ⌈ n 2 ⌉, as otherwise there would be some marker ■ in w (f ) . First, there are at most poly(n) possibilities for (w (1) , w (2) , w (d) , w (m) ). Then, w (d) contains a factor of T (β, i) for some i ≥ ⌈n/2⌉ and β ≤ α i ; to bound the number of extender sets depending on w (f ) , we consider the values of i separately (a pattern could appear in several T (β, i) for different values of i; counting the same pattern several times is not a problem when bounding from above): 1. Assume ⌈n/2⌉ ≤ i ≤ n. The only way to differentiate the extender sets of two patterns of T 1,□ by their free layer L f is for them to be different in some free bit that could be made (i, i)-periodic by putting a marker ■ laying at (m 1 , m 2 ) ∈ Z 2 \ 0, n -1 2 outside of their support. Indeed, all the other positions (p 1 , p 2 ) ∈ Z 2 \ ((m1 + iZ) × (m2 + iZ)) in such a Type 1 configuration y are such that y

(f ) (p1,p2) = 0 if y (d)
(p1,p2) = 0, or free to be either b or b ′ whenever y

(d) (p1,p2) = 1.
There are i 2 distinct (m1 + iZ) × (m2 + iZ) sets of positions for the (i, i)-periodic free bits of a Type 1 configurations. Only i 2 • β + o(i) among these i 2 grids contain a symbol 1 in their density layer w (d) (w (d) is (i, 1)-periodic, so all positions in such grids contain the same symbol), ergo only i 2 • β + o(1) such grids could actually generate distinct extenders sets. Since each of these grids could either contain b or b ′ in their layer w (f ) , there are at most 2 i 2 •β+o(i) ≤ 2 i 2 •αi+o(i) ways to distinguish these patterns by their layers L f . 2. All i > n can be counted at once: if i > n, there are at most β • n 2 + o(n) positions in w (f ) that could be made periodic in a configuration of Type 1 extending w. Since β ≤ α i ≤ α n (the sequence (α i ) i∈N is (non-strictly) decreasing), there are at most 2 n 2 •αn+o(n) distinct extender sets. So the number of extender sets for patterns of T 1 is at most

poly(n) • n i=⌈n/2⌉ 2 i 2 •αi .
Combining the extender sets of T 1 and T 2 independently, we conclude with a brief computation that:

h E (Y α ) ≤ lim n→+∞ α n = α. ◀ Proof of h E (Y α ) ≥ α.
For the lower bound, it is enough to exhibit some patterns with different extender sets. Basically, we exhibit the two-dimensional equivalents of the patterns used in the proof of Theorem A. For n ∈ N, let β = α n -2 -⌊log n⌋ . There exists some j n ≥ n such that α n,jn ≥ α n -2 -⌊log n⌋ = β.

Let W = T (β, j n ) 0,n-1 . The number of symbols 1 in W is bounded from below by

|W | 1 ≥ n • β -O(log n) = n • α n -O(log n). Now,
there are more than 2 n•αn-O(log n) patterns w ∈ T 1,□ ⊆ L n (Y α ) with w (1) ⊑ ⟨n⟩ ↑ 0 and w (2) ⊑ ⟨j n ⟩ ↑ 0 and w (d) = W ↑ . Moreover, each of these patterns w have different extender sets. Indeed, as W is T (β, j n ) 0,n-1 with β ≤ α n,jn , each of these patterns can be extended by some configurations of Type 1. Furthermore, for any such two distinct patterns w, w ′ that only differ on their layer L f at some position (m 1 , m 2 ) ∈ 0, n -1 2 , the configuration y = (⟨n⟩

↑ 0 , ⟨j n ⟩ ↑ 0 , T (β, j n ) ↑ 0 , [2n] n+m1,n+m2 , (w (f ) ) ∞ ) extends w but not w ′ . Therefore, |E Yα (n)| ≥ 2 n 2 αn-O(log n) , and so h E (Y α ) ≥ α. ◀ A.3 Soficity of Y ′ α in the proof of Theorem B
Proof of soficity. In order to prove that Y ′ α is sofic, we need to show how to construct the two layers L f and L m which verify the required conditions.

We will construct a sofic subshift Y grid whose purpose is simply to create a regular grid-like structure, which will be used to synchronize the free bits of the L f layer. A configuration of Y grid will contain three layers (see Figure 4), which verify for some i ≥ 0:

The column layer contains a (i, 1) periodic configuration of blue columns.

The red grid layer and the green grid layer contain respectively red and green square grids, of mesh 2i. The red and green grids are offset by (i, i); said differently, the red corners lie exactly at the center of the green squares (and reciprocally). By compactness, configurations containing e.g. a single blue column and/or a single corner of a square do exist. In what follows, we denote by A grid the set of symbols used to construct the subshift Y grid .

Figure 4 Configuration of Y grid . Vertical blue lines are i-periodic, red and green square grids have mesh 2i and are exactly offset by (i, i) Such a subshift is sofic: each layer taken independently is indeed sofic, placing the two grids correctly relative to one another is , and synchronizing the periods of the three layers can be done using the fact that there needs to be exactly one blue and one green column between two red columns, and one green line between two red lines.

Now consider a subshift

Y aux of Z ↑ aux × Y grid × L f × L sync , where L f is the full-shift over {0, b, b ′ }, and L sync = {0 Z 2 , b Z 2 , b ′Z 2 },
having the following conditions: Non-zero symbols of L f have to be placed exactly at positions containing a symbol 1 of L ↑ d in Z ↑ aux . Blue columns of Y grid must be placed exactly at positions containing a symbol * in L ↑ 1 from Z ↑ aux . In particular, in a configuration y such that y (1) is some ⟨i⟩ ↑ k1 , this enforces the fact that blue columns in Y grid are also i-periodic, and so the red and green square grids are (2i, 2i)-periodic. If the layer L ↑ p of Z ↑ aux is constant equal to p Z 2 , then, at all the positions that are corners of the red or green grid of Y grid , we enforce the bit of L f and of L sync to be the same. As the configuration of L sync is constant, this simply forces some bit of L f to be (i, i)-periodic.

