Françoise Pène

Univ Brest

Université De Brest

PROBABILISTIC LIMIT THEOREMS FOR CHAOTIC DYNAMICAL SYSTEMS, SOME RESULTS FOR DISPERSIVE BILLIARDS AND LORENTZ GASES

This proceeding is based on a mini-course given at the Summer School "From kinetic equations to statistical mechanics" organized by the Centre Henri Lebesgue at Saint Jean de Monts from the 28th of June to the 2nd of July 2021. After recalling classical probabilistic limit theorems for sums of independent identically distributed random variables, we consider analogous results in a dynamical context. Motivated by examples coming from statistical mechanics, we are mostly interested in the Sinai billiard and in the Z 2 -periodic Lorentz gas. We will also consider the Bunimovich stadium billiard and dispersive billiards with cusps. All these billiards are chaotic, with different behaviours. Additional explanations are given in the four independent appendices. 000000000000000000000000000000000000000000 000000000000000000000000000000000000000000

1. FCLT for sums of i.i.d. random variables and Wiener process Let (X k ) k be a sequence of centered R-valued independent identically distributed (i.i.d.) random variables defined on the same probability space (Ω, F, P).

• The Strong Law of Large Numbers (SLLN) ensures that lim n→+∞ 1 n n k=1 X k = 0 almost surely (i.e. this convergence holds true with probability one).

The Central Limit Theorem (CLT) deals with the convergence in distribution of ( n k=1 X k /a n ) n≥1 to a non-degenerate random variable for some sequence (a n ) n≥1 such that lim n→+∞ a n = +∞. Functional Central Limit Theorems (FCLT) state convergence in distribution of the sequence of processes t → a -1

n ⌊nt⌋ k=1 X k n≥1
to a non degenerate càdlàg 1 process for a n as above.

Different normalizations and limit processes may appear. In particular:

• (Standard FCLT) If E[X 2 1 ] < ∞, then t → n -1 2 ⌊nt⌋ k=1 X k n≥1 converges in distribu- tion 2 to (t → ∥X 1 ∥ L 2 W t ) with W : t → W t is standard Wiener motion • (FCLT with non-standard normalization) If lim x→∞ x 2 P(|X 1 | ≥ x) = A > 0,
then X 1 is not in L 2 but is in L p for any p ∈ [1, 2) and the family of processes t →

1 √ n log n ⌊nt⌋ k=1 X k n≥1
converges in distribution to AW, with W as above.

• (Convergence to Lévy processes)

Let α ∈]1; 2[. If lim x→∞ x α P(±X 1 ≥ x) = A ± with A + + A -> 0.
Then the family of processes t → n -1 α ⌊nt⌋ k=1 X k n≥1 converges in distribution 3 to a Lévy process Z of order α.

Date:

1 càdlàg means right continuous with left limits, in reference to 'continu à droite et limit''e à gauche' in french. 2 for the uniform topology on every compact 3 Here the convergence is in the Skorohorod space of càdlàg functions, with respect to the usual J1 metric, we won't detail this metric.

Let us say a few words about the limit processes W and Z. The standard Wiener (or Gaussian) process W has continuous trajectories t → W t , has independent increments, that W 0 = 0 and that W t -W s has centered Gaussian distribution with variance |t -s|. The Lévy processes Z of order α ∈]1; 2[ are not continuous, but càdlàg with independent centered increments and, for all s ≤ t, Z t -Z s has same distribution as |t -s|

1 α Z 1 .
Throughout this article, W = (t → W t ) will be a standard Wiener process.

Probabilistic limit theorems for dynamical systems

We consider a deterministic dynamics given by the iterations of a map. Thus, the evolution in time is completely determined by the perfect knowledge of the initial state of the system. We assume that this initial state is not perfectly known (e.g. we just know an approximation of it) and that it is chosen randomly. Formally we consider a probability (resp. σ-finite measure) preserving dynamical system, i.e.

• a probability space (Ω, F, µ) (resp. a measurable space (Ω, F) endowed with a σ-finite non negative measure µ), • a transformation T : Ω → Ω preserving the measure µ, this means that, for all

k ∈ N * , µ(T -k (A)) = µ(A) for any A ∈ F, i.e. Ω f • T k dµ = Ω f dµ for any f ∈ L 1 (µ).
Additionally, we consider

• a probability measure P which admits a density with respect to µ,

• a µ-integrable function f : Ω → R (called observable) satisfying Ω f dµ = 0.
We are interested in probabilistic limit theorems for ergodic sums (also called Birkhoff sums), that is for sums of the form:

S n (f ) = n-1 k=0 f • T k .
We are interested in limit theorems similar to SLLN and FCLT for (f

•T k ) k≥0 instead of (X k ) k≥1 .
• If µ is a probability measure, the ergodicity of the dynamical system (Ω, F, µ, T ) means that, for any µ-centered observable f :

Ω → R, Sn(f ) n n≥1
converges in distribution to 0 (this follows from the Birkhoff ergodic theorem). • If µ is σ-finite, the recurrence ergodicity of the dynamical system (Ω, F, µ, T ) means that, for any µ-centered observable f and any g ∈ L 1 (µ), Sn(f )

Sn(g) n≥1
converges in distribution to 0 (this follows from the Hopf ergodic theorem).

• The FCLT cannot hold for any square integrable observable of a same dynamical system (even bounded counterexamples with very various behaviours have been constructed). • Zweimüller proved in [START_REF] Zweimüller | Mixing limit theorems for ergodic transformations[END_REF] that CLT (or FCLT) for S n (f ) with respect to some probability measure P 0 absolutely continuous with respect to µ implies the CLT (or FCLT) for S n (f ) with respect to any P absolutely continuous with respect to µ.

A simple example of chaotic dynamical system

To enlight how a deterministic map can give rise to chaotic behaviour, we start by presenting a very simple dynamical system modeling independent and identically distributed random variables. The map T : [0, 1[→ [0, 1[ given by T (x) = 10x mod 1 preserves 4 the probability measure µ corresponding to the Lebesgue measure on [0, 1[:

1 0 f (T (x)) dx = 1 0 f (x) dx.
• If f 0 (x) = ⌊10x⌋ (first decimal digit), the random variables (X k := f 0 • T k , k ≥ 0) are i.i.d.
with uniform distribution on {0, ..., 9} and so the SLLN and also the following FCLT holds true:

  t → 1 √ n ⌊nt⌋-1 k=0 (f 0 • T k -4.5)   L -→ n→+∞ (t → ∥f 0 -4.5∥ L 2 W t ) .
• More generally this system is ergodic 5 , thus the SLLN holds for any integrable function:

∀f ∈ L 1 (λ), 1 n n-1 k=0 f • T k a.s. -→ n→+∞ 1 0 f (x) dx .
Moreover, for any f Hölder continuous and centered with respect to the Lebesgue measure, then the following FCLT holds true 6 : 4 This can be proved by changes of variables, since T is a C 1 -diffeomorphism from each ] k 10 ; k+1 10 [ onto ]0; 1[. 5 The Birkoff theorem ensures the almost sure convergence of

  t → 1 √ n ⌊nt⌋-1 k=0 f • T k   L -→ n→+∞ t → σ 2 (f )W t , with σ 2 (f ) := n∈Z E µ [f.f • T |n| ] = lim n→+∞ E (S n (f )/ √ n) 2 . Note that in general σ 2 (f ) ̸ = E[f 2 ] (contrarily to the i.i.d. case). The expression n∈Z E µ [f.f • T |n| ] is called
1 n n-1 k=0 f • T k n≥1 to E[f |I]
where I is the σalgebra of invariant measurable sets A, i.e. of measurable sets satisfying A = T -1 (A). To prove ergodicity, we may observe that limn→+∞ 1 n n-1 k=0 f •T k is measurable with respect to the asymptotic σ-algebra n≥0 σ(Xm, m ≥ k) and then conclude using Kolmogorov's 0-1 law. 6 Several strategies of proof exist, one can follow e.g. the strategy roughly presented in Appendix D.

the Green-Kubo formula. Moreover, the asymptotic variance σ 2 (f ) (and thus the limit process) is null if and only if f is a coboundary in L 2 (λ), meaning that there exists g ∈ L 2 (λ) such that f = g -g • T almost surely. 7 4. Limit theorems for smooth observables of chaotic billiards in a bounded domain

Let Q ⊂ R 2 or T 2 .
We study the dynamics at collision times of a point particle moving in Q, going straight inside Q and with elastic reflections with ∂Q (reflected angle=incident angle). Let Ω be the set of unit post-collisional vectors. The billiard map T : Ω → Ω maps a post-collisional vector x = (q, ⃗ v) to the post-collisional vector T (x) = (q ′ , ⃗ v ′ ) at the next collision time. This map T preserves the probability measure µ with density ρ : (q, ⃗ v) → 1 2|∂Q| sin ∠(T q ∂Q, ⃗ v) (with T q ∂Q the tangent line to ∂Q at q). We consider the following chaotic models. We refer to the book [START_REF] Chernov | Chaotic billiards[END_REF] by Chernov and Markarian for a general reference on chaotic billiards. The horizon of the billiard is said to be finite if the time between two collisions is uniformly bounded, and is said to be infinite otherwise. The a priori most simple case corresponding to one single obstacle (e.g. second picture of Figure 3) has infinite horizon and is much more complicated to study than the Sinai billiard with finite horizon (e.g. first picture of Figure 3). ( 2) The domain Q of the Bunimovich stadium billiard has C 1 boundary, is delimited by two semicircles and two segments of positive length (see the third picture of Figure 3). (3) The domain Q of a dispersive billiard with corners and cusps is delimited by a continuous, piecewise C 3 , closed curve, the singularity points of which are either corners (i.e. points with two different tangent lines) or cusps, the curvature is assumed to be positive (for the clockwise curvilinear absciss parametrization) except possibly at cusps (see the last picture of Figure 3).

