Sutthisak Phongthanapanich

Robert Eymard

A Comparative Study of Characteristic Finite Element and Characteristic Finite Volume Methods for Convection-Diffusion-Reaction Problems on Triangular Grids

Keywords: Characteristic Finite Element Method, Characteristic Finite Volume Method, Convection-Diffusion-Reaction, Triangular Grids

The paper aims to compare the accuracy and robustness of the characteristic finite element method (CFEM) and characteristic finite volume method (CFVM) for solving convection-diffusionreaction problems on two-dimensional triangular grids. The tests are performed on a square unit domain, to which an advective field is imposed in a domain. The results show that the CFEM gives less accurate solution than CFVM for the rotation of a slotted-cylinder and rotation of Gaussian cone problems. Moreover, CFEM gives oscillate solution while the CFVM provides an oscillationfree solution for the skew flow to the mesh problem.

Introduction

Many science and engineering applications require the simulation of time-dependent convection-diffusionreaction equations. The convection is defined as the movement of substance due to the fluid transport medium, the diffusion is referred to the dispersion of a substance involved in the process throughout the physical domain of the problem, and the reaction is the process of interaction through which the substances are generated or consumed. Generally, the scale of the diffusion is smaller by several orders of magnitude compared to the size of the advective flow field. A main feature of the convection-and reaction-dominated is the presence of the sharp layers as a part of the solution. The accurate simulation of such processes requires numerical methods such that able to compute sharp layers without an occurrence of spurious oscillations.

The standard Galerkin finite element method usually yields oscillatory solutions for convection-dominated problems [START_REF] Brooks | Streamline Upwind Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equation[END_REF]. Many methods have been proposed to overcome the instability and inaccuracy of the solution. The shock-capturing procedure [START_REF] Codina | A Discontinuity-capturing crosswind dissipation for the finite element solution of the convection-diffusion equation[END_REF] is the usual remedy for these difficulties. Among the stabilizing schemes, the characteristic Galerkin scheme has been developed by applying the time-stepping technique as the basis [START_REF] Zienkiewicz | A general algorithm for compressible and incompressible flow -part I. the split, characteristic based scheme[END_REF][START_REF] Codina | Comparison of some finite element methods for solving the diffusionconvection-reaction equation[END_REF]. The explicit characteristic Galerkin scheme based on the Taylor series expansion is an attractive one due to its simple implementation and fast calculation. The scheme uses a characteristic approximation to trace advection in time and combine with a low-order mixed finite element spatial approximation of the equation. Moreover, the scheme is locally conservative on the discrete level such that fluid is transported along with the approximate characteristics.

Recently, the author proposed an explicit characteristic finite volume method that can provide stabilized numerical solutions for pure advection and advectiondominated diffusion problems [START_REF] Phongthanapanich | A Characteristic-based Finite Volume Element Method for Convection-Diffusion-Reaction Equation[END_REF][START_REF] Phongthanapanich | Explicit characteristic finite volume method for convection-diffusion equation on rectangular grids[END_REF][START_REF] Phongthanapanich | An explicit characteristic finite volume element method for non-divergence free convection-diffusion-reaction equation[END_REF]. The idea of the characteristic-based scheme is used to discrete the Lagrangian derivative as the function of time. An explicit finite volume method based on the cell-centered scheme is employed to derive the discretized equations. The Galerkin finite element technique is applied to estimate the gradient quantities at the cell faces. The finite volume method is applied to discretize the equation because of it some of the important features. Firstly, it may be used on arbitrary geometries, using structured or unstructured grids. Secondly, it is the local conservativity of the numerical fluxes, that is the numerical flux is conserved from one discretization cell to its neighbor [START_REF] Eymard | Finite volume methods[END_REF].

The objective of this research work is to compare the computational efficiency of the CFEM and CFVM for solving two-dimensional convection-diffusion-reaction problems on structured triangular grids. The paper is organized as follows. In Section 2, the governing equations and numerical formulations are presented. Then, the schemes are examined by three benchmark test cases to evaluate their accuracy and robustness in Section 3.

The Numerical Scheme

The characteristic convection-diffusionreaction equation

The governing differential equation for the twodimensional convection-diffusion-reaction equation is where is the scalar quantity, is the given convection velocity vector, is the diffusion coefficient, is the reaction coefficient, and for .

By applying a characteristic approximation to trace advection in time, as explained in Ref. [START_REF] Phongthanapanich | A Characteristic-based Finite Volume Element Method for Convection-Diffusion-Reaction Equation[END_REF][START_REF] Phongthanapanich | Explicit characteristic finite volume method for convection-diffusion equation on rectangular grids[END_REF][START_REF] Phongthanapanich | An explicit characteristic finite volume element method for non-divergence free convection-diffusion-reaction equation[END_REF] to Eq. ( 1) yields [START_REF] Phongthanapanich | A Characteristic-based Finite Volume Element Method for Convection-Diffusion-Reaction Equation[END_REF] where all terms are to be evaluated at .

Then, by carrying out a Taylor series expansion, the convection term reappears in the equation along with an additional second-order term. This second-order term acts as a smoothing operator that reduces the oscillations arising from the spatial discretization of the convection terms.

The fully explicit characteristic convectiondiffusion-reaction equation is given by [START_REF] Phongthanapanich | Explicit characteristic finite volume method for convection-diffusion equation on rectangular grids[END_REF] Together with the use of the divergence-free assumption , Equation ( 6) could be written finally in the conservation form as (7)

The characteristic finite element method

Assuming a variation of within an element can be approximated by (8) [START_REF] Chessa | An extended finite element method for twophase fluids[END_REF] where is an element interpolation function. Applying the standard Galerkin weighting technique to Eq. ( 6) to yield [START_REF] Zalesak | Fully multidimensional fluxcorrected transport algorithms for fluids[END_REF] Because the hyperbolic nature of the convectiondominated equation, a nonlinear viscous shock capturing operator is added to minimize the oscillations at the discontinuities in the curvature of the front. The operator [START_REF] Chessa | An extended finite element method for twophase fluids[END_REF] is given by [START_REF] Wang | Development of CFLfree, explicit schemes for multidimensional advection-reaction equations[END_REF] and [START_REF] Hauke | A simple subgrid scale stabilized method for the advection-diffusion-reaction equation[END_REF] where is a parameter used to control added viscosity (typically is 0.1), and is a measure of the element e. Finally, applying the Green-Gauss theorem to Eq. ( 9) also, using Eq. ( 8) to obtain

The characteristic finite volume method

The computational domain is first discretized into a collection of non-overlapping control volumes , that completely cover the domain such that , and if . To obtain the finite volume equation, Eq. ( 7) is then integrated over the control volume to yield

(23)
Then the divergence theorem is applied to the spatial terms to yield a fully explicit characteristic-based scheme for solving Eq. ( 7) in the form For the opposite direction of velocity, the values could be computed from Eq. ( 27), but using the values from the neighboring cells according to the upwinding direction.

Moreover, in this paper, the gradient term, , is approximated by the weighted residuals method, which is commonly used in the finite element technique [START_REF] Phongthanapanich | Explicit characteristic finite volume method for convection-diffusion equation on rectangular grids[END_REF].

Finally, to ensure the stability of a numerical scheme, the CFL-like stability criterion must be fulfilled. In this paper, the time-step within each control volume i for both the CFEM and CFVM is determined from (28) where is the scaled normal velocity at , is the cell characteristic length, and .

Test Problems and Discussion

In order to illustrate the accuracy and robustness of these numerical schemes, three examples are examined by implementing a second-order accurate scheme. These examples are (1) the rotation of a slotted-cylinder, (2) the rotation of a Gaussian cone, and (3) the skew flow to the mesh. All examples presented in this section were tested on structured triangular grids.

Rotation of a slotted-cylinder

This first problem is adapted from the rotation of a slotted cylinder problem, which was introduced by Zalesak [START_REF] Zalesak | Fully multidimensional fluxcorrected transport algorithms for fluids[END_REF].

It is a challenging pure-convection problem because the sudden change of the initial conditions and the shape of the rectangular slot in the domain are difficult to capture by most numerical schemes. The rotating velocity field is imposed such that . The initial condition is given by ( 29)

where is the distance from the center of the domain . The time step for this example is set to be equal to the period required for one turn rotation. This example is performed on a uniform grid of . The exact solution at the final time step is then equal to the initial condition that is depicted in Figs. ( 1) and (2).

Figures 3 and4 show that the CFEM scheme gives a dissipative solution and inaccurate position of a slottedcylinder. Moreover, Figs. 5 and6 show that the CFVM scheme gives more sharpen and accurate solution. Finally, the comparison of the solution along is illustrated in Fig. 7. It is seen that the CFVM scheme gives a more accurate solution than the CFEM scheme. It can capture the maximum and minimum values around the slot of the cylinder accurately.

Rotation of a Gaussian cone

The second example is the rotation of a Gaussian pulse in a square domain where the scalar field is rotated around the domain . The rotating velocity field with the angular velocity of 4 rad/s is imposed as . This example has been used widely for testing the numerical artifacts that occur on different schemes [START_REF] Wang | Development of CFLfree, explicit schemes for multidimensional advection-reaction equations[END_REF][START_REF] Phongthanapanich | Explicit characteristic finite volume method for convection-diffusion equation on rectangular grids[END_REF]. The initial condition is given by (30)

where . The final time is equal to , which is the time period required for one turn rotation. this example is performed on a uniform grid of . The diffusion coefficient of , in order to evaluate the robustness and accuracy of the method on a convection-dominated diffusion problem. The 2D and 3D plots of the exact solution as shown in Figs. 8 and9. Figures 10 and11 show the numerical solution obtained from the CFEM scheme. Then the numerical solution obtained from the CFVM scheme is also shown in Figs.

12 and 13. Moreover, the comparison of the numerical and exact solutions along the line are presented in Fig. 14.