If L ↑ p is ∞ Z 2 , we enforce nothing.
In particular, in configurations y of Type 2 in Y aux that have y (1) = ⟨i⟩ ↑ k1 , and which therefore have a proper grid on Y grid , the bits of that are marked by the corners of those grids can still be completely free in y (f ) , as long as Finally, the subshift Y ′ α is the image of Y aux by the following letter-by-letter projection:

y (p) = ∞ Z 2 .
ϕ : A * × A * × A d × A p × A grid × A f × {0, b, b ′ } → A * × A * × A p × A d × A f × A m (a i , a j , a d , a p , a grid , a f , b sync ) → a i , a j , a d , a p , a f , ■ if a grid is a red corner □ otherwise
As a projection of a sofic subshift, Y ′ α is also sofic. ◀

B Proofs of Section 8 B.1 Z mixing subshifts

Proof of Proposition 15. We now prove that h E (X # ) = h E (X). First, we need to introduce the notion of follower and predecessor sets: for a subshift X ⊆ A Z , its follower and predecessor sets are respectively defined as

F X (w) = {x ∈ A N | wx ⊑ X} and P X (w) = {x ∈ A -N | xw ⊑ X}.
In other words, the follower set (resp. predecessor set) of some word w correspond to the set of right-infinite (resp. left-infinite) sequences x such that ux (resp. xu) appears in some configuration of X.

Let n ≥ 0. We first prove that:

|E X (n)| ≤ |E X # (n)| ≤ |E X (n)| + i+j<n |P X (i)||F X (j)|
The leftmost inequality holds simply because if x extends a pattern w ∈ L(X) but not w ′ ∈ L(X), then x also belongs in X # and still extends w but not w ′ in X # , so that

E X # (w) ̸ = E X # (w ′ ).
For the rightmost inequality, we need to distinguish some cases according to whether a pattern contains a # or not. Let w ∈ L n (X # ) that does not contain a symbol #. Then, we claim that E X # (w) is entirely determined by E X (w). Indeed,

E X # (w) = E X (w) ∪ l,r∈A * |lwr∈L(X) {(x # l, r # x ′ ) | x, x ′ admissible in X # } So, for w, w ′ ∈ L n (X # ) that do not contain a symbol #, E X # (w) = E X # (w ′ ) if and only if E X (w) = E X (w ′ ).
For patterns w ∈ L n (X # ) containing at least a symbol #, let i ≤ j be the first and last positions in w at which a symbol # appear, and define l, r ∈ A * as (l, r) = (w 0,i-1 , w j+1,n-1 ). Then, since # is a safe symbol, we claim that E X # (w) is exactly determined by the pair (P X (l), F X (r)). Indeed,

E X # (w) = (P X (l) × F X (r)) ∪ l ′ ,r ′ ∈A * | l ′ •l, r•r ′ ∈L(X) {(y # l ′ , r ′ # y ′ ) | y, y ′ admissible in X # }
Doing a disjunction on these two cases, and over the pairs i + j < n in the second case (and abusing notations again by denoting P X (i) = {P X (w) | w ∈ L i (X)} and F X (j) = {F X (w) | w ∈ L j (X)) we obtain:

|E X # (n)| ≤ |E X (n)| + i+j<n |P X (i)||F X (j)| As for every n, we have |P X (n)| ≤ |E X (n)| and |F X (n)| ≤ |E X (n)|, and that |E X (n)| = 2 αn+o(n) (since h E (X) = α), we obtain: |E X # (n)| ≤ |E X (n)| + i≤j |E X (i)||E X (j)| ≤ 2 αn+o(n) + i≤j 2 αi+o(i) 2 αj+o(j) ≤ poly(n) • 2 αn+o(n) Since |E X # (n)| ≥ |E X (n)|, we conclude that h E (X # ) = α. ◀

B.2 Z d block-gluing subshifts

This section is dedicated to the proof of Proposition 16. First, let us properly define block-gluingness:

▶ Definition 19. Let X ⊆ A Z d any subshift, and f : N → N a (non-strictly) increasing function. We say that X is f -block-gluing if ∀p, q ∈ L n (X), ∀k ≥ n + f (n), ∀u ∈ Z d , ∥u∥ ∞ ≥ k =⇒ (p ∪ σ u (q) ∈ L(X))

Said differently, X is f -block-gluing if for any two square patterns of size n, as long as we place them with a gap of size n + f (n) between them, there exist a configuration containing both patterns at any position. As with Definition 14, we will simply write N -block-gluing for constant gluing distance (f : n → N ).

We prove Proposition 16 in dimension 2 for simplicity, although it works in higher dimension by using hyperrectangles instead of classical rectangles in Z 2 . Sketch of proof of Proposition 16. The proof scheme follows the one of Proposition 15: we define a new subshift X # by adding a new symbol #. More precisely, since X is an effective subshift, we can assume that X is defined by the computably enumerable family of forbidden patterns F = ( m,n∈N A m×n ) \ L(X). Then, we define X # as the subshift over the alphabet A ⊔ {#} as follows: configurations of X # consist exactly of (possibly infinite) rectangles containing symbols of A on a background of symbols #, and the family of forbidden patterns F is still forbidden (which implies that the interior of any A-rectangle is a globally admissible patterns of X). The subshift X # is 1-block-gluing: it suffices to "wrap" a n × n square within a border of # to be able to put any other rectangular pattern next to it. X # is also an effective subshift.