The ergodicity has been proved by Sinai in 1970 [START_REF] Ya | Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards[END_REF] for the Sinai billiard (rough ideas are given in Appendix A) and by Bunimovich in [START_REF] Bunimovich | On the ergodic properties of nowhere dispersing billiards[END_REF] for the billiard in a stadium. Thus the SLLN holds true. Consider now f : Ω → R Hölder continuous and µ-centered. Different limit behaviour occur depending on the billiard domain.

7 Additional explanations If f = g -g • T , then Sn(f ) = g -g • T n and σ 2 (f ) = 0. Conversely, if σ 2 (f ) = 0, we can prove that (Sn(f )) n≥1 is bounded in L 2 and infer that f = g -g • T with g a weak limit in L 2 of 1 N N n=1 Sn(f ) n≥1 .
(1) Standard FCLT for Hölder observables of the Sinai billiard.

Whether the horizon is finite or not, the sequence of random processes t → 1

√ n ⌊nt⌋-1 k=0 f • T k n≥1 converges in distribution to σ(f )W, with again σ 2 (f ) = m∈Z E µ [f.f • T m ].
This result has been proved by Bunimovich and Sinai in 1981 [START_REF] Bunimovich | Statistical properties of Lorentz gas with periodic configuration of scatterers[END_REF] in the finite horizon case (see also [START_REF] Bunimovich | Statistical properties of two-dimensional hyperbolic billiards[END_REF] and [START_REF] Young | Statistical properties of dynamical systems with some hyperbolicity[END_REF]) and by Chernov in 1999 [START_REF] Chernov | Decay of correlations and dispersing billiards[END_REF] in the infinite horizon case. We will see in Section 6 that, contrarily to Hölder observables, the asymptotic behaviour of the Birkhoff sums of the free flight function Ψ depends on the finiteness of the horizon. (2) Non-standard CLT for Hölder observables of the Bunimovich stadium.

Bálint and Gouëzel [START_REF] Bálint | Limit theorems in the stadium billiard[END_REF] proved that

n-1 k=0 f •T k √ n log n n≥1
converges in distribution to a centered Gaussian random variables with standard deviation

σ f = c Q I f (q, ↑) dq + J f (q, ↓
) dq, where I and J are the bottom and top flat segments of the boundary of the stadium. (3a) Non-standard CLT for Hölder observables of the dispersing billiard with a single standard cusp.

In Case (3) with a single cusp at a point P with non vanishing curvature, Bálint, Chernov and Dolgopyat [START_REF] Bálint | Limit theorems for dispersing billiards with cusps[END_REF] proved also a CLT with non standard normalization and with σ f := c.

S 1 f (P, ⃗ v) |sin ∠(T P ∂Q, ⃗ v)| 1 2 d⃗ v.
(3b) Convergence to Lévy for Hölder observables of the dispersing billiard with higher order cusps.

Consider Case (3) with cusps at points P i with parametrizations z i,± (s)

≈ ±c i,± s β i , with c i,± ≥ 0, c i,+ + c i,-> 0. Assume β * := max j β j > 2, set α := β * β * -1 . Then the sequence of processes t → n -1 α ⌊nt⌋-1 k=0 f • T k n≥1 converges in distribution 8 to a Lévy process Z := i : β i =max j β j σ f,P i Z (i)
where Z (i) are i.i.d. Lévy processes of order α with

σ f,P i := c. S 1 f (P, ⃗ v) |sin ∠(T P i ∂Q, ⃗ v)| 1 α d⃗ v.
The CLT was proved by Jung and Zhang in 2018 in [START_REF] Jung | Stable laws for chaotic billiards with cusps at flat points[END_REF], the FCLT by Jung, Zhang and the author [START_REF] Jung | Convergence to α-stable Lévy motion for chaotic billiards with cusps at flat points[END_REF] and by Melbourne and Varandas [START_REF] Melbourne | Convergence to a Lévy process in the Skorohod M1 and M2 topologies for nonuniformly hyperbolic systems, including billiards with cusps[END_REF] (see also [START_REF] Jung | Necessary and sufficient condition for M2convergence to a Lévy process for billiards with cusps at flat points[END_REF]).

Heuristic explanations on these CLT or FCLT are given in Appendix B.

5.

A billiard in an unbounded domain: the Z 2 -periodic Lorentz gas

The Z 2 -periodic Lorentz gas describes the evolution of a point particle moving between a Z 2periodic configuration of convex obstacles. More precisely, it corresponds to the billiard system in the unbounded domain

Q = R 2 \ I i=1 ℓ∈Z 2 O i + ℓ.
The obstacles O i + ℓ are assumed to have pairwise disjoint closures, C 3 boundary and non null curvature. The horizon is said to be finite if every line of R 2 intersects the boundary of at least one obstacle and is said to be infinite otherwise. We will say that the horizon is 2-dimensionally infinite, if there exist at least two non parallel lines in R 2 meeting no obstacle boundary. This corresponds to a dynamical system preserving an infinite measure:

• We consider Ω for the set of all possible post-collisional vectors based on obstacles.

• As previously, we consider the transformation T : Ω → Ω which maps a post-collisional vector to the post-collisional vector corresponding to the next reflection time.

8 Complements for readers interested in questions of metrics in the Skorohod space: the convergence does not hold with respect to J1 because when a trajectory enters a cusp where e.g. f ≥ c > 0 it increases of at least c at each collision in the cusp, and long succession of collisions in the cusp occur. The convergence holds with respect to M1 if the function has constant sign around cusps and in some other situations, and with respect to M2 in any case (we refer to [START_REF] Whitt | Stochastic-process limits. An introduction to stochastic-process limits and their application to queues[END_REF] for a presentation of the J1, M1 and M2 metrics). • The map T preserves the infinite measure µ with density proportional to (q, ⃗ v) → sin ∠(T q ∂ Q, ⃗ v), normalized so that µ I i=1 ∂O i = 1, where C ℓ , called the ℓ-th cell, is the set of x = ( q, ⃗ v) ∈ Ω such that q ∈ I i=1 O i + ℓ.

This dynamical system is strongly related to the Sinai billiard:

• Let p : R 2 → T 2 = R 2 /Z 2 be the canonical projection. We consider the Sinai billiard (Ω, µ, T ) in the domain Q = T 2 \ I i=1 p(O i ). The dynamics of this Sinai billiard corresponds to the dynamics of the Lorentz gas modulo Z 2 for the position, i.e. T • P = P • T , with P the projection given by P(q, ⃗ v) = (p(q), ⃗ v) .

• Conversely, the Lorentz gas ( Ω, T , µ) can be modeled using the Sinai billiard, by identifying (x = (q, ⃗ v), ℓ) ∈ Ω × Z 2 with ( q, ⃗ v) ∈ Ω where p( q) = q and q ∈ ℓ i=1 ∂O i + ℓ. The Z 2 -periodicity ensures that, with this representation, T : ( q, ⃗ v) → ( q ′ , ⃗ v) corresponds to:

(x, ℓ) → (T (x), ℓ + Φ(x)) , for some Φ : Ω → Z 2 .
In other terms, the Lorentz gas ( Ω, T , µ) can be represented by the Z 2 -extension over the Sinai billiard (Ω, T, µ) by some Φ : Ω → Z 2 .

Periodic Lorentz gas: Limit theorem for the position

We choose the initial position and direction randomly with respect to some probability measure P absolutely continuous with respect to the Lebesgue measure. We are interested in the asymptotic behaviour of the position q n at the n-th collision. We observe that, if the initial state is (x = (q, ⃗ v), ℓ) ∈ Ω × Z 2 , then the corresponding q n = ℓ + n-1 k=0 Ψ(T k (x)), for some Ψ : Ω → R 2 called the free flight. We will see that this quantity is asymptotically Gaussian with either a standard or nonstandard normalization, depending whether the horizon is finite or infinite. More precisely the behaviour is the following: • If the horizon is finite, then Ψ is centered, bounded, piecewise 1 2 -Hölder continuous and satisfies a standard FCLT as the Hölder functions studied in Case (1) of Section 3:

t → q ⌊nt⌋ √ n n≥1
converges in distribution 9 to Σ(Ψ)W where W is a standard two-dimensional Wiener process 10 and where Σ(Ψ) is the nonnegative symmetric matrix square root of

Σ 2 (Ψ) := ( m∈Z E[ψ i .ψ j • T m ]) i,j=1,2 (Green-Kubo formula for Ψ). • If the horizon is infinite, then Ψ is not in L 2 (µ), but centered and µ(|Ψ| > t) ∼ At 2 .
Szász and Varjú proved in 2007 [START_REF] Szász | Limit laws and recurrence for the planar Lorentz process with infinite horizon[END_REF] that

qn √ n log n n≥1
converges in distribution to a Gaussian random variable with variance matric Σ 2 expressed explicitely in terms of the width and the periodical length of the corridors of parallel lines meeting no obstacle.

The same limit theorems holds true for (S n Φ) n≥1 since sup n≥1 ∥S n (Φ) -S n (Ψ)∥ ∞ < ∞. Let τ be the return time (i.e. number of collisions before coming back) to the initial cell. Let τ ε be the return time to the ε-neighbourhood of the initial state (recall a is a couple position-direction). Note that the function τ (q + ℓ, ⃗ v) = τ (q, ⃗ v) τ ε (q + ℓ, ⃗ v) = τ ε (q, ⃗ v) for any ℓ ∈ Z 2 . So we identify them with their quotient defined on the Sinai billiard (Ω, µ, T ).

• Lorentz gas with finite horizon.

-The fact that τ < ∞ a.e. (i.e. the recurrence of this Lorentz gas) follows from the standard CLT for the cell-change function Φ combined with a general argument (for Z 2 -extensions) by Conze [START_REF] Conze | On a two-dimensional recurrence criterion for stationary walks, applications. (Sur un critère de récurrence en dimension 2 pour les marches stationnaires, applications[END_REF], extended by Schmidt [START_REF] Schmidt | On joint recurrence[END_REF].

