Dr Victoria D S Freitas

Ana Paez

Pascal Fongarland

Dr Régis Philippe

Léa Vilcocq
email: lea.vilcocq@cnrs.fr

Dr Pr P Fongarland

Catalytic Hydrogenation of Hemicellulosic Sugars: Reaction Kinetics and Influence of Sugar Structure on Reaction Rate

Keywords: Arabinose, Galactose, Glucose, Hydrogenation, Langmuir Hinshelwood, Mannose, Ruthenium, Titanium dioxide, Xylose

Hemicelluloses are a major component of lignocellulosic biomass. Different sugars can be obtained from hemicelluloses: xylose, arabinose, galactose, mannose, glucose. Their catalytic hydrogenation produces polyols: xylitol, arabinitol, dulcitol, mannitol, sorbitol, which are valuable chemicals and platform molecules. In this paper, the hydrogenation of sugars was investigated with a Ru/TiO2 catalyst. The influence of temperature, pressure, xylose concentration and catalyst amount was studied for xylose hydrogenation and a Langmuir-Hinshelwood kinetic model was designed. The comparative study of five hemicellulose sugars showed a strong impact of sugar structure on hydrogenation rate: hexoses (glucose, mannose, galactose) react slower than pentoses (xylose, arabinose).

Scheme 1. Hydrogenation of ex-hemicelluloses sugars.

Introduction

Biomass is the only abundant renewable source of carbon for chemicals. [1,2] With the incentive of climate change mitigation and reduction of dependence on oil-based resources, the importance of bioeconomy and bio-based chemistry is gaining importance worldwide [3,4] and particularly in Europe: the European market of bio-based chemicals has recently been estimated at c.a. 177 bn €. For 2030, the target of European Green New Deal has been set at 25% of bio-sourcing in chemistry. [START_REF] Panchaksharam | Roadmap for the Chemical Industry in Europe towards a Bioeconomy[END_REF] Among various sources, lignocellulose is the most promising terrestrial and non-edible biomass. It represents the major part of fibrous plants (e.g. trees, grasses, straws). Lignocellulose is a composite material constituted of three biopolymers: cellulose (40-60%), a gluco-polysaccharide; hemicelluloses (20-40%), which are hetero-polysaccharides; and lignin (15-40%), an aromatic-and phenolicbased biopolymer. [START_REF] Vassilev | [END_REF] Therefore, hemicelluloses are major components of lignocellulose. Their valorisation can play a major role in the future development of biorefineries. Hemicelluloses tether cellulose microfibrils in the cell walls of plants and strengthen them. They present a large variety of polymeric structures (e.g. linear, branched) and of monomer units (hexoses, pentoses, acetyl groups, aldonic acids). They are usually categorized in different types according to their sugar composition: xylans, mannans, β-glucans, xyloglucans, galactans, arabinans and arabinogalactans. [START_REF] Ebringerová | Polysaccharides I[END_REF][START_REF] Scheller | [END_REF] Several hemicelluloses types can be present in a plant in various proportions: xylans are predominant in hardwoods, mannans in softwoods, xylogulcans in grasses. Five sugars are present in hemicelluloses: two pento-aldoses, xylose and arabinose; and three hexo-aldoses, glucose, mannose and galactose (Scheme 1). With the exception of glucose, all these sugars are found exclusively in hemicelluloses. They can be produced by hydrolysis with enzymatic catalysts or with mineral acids. [START_REF] Sella Kapu | Biofuels, Bioprod[END_REF][START_REF] Wyman | Polysaccharides[END_REF] Heterogeneous catalysts have also been investigated for this reaction. [START_REF] Mäki-Arvela | [END_REF]12] The reduction of hemicellulose sugars leads to the corresponding polyols: xylitol, arabitol, sorbitol (from glucose), mannitol, dulcitol (from galactose) (Scheme 1). Sorbitol and mannitol are produced at industrial scales by catalytic hydrogenation of glucose or inverted sugar (hydrolysed sucrose, i.e. mixture of glucose and fructose). Xylitol is produced industrially by catalytic hydrogenation of xylose coming from xylan (from birch or corncob). [13] Xylitol market in 2016 was estimated c.a. 190 kt/year. [14] Polyols have a high added value in pharmaceutical and agro-food industries as low-calories sweeteners and additives (sorbitol = E420; mannitol = E421; xylitol = E967). [13] Because they do not have any petrochemical alternatives, polyols can also be interesting building blocks for organic synthesis, and as such, xylitol and sorbitol have been included in several reports on the most promising molecules from biomass. [15,16] They can also be used in polymers manufacturing. [14] Catalytic hydrogenation of sugars is an industrial process but to our surprise, literature on hydrogenation catalysts for sugars other than glucose and, to a lesser extent, xylose, is rare. Xylose and glucose hydrogenation has been studied with metal supported catalysts. Historically, Ni Raney has been applied for polyol production; however, nickel leaching led to contamination issues for food applications. Among noble metals, Ru was the most active for the conversion of sugars into polyols in literature. [17] It surpassed the performances of other notable metals such as Ni, Rh, Pt and Pd. [17][18][19] Bimetallic PtSn catalysts have also been proposed. [20] Besides, Co catalyst has recently been proposed as an alternative to noble metals for xylose hydrogenation. [21] Due to their good resistance to hydrothermal conditions, the activated carbon and titanium dioxide are good support candidates for sugars hydrogenation. Few publications have compared the reactivity of different hemicellulosic sugars during hydrogenation. It was observed in a few rare studies that sugar structure could influence the hydrogenation reactivity. Zhang et al. observed that hexoses (glucose, galactose, and mannose) required higher temperatures (110°C versus 100°C for pentoses) to achieve the same conversion as pentoses (xylose, and arabinose). [18] Using NiFe nanoparticles as catalysts, Ullah et al. observed higher conversions for xylose, mannose and rhamnose than for arabinose and galactose. [22] With Ru/Al2O3 catalyst, higher yields in polyols where obtained from xylose and fructose than for mannose and galactose at full conversion. [23] Simultaneous hydrogenation of galactose and arabinose was studied by Sifontes Herrera et al. [24] and Müller et al. [25] Both teams observed that arabinose hydrogenation was faster than galactose hydrogenation and estimated similar activation energies for arabinose and galactose; the former team assumed a negligible adsorption of galactose on Ru, whereas the latter team assumed a negligible adsorption of arabinose on Ru. Simultaneous hydrogenation of xylose, glucose and arabinose over Ru catalysts indicated a slightly slower rate of hydrogenation for glucose but this difference was sometimes negligible depending on the catalytic system. [26] In a different scope, Lari et al. observed that the combination of epimerisation and hydrogenation depends on the type of sugar, with higher reaction rate for arabinose, then glucose, then xylose, over Ru/C catalyst doped with Mo. [27] Furthermore, the hydrogenation of sugars mixture produced by hydrolysis of hemicelluloses without purification has been investigated with Ni Raney catalyst on corn straw hydrolysates, [28] with ruthenium catalysts (hydrolysates from pure xylan, [29,30] sugarcane bagasse, [31] from birch [32] or from wheat bran [26] ), or with bimetallic Ru-Ni catalysts (hydrolysates from sugarcane bagasse [33] ). The presence of impurities such as furans or proteins can deactivate the catalyst. Kinetic models have been proposed for sugars hydrogenation with catalysts such as Raney nickel catalyst, [34] Ru/C. Generally, the kinetics follow a Langmuir-Hinshelwood type mechanism, considering the competition between sugar and H2 for the catalytic active site, but as far as we know, no adsorption studies have been carried out to measure the adsorption of the reactants and products. The desorption step of sugar alcohol was never considered in the literature. Kinetic modelling studies have generally been based on the hydrogenation of pure individual sugars or binary mixtures, and more rarely on the hydrogenation of hydrolysates of biomass: a Langmuir-Hinshelwood model has been used for the hydrogenation of sugars in biomass hydrolysates [31] and a pseudo-homogeneous first order model in two other studies. [30,33] In the present paper, Ru/TiO2 catalyst was prepared and applied to the hydrogenation of hemicellulosic sugars. The influence of reaction conditions was investigated for xylose hydrogenation and mass transfer limitations were estimated. A kinetic model based on Langmuir-Hinshelwood mechanism was designed and used to estimate kinetic parameters. The impact of sugar structure on initial reaction rate was investigated and discussed.