We now claim that |E

X (n)| ≤ |E X # (n)| ≤ 2 o(n 2 ) • |E X (n)|.
The left inequality is proved as in the proof of Proposition 15. For the right inequality, define, for a pattern w ∈ L n (X), its geometry G(w): removing the # from w, G(w) is the set of maximal rectangles r ∈ a,b,c,d a, b × c, d that are adjacent to the border of w. To prove the right inequality, we observe that E X # (w) is entirely determined by {E X (w| r ) | r ∈ G(w)}, and we bound the number of possible G(w) from above:

First, for a given pattern w, G(w) contains at most O(n) different rectangles adjacent to the border. Indeed, each position in the border of w is either a # or belongs to a single rectangle. Each rectangle is specified by two points (if south-west and north-east positions), so that there are O(n 4 ) such choices, and there are in total O(n 4 ) O(n) = 2 O(n log n) different possibles geometries G(w) for w ranging in the patterns of support 0, n - ≤ 2 n 2 α+o(n 2 )+O(n log n)

As such, we obtain h E (X) ≤ h E (X # ) ≤ h E (X). This concludes the proof. ◀

Z Z d , d ≥ 2 SFT

 2

Figure 1

 1 Figure 1Example configurations of the mirror and semi-mirror subshifts. The blank symbols ␣ of the semi-mirror are represented as white □.

2 . 3 .

 23 Second layer L 2 : We also set L 2 = X * . Density layer L d : Define A d = {0, 1}, and define the density layer as L d = A Z d . Consider the ruler sequence T = 12131214 . . . (see Oeis A001511). Given a real number β ∈ [0, 1) and its proper binary expansion β = i∈N β i 2 -i , let us define its Toeplitz sequence as:

Figure 2 A

 2 Figure 2 A Type 1 configuration: L d contains a Toeplitz encoding of .1010 2 = 5 8 . z = (⟨15⟩ 11 , ⟨18⟩ 1 , T ( 5 8 , 15)10,z (f ) = (b ′ 0b0b0b ′ bb0bb ′ 0b ′ b) ∞ ).The red line indicates the origin.

  L m : We define L m = G, where G is the following subshift over the alphabet A m = {□, ■}: denoting by [2i] m1,m2 the configuration defined by ([2i] m1,m2 ) (p1,p2) = ■ ⇐⇒ (p 1 = m 1 mod 2i ∧ p 2 = m 2 mod 2i) (in other words, [2i] m1,m2 draws a square lattice of ■ symbols at distance 2i from one another) and [∞] = {x ∈ A Z 2 m | |x| ■ ≤ 1} the set of configurations having at most one symbol ■, we set

Figure 3

 3 Figure 3 Projection of a (Type 1) configuration on L1 × L f × Lm. The symbols * are on L1, the symbols b are on L f and the symbols ■ on Lm. All the other bits of L f (not drawn here) are free.

  of L f marked by the ■ are equal. Computations of Theorem A show that |π L ↑ 1 ×L ↑ 2 ×L ↑ d ×Lm (T 1 )| = poly(n), there are at most poly(n) different extender sets for patterns of T 1,■ .

Figure 5

 5 Figure 5 Conditions on Yaux enforced by Y grid and Lsync. We obtain Figure 3 by removing the grids and the blue columns, and mapping red corners to ■.

Figure 6 A

 6 Figure 6 A configuration of X # . Gray rectangles are admissible patterns of X.

1 2 .

 2 Finally, for a rectangle r = a, b × c, d , denote ∥r∥ = (b -a) • (d -c) its area andE X (r) = {E X (w) | w ∈ L r (X)}. Denoting α = h E (X), we have |E X (n)| = 2 n 2 α+o(n 2 )and then:|E X # (n)| ≤ G(w)|w∈Ln(X # ) r∈G(w) |E X (r)| ≤ G(w)|w∈Ln(X # ) r∈G(w) 2 ∥r∥α+o(∥r∥) ≤ G(w)|w∈Ln(X # ) 2 r∈G(w) ∥r∥α+o(∥r∥) ≤ G(w)|w∈Ln(X # ) 2 n 2 α+o(n 2 )

  2 )| ≤ p≤n p 2 = poly(n). Since |π L ↑ )| = poly(n), there are in total poly(n) extender sets for patterns of T 2 .The extender sets of patterns of T 1,■ are entirely determined by π L ↑

	1 ×L ↑ 2	(T 2 1 ×L ↑ 2 ×L ↑ d

It will sometimes be convenient to consider patterns only up to translation of their support. Usually, context will make it clear whether we consider two patterns equal up to a Z d translation as being the same or not.

The authors define it for the case of Z subshifts only, but the definition makes sense for higher dimensional shifts.

It is a rephrasing of Inf (does a given machine halt on infinitely many inputs?), which is Π 0 2 -complete. See[START_REF] Robert I Soare | Turing computability: Theory and applications[END_REF] Theorem

4.3.2].

The layer y(m) voluntarily is (2i, 2i)-periodic instead of (i, i)-periodic, since the argument only works whenever w, w ′ do not contain the ■ marker.

Xizhong Zheng and Klaus Weihrauch. The arithmetical hierarchy of real numbers. Mathematical Logic Quarterly: Mathematical Logic Quarterly, 47(1):51-65, 2001.

It is enough to look at a single of those marked bits, since in a Type 1 configuration they are all equal.

Funding This research was partially funded by ANR JCJC 2019 19-CE48-0007-01.

A

Proofs of Section 5

A.1 Computations for the proof of Theorem A

Proof of h E (Z α ) ≤ α. We simply provide an upper bound of |E Zα (n)|. For a pattern w ∈ L n (Z α ), denoting w = (w (1) , w (2) , w (d) , w (f ) ), we say that w sees i (resp. j) at position k if w (1) (resp. w (2) ) contains two symbols * separated by i -1 (resp. j -1) symbols ␣, and w

(1)

is the first * in this layer. In that case, any configuration z ∈ Z α extending w must have z (1) = ⟨i⟩ k (resp. z (2) = ⟨j⟩ k ). We now determine the extender sets of the different patterns w.

We partition L n (Z α ) in two sets T 1 ⊔ T 2 . Patterns in T 2 will only appear in configurations of Type 2, and patterns in T 1 can appear in a configuration of Type 1.