Another proof of the recurrence has been given by Szász and Varjú [24]. This proof uses Local limit Theorem (LLT) type estimates of the form µ(S n (Φ) = 0) ∼ c 0 /n (not summable) and µ(S n (Φ) = S n+m (Φ) = 0) ∼ c 2 0 /(nm) combined with a Borel-Cantelli type argument (Lamperti's lemma).

-Dolgopyat, Szász and Varjú proved in 2008 [START_REF] Dolgopyat | Recurrence properties of planar Lorentz process[END_REF]

that µ(τ > N ) ∼ N →∞ 1 c 0 log N .
-The author and Saussol proved in 2010 [START_REF] Pène | Back to balls in billiards[END_REF] that µ(τ ε > e t 4ε 2 ρ(•) )-→ ε→0 1 1+c 0 t , ensuring the convergence in distribution of (ε 2 log τ ε ) ε>0 when ε → 0 to some random variable, showing that the return times to small sets are very long.

• Lorentz gas with 2-dimensionally infinite horizon. 9 with respect to the infinite norm 10 A standard two-dimensional Wiener process is a process t → (W

(1) t , W (2)
t ) where W (1) and W (2) are two independent standard Wiener processes -The fact that this Lorentz gas is recurrent has been proved by Szász and Varjú in [START_REF] Szász | Limit laws and recurrence for the planar Lorentz process with infinite horizon[END_REF] using again LLT type estimates:

µ(S n Φ = 0) ∼ c 0 /(n log n) and µ(S n (Φ) = S n+m (Φ) = 0) ∼ c 2 0 /(n log(n)m log(m)
). The argument by Conze or Schmidt does not apply directly to this context. The question whether the CLT with nonstandard normalization for Φ implies or not the recurrence is still open.

-The author and Terhesiu proved in [START_REF] Pène | Sharp error term in local limit theorems and mixing for Lorentz gases with infinite horizon[END_REF]

that µ(τ > N ) ∼ N →∞ 1 c 0 log log N .
Note that, both in finite and infinite horizon, the above estimates of µ(τ > N ) can be rewritten:

µ (τ > N ) ∼ N →+∞ 1 N n=0 µ(S n (Φ) = 0) = 1 E µ [N N ]
,

where N N is the number of visits to (S n (Φ)) n≥0 to 0, i.e. the number of visits to 0-cell before the N -th collision considering the particle starts from the 0-cell, and

µ(S n (Φ) = 0) ∼ c 0 a 2 n ,
where a n is the normalization of S n (Φ) in the CLT: a n = √ n if the horizon is finite and a n = √ n log n if the horizon is infinite (the presence of a square above in a 2 n comes from the fact that Φ is 2-dimensional). See Appendix C for a presentation of the proof of these results.

Pinball in finite horizon

In the Lorentz gas with finite horizon, we assume that the point particle wins β O each time it hits the obstacle O. Let Z n be the amount won up to time n. This random variable Z n is a Birkhoff sum n-1 k=0 f • T k for the Lorentz gas system Ω, µ, T . We consider the following cases:

(a) Z 2 -periodic values. Assume β O j +ℓ = β O j for all ℓ ∈ Z 2 .
Then Z n can be expressed as a Birkhoff sum S n (f ) for the Sinai billiard and, applying the results of Section 4, we obtain that (Z n /n) n≥1 converges almost surely to I 0 (β) := J j=1 β O j , and that, if converges in distribution to c 0 I(β)E, with E an exponential random variable with mean 1. (b2) The author and Thomine proved in [START_REF] Pène | Potential kernel, hitting probabilities and distributional asymptotics arXiv Ergodic Theory and Dynamical Systems[END_REF][START_REF] Pène | Central limit theorems for the Z 2 -periodic Lorentz gas[END_REF] that if I(β) = 0 and if there exists

I 0 (β) = 0, then (Z n / √ n) n≥1 converges
η > 0 such that O d(0, O) η |β O | < ∞, then Zn √ log n n≥1
converges in distribution to σ 2 (β)c 0 EW 1 , with E as above independent of W 1 , where σ 2 (β) is given by the Green Kubo formula with respect to T . (c) i.i.d. values (see [START_REF] Pène | Planar Lorentz process in a random scenery[END_REF]) If the β O are i.i.d. centered and square integrable and independent of the Lorentz gas, then

Zn √ n log n n≥1
converges in distribution to c 4 W 1 , where c 4 depends on the common distribution of β O .

Ideas of proofs for (b-c):

We prove the convergence with respect to the measure µ |C 0 ( µ restricted to the 0-cell) and conclude, by a result by Zweimüller [START_REF] Zweimüller | Mixing limit theorems for ergodic transformations[END_REF], the convergence with respect to any probability measure P absolutely continuous with respect to the Lebesgue measure on Ω.

• For (b1): Since the Lorentz gas is recurrent ergodic 11 , the Hopf ergodic theorem ensures that Z n /N n a.e.

-→ n→+∞ β O , with N n := n-1 k=0 1 S k (Φ)=0 the number of visits to 0-cell before 11 Recurrence ergodicity follows from recurrence combined with the argument presented in Appendix A.

time n. So it is enough to prove the convergence result for N n instead of Z n . This can be done by proving the convergence of every moment. The moment of order m of N n is n-1 k 1 ,...,km=0 µ(S k 1 (Φ) = ... = S km (Φ) = 0), which can be estimated by using a multi-time local limit theorem.

• For (b2): we can prove the convergence of every moment of Z n with respect to µ |C 0 (this is doable but more difficult than for (b1) since cancellations happens). another argument: Note that Z n ≈ Nn k=0 Y k , Y k being the amount won between the k-th and (k + 1)-th visit to the 0-cell. Prove, via coupling, that it behaves as if the Y k and N n were independent and so, roughly speaking using the CLT for (Y k ) k and (a2) for N n :

Z n ≈ Nn k=0 Y k ≈ N n σ 2 (β)W 1 ≈ log nc 0 E σ 2 (β)W 1 .
• For (c): Z n behaves as a random walk in random scenery n-1 k=0 ζ X 1 +...+X k : we can adapt Bolthausen's proof [START_REF] Bolthausen | A central limit theorem for two-dimensional random walks in random sceneries[END_REF], with the use of the mixing local limit theorem (see Appendix D for a proof of the mixing local limit theorem which appears also in Appendix C).

Appendix A. Scheme of Sinai's proof of the ergodicity of the Sinai billiard via hyperbolicity via Hopf's chains A.1. Hyperbolicity and Hopf's chains. Recall that the states space of the billiard is the set Ω of unit post-collisional vectors x = (q, ⃗ v) with q ∈ ∂Q = I i=1 ∂O i and ⃗ v ∈ S 1 such that ⟨⃗ n q , ⃗ v⟩ ≥ 0, where ⃗ n q is the inward unit normal vector to ∂Q at q. This space is two-dimensional (one dimension for the position in ∂Q, one dimension for the direction). A main difficulty in the study of the Sinai billiard comes from the fact that the billiard map T is discontinuous x = (q, ⃗ v) such that T (x) is tangent to an obstacle. But this system enjoys hyperbolicity:

• the action of T (resp. T -1 ) expands the length of the increasing (resp. decreasing) C 1 -curve of Ω, increasing (resp. decreasing) meaning that the angle ∠(⃗ n q , ⃗ v) increases (resp. decreases) with the counter-clockwise curvilinear absciss.

Luckily, hyperbolicity wins against discontinuities. The map T is uniformly hyperbolic: there exist C > 0, Λ > 1 such that, for µ-almost every x ∈ Ω, there exist two C 1 -curves γ (s) (x) and γ (u) (x), called respectively stable and unstable manifold, containing x and satisfying

∀n ∈ N, diam(T n (γ (s) (x)) + diam(T -n (γ (u) (x)) ≤ Λ -n .
These stable and unstable manifolds do not exist everywhere, can be arbitrarily small, but satisfy the nice following property enabling the adaptation of Hopf's argument 12 to the billiard context: For any measurable set Ω 0 ⊂ Ω of full measure, there exists a set Ω ′ 0 of full measure such that, for every i = 1, ..., I, every x, y ∈ Ω ′ 0 belonging to the connected set Ω i := {(q, ⃗ v) ∈ Ω, q ∈ ∂O i }, there exists a Hopf's chain (or zig-zag line), i.e. x 0 , ..., x n ∈ Ω ′ 0 such that: Applying this with M = N (first inequality below) and M = N ⌊log N ⌋ (second inequality below) and using the decreasingness of n → µ(τ > n), we infer that

• x 0 = x, x N = y, • ∀j = 1, ..., N -1, x j ∈ Ω ′ 0 ∩ γ (s) (x j-1 ) ∩ γ (u) (x j+1 ) or x j ∈ Ω ′ 0 ∩ γ (u) (x j-1 ) ∩ γ (s) (x j+1 ),
N n=1 c 0 a 2 n µ (τ > N ) ≲ 1 ≲ N ⌊log N ⌋-N -1 n=1 c 0 a 2 n µ (τ > N ) ≈ N n=1 c 0 a 2 n µ (τ > N ) ,
where we used also the fact that a 2 n = n or a 2 n = n log n. The estimate of µ(τ ε > ...) uses the same idea (we take for A atoms of a finer and finer partition).

Appendix D. Proof of probabilistic limit theorems using operators

We present here an important tool behind most of the results stated in this article: the study of perturbation of quasi-compact operators. For a detailed and rigourous presentation of this method, we refer to [START_REF] Hennion | Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness[END_REF] and the references therein.

(1) Consider P the dual of g → g • T (P is called the transfer operator of (Ω, µ, T )):

Ω P (h).g dµ = Ω h.g • T dµ.

(2) Prove that P is quasi-compact with only and simple dominating eigenvalue 1:

P n (h) = Ω h dµ + O(e -an ) in L(B) for some nice complex Banach space B and a > 0.

If we cannot work directly with (Ω, µ, T ), we may use auxiliary dynamical systems (In [START_REF] Young | Statistical properties of dynamical systems with some hyperbolicity[END_REF] and [START_REF] Chernov | Decay of correlations and dispersing billiards[END_REF] Young towers are constructed for the Sinai billiard adapted to this purpose). (3) Set P t (h) = P (e i⟨t,Φ⟩ h) and use characteristic functions. Observe that converges in distribution to ΣW (applies e.g. to Sinai billiard with infinite horizon).