Results and Discussion

Catalyst preparation and characterisation Ru/TiO2 catalyst was prepared by incipient wetness impregnation followed by calcination under N2 and reduction under H2 and characterised by several techniques (Table S1 in ESI). The preparation and characterisation of Ru/TiO2 catalysts was detailed in another publication from our group. [35] Briefly, characterisation results show that the catalyst is composed of nanoparticles of Ru°, weakly dispersed on a TiO2 rutile support with a very low surface area (non-porous material). Our previous results have shown that despite of the low dispersion of ruthenium, this catalytic formulation is suitable for highly selective hydrogenation.

Xylose hydrogenation with Ru/TiO2

Xylose hydrogenation was performed in batch mode, at 120°C. Without catalyst, the thermal degradation of xylose was inferior to 10 % after 4 h. In the presence of Ru/TiO2, xylose was completely converted after 2 h and xylitol was formed with a selectivity of ca. 100 % (Figure 1). For the sake of simplicity, Ii the following paragraphs the reaction advancement will be represented through xylose conversion only. Under our operating conditions, xylitol selectivity during xylose hydrogenation was always superior to 97% except when temperature was increased to 140°C (see below).

Influence of reaction conditions in xylose hydrogenation

The influence of experimental conditions on xylose hydrogenation over Ru/TiO2 was determined by varying each reaction parameter independently (Figure 2). 2-a). Varying the catalyst amount from 400 mg to 1200 mg leads to a proportional variation in conversion rate and yield. As expected, there is a clear correlation between the amount of catalyst, i.e. the number of catalytic sites, and the rate of xylose hydrogenation in aqueous phase. Stirring rate. Changing the stirring rate from 400 rpm to 1600 rpm led to a slight increase in the rate of xylose hydrogenation to xylitol, close to the uncertainty of measurement. Therefore, the experiment did not evidence a strong impact of stirring rate and thus of external mass transfer limitation at the gas-liquid interface nor at the solid-liquid interface. These results are consistent with the calculation of Froude numbers (see ESI), showing the absence of external mass transfer limitations. Pressure of H2. Reducing the pressure of H2 from 40 bar to 20 bar did not influence the rate of xylose conversion, evidencing that hydrogen availability in aqueous phase is not a limiting factor in the reaction. The determination of Froude number for gas-liquid mass transfer at the corresponding pressures did not show any limitation of the reaction by hydrogen transfer from gas to liquid (see ESI). Concentration of xylose. Decreasing xylose concentration at constant catalyst concentration (i.e. xylose/catalyst ratio was decreased) led to a drastic increase in xylose conversion rate. Temperature. The effect of temperature was investigated between 80°C and 140°C. A significant increase in reaction rate (i.e. in rate of conversion) was observed when the temperature increased (Figure 3). At 140°C, the xylitol selectivity was only 88% at complete conversion (after 20 min) but was then stable for more than 3 hours, indicating that some by-products may be formed during xylose conversion at high temperature but xylitol itself is not degraded under our reaction conditions. The nature of by-products being formed remains unknown, as no additional peak was observed during HPLC analysis (see ESI, Finally, the following parameters were applied for the rest of the study: mass of catalyst 400 mg, reaction time 4 h, stirring rate 1600 rpm and H2 pressure 40 bar, temperature 120°C and sugar concentration 0.33 M. These reaction conditions guarantee a kinetic regime.