As configurations of Type 2 have no restriction on their free layer z (f ) or their density layer z (d) , the extender set of any w ∈ T 2 is entirely determined by its first two layers (i.e. by the pair (w (1) , w (2) )): since there are poly(n) such pairs, there are at most poly(n) extenders sets of patterns of T 2 . Now, notice that in a configuration z of Type 1, layers z (1) , z (d) and z (f ) are i-periodic for some i ∈ N and z (2) is j-periodic for some j ∈ N: this implies in particular that any two distinct patterns w, w ′ ∈ T 1 have different extender sets. Let us now count the number of such patterns of size n, layer by layer: 1. If w (1) sees i, then it can see values of i ranging from 0 to n -1, and different offsets k 1 ranging from 0 to i -1; if it does not see i, then there is at most a single symbol * whose position ranges from 0 to n -1. (2) sees j, then j ranges from i to n -1 with offset k 2 ranging from 0 to j -1; if it does not see j, then there is at most a single symbol * whose position ranges from 0 to n -1. (1) sees i, then w (d) is entirely determined by T (β, i) 0,i-1 for some 0 ≤ β ≤ α i . If the first layer does not see i, then w (d) layer is a factor of T (β, i) for some i ≥ ⌈n/2⌉ and 0 ≤ β ≤ α i . In both cases, for 0 ≤ β ≤ 1, there are less than 2 ⌈log i⌉ different possibilities for w (d) . 4. For a fixed (w (1) , w (2) , w (d) ), we count the number of attributions of free bits in w (f ) . To proceed, consider whether w (d) can be extended into a configuration T (β, i) for some i ∈ N and β ≤ α i (a pattern could appear in several T (β, i) for different values of i; counting the same pattern several times is not a problem when bounding from above): a. For each 1 ≤ i ≤ n, fixing the attribution of the layer w (f ) on a single i-period is enough, since in any Type 1 configuration z extending w with z (d) = T (β, i), the layer z (f ) will be i-periodic.

If w

If w

On the other hand, by properties of Toeplitz sequences, there are i • β + O(log i) symbols 1 in an i-period of T (β, i). Since the free layer w (f ) is synchronized with the density layer w (d) , i.e. any of these symbol 1 on w (d) can either by b or b ′ in w (f ) , there are 2 i•β+o(i) possible attributions for the layer w (f ) . Since β ≤ α i , we obtain at most 2 i•αi+o(i) possibilities for w (f ) . b. If i > n, the numbers of 1s in a factor of length n in T (β, i) is bounded by n•β+O(log n), and β ≤ α i ≤ α n , so there are at most 2 n•αn+o(n) different attributions for the layer w (f ) .


Charlotte Allemand
email: charlotte.allemand@enpc.fr

Arthur Lebée

Manuel Manthey

Gilles

CHARACTERIZATION OF ROLLING AND LONGITUDINAL SHEAR CREEPS FOR CROSS LAMINATED TIMBER PANELS

Keywords: rolling-shear, creep, cross laminated timber, long-term behavior, longitudinal shear

Introduction

Everywhere around the world, high timber buildings are rising. The construction of these new buildings is possible thanks to the mainstreaming of some engineering products such as cross-laminated timber panels (CLT) and glued laminated timber (GLT). Those panels gained in popularity for several reasons. First, they are a more sustainable solution than concrete designs. Second, the prefabrication process makes them easy to assemble on site. CLT panels also have a high dimensional stability under variation of moisture content compared to other timber products. Because of all those advantages, high buildings up to 80-meters, such as the Mjøsa tower in Norway, are being erected. CLT panels are used both as walls and floors in those constructions.

Rolling and longitudinal shear deformations are involved in the mechanical response of those panels. Shear failure can occure for CLT panels loaded out-of-plane but also when they are under in-plane shear stresses [START_REF] Brandner | Cross laminated timber (CLT): overview and development[END_REF]. The value of the longitudinal shear modulus G 0 l,mean for short-term lies between 440 and 880 MPa depending on the strength class of the wood (Eurocode-5, 2004). In the rolling-shear direction, the short-term modulus G 0 r,mean for short-term lies between 50 and 200 MPa [START_REF] Ehrhart | Rolling shear: Test configurations and properties of some European soft-and hardwood species[END_REF][START_REF] Franzoni | Influence of orientation and number of layers on the elastic response and failure modes on CLT floors: modeling and parameter studies[END_REF][START_REF] Zhou | Measurement of rolling shear modulus and strength of cross laminated timber fabricated with black spruce[END_REF][START_REF] Perret | Experimental determination of the equivalent-layer shear stiffness of CLT through four-point bending of sandwich beams[END_REF]. Large differences between the studies are due to the difficulty to obtain a pure shear stress and to represent the behaviour of cross layers. Additionally, the sawing pattern and aspect ratio of the board are of importance for CLT [START_REF] Krabbe | Messungen von Gleit-und Dehnungszahlen an Holzsta¨bchen mit rechteckigen Querschnitten[END_REF][START_REF] Aicher | Basic considerations to rolling shear modulus in wooden boards[END_REF][START_REF] Ehrhart | Rolling shear: Test configurations and properties of some European soft-and hardwood species[END_REF]Perret et al., 2019).

Wood is a material subject to creep and this phenomenon must be investigate to correctly design timber builings as it amplifies the short-term deflections. The first study of creep in CLT is recent [START_REF] Park | Bending creep performances of three-ply cross-laminated woods made with five species[END_REF] and there are very few attempts to characterize the behaviour of rolling shear creep [START_REF] Pirvu | Time-dependent behaviour of CLT[END_REF][START_REF] Colling | Creep behavior of cross laminated timber in service class 2[END_REF]. In these works, bending tests of CLT panels are achieved and an overall creep factor is derived. It appears clearly that creep is larger with CLT than with GLT. A possible explanation may be a faster creep behaviour coming from rolling shear strains in cross layers.