E µ 1 A e i⟨t, Sn(Φ) an ⟩ 1 B • T n = E µ [
(7) Proof of the mixing local limit theorem using point (4) above and λ n t/an ∼ e -

µ A ∩ {S n (Φ) = 0} ∩ T -n (B) = 1 (2π) 2 [-π,π[ 2 E µ 1 A e i⟨u,Sn(Φ)⟩ 1 B • T n du = 1 (2πa n ) 2 [-πan,πan[ 2 E µ 1 A e i⟨ t an ,Sn(Φ)⟩ 1 B • T n dt ≈ µ(A)µ(B) (2πa n ) 2 R 2 λ n t/an dt ≈ µ(A)µ(B) (2πa n ) 2 R 2 e -

Figure 1 .

 1 Figure 1. A trajectory of respectively a Wiener process and a Lévy process of order α = 3/2 with A -= 0 (no down jump)

Figure 2 .

 2 Figure 2. Graph of the map T : x → 10x mod 1

Figure 3 .

 3 Figure 3. Sinai billiard in the torus with finite and with infinite horizon, Bunimovich stadium billiard, billiard with corners and cusps

Figure 4 .

 4 Figure 4. Z 2 -periodic Lorentz gas with respectively finite and with 2dimensionally infinite horizon

Figure 5 .

 5 Figure 5. A trajectory of a 2-dimensional Wiener process

Figure 6 . 7 .

 67 Figure 6. Corridors for two different periodic billiard domains (from [19])

  in distribution to σ 2 (β)W 1 . (b) Summable Values: Assume O |β O | < ∞ and set I(β) := O β O . Then (b1) Dolgopyat, Szász and Varjú proved in [10] that Zn log n n≥1

Figure 7 .

 7 Figure 7. A Hopf's chain

  with lim t→0 λ t = 1 in C, with either lim t→0 ∥Π t -E µ [•]1∥ L(B) (possible if the horizon is finite) or lim t→0 ∥Π t -E µ [•]1∥ L(B,L 1 (µ)) (if the horizon is infinite) and conclude that E µ 1 A e i⟨t, Sn(Φ) If λ u ∼ u→0 e -|Σu| 2 2 | log(|u| 2 )| , then λ n

	|Σu| 2 2 2	, then λ n t/ √	n ∼ n→+∞ e -	|Σt| 2 2 2	and so Sn(Φ) √ n	n≥1	converges in distribu-
	tion to ΣW (applies e.g. to Sinai billiard with finite horizon).
	(6) t/ √	n log(n)	∼ n→+∞ e -	|Σt| 2 2 2	and so	Sn(Φ) √ n log(n) n≥1

1 B P n t/an (1 A )] .

(4) Deduce from the quasi-compactness of P , by spectral perturbation method, that

P n t = λ n t Π t (•) + O(θ n 0 ) in L(B) an ⟩ 1 B • T n ≈ λ n t/an E µ 1 B Π t/an (1 A ) ∼ λ n t/an µ(B)µ(A) .

(5) If λ u ∼ u→0 e -

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000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 000000000000000000000000 A.2. Scheme of the proof of the ergodicity.

• Let f ∈ L 1 (µ) Hölder continuous. By Birkhoff's ergodic theorem, there exists Ω 0 ⊂ Ω so that µ(Ω \ Ω 0 ) = 0 and, for every

• Prove that h is constant on stable and unstable manifolds, and T -and T -1 -invariant.

• Infer, by the above Hopf argument, that h is µ-a.e. constant on each Ω i .

• Prove that, for any couple of connected components (Ω i , Ω j ) of Ω, there exists m ∈ Z such that µ(Ω i ∩ T -m Ω j ) > 0. • Conclude that h is constant almost everywhere and so that h

This proof can be adapted to prove the recurrence ergodicity of the Lorentz gas, up to:

• replacing Birkhoff's theorem by Hopf's ergodic theorem which ensures that, since the Lorentz gas is recurrent, for every f, g ∈ L 1 ( µ), g > 0,

-→ Case (1) (Sinai billiard) is as chaotic as the example of Section 3 and the method explained in Appendix D can be implemented to prove the standard CLT. In cases (2) (Bunimovich stadium) and (3) (billiard with cusps), the non-standard behaviour comes from the following facts:

• Let A be the set of states (q, ⃗ v) with position q belonging: to I ∪ J for (2), to a neighbourhood of cusps for (3). • The dynamics outside A is in some sense as chaotic as the example of Section 3.

• The number R of collisions during an excursion in A is not L 2 and satisfies µ(R > x) ∼ cx -α for some c > 0 (with α as defined in Case (3b) or with α = 2 in Cases ( 2) and (3a)). This explains heuristically the type of CLT. • For (2), during a trajectory of length N out of semi-disks, the direction is very close to vertical and the successive positions form a O(N -1 )-packing of I ∪ J. • For (3), during a trajectory of length N in a neighbourhood of a cusp at P , the position is very close to P and the successive directions form essentially a O(N -1 )-packing of S 1 (with, on each side of the cusp, angular increments of ⃗ v of size N -1 (sin ∠(T P ∂Q, ⃗ v)) -1 α ).

The two last points explain heuristically the integral appearing in the limit.

Appendix C. Ideas behind the proofs of quantitative recurrence

The first ingredient is a mixing local limit theorem saying roughly speaking that

with a n = √ n if the horizon is finite and a n = n log(n) if the horizon is infinite.

Then, the idea consists in adapting the following argument by Dvoretzki and Erdös in 1951 [START_REF] Dvoretzky | Some problems on random walk in space[END_REF] for planar random walks (i.e. sums of i.i.d. random variables in Z 2 ): considering the last visit time n to the 0-cell before time M and applying the mixing local limit theorem, we obtain


Anaïs Guillem
email: anais.guillem@map.cnrs.fr

Antoine Gros
email: antoine.gros@map.cnrs.fr

Kevin Reby
email: kevin.reby@map.cnrs.fr

Violette Abergel
email: violette.abergel@map.cnrs.fr

Livio Deluca
email: livio.deluca@map.cnrs.fr

RCC8 for CIDOC CRM: Semantic Modeling of Mereological and Topological Spatial Relations in Notre-Dame de Paris

Keywords: Cultural Heritage (CH), Built Heritage, Semantics, spatial, CIDOC CRM, Notre-Dame de Paris, GeoSPARQL, CRMgeo, RCC8, knowledge graph, ontology modeling, interoperability, IFC, spatial annotation, space, place, topology, mereology, mereo(topo)logy, ontology, knowledge representation, spatial relations, spatial cognition

This work aims at the conceptual and ontological modeling of the abstract spatial relations in heterogeneous cultural heritage data. This work focuses on built heritage, studying the case of Notre-Dame de Paris. The spatial information is a transversal component across the metadata and paradata collection in the datasets about Notre-Dame. The integration using spatial information is crucial for archival, query, analysis, and visualization. Cultural heritage data integration implies the use of the CIDOC CRM ontology, whereas the real-life data challenge the core model because of the complexity of spatial relations. This contribution aims at the analysis of this complexity in terms of mereological and topological spatial relations. It opens an opportunity to explore the conceptualization of space and the abstract spatial relations that go beyond the geometric or the geographic aspects. The contribution presents the conceptual and ontological modeling about the abstract spatial relations using both CIDOC CRM, its extension CRMgeo, geoSPARQL, and RCC8.

Introduction

This contribution is based on the case study of Notre-Dame de Paris as an example of big data in the cultural heritage (CH) field. The characteristics of big data for CH are: real-life data, messy, highly heterogeneous, and specialized in unstructured or semi-structured datasets. The object of study in cultural heritage is typically tied with both the materiality of objects (buildings, artifacts) and their non-materiality. The case study of Notre-Dame is no exception: on one hand, it illustrates the utmost importance of the cathedral as built work, built components, as well as archaeological artifacts. After the fire, the operations of cleaning, extracting, and sorting remains and archaeological workflow of inventory, study, and analysis enlighten the porosity between archaeological methods (excavation documentation, inventory, and documentation), with operational activities. On the other hand, all the activities for the restoration or the research on Notre-Dame have in common the characteristic of being spatialized data [START_REF] Roussel | An Approach To Build a Complete Digital Report of the Notre Dame Cathedral after the Fire, Using the Aïoli Platform[END_REF][START_REF] Gros | Faceting the post-disaster built heritage reconstruction process within the digital twin framework for Notre-Dame de Paris[END_REF][START_REF] Guillem | Faire parler les claveaux effondrés de la cathédrale Notre-Dame de Paris[END_REF][START_REF] Gros | Décrire une hypothèse au sein d'un graphe de connaissances, d'une simulation mécanique à un fait historique[END_REF]. Thus, the information about location, space, and place can be understood as shared anchors across highly heterogeneous datasets and information. The most intuitive and foundational definition of architecture is the built thing, that is the architecture qua building or built work. Human beings continuously interact with the built materiality through the non-materiality of space. Space as emptiness is formed and defined by the materiality that affects its existence. That relation between fullness and emptiness is what makes possible architecture as lived and experienced space. The cathedral itself, as architecture, is by essence a spatially complex object [Figure 1]. We will build upon this definition in the rest of this article as scope of modeling about space.

In the perspective of implementing a knowledge graph using CIDOC CRM as integration ontology for Notre-Dame's data, the integration using spatial information is crucial. Information about space and place presents itself as an entry point for the indexing and structuration of datasets, their enrichment, query, analysis, and visualization [START_REF] Bandini | A hybrid logic for commonsense spatial reasoning[END_REF][START_REF] Bandini | Common-Sense Spatial Reasoning for Information Correlation in Pervasive Computing[END_REF]. It promotes consistency within the knowledge base due to the transversality of the spatial question in the documentation of the spaces of Notre-Dame, the places, the built works and built elements. This contribution does not focus on the implementation process per se, but rather on the conceptual problems that emerge from the different models and their inherent conceptualizations of space.