Skew flow to the mesh

The last example is the skew flow problem presented by Hauke [START_REF] Hauke | A simple subgrid scale stabilized method for the advection-diffusion-reaction equation[END_REF][START_REF] Phongthanapanich | An explicit characteristic finite volume element method for non-divergence free convection-diffusion-reaction equation[END_REF]. This example considers the behavior of the numerical solutions under the convection-reactiondominated diffusion condition in a square domain , with the initial condition . The four boundary conditions are and . The velocity field is given by . The numerical solutions are examined in the steady-state condition. It is noted that the influence of boundary conditions on the algorithm is remarkable and leads to instability [START_REF] Hauke | A simple subgrid scale stabilized method for the advection-diffusion-reaction equation[END_REF]. The parameters for this test are and . The analyses of these test cases are performed on the grid size.

The 3D plots of numerical solutions of the CFEM and CFVM schemes are depicted in Figs. 15 and 16, respectively. Figure 15 shown that the CFEM scheme provides the oscillate solution along the boundaries due to the abrupt change of the solutions. Moreover, the CFVM scheme gives nonoscillate solution along these boundaries. The examples confirmed that the CFVM scheme is an accurate and robust scheme for computing a wide range of convectiondominated diffusion-reaction problems.

Concluding Remarks

The accuracy and robustness of the CFEM and CFVM schemes for solving two-dimensional convectiondiffusion-reaction problems on structured triangular grids are investigated in this paper. The characteristic convection-diffusion-reaction equation is derived. The finite element and finite volume is then used to discretize the characteristic convection-diffusion-reaction equation. Three examples are used to test the accuracy and robustness of both schemes. For the rotation of a slottedcylinder problem, the CFEM scheme provides the diffusive and inaccurate position of the slotted-cylinder solution while the CFVM scheme gives the sharpen solution with an accurate position of the slotted-cylinder solution. The CFEM scheme provides a less accurate solution than the one obtained from the CFVM scheme for rotation of a Gaussian cone problem. Finally, for the skew flow to the mesh, the CFEM scheme provides the oscillate solution along the boundaries due to the abrupt change of the solutions. Moreover, the CFVM scheme gives a non-oscillate solution. We can conclude that the CFVM scheme is an accurate and robust method for solving the convection-dominated problems.

  scalar quantity at the cell faces, , is approximated by applying Taylor's series expansion in space such that(27)

Figure 1 :

 1 Figure 1: Exact solution at the final time of problem 3.1: 2D plot.

Figure 2 :

 2 Figure 2: Exact solution at the final time of problem 3.1: 3D plot.

Figure 3 :

 3 Figure 3: CFEM solution at the final time of problem 3.1: 2D plot.

Figure 4 :

 4 Figure 4: CFEM solution at the final time of problem 3.1: 3D plot.

Figure 5 :Figure 6 :Figure 7 :

 567 Figure 5: CFVM solution at the final time of problem 3.1: 2D plot.

Figure 8 :Figure 9 :

 89 Figure 8: Exact solution at the final time of problem 3.2: 2D plot.

Figure 10 :

 10 Figure 10: CFEM solution at the final time of problem 3.2: 2D plot.

Figure 11 :

 11 Figure 11: CFEM solution at the final time of problem 3.2: 3D plot.

Figure 12 :Figure 13 :

 1213 Figure 12: CFVM solution at the final time of problem 3.2: 2D plot.

Figure 14 :

 14 Figure 14: Comparison of the solution along the line at the final time of problem 3.2.

Figure 15 :

 15 Figure 15: CFEM solution at the final time of problem 3.3: 3D plot.

Figure 16 :

 16 Figure 16: CFVM solution at the final time of problem 3.3: 3D plot.

Acknowledgments

The first author is grateful to the College of Industrial Technology, King Mongkut's University of Technology North Bangkok for funding this research work.


Dr Sophie Mouillet-Richard
email: sophie.mouillet-richard@parisdescartes.fr

ERBB2 in anti-EGFR-resistant colorectal cancer: cancer stem cells come into play

Sophie Mouillet-Richard

With over 900 000 deaths per year worldwide in 2020 according to Globocan, colorectal cancer (CRC) represents the third deadliest cancer. Over the past 20 years, the median overall survival for CRC has substantially increased, especially in patients with metastatic disease, which has doubled to reach 30 months, as a result of the advent of targeted therapies. Despite this progress, the 5-year survival rate for patients with metastatic CRC remains below 15%, calling for an improved management of these patients, which account for roughly 20%-30% of new diagnosed cases.

A consensus view is that CRC arises from uncontrolled expansion of intestinal stem cells (ISCs). [START_REF] Van Der Heijden | Stem cells in homeostasis and cancer of the gut[END_REF] By analogy with ISCs, CRC stem cells (CSCs) are endowed with self-renewal potential and display significant plasticity. They are believed to fuel CRC growth as well as metastasis, and display resistance to therapy, making their targeting one of the most critical challenges in the field of oncology. [START_REF] Van Der Heijden | Stem cells in homeostasis and cancer of the gut[END_REF] The study by Mangiapane et al 2 tackles this issue by leveraging a collection of colorectal CSCs growing as spheroids and expressing the CSC marker CD44v6, which they previously showed to define a population of CSCs with enhanced metastatic potential. [START_REF] Todaro | Cd44V6 is a marker of constitutive and reprogrammed cancer stem cells driving colon cancer metastasis[END_REF] Among those, some bear RAS or BRAF mutations and were thoroughly resistant to the anti-epidermal growth factor receptor (EGFR) antibody cetuximab, as anticipated. Unexpectedly, a significant proportion of RAS and BRAF wild-type CSCs also displayed resistance to cetuximab. To shed some light on the mechanisms involved in this resistance phenomenon, they analysed differentially expressed genes between cetuximab-resistant and cetuximabsensitive CSCs, and found an increased expression of ERBB2, encoding HER2, a member of the EGFR family, in the resistant population. Building on their previous identification of the phosphatidylinositol 3-kinase (PI3K)/AKT pathway as an important regulator of CD44v6-positive cells survival, they went on to show that ERBB2 is transcriptionally regulated by PI3K. In agreement with their previous study, 3 PI3K inhibition efficiently eliminated disseminated cancer cells after injection in the spleen, while it has low efficacy when cells are injected subcutaneously. [START_REF] Veschi | Targeting chemoresistant colorectal cancer via systemic administration of a BMP7 variant[END_REF] To account for this observation, they made the assumption that, within the tumorous tissue, additional signals emanating from the microenvironment may foster ERBB2 upregulation, and thereby protect cells from death induced by PI3K blockade. The authors focused on hepatocyte growth factor (HGF), osteopontin (OPN) and stromal-cell derived factor 1 (SDF-1), three soluble factors secreted by cancerassociated fibroblasts (CAFs), which they previously incriminated in the regulation of CD44v6 expression and in the promotion of an aggressive phenotype of CSCs on xenografting. [START_REF] Todaro | Cd44V6 is a marker of constitutive and reprogrammed cancer stem cells driving colon cancer metastasis[END_REF] Indeed, they found that these cytokines synergise to induce ERBB2 expression, making it insensitive to PI3K inhibition. This novel result actually adds another facet to the role described for these tumour microenvironment-derived molecules in supporting CSCs in CRC. The question then arose as to how design the appropriate combo to efficiently target these colorectal CSCs. Because these CAFs-derived cytokines activate the MAP kinase pathway, the authors considered the MEK kinase as potential candidate. In line with monotherapies being less efficient than drug combinations, the concomitant targeting of ERBB2 with trastuzumab, PI3K with either BKM120 or taselisib and MEK with trametinib or cobimetinib significantly hampered the tumour growth of CSCs cells injected subcutaneously. At the same time, this triplet of agents promoted a drastic reduction in CD44v6 staining in xenografts, suggesting a powerful anti-CSC activity.