This paper presents an experimental protocol to measure directly shear stiffness and creep in CLT panels. The methodology originally presented by [START_REF] Perret | Experimental determination of the equivalent-layer shear stiffness of CLT through four-point bending of sandwich beams[END_REF] is used to measure separately those characteristics in the rolling and the longitudinal direction. The paper is organized as follows. In Section 2, the sandwich-beam theory is recalled and the experimental set-up is introduced. In the following section, the short-term characterization of the rolling-shear stiffness modulus and the estimation of the Eurocode coefficient k def (Eurocode-5, 2004) by means of a power law are presented. This experiment is referred as Experiment 1 and was conducted during eight months (13/03/2020 -28/10/2020). Finally, the results of Experiment 2 (11/12/2020-11/06/2021) are presented in Section 5 and the longitudinal shear stiffness modulus for short and long time are determined.

Sandwich beam theory

The rolling shear modulus G r,mean and the longitudinal shear G l,mean are characterized by means of a four-point bending test on sandwich beams constituted of a wooden core glued between two steel skins (Figure 1). The beam is simply supported on a span d = 602 mm and two loads P/2 are applied at a distance d 0 /2 from the mid-span, with d 0 = 200 mm. The beam is under pure bending between these loads leading to a constant curvature. It is assumed that steel does not creep in time. The full methodology is detailed in [START_REF] Perret | Experimental determination of the equivalent-layer shear stiffness of CLT through four-point bending of sandwich beams[END_REF].

The sandwich beam model [START_REF] Lebée | Homogenization of cellular sandwich panels[END_REF] requires a contrast between the stiffness and the thickness of the core and skins, respectively (E a ,e a ) and (E b ,e b ) so that : e b e a and E a e a E b e b . Because of this contrast, the bending stiffness D and the shear force stiffness F are [START_REF] Allen | Analysis and Design of Structural Sandwich Panels: The Commonwealth and International Library: Structures and Solid Body Mechanics Division[END_REF]:

In this paper, two shear directions are investigated. The rolling shear, or perpendicular to the grain is tested in Experiment 1. In this experiment the wooden core orientation is pictured in Figure 2. The longitudinal shear, or parallel to the grain is tested in Experiment 2. The wooden core is then rotated as pictured in Figure 1.

Whereas for Experiment 1, the contribution of the wooden core to the bending stiffness may be neglected. This is not the case for Experiment 2. Indeed, when measuring the rolling shear (Experiment 1), the contrast assumption is satisfied with E a = 210 GPa and E b = 0.43 GPa (litterature values). Equation 1 can be written as D = E a I a + E b I b . The contribution of the steel is calculated with h = (33.5 ± 0.4) mm the total thickness, e b = (3.9 ± 0.4) mm the wood thickness, and the width of the beam b = (41.1 ± 0.6) mm. This gives E a I a = (5.06 ± 0.68) kN m 2 and the contribution of the wooden core is

E b I b = (44 ± 11) N m 2 .
From those values, the contribution of the main core to the bending stiffness can be neglected as it contributes to less than 1%. Equation 1 can be rewritten as:

D = b(h 3 -e 3 b ) 12 E a (3)
This equation can be used in Experiment 1 to calculate the bending stiffness using the litterature value of E a and the measured dimensions of the beam. The shear force stiffness F can be expressed as a function of the mid-span deflection f A and the bending stiffness D:

1 F = 4f A P(d -d 0 ) - 1 8D d 2 - 1 3 (d -d 0 ) 2 (4)
The measurement of f A gives the evolution of the rolling shear modulus G r,mean in time.

In Experiment 2, the contrast stiffness is not fullfilled but Equation 2 The contribution of the wooden core to the bending stiffness is about 16% which can not be neglected. The bending stiffness must be precisely measured, it can be estimated from the absolute rotations at B and C (respectively ϕ B and ϕ C ) as follows:

D = P(d 2 -d 2 0 ) 8∆ϕ (5)
where ∆ϕ = ϕ B + ϕ C . Equations 2 and 4 can still be used to measure the evolution of the longitudinal shear modulus G l,mean in time. For Experiment 2, an accurate measurement of D is necessary in order to calculate the longitudinal shear modulus and to correctly estimate its creep.

3 Methods

Specimen fabrication

Norway Spruce (Picea abies) boards were used to make the specimens. The layer thickness of CLT-boards in Europe are 20 mm, 30 mm and 40 mm according to Brandner (2013). It was chosen to use a 30 mm thick layer.

In Experiment 1, eighteen boards were glued together on their narrow edges with a wood glue (Titebond Ultimate 141/5). Glued narrow edges are not not a common practice in the industry but sometimes it is used to reduce the width of the gaps. It will allow here to reduce the stress concentration in the specimens. This wooden plate was then planarized to a thickness of e b = 30 mm. Then, 800 mm long specimens with a width b = 40 mm and a thickness e b = 30 mm were cut in this plate. They were cut so that wood fibres are oriented in the transverse direction. They were conditioned at a moisture content of u = (10.6 ± 0.3) % and visually graded C24. Their density was measured to be ρ =(495 ± 32) kg m -3 . The boards were oriented so that the pith is alternatively at the bottom or at the top which average their effect on the global behaviour (see Figure 2).

For Experiment 2 the boards used were conditioned at a moisture content of u = (10.2 ± 2.5) % and their density was measured to be ρ = (393 ± 53) kg m -3 . Norway Spruce board of 30 × 60 mm were planarized to be 30 × 40 mm and cut to be 800 mm long oriented in the longitudinal direction.

Carbon steel XC75 sheets of Young modulus E a =210 GPa were used. They are 800 mm long, b = 40 mm wide and e a = 1.1 mm thick. They were sanded and glued on the top and bottom faces of the wood with a two components glue including a thixotropic epoxy based impregnating resin and adhesive (Sikadur® 300). The thickness of the glue layer is about 0.5 mm. For Experiment 1, five specimens (A-E) were fabricated and six specimen (M-R) for Experiment 2. One example of each can be seen in Figure 2.

Experimental set-up

Figure 3 shows the schematic representation of the frame (0). The specimens (1) are supported on two cylinders (2) of radius R = 23 mm with a span d = 602 mm. They are loaded vertically and symmetrically by two loading fixtures (3) spaced of a length d 0 = 200 mm. The contact between the loading fixtures and the beam is made with a steel ball of diameter d b = 1.2 mm [START_REF] Allen | Analysis and Design of Structural Sandwich Panels: The Commonwealth and International Library: Structures and Solid Body Mechanics Division[END_REF]. The rotation of the motor (6) drags a metal thread ( 9) that ascends and descends the loads (5) and the lever arms (4).