The Notre-Dame dataset analysis shows the need for a foundational set of relations that consistently express spatial relationships in terms of topology and mereology. This question goes beyond the location information. The question of space cannot be limited either to its geographic concept, geometric and GIS information. Plenty of spatial and geometric data is available but there is a lack of semantics about space and place. Hence, the challenge of the complexity of spatial relations and data is a multiple level problem that this paper aims to unfold step by step. To recap, we are looking at how to semantically express the complexity of the spatial relations in order to have an accurate description of the mereological and topological spatial relations between built components, spaces, and places in Notre-Dame case study [Figure 2].

Then this work builds upon this real-life data: it aims at the knowledge representation of the complex spatial relations in heterogeneous cultural built heritage data systematically expressed. The contribution presents the conceptual modeling of these topological relations using both the CIDOC CRM (with its extension CRMgeo) and the RCC8. From the scope of the CIDOC CRM, this modeling is constructed as the interface between RCC8 and CRM to allow the expression of the needed abstract topological relations. This work is thought in analogy with existing modeling: firstly, in the CIDOC CRM model, the entity crm:E55_Type acts as a bridge to SKOS where the thesauri are externally managed. Similarly, the objective is here to investigate the compatibility between CIDOC CRM and RCC8 models. Secondly, the modeling of time properties is explicitly inspired from Allen principles: we propose to apply a similar perspective for the question of spatial relations. In brief, we posit that the RCC8 can play the role of a semantic module for the topological relationships, in combination with the CIDOC CRM as domain ontology for the integration of heterogeneous CH data.

State of the Art

The conceptualization and formalization of space and spatial relations are identified as the speciality of geomatics and geography. The geoinformation community is built around the data technical workflows and implementation of geodata. The manipulation and interoperability of geodata is made possible with the standardization effort by the OGC Standards Schemas. Organized as a technical stack, this multi-layered and multi-faceted implementations encompasses conceptual modelings (ie. Geography Markup Language (GML) [START_REF] Reed | Ogc® sensor web enablement: overview and high level achhitecture[END_REF], Keyhole Markup Language (KML) [START_REF] Bacharach | Ogc approves kml as open standard[END_REF]), geometry encodings (ie. Well-Known Text, GeoJSON), services, and standard APIs. The focus on geoinformation does not fit our scope completely because space is conceptualized as a geographic concept based mostly on 2D representations, geometry and GIS technology [START_REF] Atkinson | Roadmap for Interoperable 3D Data Models in OGC APIs and other Data Exchange Approaches, The International Archives of the Photogrammetry[END_REF].

The OGC geoSPARQL model serves as an interface to the semantic web for the geoinformation. The limits in scope of OGC Standard are acknowledged by geoSPARQL as follows: "GeoSPARQL does not define a comprehensive vocabulary for representing spatial information. Instead GeoSPARQL defines a core set of classes, properties and datatypes that can be used to construct query patterns. Many useful extensions to this vocabulary are possible, and we intend for the Semantic Web and Geospatial communities to develop additional vocabularies for describing spatial information" [START_REF]OGC GeoSPARQL -A Geographic Query Language for RDF Data[END_REF]. GeoSPARQL is designed as an open model, allowing communities to specify the model to their usage through the addition of vocabularies. In addition, this design allows an hybridization of the model, whether by the making of an extension or ontology merging. Nevertheless, the geoSPARQL model is still implicitly bound by the technical implementation of the 2D space information. Subsequently it bears the same geometric and geographic representation of the concepts of space.

In the scope of cultural heritage data integration, the CIDOC CRM is a go-to model as a starting point [START_REF] Bruseker | Cultural heritage data management: The role of formal ontology and cidoc crm, Heritage and Archaeology in the Digital Age: Acquisition, Curation, and Dissemination of Spatial Cultural Heritage Data[END_REF]. It is characterized by the central role of the temporal entities and its event-oriented modeling. The materiality of architectural objects or built works falls under the scope of E18 Physical thing and subclasses, while spaces are rather characterized as instances of E53 Place. The spatial relations are synthetically described in the introduction of the model [Figure 3]. The base model is expanded by different extensions that take into account aspects of spatial modeling. CRMgeo [START_REF] Hiebel | CRMgeo: A spatiotemporal extension of CIDOC-CRM[END_REF][START_REF] Migliorini | Enhancing CIDOC-CRM Models for GeoSPARQL Processing with MapReduce[END_REF] is to bridge geoSPARQL to CIDOC CRM. It discriminates between phenomenal and declarative places classes that help define the relation between space and geometries. To express spatial relations in CRM, CRMgeo depends on both the CRM spatial relations and the geoSPARQL interface. Still in the CRM family of models, CRMba [START_REF] Ronzino | Documenting the Structure and Adaptive Reuse of Roman Amphitheatres through the CIDOC CRMba Model[END_REF] -b.a. stands for building archaeology-considers the building as a stratigraphic object from the perspective of an archaeologist. In stratigraphic analysis, spaces are layered as stratigraphic units. The formalization of the stratigraphic units of a built works are sketched through mereological relations and an undefined topological relation.

In the Architecture, Engineering and construction (AEC) industry, the standard Industry Foundation Classes (IFC) defined for Building Information Modelling (BIM) have as powerful a bias as stratigraphic analysis. The partitioning of spaces is also done from an operational point of view: the site, building, storey, spaces and elements are differentiated in a mereological fashion [START_REF] Mcglinn | Interlinking geospatial and building geometry with existing and developing standards on the web[END_REF]. In the context of semantic web, this model is accessed via ifcOWL [START_REF] Beetz | Ifcowl: A case of transforming express schemas into ontologies[END_REF] or replicated in the Building Ontology Topology (BOT) [START_REF] Rasmussen | Recent Changes in the Building Topology Ontology[END_REF]. We can observe here a modelization pattern similar to the way in which a technical object, such as a STEP model, is partitioned. The main difference is that parts of technical objects are interrelated by (mechanical/physical) relations, defining the interaction, while IFC parts do not refine the interface.

Since the CIDOC CRM is a domain ontology that has been developed bottom-up, the modeling reflects what is the most commonly documented in specific CH fields. We showed the need for a more generic representation of space or spatial relations. In this direction, the work of [START_REF] Danash | On the Analysis of FORT; arguments, alignment to FOs, and CLIF validation[END_REF][START_REF] Danash | A Parthood Approach for the conceptual modelling of Tangible Objects Composition (TOC)-an application on Cultural Heritage (CH)[END_REF] investigates a modeling of foundational relations (FORT) in relation to the most known foundational ontologies (BFO, DOLCE, UFO). They point out the foundational relational aspect that is key in spatial relations: Entity-Location, Location, Connection, Parthood, Dependence, Constitution, Membership, Unity. We identified the need for a similar level of genericity in relations as in [START_REF] Danash | On the Analysis of FORT; arguments, alignment to FOs, and CLIF validation[END_REF][START_REF] Danash | A Parthood Approach for the conceptual modelling of Tangible Objects Composition (TOC)-an application on Cultural Heritage (CH)[END_REF] but with the specificity of application domain in cultural heritage, that is the scope of the CIDOC CRM ontology.

IFC, CRMgeo, CRMba, GIS related models are known models, that means they are operational and used by specific communities. They carry their own bias in the definition of space and put their focus on spatial relations as geometry management. The mereological aspect is systematically taken into account as a hierarchy that can be represented as a tree-like structure. The topological relations are developed according to operational needs and context, they are more prone to the modelisation bias. An abstract way to represent them is a graph-like structure. In built works, entities considered by these relations are heterogeneous (space, built work, elements. . . ), defined by both a geometry or abstract from it. In the next part, we will then look at the method to express mereological and topological relations between heterogeneous elements composing space. It will build on the alignment of RCC8 relations with the CRM model for phenomenal places.

Methodology

The objective of this contribution is to model spatial semantics where space is not just understood as geographic or geometric information as in [START_REF] Casati | Parts and Places: The Structures of Spatial Representation[END_REF].

We can reason about the relative positions of the entities that make the accounted spaces up. There are three main categories of spatial relations [START_REF] Egenhofer | A formal definition of binary topological relationships[END_REF]: metric relations, topological relations and order relations. In this work, we are interested in topological relations. Topology can be defined as the set of perceived relations that enable us to situate objects in relation to one another. To be more specific, topology is the study of properties of spaces that are invariant under any continuous deformation. Thus, topological relationships are the subject of an abundant scientific literature using a wide range of sound mathematical models [START_REF] Chen | A survey of qualitative spatial representations[END_REF]. The dominant models are: the 9-IM model (9-Intersection Model), Egenhofer, and the RCC model (Region Connection Calculus). Basically, these models distinguish several fundamental topological relationships identified by the Region Connection Calculus (RCC) between spatial entities [START_REF] Randell | Modelling topological and metrical properties in physical processes[END_REF][START_REF] Randell | A spatial logic based on regions and connection[END_REF]. The main difference between them lies in the dimension of the handled entities. The characterisation of spaces in buildings consists in identifying 3D regions, whether empty or filled, often presented in orthogonal projections. The first region contains the second region and their boundaries do not touch

The RCC8 formalism defines eight elementary relationships [Figure 5] to describe spatial relations between entities whose primitives are regions [Table 1]. [START_REF] Kontchakov | Spatial reasoning with RCC8 and connectedness constraints in Euclidean spaces[END_REF] state that RCC8 formalism is dimension independent, applicable in R 𝑛 , and then demonstrate each of the axioms and subsequent theorems: "The language RCC8 is a widely-studied formalism for describing topological arrangements of spatial regions. The variables of this language range over the collection of non-empty, regular closed sets of n-dimensional Euclidean space, here denoted (𝑅𝐶 + R 𝑛 ), and its non-logical primitives allow us to specify how the interiors, exteriors and boundaries of these sets intersect" [START_REF] Kontchakov | Spatial reasoning with RCC8 and connectedness constraints in Euclidean spaces[END_REF][START_REF] Mark | Modeling spatial relations between lines and regions: Combining formal mathematical models and human subjects testing[END_REF]. However, the RCC system does not distinguish between open and closed geometries. Conceptually, human thought is capable of manipulating abstract notions of openness, such as the interior of a room, a building, etc. [START_REF] Larvor | Notions de méréogéométrie: description qualitative de propriétés géométriques du mouvement et de la forme d'objets tridimensionnels[END_REF]. Moreover, Dia Miron points out that inference procedures based on this formalization are not the most efficient, and reasoning can sometimes turn out to be incomplete or undecidable [START_REF] Miron | Découverte d'associations sémantiques pour le Web Sémantique Géospatialle framework ONTOAST[END_REF] [START_REF] Vandecasteele | Modélisation ontologique des connaissances expertes pour l'analyse de comportements à risque: application à la surveillance maritime[END_REF].