Altogether, these enticing data consolidate the notion that activation of HER2 signalling contributes to cetuximab resistance in CRC, and that this may arise as a result of increased ERBB2 expression, beyond ERBB2 amplification 5 6 or mutations, [START_REF] Bertotti | The genomic landscape of response to EGFR blockade in colorectal cancer[END_REF] as also recently demonstrated by Lupo et al. [START_REF] Lupo | Colorectal cancer residual disease at maximal response to EGFR blockade displays a druggable Paneth cell-like phenotype[END_REF] These results and the Lupo et al study [START_REF] Lupo | Colorectal cancer residual disease at maximal response to EGFR blockade displays a druggable Paneth cell-like phenotype[END_REF] thus advocate for a comprehensive analysis of ERBB2, including mRNA expression, in EGFR-agents refractory settings, and may set the stage for a wider use of anti-ERBB2-based therapies to treat metastatic CRC. Interestingly, the study by Lupo et al documents that ERBB2 is upregulated in colorectal residual disease and that the corresponding drug tolerant cells display hallmarks of DNA labelretaining cells, themselves being secretory precursors committed to mature into Paneth cells. [START_REF] Lupo | Colorectal cancer residual disease at maximal response to EGFR blockade displays a druggable Paneth cell-like phenotype[END_REF] The question then arises as to whether the ERBB2-positive cell population described by Lupo and colleagues [START_REF] Lupo | Colorectal cancer residual disease at maximal response to EGFR blockade displays a druggable Paneth cell-like phenotype[END_REF] shares some common features with the ERBB2-expressing CSCs of the Mangiapane et al study. [START_REF] Mangiapane | PI3K-driven HER2 expression is a potential therapeutic target in colorectal cancer stem cells[END_REF] Such a hypothesis would notably fit with the well-established reciprocal relationship between CSCs and drug resistance. Another noteworthy point of convergence in the two studies relates to the signalling pathways involved: both incriminate MEK and PI3K, which may operate both upstream and downstream from ERBB2 and thereby maintain a positive feedback loop. Thus, the study by Lupo et al [START_REF] Lupo | Colorectal cancer residual disease at maximal response to EGFR blockade displays a druggable Paneth cell-like phenotype[END_REF] directly supports the triplet strategy-combined targeting of ERBB2, MEK and PI3K-proposed by Mangiapane and colleagues. [START_REF] Mangiapane | PI3K-driven HER2 expression is a potential therapeutic target in colorectal cancer stem cells[END_REF] Of note, the synergistic use of PI3K and pan-ERBB inhibitors has previously been reported to be efficient in the context of mutant KRAS CRC. [START_REF] Belmont | Resistance to dual blockade of the kinases PI3K and mTOR in KRASmutant colorectal cancer models results in combined sensitivity to inhibition of the receptor tyrosine kinase EGFR[END_REF] However, in contrast to, 2 the rationale for the dual targeting of PI3K and ERBB in the latter study was based on the increased expression and activity of ERBB family members in mutant KRAS CRC resistant to PI3K/mTOR inhibition, as a result of FOXO3 activation. [START_REF] Belmont | Resistance to dual blockade of the kinases PI3K and mTOR in KRASmutant colorectal cancer models results in combined sensitivity to inhibition of the receptor tyrosine kinase EGFR[END_REF] Nevertheless, in line with, 2 PI3K/mTOR resistant, ERBBhyper-activated CRC cells were found to exhibit stem-cell like characteristics. [START_REF] Belmont | Resistance to dual blockade of the kinases PI3K and mTOR in KRASmutant colorectal cancer models results in combined sensitivity to inhibition of the receptor tyrosine kinase EGFR[END_REF] However, attractive the combined ERBB2, MEK and PI3K therapeutic strategy may appear, some open questions remain. Should not the triplet strategy be compared with the dual targeting of MEK and PI3K, since the latter appears to be more efficient according to cell-based xenograft assays? [START_REF] Belli | Combined blockade of MEK and PI3KCA as an effective antitumor strategy in HER2 gene amplified human colorectal cancer models[END_REF] How can we reconcile the observation that CRC cells engineered to overexpress ERBB2 lose markers of epithelial to mesenchymal transition (EMT) as well as invasive and migratory properties [START_REF] Belli | Combined blockade of MEK and PI3KCA as an effective antitumor strategy in HER2 gene amplified human colorectal cancer models[END_REF] with the report that CD44v6positive cells, which express abundant levels of ERBB2 2 display hallmarks of EMT and are endowed with metastatic properties? [START_REF] Todaro | Cd44V6 is a marker of constitutive and reprogrammed cancer stem cells driving colon cancer metastasis[END_REF] At a mechanistic level, it is unclear what other ERBB receptor ERBB2 dimerises with in CD44v6-positive cells. Answer to this question may help further refine therapeutic schemes.

In summary, notwithstanding these issues, the study by Mangiapane et al [START_REF] Mangiapane | PI3K-driven HER2 expression is a potential therapeutic target in colorectal cancer stem cells[END_REF]

(ABES). Protected by copyright. on February 28, 2021 at Agence Bibliographique de l Enseignement Superieur http://gut.bmj.com/ Gut: first published as 10.1136/gutjnl-2020-323924 on 22 February 2021. Downloaded from

Commentary

brings to light an unprecedented link between resistance to cetuximab and the upregulation of ERBB2 in the CSC compartment of CRC. Most importantly, it clearly offers new therapeutic prospects for the management of patients who have become refractory to EGFR-targeted agents. In addition, it may give a second wind to ERBB2-based clinical trials such as the HERACLES study. [START_REF] Siena | Targeting the human epidermal growth factor receptor 2 (HER2) oncogene in colorectal cancer[END_REF] The combination of trastuzumab to MEK and PI3K inhibitors advocated by Mangiapane and colleagues 2 may notably provide an adequate response to the development of resistance to ERBB2 blockade, potentially involving these effectors as recently suggested. 11

Contributors SM-R wrote the article.

Funding The author has not declared a specific grant for this research from any funding agency in the public, commercial or not-for-profit sectors.

Competing interests None declared.

Patient consent for publication Not required.


Anna Florio
email: anna.florio@imj-prg.fr

Patrice Le Calvez

Torsion of instability zones for conservative twist maps on the annulus

published or not. The documents may come

Introduction

Let us denote T = R/Z and A = T × R. Fix the standard Riemannian metric and the standard trivialization on A. This enables us to define the notion of oriented angle between two non-zero vectors in the tangent plane of a point of A.

Let R 2 (x, y) → π(x, y) = (x + Z, y) ∈ A be the covering projection on A. Let p 1 : A → T and p 2 : A → R denote the projections over the first and the second coordinate respectively. With an abuse of notation, we denote as p 1 , p 2 : R 2 → R also the projections over the first and the second coordinate on the plane.

Denote the area form ω = dx ∧ dy and let Leb be the associated Lebesgue measure on A. Let α = y dx be the Liouville 1-form on A. Let T 1 A be the bundle of half-lines, that is

T 1 A = {(z, w) ∈ T A : w = 0}/ ∼
where (z, w) ∼ (z , w ) if and only if z = z and w = λw for some λ ∈ R, λ > 0. For every z ∈ A, T 1 z A is the fiber of z and for a vector w ∈ T z A we will denote, with an abuse of notation, as w ∈ T 1 z A the equivalent class to which w belongs. For every X ⊂ A denote as T 1 X the restriction of the bundle to X, that is T 1 X = z∈X T 1 z A. For every z ∈ A denote as v the class of the vertical vector v = (0, 1). For every z ∈ A and every w, w ∈ T 1 z A let ≺ (w, w ) be the oriented angle between w and w divided by 2π. We consider angles in T = R/Z, after rescaling R/2πZ. Define the continuous function θ :

T 1 A -→ T (z, w) -→≺ (v, w).

(1) It induces a closed 1-form dθ on T 1 A, see [START_REF] Godbillon | Éléments de topologie algébrique[END_REF]. For any continuous path [0, 1] t → γ(t) ∈ T 1 A denote (2) γ dθ = Θ(1) -Θ(0), where Θ : [0, 1] → R is a lift of θ • γ (the definition in (2) does not depend on the chosen lift Θ). Let f : A → A be a C 1 diffeomorphism isotopic to the identity and I = (f t ) t∈[0,1] Date: September 15, 2020.

an isotopy in Diff 1 (A) joining the identity to f 1 = f . Denote as DI = (Df t ) t∈[0,1] the continuous isotopy such that for every t ∈ [0, 1] Df t :

T 1 A -→ T 1 A (z, w) → Df t (z, w) = (f t (z), Df t (z)w) .

Extend the isotopy for t ∈ R + so that Df 1+t = Df t • Df 1 .

We are interested in the notion of torsion of a point: roughly speaking it is the asymptotic rotational behavior of vectors in the tangent space of the point under the action of the linearized dynamics. See [START_REF] Ruelle | Rotation numbers for diffeomorphisms and flows[END_REF] and [START_REF] Béguin | Existence of orbits with non-zero torsion for certain types of surface diffeomorphisms[END_REF].

More precisely, let (z, w) ∈ T 1 A. For every n ∈ N * denote DI n (z, w) the continuous path [0, 1] t → Df nt (z, w). The torsion at finite time n ∈ N * at (z, w)

∈ T 1 A is (3) Torsion n (f, z, w) := 1 n DI n (z,w) dθ = 1 n n-1 i=0
Torsion 1 (f, Df i (z, w)).

It does not depend on the choice of the isotopy joining the identity to f and can be equivalently given by using the universal covering space of T 1 A and a continuous lift θ : T 1 A → R of the function θ in (1), see [START_REF] Béguin | Existence of orbits with non-zero torsion for certain types of surface diffeomorphisms[END_REF] and [START_REF] Florio | Torsion and linking number for a surface diffeomorphism[END_REF].

The asymptotic torsion at z ∈ A is, whenever the limit exists, (4)

Torsion(f, z) := lim n→+∞ Torsion n (f, z, w).
The definition of asymptotic torsion at z ∈ A coincides with Ruelle's definition of rotation number, see [START_REF] Ruelle | Rotation numbers for diffeomorphisms and flows[END_REF]. It does not depend on w ∈ T 1 z A. Indeed, let z ∈ A, n ∈ N * and w, w ∈ T 1 z A. Consider the quantity (5)

DI n (z,w) dθ - DI n (z,w )
dθ.

If ≺ (w, w ) ∈ (0, 1 2 ) + Z, then ≺ (Df n (z)w, Df n (z)w ) ∈ (0, 1 2 ) + Z because Df (z) preserves the orientation and so the quantity (5) cannot be equal to 1 2 + k, k ∈ Z. This is also true if ≺ (w, w ) ∈ (-1 2 , 0) + Z. Moreover (5) vanishes if ≺ (w, w ) ∈ {0, 1 2 } + Z, meaning w = w or w = -w . By connectedness, for any n ∈ N * and for every w, w ∈ T

1 z A it holds (6) - 1 2 < DI n (z,w) dθ - DI n (z,w ) dθ < 1 2 .
Observe that the function

T 1 A × Z (z, w, n) → DI n (z,w) dθ ∈ R is a cocycle over Df .
Assume that f admits an invariant Borel probability measure µ on A with compact support. There exists a Df -invariant probability measure ν on T 1 A whose projection on A is µ. Indeed, if ν 0 is a probability measure of T 1 A that projects onto µ, any limit point ν of the sequence 1 n n-1 i=0 Df i * (ν 0 ) n≥1 is Df -invariant and projects onto µ. Since the asymptotic torsion is the limit of the Birkhoff's sum in (3), by Birkhoff's Ergodic Theorem, the limit exists at ν-almost every (z, w) ∈ T 1 A. Since the asymptotic torsion does not depend on the chosen w ∈ T 1 z A and since ν projects onto µ, we deduce that the asymptotic torsion exists at µ-almost every z ∈ A, see [START_REF] Ruelle | Rotation numbers for diffeomorphisms and flows[END_REF].