Mass of catalyst (Figure

Kinetics of xylose hydrogenation

The experimental results were used to determine kinetics parameters for xylose hydrogenation over Ru/TiO2. The experiments were carried out by varying the experimental conditions described in Table 1. Mass balances. The reaction was assumed to be 100% selective with xylose as the only reactant and xylitol as the only product. The mass balance in the liquid phase is given by Equation ( 1):

𝑑𝐶 𝑑𝜏 = - 𝑑𝐶 𝑑𝜏 = - 𝑛 𝑉 . 𝑟 = 𝑛 𝑉 . 𝑟 (1)
where CXy and CXoh are the reactant and product concentrations, respectively, VL is the liquid volume inside the reactor, and r is the reaction rate in molxylose.molRu -1 .min -1 . τ stands for the contact time in molRu.min.molxylose -1 , calculated using the number of ruthenium moles (nRu, in molRu) the initial number of xylose moles (nxylose0, in molxylose), and the reaction time t, as given by Equation (2):

𝜏 = 𝑡. 𝑛 𝑛 ( 2
)
Hypothesis for modelling xylose hydrogenation. (i) The absence of internal and external limitations was verified (see above). (ii) The H2 pressure did not influence the reaction rate during the experimental tests (see above) and the reactor was open in H2. Therefore, H2 concentration in water was assumed to be constant and equal to the concentration at thermodynamic equilibrium during the reaction.

(iii) The reaction rate was assumed to follow the Arrhenius law. Kinetic model. Based on these hypotheses and previous kinetic studies from literature, a Langmuir Hinshelwood formalism was considered to set the reaction rate. This rate equation suggests that a surface bimolecular reaction is the rate determining step with an adsorption competition between the sugar and the polyol adsorbed on the active sites of the catalyst. The absence of H2 term in the denominator of the rate expression is justified according the literature where H2 is founded to have little influence on the sugars hydrogenation at high pressure (above 30 bar), presenting low adsorption constants [24] or even not being considered in the kinetic model. [25] For example, Sifontes Herrera et al. estimated KH2 for the galactose or arabinose hydrogenation at least 100 times lower than the adsorption constants found for sugars and polyols. [24] Finally, according to our experimental observation of no influence of partial pressure of H2 on reaction rate (as already mentioned in the paper), we have assumed a zero-order for the numerator term in our kinetic rate expression. Consequently, seeking to improve the estimation of the kinetic parameters and considering the high pressure (40 bar) applied in our experiments, the following reaction rate was proposed. The xylose hydrogenation reaction rate was then defined by Equation ( 3):

𝑟 = 𝑘. 𝐾 . 𝐶 1 + +𝐾 𝐶 + 𝐾 𝐶 ( 3
)
where r is the reaction rate, defined in molxylose.molRu -1 .min -1 , KXy and KXoh are the adsorption equilibrium constants of xylose, and xylitol, respectively, and CXy and CXoh represent the concentrations of xylose and xylitol, respectively, in mol.L -1 .

The rate constant k follows the Arrhenius law, and the adsorption equilibrium constants follow the Van't Hoff equations. To decrease the correlation between the pre-exponential factor/adsorption constant and the activation energy/adsorption enthalpy, a reparametrisation of the Arrhenius and the Van't Hoff equations was applied [36] , as defined by the Equation ( 4), which represents the Arrhenius equation:

𝑘 = 𝑘 𝑒𝑥𝑝 - 𝐸 𝑅 1 𝑇 - 1 𝑇 ( 4
)
where kTref is the frequency factor at the reference temperature (Tref) (mean value on the temperature range of this study). Ea is the activation energy in kJ.mol -1 . T is the temperature in K, and R stands for the universal gas constant (8.314 J.K -1 .mol -1 ). k0 is then calculated from kTref by the Equation ( 5):

𝑘 = 𝑘 𝑒𝑥𝑝 - 𝐸 𝑅𝑇 ( 5
)
The same reparametrisation was applied to adsorption constants following Van't Hoff equations.

Estimation of kinetic parameters. The estimated parameters are presented in Table 2. The estimated activation energy is in the same range of values as those found in the literature with different ruthenium catalysts. Ea [a] 53.1 kJ.mol -1 [37] Ea [b] 21 kJ.mol -1 [17] Ea [c] 46.8 kJ.mol -1 [38] [a] Catalyst 3%Ru/carbon foam, 100-120°C (data from literature) [b] Catalyst 5%Ru/C, 80-125°C (data from literature) [c] Catalyst 1%Ru/zeolite HY, 100-140°C (data from literature)

The parity plot is presented in The selectivity towards xylitol decreases with increasing temperature, suggesting increased by-products formation at high temperatures. This behaviour is observed at 140°C, where the selectivity reaches only 88% at long reaction times, compared to the 100% of selectivity towards xylitol at 120°C. This selectivity decrease is not taken into account in the model, thus leading to a drift in the modelled values at high temperature. It could be interesting to add a reaction to represent the formation of the by-products and therefore extending kinetic modelling to a larger range of temperature.