As explained by [START_REF] Nys | Spatio-temporal reasoning in CIDOC CRM: An hybrid ontology with GeoSPARQL and OWL-Time[END_REF], "Besides CIDOC CRM spatial classes (E53 Place, E44 Dimension, E47 Spatial coordinates and E94 Space Primitives), the model offers properties which fulfill most common topological spatial relations (Dimensionally Extended nine-Intersection Model (DE-9IM), Region Connection Calculus (RCC8))[...]. Finally, CIDOC CRM defines class E92 Space Time Volume that designates four dimensional point sets and has temporal (CIDOC:P160) and spatial (CIDOC:P161) projections. Besides this, the CRMgeo extension provides spatial and temporal classes and properties dedicated to formulate declarative information. It also provides links with GeoSPARQL. Indeed, these links with the OGC GeoSPARQL standard are necessary to make use of the conceptualization and formal definitions that have been developed in the Geoinformation community" [START_REF] Nys | Spatio-temporal reasoning in CIDOC CRM: An hybrid ontology with GeoSPARQL and OWL-Time[END_REF]. Building upon the same observation, our approach bears some major differences: this contribution looks only at spatial relations (instead of spatio-temporal) for a rather diverse community of architects, archaeologists, conservators,

Table 2

Analysis of the CIDOC CRM properties against the RCC8 relations: we check whether the CIDOC CRM properties express some topological relationships. We specify if the topology is rather defined in (a) the scope note, (b) the examples or if it is (c) unspecified.

Domain Property label

Range DC EC PO EQ TPP TPPi nTPP nTPPi

E18 P53 has current or former location E53 x(a) x(b) x(b) E18 P59 has section E53 x x E18 P156 occupies E53 x(a) E18 P157i provides reference space for E53 x x x x x x x x E19 P55 has current location E53 x(a) x(b) x(b) E53 P53i is former or current location of E18 x(a) x(b) x(b) E53 P55i currently holds E19 x(a) x(b) x(b) E53 P59i is located on or within E18 x x E53 P89 falls within E53 x x E53 P89i contains E53 x x E53 P121 overlaps with E53 x x(c) x(c) x(c) x(c) x(c) E53 P122 borders with E53 x x x E53 P156i is occupied by E18 x x x E53 P157 is at rest relative to E18 x x x x x x x x E53 P168 place is defined by E94 x x(a) x(a) E53 P171 at some place within E94 x x x E53 P172 contains E94 x x x E53 P189 approximates E53 x x x x x x x x E92 P10 falls within E92 x x x E92 P10i contains E92 x x x E92
P132 spatiotemporally overlaps with E92

P133 spatiotemporally separated from E92 x E94 P168i defines place E53 x x(a) x etc. and not for the geoinformation community specifically. As shown in the state of the art, the conceptualization of space and spatial objects differs in terms of scope of application. Similarly as the modeling about time based on Allen's principles in CRM or FORT model [START_REF] Danash | On the Analysis of FORT; arguments, alignment to FOs, and CLIF validation[END_REF][START_REF] Danash | A Parthood Approach for the conceptual modelling of Tangible Objects Composition (TOC)-an application on Cultural Heritage (CH)[END_REF], we propose to use high level relations to express systematically and consistently the spatial relations between heterogeneous entities that compose space. For that purpose, we choose to analyze the compatibility and the possible alignment between the RCC8 and the CIDOC CRM base model.

Results

The results of this work are twofold: first, the analysis of the CIDOC CRM properties in regard with the RCC8 topological model. Second, the consistency of the modeling is validated against a sample of Notre-Dame de Paris' data. We showed that RCC8 relations are compatible with the heterogeneity of the spatial entities in cultural heritage built works. We present a survey and analysis of the direct properties of the CIDOC CRM model in regards to the RCC8 model to highlight the expressivity of the CRM base model [Table 2]. This survey is grouped by the domains and ranges of the properties: it 6] and [ Table 2] shows that spatial properties link few dedicated classes in the CRM model: E18 Physical Thing, E53 Place, E94 Space Primitive and E92 SpaceTime Volume classes. Only E92 and E53 disposes of self-referential properties, E53 is both linked with E18 and E94, E92 is isolated from the other ones. The resulting 4 classes reflect the heterogeneity of elements that we posited in our initial definition of space.

The scope of CIDOC CRM is sufficient in most cases [Figure 3]. More comprehensive cases can arise with cultural heritage built works: we present a sample from Notre-Dame de Paris cathedral data that illustrates this spatial complexity that comes from both the array of spatial entities and their mereo-topological relations. As a representative sample, it features a collatéral (fr) / side-aisle (en) [CI28] as an empty space 3-dimensional region, two files (fr) / axes (en) [F28, F30] as abstract 2-dimensional regions, one travée (fr.) / span (en) [T28] and two niveaux (fr) / storeys (en) as abstract 3-dimensional region, building elements pile intérieure (fr) / interior column (en) [PI28] and its chapiteau (fr) / capital (en) [PIc28] physical as 3-dimensional regions [Figure 6]. This data sample shows the array of spatial relations considered: mereological and topological between place-place, place/object and object/object [Figure 7].

This example is a proof of concept for consistent modeling of abstract spatial relations about built work entities. The aforementioned analysis prevents us from propagating the initial heterogeneity of CRM spatial entities and properties to the application profile. The CRMgeo extension provides an in-between for CRM and geoSPARQL that allows us to reach the RCC8 model in geoSPARQL. The sample data and model is available at: https://gitlab.huma-num.fr/gtcidoc-crm/architecture-and-built-works-abcrm

Conclusion

This contribution was initiated from the case study of Notre-Dame de Paris: the information about space is crucial in the data integration. While in CH datasets, the information about space is mostly homogeneous, the Notre-Dame's spatial data range from microscopic (ie. sample location) to the scale of an object (ie. a built element), a portion of space, or part of the cathedral. The scale of the considered spatial objects depends on the type of research question, method, analysis that are relevant for the dataset. Thus, this range in scale can be seen as a different level of detail in spatial indexing. This led us to go beyond the expression of space as its geometrical representation. We explored specifically abstract spatial relations as a transversal component for archeological, restoration, and analysis data. The proposed modeling with RCC8 and CIDOC CRM is checked against a sample of data representing mereo-topological relations between architectural spaces and built components for a span with collateral and sexpartite vault of the nave in Notre-Dame de Paris. This subset dataset is used as a proof-of-concept that illustrates the conceptual modeling as hybridization/composition between models. The exploration is carried out using an ontological analysis of CIDOC CRM, CRMgeo, geoSPARQL (OGC standard) focusing on the semantics about space, but not about its geometry. The mereological and topological relations in the spaces of a built work, as expressed in 2D (plan), but also as volumes in 3 dimensions and nomenclature.

This article proposed a conceptualization of space from an anthropological perspective of the lived space. Architecture and space are considered as an experienced built environment and thus a primordial substrate of material culture. From an operational viewpoint, the modeling of space as a transversal component enables further operational investigation: interlinking the dense spatialised information as a network, the organization of the perceived space, description of engineering system boundaries (ie. thermics, mechanical analysis) and the alignment of expert systems. The application of this modeling showed its usefulness to the spaces of the cathedral of Notre-Dame but can be transferred to any built environment and architecture.

Figure 1 :

 1 Figure 1: Plan of the ground floor of the cathedral with the nomenclature entities as defined by the architects of the restoration project (@RNDP).

Figure 2 :

 2 Figure 2: (a) A photograph taken from the collateral (@Komenda/C2RMF), (b) extract of the nomenclature (@RNDP), (c) section on the interior (@E. Viollet-le-Duc).

Figure 3 :

 3 Figure 3: Basic CIDOC CRM Properties and classes about spatial relations in CIDOC CRM 7.1.2 [11].

Figure 4 :

 4 Figure 4: Overview of CRMgeo and GeoSPARQL [13].

Figure 5 :

 5 Figure 5: Schema representing the RCC8 [26] modified (translated).

Figure 6 :

 6 Figure 6: Example of spaces mereotopologically related within the Notre-Dame de Paris cathedral showing the collatéral intérieur 28 [CI28] / interior side of the aisle 28. It is a portion of space in relation with: files (fr)/ axes (en) [F28] and [F30], travée (fr)/ span (en) [T28], and niveaux (fr)/ storeys (en) [RDC].