Definition 1.1. A positive (respectively negative) twist map is a C 1 diffeomorphism f : A → A isotopic to the identity such that for any lift F : R 2 → R 2 of f and for any x ∈ R the function

R y → p 1 • F (x, y) ∈ R is an increasing (respectively decreasing) diffeomorphism of R.
It can be easily proved that f is a positive twist map if and only if f -1 is a negative twist map. Concerning the torsion for positive (respectively negative) twist map, for every z ∈ A and for every n ∈ N * it holds Torsion n (f, z, v) ∈ (-1 2 , 0) (respectively (0, 1 2 )), see [START_REF] Florio | Torsion and linking number for a surface diffeomorphism[END_REF]. It is an outcome of the fact that, for a positive twist map, Torsion 1 (f, z, v) ∈ (-1 2 , 0) and that for every isotopy (f t ) t∈[0,1] from Id to f and for every non colinear vectors w, w ∈ T z A, the vectors Df t (z)w, Df t (z)w ∈ T ft(z) A are non colinear and if ≺ (w, w ) ∈ 0, 1 2 + Z, then ≺ (Df t (z)w, Df t (z)w ) ∈ 0, 1 2 + Z. Definition 1.2. Let f : A → A be a positive twist map. A point z ∈ A has conjugate points if there exist n, k ∈ N * such that DI n (z,v) dθ = -k 2 , it has over-conjugate points if there exists n ∈ N * such that DI n (z,v) dθ < -1 2 . Observe that if a point z ∈ A has negative asymptotic torsion, then z has over-conjugate points. Actually, the existence of points with conjugate points and with over-conjugate points is equivalent, see Lemma 3.1.

Let us state now the main results of the article. The first result, that will be fundamental, asserts that for a twist map that preserves the Lebesgue measure, the existence of points having over-conjugate points in a f -periodic open domain U of finite measure is equivalent to the fact that the set of points in U of non-zero torsion has positive Lebesgue measure. More precisely: Proposition 1.1. Let f : A → A be a positive twist map that preserves the Lebesgue measure. Let U ⊂ A be a f -periodic open set of finite Lebesgue measure. Then the following conditions are equivalent.

(i) The asymptotic torsion is defined and equal to zero at every point z ∈ U .

(ii) The asymptotic torsion is defined and equal to zero at Leb-almost every point z ∈ U .

(iii) No point z ∈ U has over-conjugate points.

Recall that an essential curve γ : T → A is a topological embedding that is not homotopic to a point, and that an open subset U of A is essential if it contains an essential curve. We show that any f -periodic open subset U of finite measure which is not essential must contain points with over-conjugate points. Actually, we prove the following Theorem 1.1. Let f be a positive twist map that preserves the Lebesgue measure. Let U be a f -periodic non essential open set of finite measure. Then the set of points of U of negative asymptotic torsion is dense and has positive measure.

Question 1.1. Let U be a f -periodic non essential open set of finite Lebesgue measure. Is the measure of the set of points of non zero torsion in U equal to the measure of U ?

Definition 1.3. Let U ⊂ A be an f -invariant essential subannulus. The dynamics f |U is C 0 -integrable if f |U admits a partition into essential f -invariant curves.
Observe that the definition of C 0 -integrability is more restrictive than the usual notion of integrability. In higher dimensions, for Tonelli Hamiltonian flows on the cotangent bundle T * T n of the n-dimensional torus, the analogous definition appears in [START_REF] Arnaud | Tonelli hamiltonians without conjugate points and C 0 integrability[END_REF].

Recall that a C 1 diffeomorphism f : A → A that preserves the Lebesgue measure is exact symplectic if f * α -α is exact, or equivalently if for every essential curve γ : T → A, the algebraic area contained between γ(T) and f • γ(T) is zero (zero flux condition).

We provide a simple proof of a result by Cheng and Sun, see Theorem 1 in [START_REF] Cheng | A necessary and sufficient condition for a twist map being integrable[END_REF].

Theorem 1.2. An exact symplectic twist map f : A → A is C 0 -integrable if and only if f does not have points with conjugate points.

In the continuous case, the same result holds for Tonelli Hamiltonian flows on the cotangent bundle of the n-dimensional torus, see Theorem 1 in [START_REF] Arnaud | Tonelli hamiltonians without conjugate points and C 0 integrability[END_REF]. Cheng and Sun's proof uses Aubry-Mather theory as presented by Bangert in [START_REF] Bangert | Mather sets for twist maps and geodesics on tori[END_REF], that is considering minimizing configurations. Their ideas can be easily adapted to any bounded invariant subannulus, see [START_REF] Florio | Asymptotic Maslov indices[END_REF]. Our proof is a geometrical one and it works both for exact symplectic twist maps on the unbounded annulus and for conservative twist maps on bounded essential subannuli, see Remark 4.5. Thus, thanks to Proposition 1.1, we deduce the following Corollary 1.1. Let f : A → A be a twist map that preserves the Lebesgue measure. Let U be an f -invariant bounded essential subannulus. Then, f |U is C 0 -integrable if and only if Leb-almost every point of U has zero torsion. [START_REF] Arnaud | Hyperbolicity for conservative twist maps of the 2-dimensional annulus[END_REF]. Thus, from Proposition 1.1 and Theorems 1.2 and 1.1, we deduce the following outcome.

Denote as

I (f ) the union of all f -invariant essential curves in A. A bounded connected component of A \ I (f ) is either an f -invariant essential subannulus or a f -periodic disk, see
Corollary 1.2. Let f : A → A be a positive twist map that preserves the Lebesgue measure. Then every bounded connected component U of A \ I (f ) contains a set of positive measure of points with non-zero torsion. In particular, it holds

i∈Z f i (U ) Torsion(f, z) dLeb(z) < 0.

Torsion and over-conjugate points: proof of Proposition 1.1

The implications (iii) ⇒ (i) and (i) ⇒ (ii) of Proposition 1.1 are obvious. The implication (ii) ⇒ (iii) is an immediate consequence of the following result.

Proposition 2.1. Let f : A → A be a positive twist map that preserves the Lebesgue measure. Let z 0 ∈ A be such that (i) z 0 has over-conjugate points;

(ii) z 0 has a neighborhood U such that Leb( n∈Z f n (U )) < +∞.

Then there exists an open neighborhood W ⊂ U of z 0 such that for Leb-almost every z ∈ W it holds that Torsion(f, z) < 0.

Proof. By the definition of points having over-conjugate points, there exists m ∈ N * such that DI m (z 0 ,v) dθ < -1 2 . By the continuity of finite-time torsion, there exists a neighborhood W ⊂ U of z 0 such that for every z ∈ W it holds that DI m (z,v) dθ < -1 2 . For every z ∈ W and for every n > m, we have that DI n (z,v) dθ < -1 2 . This fact can be shown by induction. Indeed, for n = m + 1, since Df (z) preserves the orientation and since DI(f m (z),v) dθ < 0, we deduce that, if DI m (z,v) dθ < -1 2 , then

DI m+1 (z,v) dθ ≤ - 1 2 + DI(f m (z),v) dθ < - 1 2 .
We argue similarly for n > m + 1. Actually, it holds that

DI n (z,v) dθ = DI m (z,v) dθ + DI n-m (Df m (z,v)) dθ ≤ - k 2 + DI n-m (f m (z),v) dθ < - 1 2 ,
where k ∈ N, k > 0 is the integer part of -2 DI m (z,v) dθ.

Thus, by (6), for every (z, w) ∈ T 1 W and for every n ≥ m it holds that (7)

DI n (z,w) dθ < 0.
There exists a finite measure Leb defined on T 1 (∪ n∈Z f n (U )) that projects onto the restriction of Leb to n∈Z f n (U ). Indeed, the argument given in the introduction can be extended to the case of an open set of finite measure: we can either consider the Alexandroff compactification of T 1 ( n∈Z f n (U )), or we can use Prohorov's Theorem (see Proposition 2.4.1 in [START_REF] Viana | Foundations of ergodic theory[END_REF]). Since f is a positive twist map, one has

-1 < Torsion 1 (f, z, w) < 1 2 for every (z, w) ∈ T 1 A. By Birkhoff's Ergodic Theorem, lim n→+∞ Torsion n (f, z, w) is defined Leb-almost everywhere on T 1 ( n∈Z f n (U )
) and so Torsion(f, z) is defined Lebalmost everywhere on n∈Z f n (U ).

We want to prove that Torsion(f, z) < 0 for Leb-almost every point in W . Set g = f m and J = I m . We will prove that Torsion(g, z) is well-defined and negative for Leb-almost every z ∈ W , which will imply the proposition because Torsion(f, z) = Torsion(g, z)/m.

Let ψ : z → g τ (z) (z) be the first return map of g on W , where τ : W → N * is the first return time map. These maps are defined Leb-almost everywhere on W . The map

Dψ : (z, w) → (g τ (z) (z), Dg τ (z) (z)w) is defined Leb-almost everywhere on T 1 W . We know that (8) -mτ (z) < DJ τ (z) (z,w) dθ = τ (z)-1 i=0 DJ(Dg i (z,w))
dθ < 0, the right inequality occuring because DI n (z,w) dθ < 0 if n ≥ m and DJ τ (z) (z, w) = DI mτ (z) (z, w) (see ( 7)) and the left inequality occuring because Torsion 1 (f, z, w) > -1 for every (z, w) ∈ T 1 A. By Kac's Lemma we know that τ is Leb-integrable and so that φ : (z, w) → DJ τ (z) (z,w) dθ is integrable and negative. By Birkhoff's Ergodic Theorem, we deduce that for Leb-almost every (z, w) ∈ T 1 W , one has

(9) lim n→+∞ 1 n n-1 i=0 φ(Dψ i (z, w)) = φ(z, w) ∈ (-∞, 0)
and ( 10)

lim n→+∞ 1 n n-1 i=0 τ (ψ i (z)) = τ (z) ∈ [1, ∞).
Recall that the fact that φ(z, w) < 0 in (9) is a classical application of Birkhoff's Ergodic Theorem. Consequently, from (8), ( 9) and (10), for Leb-almost every (z, w) ∈ T 1 W , it holds that φ(z, w)/τ (z) ∈ (-∞, 0). Note now that

n-1 i=0 φ(Dψ i (z, w))/ n-1 i=0 τ (ψ i (z)) = Torsion Nn (g, z, w)
where N n = n-1 i=0 τ (ψ i (z)). So Torsion(g, z) = lim n→+∞ Torsion Nn (g, z, w) < 0 for Lebalmost every point in W .