Effect of sugar type on the rate of hydrogenation reaction

Hemicellulose is a heteropolymer containing various pentoses and hexoses. The main sugars in its composition are xylose, arabinose (pentoses) and glucose, mannose, galactose (hexoses), all of which are aldoses, i.e. the unsaturated carbon is C1, forming alternatively a carbonyl C=O bond or an ether C-O-C bond depending on the conformation of the sugar. Arabinose, glucose, mannose and galactose were hydrogenated individually under the experimental conditions previously optimised for xylose (Figure 5). Xylose reached 93 % conversion after 4 h, arabinose reached 90 % after 4 h with a conversion trend very similar to the one of xylose; glucose reached only 50 % conversion after 4 h, galactose 56 %, mannose 52 %, the three hexoses presenting similar curves of conversion. Each sugar produced only the corresponding polyol, namely xylitol, arabitol, sorbitol, mannitol, dulcitol (see Scheme 1). Surprisingly, the rate of conversion was correlated with the number of carbon atoms: arabinose and xylose were more than 50 % converted after 1 h whereas glucose, mannose and galactose reached 50 % conversion only after 4 h. Initial reaction rates were calculated from the linear regression of sugars concentration vs time at 120°C for 60 min and depicted on Figure 6. The highest initial reaction rate was obtained with xylose hydrogenation (238 molXylose.min -1 .molRu -1 ). Initial rate of arabinose hydrogenation was only 85 % of this value; initial rates of glucose, galactose and mannose hydrogenations were only 42 %, 48 % and 53 % of xylose hydrogenation rate, respectively, demonstrating the impact of carbon atoms number on initial reaction rate but also, to a lesser extent, the role of stereochemistry of the sugars on hydrogenation. Indeed, arabinose and xylose differ only in the conformation of carbon n°4 and glucose, galactose and mannose differ only in the conformation of carbon n°4 (galactose) or n°2 (mannose). The diffusion of sugars into the catalyst pores is the rate-limiting step, or ii) The adsorption of sugar is the rate-limiting step and depends on the sugar structure, or iii) The hydrogenation itself is the rate-limiting step and depends on the sugar structure. Hypothesis i) can be disregarded when comparing xylose and glucose: their diffusion coefficients are very similar [39] and thus have very little impact on the Weisz-Prater modulus used to determine internal diffusion limitations. Indeed, the value of ϕ'xylose is 9.3.10 -5 (see ESI) and the value of ϕ'glucose is 3.9.10 -5 , evidencing an absence of internal mass transfer resistance. However, in cases ii) and iii), an impact of sugar structure in an open-chain form is improbable, since all aldoses would have very similar conformations with an aldehyde group at one end of the carbon chain and a primary -OH group at the other end. It is well known that in aqueous phase all the studied sugars are in large majority (>95%) present in pyranoside form at thermodynamic equilibrium (see ESI for details). [40] In pyranoside conformation, pento-aldoses and hexo-aldoses have different geometries: for the former ones, all carbon atoms are included in 6 members-cycle, whereas for the latter ones, a -CH2OH group is placed at the exterior of the cycle, in  position in relation to the ether bond, and could generate steric hindrance effects slowing down adsorption and/or hydrogenation steps. Further theoretical investigation (for example DFT calculation) would be necessary to support this hypothesis.

Conclusion

The hydrogenation of hemicellulose sugars over a solid Ru/TiO2 catalyst was investigated. Hydrogenation of xylose resulted in the selective production of xylitol. The absence of internal or external mass transfer limitations was verified under the studied reaction conditions. The influence of several operating conditions was established: increasing the concentration of catalyst led to increase the reaction rate; stirring rate and H2 pressure did not have a strong influence on reaction rate; increasing the temperature led to increase the reaction rate, without any impact on selectivity; increasing xylose concentration led to decrease reaction rate. Kinetic modelling was used to determine the kinetic parameters for xylose. The proposed kinetic model followed a Langmuir-Hinshelwood mechanism and was able to predict the results obtained experimentally on the range 80 and 140°C and initial xylose concentrations between 0.08 and 0.33 M. The kinetic model included the competition between the reactants and the products for the active sites of the catalyst. Kinetic parameters were obtained. They allow a good fit between calculated and experimental data and are decorrelated, except for two sets of parameters. More experimental data (varying the initial concentration in xylose, and xylitol for example) could help to resolve this correlation. In addition, it would be interesting to measure experimentally the adsorption of reactants and products on the ruthenium surface to refine the kinetic model. For the first time, a comparative study of hydrogenation of five sugars composing hemicelluloses was performed. We showed that sugar structure has a strong impact on hydrogenation rate over Ru/TiO2: aldo-pentoses react 2 times faster than aldo-hexoses under identical reaction conditions. The presence of α-CH2OH group outside the pyranoside ring for hexo-aldoses is suggested to be responsible for this difference in reaction rates. Further investigation is needed to clarify this point. This result is interesting for the development of future biorefinery processes where different types of sugars can be produced and transformed.

Experimental Section

Materials Ruthenium (III) chloride (RuCl3•xH2O), titanium dioxide rutile and xylose were purchased from Sigma-Aldrich, arabinose, galactose, mannitol, arabitol were purchased from Alfa Aesar, xylitol, mannose, dulcitol, sorbitol, xylitol were purchased from Acros Organics, and glucose from Fischer Chemicals, with purity higher than 98%. All materials were used without further purification.

Catalyst preparation

The catalyst was prepared by incipient wetness impregnation. TiO2 powder was dried 2 h at 120°C. The precursor solution was prepared using the retention volume of the support for ethanol and the amount of RuCl3•xH2O necessary to obtain 1%Ru/TiO2. The solution was added drop by drop to the dry support with a continuous manual stirring until the formation of a homogenous paste. Finally, the paste was dried overnight at 120°C and crushed afterward. The powder was calcined at 250°C under N2 flow and reduced at 350°C under H2 flow in a tubular oven.

Catalytic test

The catalytic hydrogenation of xylose was performed in a 120 mL Top Industrie autoclave batch reactor heated by a jacket and stirred with a Rushton turbine. The reactor is equipped with a gas cylinder and a continuous feed of H2. 80 mL of 0.0825 to 0.33 M sugar solution and 400 to 1200 mg of catalyst (molar ratio Ru/sugar 0.15% to 0.45%) were introduced in the reactor, which was closed and purged with N2 before heating to the reaction temperature. Time zero was determined as the moment when the reaction mixture reached the programmed temperature and the reactor was pressurized with 20 to 40 bar H2. Samples were taken regularly through a sampling valve and filtered with 0.2 µm syringe filters.

Analytical methods

HPLC analysis was performed on a Shimadzu apparatus equipped with a RID detector and a Phenomenex Rezex RPM column at 80°C with pure filtered water as a mobile phase, 0.6 mL.min -1 . Additionally, a Phenomenex Rezex ROA column at 50°C with acidified water as mobile phase (0.005 N H2SO4) was used to confirm the identification of sugars and polyols by comparing the retention times of commercial standards. External calibration with four levels was used for quantification of sugars and polyols. Sugars conversion and polyols yield at time t were calculated from molar concentrations as follows: Concentrations at time = 0 came from the first sample taken at reaction temperature immediately after H2 feeding.