Figure 7 :

 7 Figure 7: Array of spatial relations expressed using the geoSPARQL RCC8 model between heterogeneous spatial entities with the example presented in [Figure6] and [Table 2]

Table 1

 1 The eight spatial relations in RCC8 with labels and definitions.

	Symbol	Name	Note
	DC	Disconnected	The two regions are completlydisconnected (there are no
			common pieces)
	EC	Externally Connected	The boundaries of the regions touch, but their interiors
			are disjoint
	PO	Partially Overlapping	The two regions partially overlap (there are disjoint sub
			parts, and some part of one is included in some part of
			the other)
	EQ	EQual	Two regions have the same spatial extent
	TPP	Tangential Proper Part	The first region is entirely inside the second region and
			their boundaries touch each other from the inside
	TPPi	Tangential Proper Part In-	The first region contains the second region and their
		verse	boundaries touch each other from the inside
	NTPP	Non-Tangential Proper Part The first region is entirely inside the second region and
			their boundaries do not touche
	NTPPi	Non-Tangential Proper Part
		Inverse

Acknowledgments

This work was supported by: the projects REPERAGE and E-RIHS funded by the Fondation des Sciences du Patrimoine (France), the projects ASTRAGALE and TEATIME funded by the Mission pour les Initiatives Transverses et Interdisciplinaires of the Centre National de la Recherche Scientifique (France), and the ERC Ndame Heritage funded by ERC advanced Grant 2021. The authors wish to acknowledge the help and collaborative support from: the chief architects of historical monuments in charge, Philippe Villeneuve, Pascal Prunet and Rémi Fromont, the Établissement public chargé de la conservation et de la restauration de la cathédrale Notre-Dame de Paris (RNDP), and the heritage conservators. The authors thank the numerous scientific partners and collaborators from the research groups working on Notre-Dame de Paris cathedral, with special recognition for the Digital Data working group.


Anaïs Guillem
email: anais.guillem@map.cnrs.fr

Antoine Gros
email: antoine.gros@map.cnrs.fr

Violette Abergel
email: violette.abergel@map.cnrs.fr

Livio De Luca
email: livio.deluca@map.cnrs.fr

Livio De

Reconstruction beyond Representation in Notre-Dame de Paris

est destinée au dépôt

I. CONTEXT

There is a saying, a picture is worth a thousand words, meaning that the images have the power of conveying realistic, spatial, social information implicitly. Despite their aesthetic expressivity, hyper-realistic images of reconstructed pasts tend to withhold the scientific work of reconstruction. While the study of reconstruction is based on archaeological evidence and argumentation, the produced visualizations fail to provide scholarly reusable documentation [START_REF] Manuel | The Principles of the Seville Charter[END_REF][START_REF] Pletinckx | An EPOCH Common Infrastructure Tool for Interpretation Management[END_REF]. This contribution builds on a previous reconstruction work of the authors on the archaeological reconstruction of the collapsed arch of Notre-Dame de Paris' cathedral [START_REF] Gros | Faceting the Post-Disaster Built Heritage Reconstruction Process within the Digital Twin Framework for Notre-Dame de Paris[END_REF] after the spire collapsed on the nave, sweeping the vaults away at two places during the fire on April 15th 2019. The object of this article is to look specifically at the role of scientific visualizations in the reconstruction argumentation and hypotheses.

II. PROBLEM

A common pitfall of 3D visualization democratization is that a 3D model of a reconstruction is too often reduced to an illustration. Traditionally drawings or engravings used to accompany the reconstruction publications as visual aid and illustration of the argumentation in the text. With the accessibility of 3D modeling tools and the development of virtual archaeology, the representation of reconstruction is dissociated from the reconstruction hypothesis formulations. The image of reconstruction is decoupled from the scientific research that produces it. The interpretive and analytical informative elements present in a scientific image are sacrificed for the sake of cultural consumption for their aesthetic quality. Looking at different visualization outputs in relation to the reconstruction project, we demonstrate the two-edged sword of visualization in reconstruction research. What makes a visualization a good scientific visualization of a reconstruction? What are the potential and limits of such a representation? How to overcome or mitigate these limits?

First, we porpose a method for a visual semiotic analysis for reconstruction images. Second, we will apply this semiotic framework to reconstruction images from the case-study of Notre-Dame de Paris to show the dissociation reconstruction-representation. The semiotic analysis of reconstruction images from the authors' previous work shows that no matter the medium used for the visualization, representation and reconstruction are problematic. Third, from a theoretical perspective, the double problem of reconstruction and representation is unraveled looking at the compressions in the reconstruction discourse and representation. They are both based on compressions of meaning and discourse. Finally, we argue that the linking of representation and reconstruction is possible through consistent documentation practice. We will show that it is possible to enrich the metadata and paradata of the reconstruction following the method in [START_REF] Bruseker | Semantically Documenting Virtual Reconstruction: Building a Path to Knowledge Provenance[END_REF][START_REF] Guillem | Building an Argumentation Platform for 3D Reconstruction Using CIDOC-CRM and Drupal[END_REF]. The argumentation developed in the reconstruction can be linked to its representation using Linked Open Data and mapping with CIDOC CRM and CIDOC CRMinf. III. METHOD This part will define the analysis method we will apply to analyze reconstruction representations. We borrow from the theoretical framework of semiotics [barthesChambreClaireNote1980]. We reclaim the distinction between (1) aesthetical analytical representation and (2) technical analytical representation that Fritz and Tosello use to describe survey as process [START_REF] Fritz | The Hidden Meaning of Forms: Methods of Recording Paleolithic Parietal Art[END_REF][START_REF] Fuentes | Transferts méthodologiques 3D appliqués à l'étude de l'art paléolithique : une nouvelle dimension pour les relevés d'art préhistorique[END_REF]. We choose to adapt it for the analysis of reconstruction visualizations. The former (1) uses the morphology and geometry of the reconstructed object as a starting point. In this case, the illustration becomes an end in itself: the graphical expression conveys the interpretation. In the latter (2), the illustration is built as an open medium for discussion and interpretation. While the first is an accurate witness of the interpretation about a reconstructed past reality, the second is rooted in an accurate record of reality (ie. measurement). The two types of representations present different sets of qualities: aesthetic, technical and analytical. We propose to synthesize and reformulate Fritz and Tosello's approach in [Figure 1] as an abstract triangle representing the 3 opposed qualities.

This simple framework allows to highlight qualitative key points for the assessment of reconstruction representation:

• the role of information interpretation and curation in image production; • to expose the image production bias; • the volume and the diversity of information carried by the image regarding the original data. Conversely, it allows us to weigh the risk of information overload and evaluates how the interpretative layer filters the noise it builds up in the original data; • the fascination for the visual and aesthetic value of reconstruction.

IV. RECONSTRUCTION IMAGES ANALYSIS In this part, the analysis through practical examples [Figure 2, 3, 4 and 5] will show different roles that visual-Fig. 2. The hybrid hypothesis on the online 3D viewer [START_REF] Abergel | Aïoli -Public Viewer[END_REF]. The arch and the cathedral are represented by point clouds or meshes that evoke placement of voussoirs in the reconstruction. This image is technical and aesthetic.

izations assume in the reconstruction study. We purposefully choose to look at visualizations separated from the reconstruction data. The semiotic analysis of these decontextualized visualizations will demonstrate how the visual representation bears specific aspects of the reconstruction process and reasoning. Let us consider 5 different images about the reconstruction of the collapsed arch of Notre-Dame de Paris [START_REF] Gros | Faceting the Post-Disaster Built Heritage Reconstruction Process within the Digital Twin Framework for Notre-Dame de Paris[END_REF] through the lens of the semiotic framework described in the previous part. The objective is to analyze qualitatively these images according to the aesthetic, analytical and technical criterias [Figure 1] presented in part iii..

In the case of 3D photorealistic images, we can argue that the decontextualization of the signifier and its separation from the signified is the reason that visualization tends to supersede reconstruction. Using Fritz et Tosello's terminology, the visualization can be characterized as both aesthetical and technical but it has lost its analytical end. The visualization becomes the object of understanding before what it represents. In contrast, the use of the schema like in figure 4

V. THE DOUBLE PROBLEM OF RECONSTRUCTION AND REPRESENTATION

The problem around the question of reconstruction has been clearly identified and stated in [START_REF] Clark | The Fallacy of Reconstruction[END_REF]: "reconstruction" is problematic as a term itself and as what it means in the practice of archaeology. Although the problem can be intuitively appreciated, it has rarely been detailed, and even fewer propositions and tangible research have ensued. Why is this problem still open? The aspect of argumentation in the reconstruction exercise has proved all the more crucial [START_REF] Denard | A new introduction to the London Charter[END_REF] with the evolution of 3D modeling tools and virtual archaeology. The question of argumentation or documentation of the reconstruction did not follow the same pace as the increasing interest for virtual archaeology. Then, the literature in archaeology related fields iden- tifies the problem, but it is necessary here to make a little digression in philosophy to dig deeper. Building on the chapter "the construction of the unreal" [START_REF] Fauconnier | The way we think: Conceptual blending and the mind's hidden complexities[END_REF], the exercise of reconstruction can be defined as the shift in the object of study from the "actual" (ie. archaeological remains) to the "unreal" and "counterfactual conditionals" reasoning (reconstruction hypothesis of what a past might have been). The methodology about reconstruction sends us back to a fundamental open philosophical question that Goodman formulated as follows:

"The analysis of counterfactual conditionals is no fussy little grammatical exercise. Indeed, if we lack the means for interpreting counterfactual conditionals, we can hardly claim to have any adequate philosophy of science" [START_REF] Goodman | The Problem of Counterfactual Conditionals[END_REF].

If we identified the root of the problem, how does it apply in the practice of reconstruction? Fauconnier and Turner give some insights: "But as Goodman first recognized, changing any one element opens up complicated questions of what else would need to be changed in order for that one to differ. Counterfactual scenarios are assembled mentally not by taking full representations of the world and making discrete, finite, known changes to deliver full possible worlds but, instead, by conceptual integration, which can compose schematic blends that suit the conceptual purpose at hand" [START_REF] Fauconnier | The way we think: Conceptual blending and the mind's hidden complexities[END_REF].

Hence, reconstruction can be defined as examples of intricate counterfactual blends that are built on compressions of space and time. Counterfactual is defined by Fauconnier and Turner as when "one space has forced incompatibility with respect to another" [START_REF] Fauconnier | The way we think: Conceptual blending and the mind's hidden complexities[END_REF].

The reconstructions in archaeology are then tied both in the "actual" of archaeological data and "conditional scenarios" as a compression between fact and hypothesis in the discourse about the past. If the reconstruction is a hypothetical compression, the reconstruction images showed that there is a second level of compression in the act of representation itself as well. The analysis of reconstruction representations in part iv. led us to consider the different qualities of the image in their ability to convey some analytical, aesthetic or technical information. We will explore here 3 possible aspects of the representation compression.