Several sensors are positioned to measure the different variables:

• Three Orbit® linear variable differential transformers (DP20S) are placed on each span of the frame (8). Two of them measure the horizontal displacement of the steel flat angles. They are represented Figure 3 as arrows L B and L C . The last linear variable differential transformer (LVDT) measures the vertical deflection of the beam at mid-span (f A : arrow L A ). They have an expanded measurement uncertainty U = 16 × 10 -3 mm.

• Two AEP transducers ® S-type load cell (TS) [START_REF] Aicher | Basic considerations to rolling shear modulus in wooden boards[END_REF] are placed in the loading fixture (3). They measure the load on points D and E. Their expanded measurement uncertainty is U = 4.8 N.

• Two Sensel Measurement® single axis inclinometers (SM-NA) are placed on both sides of the specimen (11). They measure the rotations ϕ B and ϕ C and have an expanded measurement uncertainty U = 5 × 10 -3 rad.

Figure 3. Framework design for the tests

In Experiment 2 the measure of the bending stiffness D is crucial to determine the instantaneous and the long term modulus. Since the inclinometers are not specifically designed for long term experiment, the LVDT sensors will be used to measure the long term displacement.

Test method

The viscoelastic limit has been well studied in the longitudinal direction [START_REF] Hoyle | Primary Creep in Douglas-fir beams of commercial size and quality[END_REF][START_REF] Nakai | Deflection of wood under intermittent loading[END_REF][START_REF] Foudjet | Creep of four tropical hardwoods from Cameroon[END_REF][START_REF] Hayashi | Wood viscoelastic compliance determination with special attention to measurement problems[END_REF][START_REF] Bhatnager | Creep in wood in tension parallel to grain[END_REF][START_REF] King | Time-Dependent Strain Behavior of Wood[END_REF] and observed to be between 40% to 50%. It is a priori not the same for the rolling-shear stiffness.

Therefore, Specimens A-E were tested under different load conditions ranging from 27 to 52 % of the characteristic shear strength f r,mean = 1.88 MPa [START_REF] Ehrhart | Rolling shear: Test configurations and properties of some European soft-and hardwood species[END_REF]. These stress levels are approximate since the shear strength varies from one board to another. The specimens M-R loaded between 11 and 14% of the characteristic shear strength f l,mean = 7.8 MPa (Wood-Handbook, 2010).

Under the viscoelastic limit, D. G. [START_REF] Hunt | A unified approach to creep of wood[END_REF] proposed to separate creep into three different components: pure viscoelastic (time dependent creep), mechano-sorptive creep and pseudo-creep followed by a recovery phenomenon. These components are independent but coupling effects may appear. The present experiment will be achieved such that the pure viscoelastic behaviour is isolated. Therefore, the tests were performed in a climate-controlled room with constant temperature of (19 ± 1) °C and a relative humidity in the air of (58 ± 8) % in the first experiment. In the second experiment the temperature was (20 ± 1) °C and the relative humidity in the air of (54 ± 6) %. Both times the specimens were conditioned at least one week in this room before the tests.

4 Rolling shear results

Short term

Results are plotted for beam C but are similar for all the specimens. In Figure 4 the load P is plotted as a function of the deflection f A for the first two minutes of the experiment.

A linear regression on the linear part of this curve gives an estimation of the modulus G 0 r,mean using Equations 2 and 4 and the value of the bending stiffness calculated section 2. Table 1 summarizes the identified moduli. The cross-layer shear modulus G 0 r,mean = 109 MPa with a coefficient of variation of 8.6% is consistent with the literature values given in Table 2. A rather high value is consistent with the sawing pattern presented in Figure 2. The coefficient of variation found is very low compared to the literature and comparable to the one already found by [START_REF] Perret | Experimental determination of the equivalent-layer shear stiffness of CLT through four-point bending of sandwich beams[END_REF]. Indeed, the four-points bending test averages the rolling-shear stiffness on several boards.

Long term

The mid-span deflections can be separated in two contributions; the bending and the shear deflection. The total mid-span deflection varies from (1.2848 ± 0.0033) mm dur-ing the experiment. The symmetric part of the rotations varies from (1.3 ± 1.7) × 10 -3 rad. This variation falls within the accuracy of the LVDT. Hence, we can neglect the variations of the rotations in this experiment and calculate D.

The measured relative creep of the specimens during this experiment are plotted Figure 5 for the rolling-shear modulus G ∞ r,mean . It can be observed that those curves are close to linear in a log-log scale at long time, therefore the creep can be modelled with a power law:

J(t) J 0 -1 = mt n (6)
where n, m and J 0 are experimental parameters and t the time in seconds. They are independent from the load as long as it is in the linear viscoelastic domain. Indeed, wood creep has been modelled with empirical and mechanical models. Mechanical models come from the thermodynamics [START_REF] Schapery | An engineering theory of nonlinear viscoelasticity with applications[END_REF]. They are systems composed of Kelvin and Maxwell elements. Those models are useful to differentiate the several creep components: elastic, viscous and viscoelastic. They are mostly used to model creep occurring with climate changes [START_REF] Varnier | Comportement thermo-hygro-mécanique différé des feuillus[END_REF][START_REF] Mukudai | Evaluation on non-linear viscoelastic bending deflection of wood[END_REF][START_REF] Hunt | Possible contribution of fibre slippage to the longitudinal creep of wood[END_REF]. Empirical models are power or logarithmic functions and they fit and predict well those phenomena. [START_REF] Youngs | The Perpendicular-to-grain mechanical properties of red oak as realated to temperature, moisture content and time[END_REF] was the first to model wood creep from a 3days experiment as a power law similar to the Equation 6. [START_REF] Clouser | Creep of small wood beams under constant bending load[END_REF] and [START_REF] Gressel | Zur Vorhersage des langfristigen Form inderungsverhaltens aus Kurz-Kriechversuchen[END_REF] used this same equation to fit their 10-year creep tests. It has been widely used since then to fit and predict creep wood behaviour [START_REF] Sugiyama | The creep deflection of wood subjectied to bending under constant loading[END_REF][START_REF] Schniewind | On the influence of moisture content changes on the creep of beech wood perpendicular to the grain including the effects of temperature and temperature changes[END_REF][START_REF] Hayashi | Wood viscoelastic compliance determination with special attention to measurement problems[END_REF]. In international standards the design of buildings has to be estimated for 50-years. Therefore, a prediction model is required to extrapolate a value of the deflection of wood at such time period. In the Eurocode, this is taken into account by means of the coefficient k def which corresponds to the relative creep at 50-years. Considering exclu-sively rolling shear deformations, we define k GR def as: [START_REF] Aicher | Basic considerations to rolling shear modulus in wooden boards[END_REF] with G 0 r,mean the short-term rolling shear modulus and G ∞ r,mean the secant rolling-shear modulus in 50 years. A linear regression was done on the logarithmic values. The k GR def values found are summarized in Table 3 with a mean value of 2.43 and a coefficient of variation equal to 6.18 % which corresponds approximately to the coefficient of variation of the instantaneous modulus. The stress level does not seem to have a significant influence on k GR def which comforts the assumption of linear visco-elasticity. The values found for the coefficient of Equation 6 are shown in Table 4. They fall into the range of coefficients found in the literature and summarized in [START_REF] Tong | Wood creep data collection and unbiased parameter identification of compliance functions[END_REF].

G ∞ r,mean = G 0 r,mean 1 + k GR def
5 Longitudinal results

Short term

The value of the instantaneous bending stiffness is measured with the inclinometers during the loading of each specimen. The load is plotted against the symmetric part of the rotations during the loading Figure 6. Results are plotted for beam N and are similar for all specimens. A linear regression on the linear part of this curve allows to determine the bending stiffness D. The results are summerized Table 5. They are in the scope of the calculus that was presented in Section 2.

Using Equations 2 and 4 the value of the instantaneous longitudinal shear modulus can be calculated. It is deduced from the slope of the function between the load P and the mid-span deflecion f A . The values found are summerized in Table 5 and, except for beam O, the results are consistent. The coefficient of variation found for the longitudi-

Long term

The long term part of the long term mid-span deflection is (0.2959 ± 0.0033) mm while it was (1.2848 ± 0.0033) mm for the rolling-shear. It is much smaller than in the previous experiment and therefore require a more precise analysis. The bending and shear contribution to the mid-span bending can be separated as:

f A = k 1 F + k 2 D (8)
where k 1 and k 2 are two coefficients that can be determined with Equation 4 and are constant during the whole experiment.

The long term symmetric part of the rotations is (1.4 ± 1.7) × 10 -3 rad. This variation falls into the uncertainty of the sensors and could be neglected. If this variation is not neglected the bending contribution on the mid-span creep deflection is then about 45%. This contribution may be associated to the creep of the wood Young modulus E b in the longitudinal direction with Equation 1.

Hence, two estimations of k GL def are proposed here ; one considering that only the shear contributes to the creep of the long term mid-span deflection f A and one considering that there is a bending creep as well. In the first case, D is assumed constant in Equation 8, while in the second case, D is taken from the long term measurement. In both cases, considering the deformations in the longitudinal direction, we define a creep coefficient k GL def as :

G ∞ l,mean = G 0 l,mean 1 + k GL def (9)
with G 0 l,mean the short-term longitudinal shear modulus, G ∞ l,mean the longitudinal shear modulus in 50 years.

Using the power law Equation 6, a linear regression was fitted on the logarithmic values. Figure 7 shows the fitting of the data when assuming that there is no bending creep (D is constant). The cycles visible corresponds to those of the air conditioning that regulated the temperature of the room during the experiment. Those variations correspond to the sensitivity of the LVDT sensors to temperature. They exist in Experiment 1 but are not visible in Figure 5 as the creep deflection is 4 times larger. The values of k GL def are summerized Table 6 with a mean value of 1.1 (CoV = 15%). A similar fitting was done when considering that the bending contributes to the creep of the mid-span. There was a power outage the second month of the experiment and the measure of the rotation of the beam M was lost. Therefore no creep results are presented for this specimen in this case. The results are in Table 6 and k GL def is calculated with a mean value of 0.6 (CoV = 26%). In both estimations, the stress level seems not to not have any influence on the creep which validates the hypothesis of linear visco-elasticity. Experiments are still going on to better understand those results and obtain a more precise value of k GL def .

The value found in the current version of the Eurocode of the creep coefficient k def is 0.6. This value is the same for all orthotropic elastic moduli (Eurocode-5, 2004). The value of the rolling shear coefficient k GR def identified in the present experiment is 2.43 while the value of the longitudinal shear coefficient k GL def is between 0.60 and 1.1. This gives an approximated range of k GL def but seems to indicates that this creep is comparable to the creep of the bending stiffness in the longitudinal direction E b . The rather large value of the rolling-shear coefficient is not inconsistent and was expected considering previous attempts to estimate k def for CLT panels [START_REF] Pirvu | Time-dependent behaviour of CLT[END_REF][START_REF] Colling | Creep behavior of cross laminated timber in service class 2[END_REF]. Indeed, the deflection of a CLT panel is the superposition of the bending and the shear deflection. The shear contribution in the global creep depends on the slenderness of the panel and varies from 30% of the total deflection for thick panels to few percent for slender panels. Hence, a distinction between the different deformation types might be necessary to have a better estimation of the long term deflection in CLT panels. Indeed, the present results clearly indicates that rolling shear creeps faster than longitudinal shear. More experimental campaigns need to be performed to obtain statistically significant results. If the present value is confirmed it could have consequences for the design of CLT and other timber products. For instance, this may lead to more accurate design guidelines either for serviceability limit state (SLS) or long-term buckling strength of CLT walls, which completes recent recommendations for more accurate modelling (Franzoni, Lebée, Lyon, et al., 2017;[START_REF] Franzoni | Advanced modelling for design helping of heterogeneous CLT panels in bending[END_REF][START_REF] Perret | Experimental determination of the equivalent-layer shear stiffness of CLT through four-point bending of sandwich beams[END_REF].