Kinetic modelling tools

Xylose hydrogenation kinetics were modelled in MATLAB® R2020b. Kinetic parameters were estimated by non-linear data fitting, regressing the objective function, based on the sum of least-squares deviations between calculated and experimental data.

Figure 1 .

 1 Figure 1. Xylose hydrogenation over Ru/TiO2 catalyst. Reaction conditions: 120°C, 40 bar H2, 0.33 M xylose, ratio Ru/xylose 0.45%, stirring rate 1600 rpm.

Figure 2 .

 2 Figure 2. Influence of catalyst amount (a), stirring rate (b), H2 pressure (c), xylose concentration (d) on xylose hydrogenation over Ru/TiO2. Reaction conditions: 40 bar H2, 0.33 M xylose, ratio Ru/xylose 0.15%, stirring rate 1600 rpm.

  Fig S2).

Figure 3 .

 3 Figure 3. Influence of temperature on xylose hydrogenation over Ru/TiO2. Reaction conditions: 40 bar H2, 0.33 M xylose, ratio Ru/xylose 0.15%, stirring rate 1600 rpm.

Figure 4 ,

 4 together with an example of model-experiment fit. The kinetic model presented a good fit to the experimental data (R 2 = 0.9818), being able to represent well the effect of the temperature on the conversion of xylose.

Figure 4 .

 4 Figure 4. a) Parity plot of the experimental concentrations and the simulated concentrations. R 2 = 0.9833. b) Example of fitting experimental and calculated data. Reaction conditions: 40 bar H2, 0.33 M xylose, ratio Ru/xylose 0.15%, stirring rate 1600 rpm.

Figure 5 .

 5 Figure 5. Hydrogenation of ex-hemicellulose sugars over Ru/TiO2. Reaction conditions: 120°C, 40 bar H2, 0.33 M sugar, ratio Ru/sugar 0.15%, stirring rate 1600 rpm.

Figure 6 .

 6 Figure 6. Initial reaction rate of sugars hydrogenation over Ru/TiO2. Reaction conditions: 120°C, 40 bar H2, 0.33 M sugar, ratio Ru/sugar 0.15%, stirring rate 1600 rpm.

Table 1 .

 1 Experimental parameters of xylose hydrogenation.

	Parameter	Value
	Number of experimental points	70 (6 or 8 per catalytic test)
	mcata -(0.77 %) Ru/TiO2 (g)	0.4, 0.8, 1.2
	Temperature (°C)	80, 100, 120, 140
	Pressure (bar)	40
	Liquid volume (L)	0.08
	Initial solution	Pure xylose (0.08, 0.19, and 0.33 M) or xylitol (0.08 M) + xylose (0.33 M)
	C0,xylose (mol.L -1 )	0.08, 0.16, 0.33, 0.5
	C0,xylitol (mol.L -1 )	0 or 0.08

Table 2 .

 2 Estimated parameters for the xylose hydrogenation into xylitol kinetic model in batch mode.

	Parameter	Value	Unit
	kTref	22.37 ± 6.4 %	molxylose.molRu -1 .min -1
	KXy, Tref	6.14 ± 14 %	L.mol -1
	KXoh, Tref	3.67 ± 18 %	L.mol -1
	HXy	31.59 ± 66 %	kJ.mol -1
	HXoh	1.45 ± 100 %	kJ.mol -1
	Ea	71.2 ± 21 %	kJ.mol -1 (this work)

Acknowledgements

Stéphanie Pallier (CP2M), Laurent Veyre (CP2M), Ruben Vera (CDHL) and IRCELYON characterization platform are acknowledged for catalyst characterization. This project is funded by ANR grant CHICHE (ANR-18-CE43-0004). Marie-Line Zanota and Vincent Bernardin are thanked for their help on kinetic modelling.

Electronic Supplementary Information

Catalyst characterization; calculations of mass transfer resistance; HPLC chromatograms; thermodynamic equilibrium of sugar forms. Additional references cited within the Supporting Information: [39,[41][42][43][START_REF] Trambouze | Chemical reactors : from design to operation[END_REF][START_REF] Goldberg | [END_REF][46]


Olivier Morin
email: morin@shh.mpg.de.

James Winters

Thomas Müller

Tiffany Morisseau

An overview of the "Color Game" App project

The Color Game gaming app (2018-2019) invited players from all over the world to invent a visual language without words. Participants took part in a referential communication task where a Sender had to indicate a colour to a Receiver, with the help of black and white symbols.

General description

The Color Game app was a gaming app designed to investigate the creation of an artificial language on a large scale, while allowing large numbers of participants to interact synchronously as often as they choose and with a wide variety of partners. At its core is a standard referential communication task (sensu [START_REF] Yule | Referential Communication Tasks[END_REF]. The task was piloted in the lab and used to generate results published in [START_REF] Müller | The Influence of Shared Visual Context on the Successful Emergence of Conventions in a Referential Communication Task[END_REF]) before we turned it into an online game.

The task involves two players. The first player (the Sender) is presented with a "target colour", and his goal is to communicate with the second player (the Receiver) to help them pick the target colour in an array of four, thereby earning points. Senders communicate using black and white symbols that bear no straightforward association with any single hue of colour (Figs. 1 &2). These symbols have been experimentally tested to make sure that they would be neither too easy (evoking too narrow a range of colours), nor too difficult (allowing no colour associations whatsoever). Laboratory experiments show that the Color Game symbols are as ambiguous as desired, since different pairs of participants can use them to solve the communication task above chance, but distinct pairs will associate the same symbol with different colours [START_REF] Müller | The Influence of Shared Visual Context on the Successful Emergence of Conventions in a Referential Communication Task[END_REF].