First, to understand the compression happening because of the representation, the work of Barthes' on the analysis of photography in [START_REF] Barthes | La Chambre claire : Note sur la photographie[END_REF] shows the "duplicity of this medium, which is able to function in both registers of memory: the first being the capability to suspend and embody the moment in the time and space of the photograph and the second being the capability to bring a dead past to life" [START_REF] Guillem | Global Heritage, Knowledge Provenance, and Digital Preservation: Defining a Critical Approach[END_REF]. This duplicity in photography explains why the representation gives almost life to what it represents. The representation becomes the condition of existence of the represented. The representation can be understood as a compression that allows something unreal (it is only represented) to become a new incarnated existence through the representation.

Second, the representation compression goes beyond the representation process and the creation of images. The act of reading is another compression aspect. In the decoding of the visual information, the interpretation of the meaning is left to the person looking at the image. The reading of an image is not univocal [START_REF] Minguet | L'isotopie de l'image[END_REF]. The compression of reading happens between the reader and the image in the appropriation and interpretation of the image.

Third, this aspect is a common compression for both the reconstruction and the representation. The double meaning of representation is useful to understand this aspect: "Representation in the humanities rests on a denotative philosophy of language according to which names represent things and maps represent territories. There is a second meaning of representation, more common in the social sciences, used to refer to persons speaking or being in place of others" [START_REF] Mignolo | The darker side of the Renaissance: Literacy, territoriality, and colonization[END_REF].

The reconstruction aims at building knowledge about a lost past, but we do so from our present time, our own cultures, societies, communities, and bias. Doing reconstruction, one speaks for people and communities who are long dead. It informs about the us/here/now more than the "other"/there/past [START_REF] Said | Orientalism: Western concepts of the Orient[END_REF]. The decolonial studies were precursors for these questions and critical postcolonial think- ing and theory can be applied to reconstruction works.

We showed here that the problem of reconstruction and of representation operate like a double layer of compression: the one due to the layers of representation and the ones of the reconstruction itself. The next part will tackle some proposition to unblend what is of representation and what is of reconstruction and how it can be documented. The counterfactual or the analytical elements of images are part of the argumentation and need to be decompressed and documented as such.

VI. ARGUMENTATION DOCUMENTATION AND

REASONING Secondary questions arise at this point: first, do the visualizations have this ability to convey explicitly the reconstruction study process? Second, what is this process made of?

To push the boundaries of reconstruction study and discipline bias, one needs to understand how scientific visualization is defined in so-called hard sciences [START_REF] Johnson | Top scientific visualization research problems[END_REF]. The challenge related to the quality of information can be formulated as the need to make visible something that is currently invisible in the amount of data. In the case of reconstruction, the expressivity of uncertainty and degree of fuzziness applied to the data is what makes the quality of the argumentation deployed in the reconstruction process, and it is expected in the resulting visualization. The granularity of data becomes a top-notch challenge: researchers differentiate what data is considered as certain, from what is questionable or what is an object of conflict of interpretations [START_REF] Ricoeur | Le conflit des interprétations. Essais d'herméneutique[END_REF].

The conflicting interpretation of the same element is rather a common issue in the study of the past. This entanglement of adverse interpretations is a central part of the research in reconstruction: it constitutes its scientific, ethical, and epistemological dimensions. The intellectual probity of the archaeologist or historian is at play. The entanglement of interpretations intertwines argumentative Fig. 6. Workflow diagram in the reconstruction reasoning and documentation [START_REF] Guillem | Building an Argumentation Platform for 3D Reconstruction Using CIDOC-CRM and Drupal[END_REF].

threads that come from the same observations. Then the interpretations and analyses differ thanks to methodological reasons or to equal probability of opposite solutions [Figure 6].

The presentation demonstrates through the case of Notre-Dame's arch how the reconstruction argumentation can be tied together with the visualizations and the research process [Figure 6]. The goal is to document the reasoning and hypothesis in the process of reconstruction as in [START_REF] Bruseker | Semantically Documenting Virtual Reconstruction: Building a Path to Knowledge Provenance[END_REF][START_REF] Guillem | Building an Argumentation Platform for 3D Reconstruction Using CIDOC-CRM and Drupal[END_REF]. Traditionally a text accompanies the visualization to support the reconstruction argumentation, but in our case, the documentation of the argumentation targets the metadata enrichment, mapping with CIDOC CRM, and transformation in LOD [START_REF] Guillem | Faire parler les claveaux effondres de la cathedrale Notre-Dame de Paris[END_REF]. The methodology of documentation is complementary to the approach developed by [START_REF] Demetrescu | From Field Archaeology to Virtual Reconstruction: A Five Steps Method Using the Extended Matrix[END_REF][START_REF] Schmidle | MS37 Unified CIDOC CRM mapping of the Extended Matrix and the LOD scenario[END_REF]. The main difference is that the focus of our modeling is on the argumentation patterns and the conflict of interpretations rather than the geometry reconstruction. The reconstructions in archaeology are then linked at the data level both with the archaeological data and reconstruction hypothesis using CRMinf for the mapping of the arguments. The semantic nodes are mapped in order to make explicit the links between the data of reconstruction, the visualizations produced and the documentation of the process. Our contribution aims at leveraging documentation of the conflicts of interpretations in relation to the visualization and the reconstruction argument and data.

VII. DISCUSSION

In the previous part, we showed research directions towards the argumentation documentation of reconstruction to overcome the entanglement of representation. Make explicit both the geometry creation and the reasoning process about reconstruction. Working on paper or with 3D modeling software does not change the underlying problem. Once again Fauconnier and Turner are helpful in this discussion. By asking about modus tollens, it applies nicely to our problem of argumentation about reconstruction: "Question: Isn't all this just the simple logical operation of modus tollens, in which, given that p implies q and given that not-q, then it follows that non-p? [Fauconnier and Turner's] Answer: It is true that if an assumption p implies a contradiction q [...], it follows that non-q is true tautologically and therefore non-p is true, by modus tollens. Clearly, in that sense, the logical law of modus tollens captures something true and important. But modus tollens has nothing to say about how q could possibly be found. This is where the counterfactual blend is crucial, and it takes considerable running of the blend before q pops out.

[...] The long work of the mathematician or logician who finally finds the contradiction q can, in retrospect, be compressed into a modus tollens statement, but the logical law of modus tollens is not in itself a way to discover contradictions. In fact, even more simply, inference, as in 'p implies q' is after-the-fact shorthand for what may be a long, difficult, imaginative cognitive process. Constructing the contradiction is a cognitive achievement. [...] The modus tollens formula is only a superficial report that such a process has taken place (somehow) and delivered a certain result" [START_REF] Fauconnier | The way we think: Conceptual blending and the mind's hidden complexities[END_REF].

The documentation of the proposition sets, arguments and inference logic that were used in the argumentation of the reconstruction is, as Fauconnier and Turner put it, "a long, difficult, imaginative cognitive process". This effort of formalizing these arguments using an ontological model like CIDOC CRMinf opens the possibility of reasoning and querying the reconstruction arguments. VIII. CONCLUSION Paradigms of representation are inherited from Renaissance knowledge and scholarship. The technological shift (2D to 3D) perpetuates the way one reads, understands and produces representations. It ensues the continuing challenges of linking argumentation, reasoning and visualization in the making. The issue is rather not the medium as signifier (3D or not 3D) but the documentation of scientific context and reasoning (ie. signified) in which the visualization is produced. We need to be able to operate a reduction on the overwhelming aesthetic information present in 3D realistic images to be able to let appear the relevant argument aspects of the reconstruction. Doing so, the visualization moves away from a static and finalized illustrative output of the reconstruction study and toward an open dynamic visualization of reconstruction data.

Fig. 1 .

 1 Fig.1. Triangle with criteria for visualization analysis based on Fritz and Tosello's work[START_REF] Fritz | The Hidden Meaning of Forms: Methods of Recording Paleolithic Parietal Art[END_REF]

  abstracts the signifier, focusing on the signified. It avoids shortcuts to the aesthetical value through the schematic representation. Inversely, figure6illustrates the archaeological predicates that guide the reconstruction hypothesis. The figure leans towards the aesthetical and analytical sides of the triangle. The analysis of the figures 4 et 5 shows that aesthetics is uncorrelated with the curated documentation of reconstruction. Between aestheticanalytical and techincal-analytical, there is an array of scientific visualization and data visualization with the possibility of interaction and the capacity to aggregate large volumes of information.The triangle for representation analysis helps clarify to what degree each representation eludes aspects of the reconstruction by its aesthetical, analytical or technical quality in the representation. It is because of the fundamen-

Fig. 3 .

 3 Fig. 3. An excerpt from the cathedral point cloud provides support for a parametric volumetric model of the arch with arbitrary colors. This image is technical and aesthetic.

Fig. 4 .

 4 Fig. 4. The schematic representation filters the information represented towards the analytical elements and exacerbates its technical character.

Fig. 5 .

 5 Fig. 5. Illustration of archaeological predicates for the reconstruction. This figure is analytical: it synthesizes the criteria for the reconstruction. The representation has an aesthetical value (architectural line drawing).

IX. ACKNOWLEDGEMENTS

This work was supported by: the project E-RIHS and REPERAGE funded by the Fondation des Sciences du Patrimoine (France), the projects ASTRAGALE and TEATIME funded by the Mission pour les Initiatives Transverses et Interdisciplinaires of the Centre National de la Recherche Scientifique (France), and the ERC Ndame Heritage funded by ERC advanced Grant 2021. The authors wish to acknowledge the help and collaborative support from: the chief architects of historical monuments in charge, Philippe Villeneuve, Pascal Prunet and Rémi Fromont, the Établissement public chargé de la conservation et de la restauration de la cathédrale Notre-Dame de Paris (RNDP), and the heritage conservators. The authors thank the numerous scientific partners and collaborators from the research groups working on Notre-Dame de Paris cathedral, with special recognition for the Digital Data working group.