Density of over-conjugate points in non-essential periodic bounded

open sets: proof of Theorem 1.1

In the introduction, we give the definition of conjugate and over-conjugate points. Let us show that the existence of the first ones is equivalent to the existence of the second ones.

Lemma 3.1. Let f be a positive twist map. There exists a point with conjugate points if and only if there exists a point with over-conjugate points.

Proof. On one hand, the first implication is an outcome of the twist property. Indeed, if there exists z ∈ A such that DI n (z,v) dθ = -k 2 for some n, k ∈ N * , then Df n (z)v = ±v and so DI(f n (z),Df n (z)v) dθ ∈ (-1 2 , 0). Consequently, one has

DI n+1 (z,v) dθ = DI n (z,v) dθ + DI(f n (z),Df n (z)v) dθ < - k 2 ≤ - 1 2 .
On the other hand, it can be proved that, for every n ∈ N * , there exists z ∈ A such that [Flo]. More precisely, selecting a lift F of f , the linking number at any finite time n ∈ N * of the couple of points (x, r 0 ), (x + 1, r 0 ), for some x ∈ R, r 0 ∈ R, is zero. Thanks to the relation between torsion and linking number shown in [START_REF] Florio | Torsion and linking number for a surface diffeomorphism[END_REF], we deduce that, for any n ∈ N * there exists z = z(n) ∈ T × {r 0 } such that DI n (z,u) dθ = 0, where u is the class of the horizontal vector u = (1, 0). By (6) and since f is a positive twist map, we so have that DI n (z,v) dθ ∈ (-1 2 , 0). By this fact and by the Intermediate Value Theorem, we conclude that the existence of a point with over-conjugate points implies the existence of a point with conjugate points.

DI n (z,v) dθ ∈ (-1 2 , 0), see
The equivalence between the existence of points with conjugate points and with overconjugate points still holds for composition of positive twist maps and even to the restriction of the dynamics to a f -invariant essential subannulus.

This section is devoted to the proof of Theorem 1.1.

Lemma 3.2. Let f be a positive twist map, F a lift of f to R 2 and K ⊂ A a compact set that does not contain any point with over-conjugate points. Then there exists β ∈ (0, 1 2 ) so that for every z, z ∈ π -1 (K) satisfying (i) the angle ≺ (v, z -z) belongs to [0, β] + Z, (ii) the segment joining z to z is contained in π -1 (K), we have that for every n ≥ 1

p 1 • F n (z ) -p 1 • F n (z) > 0.
Proof. The map f being a positive twist map, we have ≺ (v, Df (z)v) ∈ -1 2 , 0 + Z, for every z ∈ A. By compactness of K, there exists β ∈ 0, 1 2 such that for every (z, w) ∈

T 1 K such that ≺ (v, w) ∈ [0, β] + Z it holds that ≺ (v, Df (z)w) ∈ -1 2 , 0 + Z. Moreover we know that ≺ (Df (z)v, Df (z)w) ∈ 0, 1 2 + Z.
We deduce that for every n ≥ 0 we have ≺ (Df n (z)v, Df n+1 (z)w) ∈ -1 2 , 0 + Z and ≺ (Df n+1 (z)v, Df n+1 (z)w) ∈ 0, 1 2 + Z. Since there are no points with over-conjugate points in K, we have ≺ (v, Df n+1 (z)v) ∈ -1 2 , 0 + Z for every n ≥ 0. Consequently we have ≺ (v, Df n+1 (z)w) ∈ -1 2 , 0 + Z for every n ≥ 0.

Let z, z ∈ π -1 (K) be such that ≺ (v, z -z) ∈ [0, β] + Z and the segment γ : [0, 1] t → (1 -t)z + tz is contained in π -1 (K). Consequently, for every t and for every n ≥ 1

≺ (v, (F n • γ) (t)) ∈ (-1 2 , 0) + Z. So for every n ≥ 1 it holds that p 1 • F n (γ(1)) -p 1 • F n (γ(0)) = p 1 • F n (z ) -p 1 • F n (z) > 0.
Proof of Theorem 1.1. Let U be a f -periodic non essential open set of period M ∈ N * . Let F : R 2 → R 2 be a lift of f . Let U = π -1 (U ). By Proposition 2.1, to get Theorem 1.1, it is sufficient to prove that the set of points with over-conjugate points is dense in U .

Argue by contradiction and assume there is a closed ball B ⊂ U which does not have any point with over-conjugate points.

Let β ∈ (0, 1 2 ) be given by Lemma 3.2 applied at the closed ball B. We can find two balls B 1 , B 2 ⊂ B such that for every z ∈ B 1 , z ∈ B 2 the oriented angle ≺ (v, z -z) belongs to [0, β] + Z and so that the segment joining z to z is contained in B. On one hand, by Lemma 3.2, for every n ≥ 1 we have that

p 1 • F n (z ) -p 1 • F n (z) > 0.
On the other hand, on the annulus, we apply Poincaré's Recurrence Theorem to the map f × f and the set π(B 1 ) × π(B 2 ). We find

n ∈ N * , z ∈ B 1 , z ∈ B 2 such that f M n (π(z)) ∈ π(B 1 ) and f M n (π(z )) ∈ π(B 2 ). Thus, there exists k 1 , k 2 ∈ Z such that F M n (z) + (k 1 , 0) ∈ B 1 , F M n (z ) + (k 2 , 0) ∈ B 2 . Since B ⊂ U and since U is non essential, we deduce that k 1 = k 2 and, by the choice of B 1 , B 2 it holds that p 1 • F M n (z ) < p 1 • F M n (z), obtaining the required contradiction. B z z F F F(z ) F(z ) F (z ) F (z ) 1 2 1 2 1 2 2 2 B (z) B (z') ) ) , , ) , B B 1 2 ) )
Figure 1. Each vector DF n (γ(t))γ (t) lies in the right half-plane.

Remark 3.1. The idea of the proof of Theorem 1.1 can be pushed further. Actually, for a bounded invariant topological open disk the absence of points with over-conjugate points allows to build a partition into partial graphs of Lipschitz functions which are periodic. See [START_REF] Florio | Asymptotic Maslov indices[END_REF] for such a construction.

We remark also that an alternative proof of Theorem 1.1 can be given by using the notion of linking number (see [START_REF] Béguin | Existence of orbits with non-zero torsion for certain types of surface diffeomorphisms[END_REF] and [START_REF] Florio | Torsion and linking number for a surface diffeomorphism[END_REF]) and thanks to the relation between torsion and linking number proved in [START_REF] Florio | Torsion and linking number for a surface diffeomorphism[END_REF]. Indeed, since U is non essential, denoting as n ∈ N the period of U , we can select a lift F of f such that F n fixes the lifts of U . In the lifted framework, we can consider the linking number at finite-time of a couple of points, calculated with respect to an isotopy obtained by lifting an isotopy on A.

Apply then Poincare's Recurrence Theorem at any small ball contained in U with respect to the dynamics of f |U × f |U . Thanks to the twist property, we find a couple of points whose linking number at finite-time is almost -1. Consequently, by the result in [START_REF] Florio | Torsion and linking number for a surface diffeomorphism[END_REF], there exists a point belonging to the starting small ball whose torsion at finitetime is almost -1. That is, any arbitrary small ball contains points with over-conjugate points.

4.

A geometric criterion for C 0 -integrability: proof of Theorem 1.2

The aim of this section is giving a simple, geometrical proof of the result due to Cheng and Sun, see [START_REF] Cheng | A necessary and sufficient condition for a twist map being integrable[END_REF] and here Theorem 1.2. We prove here the non trivial implication, i.e. if there are no points having conjugate points, then f |U is C 0 -integrable.

Remark 4.1. The definition given in [START_REF] Cheng | A necessary and sufficient condition for a twist map being integrable[END_REF] of points with conjugate points is equivalent to our definition of conjugate points (here Definition 1.2).

Let F be a lift of the exact symplectic twist map f and let h : R 2 → R be a generating function for [START_REF] Aubry | The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states[END_REF] or [START_REF] Mackay | Lectures on orbits of minimal action for area-preserving maps[END_REF]. Consequently, when looking at the linearized dynamics, a sequence (x n , y n ; ξ n , η n ) n∈N ⊂ (T R 2 ) N is a trajectory of the linearized map DF if and only if the sequence (ξ n ) n∈N is a Jacobi vector field over (x n ) n∈N , i.e.

F . A sequence (x n , y n ) n∈N ⊂ (R 2 ) N is a trajectory for F if and only if the sequence (x n ) n∈N satisfies ∂ 1 h(x n , x n+1 ) + ∂ 2 h(x n-1 , x n ) = 0 ∀n ∈ N, see [Ban88],
∂ 1 ∂ 2 h(x n-1 , x n )ξ n-1 + [∂ 2 1 h(x n , x n+1 ) + ∂ 2 2 h(x n.1 , x n )]ξ n + ∂ 1 ∂ 2 h(x n , x n+1 )ξ n+1 = 0.
In [START_REF] Cheng | A necessary and sufficient condition for a twist map being integrable[END_REF], a point (x 0 , y 0 ) has conjugate points if there exists a non-zero Jacobi vector field (ξ n ) n∈N over (x n ) n∈N = (p 1 • F n (x 0 , y 0 )) n∈N such that ξ 0 = ξ N = 0. Equivalently, DF N (x 0 , y 0 )v = ±v. Since f is a positive twist map, it means that DI N ((x 0 ,y 0 ),v) dθ = -k 2 for some k ∈ N * , i.e. (x 0 , y 0 ) has conjugate points according to Definition 1.2.