Independently of any communication task (in the lab or with the app), we have collected data on prior associations between the game's 35 symbols and its 32 colours, by asking 960 participants to freely associate one given symbol with a colour, or the reverse, in one-shot tasks. When confronted with the data generated by the app, this prior association data will allow us to observe and quantify the extent to which communicative conventions may strengthen or override a symbol's pre-existing meanings. To maximise the variability in symbol use, as well as provide the game with a reward structure, the players who start the game are only provided with a random sample of 10 symbols (out of 35), earning the right to use additional symbols progressively as they earn points and ascend to new levels. Fig. 1. A trial in synchronous mode. The Sender (her screen is shown on the left panel) communicates with the Receiver (right panel) to help the Receiver find the target colour (here, the lighter shade of blue), marked (for Receiver) by a dot. The Sender selects symbols from her keyboard on the bottom of the screen. The symbols are arranged in random order; scrolling down would reveal additional symbols. The symbols tapped by the Sender show in the rectangle window in the middle. They also show instantly on the Receiver's screen. In the black bar on top, both players can see the identity of the Sender or Receiver in the language they chose, as well as two symbols, "!" and "?". Either player can tap of one of these symbols. This will highlight and enlarge the symbol for both players and produce a sound. In asynchronous mode, the set-up is similar, but the "!" and "?" symbols are not available.

Unlike most language evolution experiments, our app does not provide players with trialby-trial feedback on the success or failure of communication. A block of 10 trials must be played by both Sender and Receiver for either of them to earn the corresponding points. After every block, the Receiver is told how many of the 10 they got correct, but not given specific information about which ones. Our reason to avoid trial-by-trial feedback is that it would let Receivers know instantly which symbol their Sender associates with which colour, allowing Receivers to learn a Sender's code by mere association. Instead, players must leverage the symbols' pre-existing connotations (vague as they are) to build shared conventions. Our laboratory experiments show that most participant pairs play above chance, and that all above-chance participants achieve significant progress with time, implying that the symbols acquire informative meanings they lacked at the start of the game.

Fig. 2. The 35 symbols used in the game (first four rows).

Bottom row, in grey: the five symbols used for the tutorial and for the videos advertising the game (these symbols are for tutorials only). The tutorial symbols were also used in the advertisement video we released to promote the game. The game's colour space was designed to make all trials as comparable as possible, save for one randomized intervention. Each of the game's 32 colours is drawn from the CIE2000 colour space [START_REF] Luo | The Development of the CIE 2000 Colour-Difference Formula: CIEDE2000[END_REF], chosen because it provides a metric for distance between colour hues ("Delta E") that was built to reflect perceptual distance, as opposed to merely physical quantities. The colours were equal in luminance (L = 55) and saturation (S = 85), with a constant perceptual distance between any colour and its two neighbours of Delta E = 7.8 (Fig. 3). Thirty-two arrays of four colours were formed from this set of 32 colours, by picking every fourth colour along the dimension of hue, using each of the 32 colours as starting point (Fig. 4.). Each colour thus appeared in four different arrays. The arrays were randomly generated in terms of what portion of a Receiver's array is visible to a Sender. In addition to the target colour, a Sender could see some or all of the colours visible in the Receiver's array. This quantity varied from one (only the target) to four (the full array), and was randomised. Every new player, on their first opening of the app (but not later) was greeted with a short tutorial explaining the basics of the game. The tutorial simulated a referential communication game, using dummy symbols that would never be re-used in the normal course of the game (Fig. 2, bottom row), and a colour array randomly picked from the 32 possible ones. The player was presented with a symbol and an array, and asked to point the colour it could refer to. Next, the player was asked to play as Sender and to use one of the dummy symbols to refer to a target colour. They were then told they had completed that step of the tutorial. After this introduction, the players were given a guided tour of the home screen, with pop-up messages in their chosen language.

The app left players free to choose their interlocutors and the format of their interaction. It allowed for both "synchronous" (i.e., live, or in real time) and "asynchronous" play. Senders playing asynchronously simply typed black and white symbols corresponding to the target colour. The symbols were then sent to Receivers along with the colour array, and Receivers figured out the target from these symbols. Their message was always accompanied by the corresponding array of four colours, out of which the Receiver must pick the target. These asynchronous puzzles remained available inside the app indefinitely as long as no one had played them, but each puzzle disappeared as soon as one Receiver had played it. When playing synchronously, both players received feedback on their performance. When playing asynchronously, the Receiver received similar feedback, but the Sender was simply notified that someone had played with their messages, earning them the corresponding points.

Synchronous play required the two players to contact each other and stay connected for as long as they played. This enabled them to communicate in real time, and exchange repair signals consisting of the signs "?" and "!". The players were not told what these punctuation marks precisely meant in the context of the game, and we expected variation in the way they were used. Asynchronous play made it possible for players to interact with a vast number of players at the time of their choosing, greatly enhancing the number of interactions we could observe, while synchronous play allowed us to study interactional properties of communication, such as repair. A "notebook" feature allowed players to write down the meaning of particular symbols (as they figured it). The app's players were free to choose their partners from a vast pool of players (Fig. 5).

To play with another contact, a player either invited them for synchronous play and waited for the invitation to be accepted, or sent an asynchronous message, which could be broadcast to the whole group or sent to a specific individual contact who could open it at a later time. Players were incentivized to play as Senders, so that the app would always contain a sufficient number of high-quality asynchronous puzzles. Receivers had to pay a number of points to the Sender whose puzzle they wanted to play, and Senders received these points regardless of the Receiver's performance. Extra points could be earned by playing an especially difficult, time-limited "speed mode", available to players after a certain level.

Fig. 5. The app's home screen (left) with legend (right).

The colourful logos that identify each contact are randomly generated from a set of black and white pictures and a set of colours.

We expected to see the emergence of a market dynamic, whereby the most skilful players would get more proposals for synchronous play and more takers for their asynchronous puzzles. The app made this more likely by putting each player's last-played-with contacts on top of the list, showing their level, and attaching a distinctive picture to each. The pricing of puzzles, coupled with quality indicators, was meant to discourage the creation of lowquality puzzles or the absorption of high-quality puzzles by negligent players.