Figure 1 .

 1 Figure 1. Four-point bending configuration on a sandwich beam, for the rolling shear configuration (experiment 1)

  is still valid. The contributions of the steel and the wood are calculated with h = (31.8 ± 0.2) mm, e b = (29.6 ± 0.2) mm, and b = (40.6 ± 0.2) mm. This gives E a I a = (4.48 ± 0.49) kN m 2 and with E b = 10 GPa ; E b I b = (0.88 ± 0.21) kN m 2 .

Figure 2 .

 2 Figure 2. View of the specimens (above : Experiment 1, below : Experiment 2)

Figure 4 .

 4 Figure 4. Applied force as a function of the mid-span deflection f A

Figure 5 .

 5 Figure 5. Rolling-shear relative creep curves and power law fittings

Figure 6 .

 6 Figure 6. Load P function of the symmetric part of the rotation ∆ϕ, beam N

Figure 7 .

 7 Figure 7. Fitted data, longitudinal creep of G l,mean considering that there is only a shear creep

Table 1 .

 1 Instantaneous values

	Name	G 0 r,mean (MPa)
	A	98.4
	B	120
	C	99.9
	D	115
	E	110
	Moyenne	109
	CoV * (%)	8.6

* Coefficient of Variation

Table 2 .

 2 Short term rolling shear modulus from other experiments

	Reference	MC 1 (%) G 0 r,mean (MPa) CoV 2 (%)
	Aicher and Dill-Langer (2000)	12	50	20
	Keunecke et al. (2007)	12	53
	Franzoni, Lebée, Lyon, et al. (2017)		110	27
	Perret et al. (2018)	10-13	124	6.7
	1 Moisture Content 2 Coefficient of Variation

Table 3 .

 3 Values of k GR def

	Name k GR def
	A	2.59
	B	2.44
	C	2.22
	D	2.34
	E	2.54
	Mean 2.43
	CoV * (%) 6.18

* Coefficient of Variation

Table 4 .

 4 Fitting coefficients values

	Name J 0	n	m
	A	92.8 0.210 0.0302
	B	113 0.212 0.0274
	C	88.9 0.217 0.0224
	D	107 0.211 0.0266
	E	106 0.194 0.0422

Table 5 .

 5 Instantaneous values for longitudinal shear

	Name	M	N	O	P	Q	R	Mean CoV * (%)
	D 0 (kN m 2 )	5.71 5.26 5.81 5.13 5.21 4.98 5.35	6.2
	G 0 l,mean (MPa) 425 385 229 359 384 355	356	19

* Coefficient of Variation

Table 6 .

 6 Values of k GL

	def
	Name	M	N	O	P	Q	R	Mean CoV * (%)
	k GL def	-0.84 0.52 0.68 0.46 0.51 0.60	26
	k GL def (without bending creep) 1.4 1.1 0.88 1.2	1.1	1.1	1.1	15

* Coefficient of Variation


Mebarek Belkadi

Anne Sergent

Berengere Podvin

Yann Fraigneau

Yann Fraigneau Heat

Transfert At

Belkadi Mebarek

Sergent Anne

Podvin Bérengère

Fraigneau Yann

HEAT TRANSFERT AT THE ROUGHNESS SCALE IN TURBULENT RAYLEIGH-BÉNARD CONVECTION

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Turbulent convection is a spontaneous physical process present in many natural systems as well as in engineering applications. However most of these systems are non-ideal in terms of underlying surfaces involving specific topography or small-scale roughness. Moreover interactions between buoyant flow and plate roughness result in a global heat transfer enhancement and in some cases, in an increase of the heat transfer scaling exponent (see for ex. [START_REF] Rusaouën | Thermal transfer in Rayleigh-Bénard cell with smooth or rough boundaries[END_REF]). In this study, we consider thermal convection over a regularly roughened plate in a Rayleigh Bénard cell. The objective is to clarify interactions between the large scale circulation filling the box and the flow around the roughness elements by means of direct numerical simulations. Three successive heat transfer regimes are investigated from inactive roughness to a regime where the heat transfer relative increase is larger than the relative surface increase induced by roughness addition.

The simulations are performed in a box-shaped cell of aspect ratio (depth over cell height) equal to 1/2 with water as working fluid for a Rayleigh number Ra ranging from 5 × 10 5 to 5 × 10 9 . The roughness is introduced on the bottom plate by a set of square based obstacles regularly spaced, modelled by using an immersed boundary method, while the top plate is kept smooth.

Results are validated against experimental and numerical data from literature. As expected, it exhibits an enhancement of heat transfer measured by the Nusselt number N u, depending on the relative sizes of the mean boundary-layers and the obstacle height in agreement with previous studies from the literature [START_REF] Stringano | Mean flow structure in thermal convection in a cylindrical cell of aspect ratio one half[END_REF][START_REF] Liot | Velocity fluctuations and boundary layer structure in a rough Rayleigh-Bénard cell filled with water[END_REF][START_REF] Zhu | Roughness-facilitated local 1/2 scaling does not imply the onset of the ultimate regime of thermal convection[END_REF]. Then, we investigate the different heat transport mechanisms at the roughness scale for three successive regimes (fig. 1). Our study reveals that the change of heat transport regime is mainly governed by physical features of fluid zone surrounding the obstacles. Flow dynamics and boundary layer structure near rough surface are also reported.

Figure 1 :

 1 Figure 1: Instantaneous temperature fields obtained for (a) Ra = 10 6 , (b) Ra = 5 × 10 7 and (c) Ra = 10 9 . Isosurfaces correspond to θ = ( 0.2, 0.45, 0.65, 0.8).