We recall the definition of rotation number for a homeomorphism of the annulus. Definition 4.1. Let F : R 2 → R 2 be the lift of a homeomorphism f of A isotopic to the identity. Let z ∈ R 2 . The rotation number for F of z is, whenever it exists, the limit

ρ F (z) = lim n→+∞ p 1 • F n (z) -p 1 (z) n .
Remark 4.2. Let ψ : T → R be a continuous map whose graph is f -invariant and let Ψ = ψ • π , where R x → π (x) = x + Z ∈ T is the covering projection on T. Then, we can define the rotation number ρ F (Ψ) for F of Ψ as the rotation number ρ F (x, Ψ(x)) for any x ∈ R. Indeed, for every x ∈ R the rotation number ρ F (x, Ψ(x)) exists and moreover it is independent from x ∈ R.

Lemma 4.1. Let F : R 2 → R be the lift of a positive twist map f . The set

(11) CF := {z ∈ R 2 : p 1 • F (z) = p 1 (z)}
is the lift of the graph of a function

ψ 1 : T → R of class C 1 .
Proof. For every given x ∈ R, by the twist property, the function R y → p 1 •F (x, y) ∈ R is an increasing diffeomorphism and so there exists a unique y ∈ R such that p 1 •F (x, y) = x. Let us denote such y ∈ R as Ψ 1 (x). The set CF is the graph of the function Ψ 1 : R → R.

To prove that Ψ 1 is of class C 1 , it is sufficient to apply the Implicit Function Theorem because ∂ ∂y (p 1 • F (x, y) -x) > 0. As Ψ 1 is clearly periodic of period 1, it lifts a function

ψ 1 : T → R of class C 1 .
Remark 4.3. Lemma 4.1 holds true for negative twist maps too. In particular, we denote the set

CF -1 = {z ∈ R 2 : p 1 • F -1 (z) = p 1 (z)} as the graph of the function Ψ -1 : R → R of class C 1 that lifts ψ -1 : T → R. Note that CF -1 is nothing but the image of CF by F .
Remark that if a point z ∈ R 2 lies below (respectively above) CF , then F (z) lies on the left (respectively on the right) of the vertical passing through z. Similarly, if z lies below (respectively above) CF -1 , then F -1 (z) lies on the right (respectively on the left) of the vertical passing through z.

Lemma 4.2. Let f : A → A be a positive twist map with no point with conjugate points. Then for every n > 0 the function f n (respectively f -n ) is a positive (respectively negative) twist map.

Proof. Let us prove this fact by induction. Suppose that f n is a positive twist map and fix a lift F of f . Consider x, x ∈ R. The map y → p 1 • F n (x, y) is an increasing diffeomorphism of the real line, while the map y → p 1 • F -1 (x , y) is a decreasing one. Moreover, lim y→±∞ p 2 •F n (x, y) = lim y→±∞ p 2 •F -1 (x , y) = ±∞. This implies that there exist y, y ∈ R such that F n (x, y) = F -1 (x , y ). Equivalently, it holds that F n+1 (x, y) = (x , y ). Thus, the map y → p 1 • F n+1 (x, y) is surjective. Since there are no points with conjugate points, by Lemma 3.1, there are no points with over-conjugate points. In particular for any y ∈ R it holds that ≺ (v, DF n+1 (x, y)v) ∈ (-1 2 , 0) + Z. Thus d dy p 1 • F n+1 (x, y) > 0 and consequently the map y → p 1 • F n+1 (x, y) is an increasing diffeomorphism onto its image. So f n+1 is a positive twist map.

Lemma 4.3. Let f : A → A be a positive twist map with no point with conjugate points and F : R 2 → R 2 be a lift of f . Let x ∈ R.

(i) If Ψ 1 (x) < Ψ -1 (x), then for every y ∈ (Ψ 1 (x), Ψ -1 (x)] the sequence (p 1 •F n (x, y)) n≥0 is strictly increasing and for every y ∈ [Ψ 1 (x), Ψ -1 (x)) the sequence (p 1 •F n (x, y)) n≤0 is strictly decreasing. Moreover, (p 1 • F n (x, Ψ 1 (x))) n≥1 is strictly increasing and

(p 1 • F n (x, Ψ -1 (x))) n≤-1 strictly decreasing. (ii) If Ψ -1 (x) < Ψ 1 (x), then for every y ∈ [Ψ -1 (x), Ψ 1 (x)) the sequence (p 1 •F n (x, y)) n≥0
is strictly decreasing and for every y ∈ (Ψ -1 (x), Ψ 1 (x)] the sequence (p 1 •F n (x, y)) n≤0 is strictly increasing. Moreover, (p 1 • F n (x, Ψ 1 (x))) n≥1 is strictly decreasing and (p 1 • F n (x, Ψ -1 (x))) n≤-1 strictly increasing.

Proof. Let us consider x ∈ R such that Ψ 1 (x) < Ψ -1 (x), the case (ii) can be treated similarly. Recall that f n is a positive twist map for every n ∈ N * . So, for every n ≥ 1, the image by F n of the oriented vertical segment {x} × [Ψ 1 (x), Ψ -1 (x)] is a path that projects injectively on the horizontal axis, with the first coordinate increasing, and that joins F n (x, Ψ 1 (x)) to F n (x, Ψ -1 (x)) = F n+1 (x, Ψ 1 (x)). We deduce that for every y ∈ (Ψ 1 (x), Ψ -1 (x)], the sequence (p 1 • F n (x, y)) n≥0 is strictly increasing and that (p 1 • F n (x, Ψ 1 (x))) n≥1 is strictly increasing. See Figure 2. Since f n is a negative twist map if n < 0, we prove similarly that (p 1 • F n (x, y)) n≤0 is strictly decreasing if

y ∈ [Ψ 1 (x), Ψ -1 (x)), as is (p 1 • F n (x, Ψ -1 (x))) n≤-1 . C F F(C ) ~F ={x} [ (x), (x)] x 1 -1 F( ) F ( ) 2 F ( ) -1 Figure 2. If Ψ 1 (x) < Ψ -1 (x)
, then, for every n ≥ 1, the image by F n of {x} × [Ψ 1 (x), Ψ -1 (x)] projects injectively on the horizontal axis.

Notation 4.1. Denote (12)

X -:= (x, y) ∈ R 2 : Ψ 1 (x) < y < Ψ -1 (x) , (13)
X + := (x, y) ∈ R 2 : Ψ -1 (x) < y < Ψ 1 (x) .
Remark 4.4. Thanks to Lemma 4.3, if z is not fixed by f , then:

(1) either the sequence (p 1 • F n (z)) n∈Z is strictly monotone;

(2) or there exists n 0 ∈ Z such that the sequences (p 1 • F n (z)) n≤n 0 and (p 1 • F n (z)) n≥n 0 are strictly monotone, with different monotonicity;

(3) or there exists n 0 ∈ Z such that the sequences (p 1 •F n (z)) n≤n 0 and (p 1 •F n (z)) n≥n 0 +1 are strictly monotone, with different monotonicity, while p 1 • F n 0 (z) = p 1 • F n 0 +1 (z). Note that the set of points satisfying the second item is nothing but the union of n∈Z F n (X -) and n∈Z F n (X + ) and that these two sets are disjoint because they correspond to complementary monotonicity. Note also that the sets X -and X + are wandering open sets: the sets F n (X -), n ∈ Z, are pairwise disjoint, as are the sets F n (X + ), n ∈ Z.

Let us write CF = C+

F C- F Fix(f ), where (x, Φ 1 (x)) ∈ C+ F if Ψ -1 (x) > Ψ 1 (x) and (x, Ψ 1 (x)) ∈ C- F if Ψ -1 (x) < Ψ 1 (x).
The set of points satisfying the third item is the (disjoint) union of n∈Z F n ( C+ F ) and n∈Z F n ( C-F ). Note also that the last two sets are disjoint from n∈Z F n (X + ) and n∈Z F n (X -).

Lemma 4.4. Let f : A → A be a positive twist map with no point with conjugate points and F : R 2 → R 2 be a lift of f . Let x 1 < x 2 and suppose that Ψ 1 (x 1 ) < Ψ -1 (x 1 ) and Ψ -1 (x 2 ) < Ψ 1 (x 2 ). Then (a) for every y ∈ [Ψ 1 (x 1 ), Ψ -1 (x 1 )] and for every n ∈ N it holds that p 1 • F n (x 1 , y) < x 2 or (b) for every y ∈ [Ψ -1 (x 2 ), Ψ 1 (x 2 )] and for every n ∈ N it holds that p 1 •F -n (x 2 , y) > x 1 .

Proof. As explained in the proof of Lemma 4.3, the union of the images by F n , n ≥ 1, of the vertical segment {x 1 } × [Ψ 1 (x 1 ), Ψ -1 (x 1 )] is the graph of a continuous function χ 1 defined on an interval [x 1 , x 1 ), where x 1 ∈ (x 1 , +∞]. This graph is strictly above Suppose now that f has no point with conjugate points. For every ρ ∈ Q, define G ρ = F q -(p, 0), where ρ = p q , p ∈ Z, q ∈ N * is written in an irreducible way. Applying Proposition 4.1 to G ρ , one deduces that Fix(G ρ ) is the graph of a continuous map ψ ρ : T → R. This graph is invariant by f and it is the set of periodic points of f of rotation number ρ. A periodic point is above this graph if its rotation number is larger that ρ and below if it is smaller. In particular one has ρ < ρ ⇔ ψ ρ < ψ ρ .