Players reached new levels as they earn more points (Table 1). In addition to being symbolic rewards, higher levels allowed players to add new symbols to their keyboard. The allocation of symbols was always random: a beginner would start at level 0 with 15 randomly assigned symbols, rising gradually to a full keyboard of 35. The levels achieved by players of the Color Game, with the number of points required to ascend to each level and the number of symbols that each level gave access to the player (symbols that the player could use when playing as Sender).

Level

Number of points

Level 10 was notable since players at that level and above could choose to play as Senders in stress mode.

Strict player anonymity was ensured by a system of pseudonyms and "cover names". The app recorded no personal identifiers (such as names), but gave each player a public pseudonym, displayed on the "top players" score board. Crucially, however, that pseudonym could not be used to identify a player as a contact to play with. Instead, contacts were only ever known to one another by means of "cover names" that varied from contact to contact and from player to player. Player X knew her contact through a list of randomly generated names that reliably identified contact A, contact B, contact C, and so on. Player Y could also identify the same contacts A, B, C, but by completely different names, so that A's name for X did not correspond in any way to A's name for Y, even though both players could reliably identify and contact A. This ensured that players could not circumvent the app to contact one another in real life, on social networks, etc.

Players were divided, when the app was launched, between two pools of players called "Halves", with each new player being assigned randomly to one half. All interactions between players were restricted to their half, and this primary division stood in place indefinitely. We planned to further divide our pool of players into more plastic subgroupings, to create population divergences or merging events, an important condition of one of our projects (TREES). However, the pool of players was not big enough to make this a safe option, and this project was not carried out.

Participants agreed to have their data collected in an anonymous format and for research purposes in a consent form they filled out at the start of the game. The form and the app itself were approved by the Max Planck Society headquarters' ethical committee (advice n° 2017_05). (See our full statement about data privacy, section 8). In addition to recording all the players' moves as Sender or Receiver, the app also recorded notes taken by the players who made use of "the notebook".

In order to attract the widest possible number of players, the app was translated into four languages (plus English): Chinese, French, Spanish, and German. Later on (September 2018), we added Portuguese (Brazil), Japanese, and Russian (see our deployment log, section 4).

Following an extensive period of alpha-and beta-testing (all beta players involved in the piloting and debugging, most of them friends and collaborators of the authors and developers, were barred from taking part in subsequent stages), the App was launched in two steps. A technical launch (April 26 th , 2018) opened a "seeding period" where the app was restricted to a group of guests recruited through social networks (and excluding anyone connected in any way with the app's development). 80 players were thus brought in. They allowed us to fine-tune the app's points system, and seeded the app with puzzles that new players would be able to solve. Two weeks later (May 15 th ), the app was officially launched and opened to the broader public through three main media: the Google Play store, the App Store (for iPhone), and the website colorgame.net. We set up a mailing contact and a "subreddit" forum to address possible questions and conversations from the players. Since we were moderating this forum, we could make sure that players would not use it to coordinate around shared codes outside the game. We also kept a Twitter account. The deployment log in section 4 below provides a complete chronology of the app's life cycle.

Preregistered predictions & projects

Preregistration process

The Color Game project and the six studies that constitute it were preregistered on the Open Science Framework on April 11 th , 2018. The full registration can be found here: https://osf.io/9pdzk. It includes the source code for the app, a plan for data collection along with criteria for including and excluding data points, and the specific predictions for each of the six projects. Since we did not want the app's users to be aware of our expectations, the hypotheses and methods for our six studies were not made public when the app was launched. They were listed on the Open Science Framework with nicknames that we also use for cross-referencing in the registration documents: FRIENDS, INFORMATION, LANGUAGE, PRIORS, SALIENCE, TREES. At the time the registration took place, we did not communicate with the public about our particular project, but did a presentation of the Color Game as a whole in [START_REF] Morin | What Smartphone Apps May Contribute to Language Evolution Research[END_REF].

The projects

For each project we provide a link to a public OSF repository, which contains -The project's preregistration (or preregistrations) (under "Registrations"); -A link to a preprint of the paper taken from the project, the preprint itself, or a report (except for the PRIORS project, still in progress); -The data and code needed to replicate the preregistered analyses.

2.2.1. FRIENDS (https://osf.io/y2vak/).

(Authors: Morin, Müller, Morisseau, Winters.) This project tested the hypothesis that the precision of symbol-colour mappings would increase through time. Symbols' meanings were defined as the distribution of colours a given symbol was associated with for each Sender. We measured the precision of symbol meaning using entropy (more precisely, a circular measure of entropy specifically designed for this project by Winters). Our prediction was validated, although not in its original preregistered version. This project became a paper that is currently submitted to a journal. 2.2.2. INFORMATION (https://osf.io/7y9pn/).

(Authors: Morin, Winters, Müller, Morisseau.) This project aimed to find out whether the amount of information that a Sender possessed concerning the colour array seen by the Receiver would influence performance in referential communication. Starting from the fact that the number of colours seen by Senders randomly varied between 1 and 4, we calculated the amount of information that each colour shown to Sender revealed about the full array. We hypothesized that this quantity of information would predict performance better than the absolute number of colours seen by Senders. This hypothesis was verified. However, we also found that this effect was crucially modulated by the position of the colour target inside the array (whether it was peripheral or central), something that our model did not predict. This project became a report, not currently submitted to a journal. The goal of this project was to investigate a possible impact of the colour categories present in the players' native language on the categorization of colours through symbol use within the Color Game. Using data from a separate survey on colour naming, we determined the semantic structure of three languages with respect to colours. Confirming our predictions, native and artificial language structure overlap at least moderately. Communicative behaviour and performance were influenced by the shared semantic structure, but only for English-speaking pairs. This project became a paper that is currently submitted to a journal. 2.2.4. PRIORS (https://osf.io/dqhtv/).