Fix n ≥ 0. The maps ψ ρ , |ρ| ≤ n, are uniformly bounded, but they are also uniformly Lipschitz by (vi). So, by Ascoli-Arzelà's theorem the set F n := {ψ ρ : ρ ∈ Q ∩ [-n, n]} is relatively compact in C 0 (T, R). Moreover, it holds that F n ⊂ G(f ) and that the image of F n by ρ, being compact, is equal to [-n, n]. One knows by (iii) and (iv) that ρ is injective on F n and consequently induces a homeomorphism from F n to [-n, n]. We note ρ → ψ ρ its inverse. The set F := {ψ ρ : ρ ∈ Q} can be written as F = n≥1 F n . This implies that our original family (ψ ρ ) ρ∈Q can be extended to a continuous and injective family (ψ ρ ) ρ∈R . The function z → ρ(z) is locally constant on Fix(f ) and so bounded on every compact set of A ∩ Fix(f ). This implies that lim n→+∞ ψ n (x) = +∞ and more generally that lim ρ→+∞ min ψ ρ (x) = +∞ because ψ ρ < ψ ρ . Similarly, it holds that lim ρ→-∞ max ψ ρ (x) = -∞. This means that the map ρ → ψ ρ is proper and so defines a homeomorphism from R to F = n≥1 F n . Let us fix x ∈ T. The map ρ → ψ ρ (x) is continuous, increasing and proper. So it defines a homeomorphism from R to R. More precisely, the map A (x, ρ) → (x, ψ ρ (x)) ∈ A is a homeomorphism, which means that f is C 0 -integrable.

Remark 4.5. The proof of Theorem 1.2 can be adapted to any f -invariant bounded essential subannulus. Let ψ : T → R be an f -invariant essential curve and let U be the essential subannulus that is the connected component of A \ ψ(T) lying above the curve. Let ρ be the rotation number of ψ. We can consider the set F of functions ψ ρ only for ρ ∈ Q, ρ > ρ. Then the exact symplectic twist map restricted to U f |U : U → U is C 0 -integrable if and only if f |U does not have points with conjugate points. The adapted argument holds for the connected component lying under the f -invariant curve. Thus, the result holds for any f -invariant bounded essential subannulus by considering it as intersection of a lower and an upper f -invariant unbounded essential subannulus.

In the end, we want to present an alternative proof for the result in [START_REF] Cheng | A necessary and sufficient condition for a twist map being integrable[END_REF] that is due to an anonymous referee. His/her proof uses the results in [START_REF] Bangert | Mather sets for twist maps and geodesics on tori[END_REF] and so a different approach from our one. In Proposition 1 in [START_REF] Cheng | A necessary and sufficient condition for a twist map being integrable[END_REF], Cheng and Sun prove that if an area preserving twist map has no points with conjugate points, then the Hessian matrix of the second partial derivatives of the associated action functional is positively definite. Using Bangert's notations (see [START_REF] Bangert | Mather sets for twist maps and geodesics on tori[END_REF]), the referee observes that for each rational p q , it holds that p 0 (M per p/q ) = R1 . Otherwise, there exist neighbouring elements x -< x + in M per p/q and, by the Mountain Pass Lemma, we find a minmax (p, q)-periodic x * between x -and x + : this would contradict Proposition 1 in [START_REF] Cheng | A necessary and sufficient condition for a twist map being integrable[END_REF]. Repeating then the same arguments used in the proof of Theorem 1.2, one can conclude.

Here M per p/q corresponds to the set of periodic orbits that minimize the action functional and have rotation number equal to p/q.

Sorbonne Université, Université de Paris, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, F-75005 Paris, France et Institut Universitaire de France E-mail address: patrice.le-calvez@imj-prg.fr

Acknowledgements. The first author is very grateful to prof. Marie-Claude Arnaud and Andrea Venturelli for precious advices and discussions. The authors thank the anonymous

the graph of Ψ 0 = max(Ψ 1 , Ψ -1 ) except at its left end. Similarly, the union of the images by F -n , n ≥ 1, of the vertical segment {x 2 } × [Ψ -1 (x 2 ), Ψ 1 (x 2 )] is the graph of a continuous function χ 2 defined on an interval (x 2 , x 2 ], where x 2 ∈ [-∞, x 2 ), and this graph is strictly above the graph of Ψ 0 except at its right end. These graphs are contained in n∈Z F n (X -∪ C-F ) and n∈Z F n (X + ∪ C+ F ) respectively and so are disjoint. By the Intermediate Value Theorem applied at χ 1 -χ 2 , one deduces that x 1 ≤ x 2 or x 1 ≤ x 2 . Proposition 4.1. Let f : A → A be an exact symplectic positive twist map with no point having conjugate points and F : R 2 → R 2 be a lift of f . Then it holds that CF = {z ∈ R 2 : F (z) = z}.

We want to prove that Ψ 1 = Ψ -1 to get the proposition. We will argue by contradiction supposing that

is on the left of the vertical {x 2 }×R (oriented as y increasing) or n≥0 F -n (X + 2 ) is on the right of {x 1 }×R. We will suppose that we are in the first case, the other case can be treated similarly. We also know that n≥0 F n (X - 1 ) is on the right of {x 1 }×R and so n≥1 F n (X - 1 ) is below the graph F ({x 1 }×R) because F is a positive twist map. But we have seen in the proof of Lemma 4.4 that n≥1 F n (X - 1 ) is above the graph of Ψ 0 = max(Ψ 1 , Ψ -1 ). We conclude that n≥1 F n (X - 1 ) is bounded. We have seen in Remark 4.4 that the F n (X - 1 ), n ≥ 1, are pairwise disjoint. The conclusion comes from the fact that F preserves the Lebesgue measure.

Proof of Theorem 1.2. Theorem 1.2 can be deduced easily from Proposition 4.1 and from classical results about twist maps that we will recall and that can be found, for example, in [START_REF] Michael | Sur les courbes invariantes par les difféomorphismes de l'anneau[END_REF]. Let f : A → A be an exact symplectic positive twist map. We fix a lift F : R 2 → R 2 of f and will refer to this lift, while talking about rotation numbers. We endow the space C 0 (T, R) of continuous maps ψ : T → R with the uniform topology. We denote G(f ) the set of functions ψ ∈ C 0 (T, R) such that the graph of ψ is invariant and define ρ(ψ) as being the rotation number of the graph of ψ.

(i) For every ψ, ψ ∈ G(f ), we have

(iv) If there exists ψ ∈ G(f ) such that ρ(ψ) = ρ ∈ Q and such that its graph contains only periodic points, there is no map ψ = ψ in G(f ) such that ρ(ψ ) = ρ. (v) The map ψ → ρ(ψ) is continuous in G(f ). (vi) By Birkhoff's theory, for every compact set K ⊂ A, there exists M K > 0 such that ψ is M K -Lipschitz if ψ belongs to G(f ) and if its graph is included in K.

referees for their suggestions and comments which helped to improve the presentation of the paper.


An accessible and transparent pipeline for publishing historical egodocuments

Alix Chagué1 and Floriane Chiffoleau 1 1 ALMAnaCH, Inria, Paris, France, name.surname@inria.fr March 17, 2021

Presentation script

Why a pipeline ?

When working on the transcription and edition of a corpus of archival documents, we often find ourselves in two types of situations:

• Either we're running a blackbox that does almost everything we want, but gives us no power to intercept data anywhere in the pipeline except at the end, and doesn't let us customize a specific step that would be of importance to the project.

• Or we're led to use what we can call a scattered toolbox, made of various software, which exposes us to several risks:

the obsolescence of a single one of these tools would question the integrity of the whole scenario;

the portability of data from one software to the next is not guaranteed and might force us either to waste time or to sacrifice a portion of our (meta)data.

Finding a middle ground between these two extreme scenarios is key to ensure an efficient processing of a corpus where the integrity of the data is guaranteed and where the user has complete power over every step of the pipeline. So within the DAHN project, we decided to develop and document a turnkey pipeline for scientific edition which meets these stakes.

The DAHN project

In 2020, Inria, the EHESS and the University of Le Mans created a scientific and technological partnership, with the support of the French Ministry of Higher Education, Research and Innovation (MESRI), under the project named DAHN (Dispositif de soutien à l'Archivistique et aux Humanités Numériques) 1 .

Its goal is to facilitate the digitization of data extracted from archival collections and their dissemination to the public in the form of digital documents in various formats and/or as an online edition.

One of the main productions of this project is indeed the documentation of a complete scenario to automatically create a scientific digital edition starting from a corpus of physical documents. As a way to emphasize the importance of intermediary products and modularity, attention is paid to the reusability of the data at any point in the scenario.

A most remarkable trait of the project is also its interdisciplinarity. By mixing these three institutions, the group is made of engineers and researchers, who provide different points of view on the same project, definitely rooting it in the field of Digital Humanities.

The mission of the ALMAnaCH team is to choose, develop and enhance the tools that will be part of the pipeline created for the transformation of the corpus. We have to think about these tools in terms of adequation to the user's goals, ergonomics, accessibility, and, obviously, the portability of formats.

The corpus I: Paul d'Estournelles de Constant correspondence

To elaborate and test our pipeline, we use a study corpus made of a correspondence between two diplomats dating from the First World War period, and continuing a few years after. During this period, the French senator, Paul d'Estournelles de Constant, wrote to his American friend and colleague, Nicholas Murray Butler and exposed the French situation as well as his defense for Peace. One of the major points that determined our choice is the duality in this corpus: these letters are from the private sphere, they're a personal correspondence where d'Estournelles shares his experience and his opinion. But they bear a strong public dimension, because they were intended, beyond their initial recipient, to be published in American newspapers as a report of the war and a motivation for the United States to engage in it.

This corpus is of genuine historical interest but it is also a perfect test subject because of its size and content. The correspondence is made of about 1 500 letters, 430 from the war period alone (from April 15, 1914to November 19, 1918), with lengths varying from one page to several tens which means that it is a big enough corpus for automatic transcription to be relevant. 2

The corpus II: Digitization of the corpus

The first step of our pipeline and as a matter of fact, of every digital processing of an ensemble of documents, is its digitization. Our corpus is peculiar because its digitization is not homogenous. We started our experiments with a set of photographs taken for another use by a member of the project prior to its creation. It sometimes results in cropped images, blurred lines and incomplete pages which can slow or complicate the process, but it also shows that for the most part, the pipeline does not require high standards images to function.

We are currently in discussion with the Archives Départementales de la Sarthe, the institution where the collection is kept, to start an institutional and exhaustive digitization campaign with proper material. The goal here is to be provided with high quality images for the subsequent publication. Once we have those images and their data, we will import them into a IIIF server, in order to have them available for our web application.

The pipeline

The pipeline we are developing aims at going from point A (the document is digitized) to point B (the document is published). To do so, there is a few steps that need to be executed:

• digitization,
• segmentation,

• transcription,

• post-process correction,

• encoding ; and

• publication. In our scenario, we want to execute these steps one after the other without encountering problems, such as the ones we could have with a blackbox or a scattered toolbox.

The core software in our pipeline is eScriptorium, but we also rely on several tools and frameworks such as IIIF for the distribution of images, TEI XML for the encoding of the transcriptions and TEI Publisher for their publication. For now, Python programs -which is the language used to develop the eScriptorium application-are used at different points to serve the data from one step to the next.

The OCR engine: a capstone

The capstone in our scenario is thus the use of an OCR engine to produce a transcription. Various software are available, but we decided to base our pipeline on the eScriptorium and Kraken solution.

eScriptorium is a web interface for collaborative and automatic transcription projects: it enables the user to load sets of images, to associate them with metadata, to perform segmentation and layout analysis, to perform transcription, to train models for these two steps ; and to export the resulting data in various formats as well as to export the models. The platform currently uses Kraken as a core OCR engine but it is intended to be an agnostic shell for any OCR engine if one wishes to use another solution.

There are several reasons for our choice:

• there are historical ties between the ALMAnaCH team and SCRIPTA (the EPHE project team behind eScriptorium);

• the platform gives the user a lot of freedom in terms of sharing data, documents and models and its interface is rather intuitive;

and most importantly:

• it uses up-to-date import and export standards such as ALTO XML and PAGE XML;

• it is entirely open-source, so not only are we sure never to be stuck with a blackbox, but we're also able to build adds-on for the platform to gather into one place every tool we need.

Overall we've designed a pipeline relying exclusively on open-source services and software and we intend it to comply with the FAIR principles3 .

The pipeline step by step Segmentation and Layout Analysis

The segmentation and annotation of the layout is taken on by a system of line and region tagging coupled with an ontology. It is therefore possible to train with Kraken a segmentation model that automatically tags the types of zones and lines according to a modelization of the layout which fits exactly the needs of a specific project. Of course it is possible to rely on eScriptorium's default segmentation model as well.

It is to be able to build our own modelization and pass it on to a segmentation model because it makes possible the integration of a framework such as the one provided by TEI standards as early as in the lay-out analysis step, which then facilitates the elaboration of a TEI output down the pipeline.

On this note, the DAHN project participates in the SegmOnto4 initiative which is a working group aiming at creating a general TEI-based ontology for HTR projects, within eScriptorium.

Transcription strategy

Then, to generate our transcription model, we coupled ground truth based on our own corpus with a generic model for French typewritten text trained on a corpus of ground truth5 which we created besides6 .

Let's briefly go over a few key rules we applied while creating the ground truth for our transcription model. Our documents contain mostly typewritten text, but they also include handwritten insertions and manual strokes. For these manually added components, we trained our model not to transcribe them but merely to signal them with a symbol (£ or e). Then handwritten insertions are transcribed manually (because there are very few) in a post-processing step.

Post-process corrections 7

The next step is the post-process corrections, which are essential to finish the transcription and have it completely ready for encoding. This phase includes several tasks: some of which are automatically performed, the rest manually (nb: we already mentioned the additions of handwritten insertions).

We use Python programs which include tools and libraries such as pyspellchecker8 , which as suggested by its name, spell-checks a text and suggests corrections based on the use of the Levenshtein distance and a dictionary. The program processes the words in the text, and automatically replaces the misspelled ones with their correction. The corrected text is then reimported into eScriptorium.

At the moment, these Python programs are peripheral to eScriptorium but they can be fully integrated within the platform's workflow thanks to its API.

Encoding

We have now fully transcribed texts, in TEXT format after exportation from eScriptorium, so we can proceed to the encoding. Our corpus is made of egodocuments, which are, to put it simply, documents written in first person reporting personal events. In a case like our corpus of letters, there is a pattern in the layout and the logical structure of the documents which can normally easily be rendered thanks to the TEI framework.

The header is not really specific to egodocuments. It encodes texts as usual, with the addition of a manuscript description to detail the information inherent to the corpus. The only specificity is in the profileDesc, which contains few tags dedicated to correspondence, that are specified and explained in the TEI Guidelines.

For most of the body, we use default text structure. The text is consequently encoded by using tags such as titles and paragraphs, as well as tags to encode the changes made in the text (addition and deletion), the difficulties presented by the corpus (unclear, gaps) and the named entities (person, place, organization). Then, we have to encode the correspondence part and that's where it can become more difficult. There are easy tags such as opener and closer and the tags nested in it like address, signature or postscript but there also are more complicated parts that require choices from us, like letterhead. To develop this XML tree, we used information from TEI Guidelines9 , the TEI mailing-list10 and the Special Interest Group dedicated to Correspondence11 and we wrote an ODD for egodocuments, documenting all the choices we made.

Furthermore, to help with the encoding, we develop Python scripts to encode the metadata using an inventory filled with specific information and the body using regular expressions, helped by the pattern in the layout.

Publication

At this point in the pipeline, we possess images and TEI XML files ready to be published.

The final step consists of importing these TEI files in a specific application developed with TEI Publisher. TEI Publisher is an open-source application based on exist-db, which loads TEI XML files and generates a complete web application using transformations files and templates, to populate web pages. It's a very convenient way to quickly deploy an online edition of a corpus once the TEI edition is ready. Furthermore, it is easily customizable, and extensively documented.

Favoring the use of the pipeline: Documentation

Speaking of, one of the ways to ensure the transparency of a workflow is obviously to document it extensively. This way any person wishing to adapt it to their own corpus and objectives, can easily do so. The presence of documentation was one of our main concerns when developing the pipeline. There is on the one hand the initial documentation provided with the software, standards and framework we use, and then there is the documentation we produce ourselves. As we said before, we developed an ODD in TEI XML and wrote a comprehensive series of guidelines. All of our Python scripts currently existing outside of eScriptorium are properly commented and we publish them in the form of interactive Jupyter notebooks where it is possible to replicate a specific step of transformation. Finally, the data we produce is published on a dedicated Github repository where one can see what the data looks like at any point in the pipeline. It is therefore possible to replicate the entire pipeline from the beginning.

Generalisability & modularity

An experimentation we did with another corpus, the Berlin intellectuals12 , which is a corpus made of a correspondence from the start of the nineteenth century, enabled us to confirm two key aspects of our scenario: its modularity and its capacity to be generalized:

• It is possible to plug data in down the pipeline without following the steps from the beginning: as long as the corpus complies with the encoding rules we established, it is not necessary to have generated the TEI XML with eScriptorium and our scripts, to be able to publish them on our web application.

• Our encoding rules and the publication are not limited by space and time: the Berlin Intellectuals corpus and d'Estournelles' letters span over more than a hundred years and were written in two different languages.

Furthermore, when addressing the question of sustainability, we wish to point to the fact that, on top of using standards widely adopted by the community, we ensure the dissemination of the data in useful formats. This does not simply relate to TEI XML files but also to the ground truth created to generate our transcription models. We contribute to the HTR-United project 13 which is a Commons project for HTR Ground Truth.

Key takeaways

So in terms of what we've gone through during this presentation, here are the key takeaways we would like to reiterate:

• in order to avoid a blackbox or scattered toolbox scenario, we rely on a scenario fully documented and based on open-source software and widely adopted standards;

• we build on the adaptability of the input output formats of software like eScriptorium and TEI Publisher to go beyond our case scenario and consider the importance of modularity and the capacity of the scenario and tools to be generalized;

By way of conclusion, if you wish to use our scenario, all you need is:

• digitized images;

• a transcription system such as eScriptorium/Kraken;

• a TEI modelisation of the structure of your documents;

• and a server to deploy a TEI Publisher application.

13 HTR-United organization: https://github.com/HTR-United/

Figure 1 :

 1 Figure 1: Pipeline for the scientific digital edition of a corpus

The project has a repository: https://github.com/FloChiff/DAHNProject

Schematic representation of the corpus are available in the annexes with the figures 2 and

3 Findability, Accessibility, Interoperability and Reusability

SegmOnto organization: https://github.com/SegmOnto

Tapuscorpus dataset: https://github.com/HTR-United/tapuscorpus

A first model trained from scratch on 84 pages gave unsatisfying results. We partially transcribed 100 more pages focusing on building a train set more representative of the whole corpus. This second model, trained from scratch as well, reached an accuracy of 92.74 %. Lastly, fine-tuning a generic model such as the Tapuscorpus one enabled us to reach an accuracy of 93.90 %. While these accuracy levels suggest that fine-tuning is not more efficient than training from scratch when working with typewritten text, we believe our third model is likely to give more constant results and to show a better robustness to image/text variations.

A schematic representation of the steps is available in the annexes with the figure4

https://pyspellchecker.readthedocs.io/en/latest/

TEI Guidelines: https://tei-c.org/release/doc/tei-p5-doc/en/html/index.html

TEI mailing-list: https://tei-c.org/support/

Correspondence SIG: https://tei-c.org/Activities/SIG/Correspondence/

Berlin intellectuals' website: https://www.berliner-intellektuelle.eu/?en

Annexes