(Authors: Winters, Müller, Morin.) This project investigated conventions and biases in the emergence and evolution of symbol-colour mappings. Using a separate task in which participants were asked to associate one of the game's 35 symbols with each of its 32 colours, we were able to assess "prior" values for the probability of associating a colour with a symbol, independently of any communicative interactions. Our predictions concerned the influence of these prior associations over the conventional associations that arose in the Color Game. Analyses for this project are still in progress.

2.2.5. SALIENCE (https://osf.io/f9xzq/).

(Authors: Morin, Müller, Morisseau, Winters.) This project tested the prediction that focal colours (focal meaning here, centrally associated with basic colour terms) are easier to communicate. The association of focal colours with basic colour terms in five languages was asserted with an online survey independent of the app. We predicted that focal colours would be easier to communicate, and that codes indicating focal colours would be more stable through time. These predictions were validated. This project became a paper that is currently submitted to a journal.

2.2.6. TREES (https://osf.io/r7n32/).

(Authors: Morin, Morisseau, Winters, Müller, Greenhill.) The goal of this project was to simulate the evolution of distinct dialects inside the Color Game, and to retrace their emergence with a phylogenetic algorithm. A necessary first step in this project was to "grow" different dialects inside the Color Game. To this end, we had separated the game's players into two "Halves". The app did not allow players from one Half to play with a player identified by their device as belonging to the other Half. We expected that distinct dialects would emerge in each Half, and that different symbols would be paired with different colours. We observed dialectal differences for a small minority of symbols (between 2 and 5 out of 35). Even for these symbols, the differences are subtle. Thus, dialectal differences were not important enough to make the TREES project viable. Carrying out to completion would have involved dividing the two Halves into sub-groups, with the risk of losing too many players and no guarantee of exploitable data. A report on the preliminary results was uploaded.

Open-ended exploration

In addition to the preregistered analyses, we also carried out some non-preregistered descriptive analyses, to see how well players succeeded, when and with whom they played, etc. These analyses were exhaustively recorded in our research diary, reproduced in section 5.

Open data & code

The Color Game dataset repository

The open Color Game dataset is available on the Open Science Framework (https://osf.io/9yc25/). This page contains all the information necessary to explore, use, and cite the data. We provide both the raw data as outputted by the app, and the cleaned-up dataset which we used for all the analyses involved in our six projects (the exact same dataset was used systematically). We provide the code that we used to get from the raw version to the cleaned-up one, and a comprehensive description of all the deletions we made, and why they were made.

Exclusion and inclusion criteria: preregistered rules

We preregistered a number of rules to create, select, or exclude data points. They are copied below:

"The data collection period runs for one year from the official launch of the game. After this period has passed, the data collected by the app will no longer be taken into account for the purpose of testing our original preregistered hypotheses. The app may no longer be maintained or advertised, although it will still be possible to download it for a period of time, and it will keep recording data.

The following players and trials will be excluded from the data: -IP addresses that show cues of bot behaviour: for instance, if an IP plays an unnaturally high number of trials per day; or if a number of machines enter the game at the same hour from one single city, in a country known to host anomalous behaviour on the cyberspace, each machine playing one or a few trials. Such incidents will be signalled in the diary.

-Trials where Sender did not send any symbol will be excluded (this may happen in speed mode).

-All players have the legal right to request the removal of their data from our dataset without giving a reason (see the document on data protection attached to the registration, also here: https://colorgame.net/downloadables/data-privacy-full.pdf).

-Our technical report, finalised after the end of the one-year period, may uncover sources of corrupt data that warrant further exclusion measures. These exclusions will be decided on a technical basis and will apply to all six studies (regardless of what they predict)."

Exclusion and inclusion criteria: departures from the preregistered rules

The main departure from our preregistered rules was that we ended the App's lifespan a month earlier than planned (April 10 th , 2019). Activity had slowed down considerably at that point (see our deployment log, section 4). Other departures were motivated by unplanned bugs (see the report on the Clean-up script for a full description of exclusions).

Other datasets

In addition to the main dataset, we constituted two other independent datasets to be used for specific projects (PRIORS, SALIENCY, and LANGUAGE). These tasks were carried out online with completely independent pools of subjects recruited through Prolific or SurveyMonkey. The PRIORS project used a task that asked participants to pair colours and symbols according to their intuitions, independently of any communicative interactions, to obtain data on prior associations between symbols and colours. The SALIENCY and LANGUAGE projects both made use of data from an online experiment whose participants had to name colours in their native language. These datasets are available on the page of their respective projects.

Open code

The source code for the Color Game app is available on GitHub (https://github.com/ColorGameMPG/ColorGame) and copied on the project's main OSF page (https://osf.io/wq3uz/). The R scripts used to analyse the data for various projects is uploaded on these projects' respective OSF pages. The R script used to clean up the data is provided on the Color Game dataset OSF page.

Color Game deployment log

We kept on the Open Science Framework an online diary of the app's functioning, to document major events (or possible changes) in the game. It is reproduced here. It is also accessible at this address: https://osf.io/eaxkj/ 26.04.2018: The game's "soft launch" has started. Between now and the 15th of May, we will invite around 100 journalists and friends to "seed" the game and explore it before the full public launch. Anyone who has already played the Color Game has received a message discouraging them from playing this version. 27.04.2018: Thomas Müller & James Winters promote the game at a games conference in Berlin, allowing visitors to test the app.

27.04.2018:

A press release was sent to around 50 journalists (English and Frenchspeaking mostly). It included an invitation to access the app before its launch.

04.05.2018:

The "soft launch" has brought around 80 players to download the app. I (OM) have been regularly tuning the game's parameters. [The] threshold of successful trials

21.05.2018:

The affluence of players this week proved taxing for our system. Smartphones need to refresh regularly a set of several hundred profiles, which occasions delays. To avoid this, as a temporary fix, we decided to split the two Halves, Half A and Half B, into two sub-halves (A1, A2 from A, B1, B2 from B). The two splits took place at 9:12 AM CET (recorded as 7:12 in the data, which gives time as GMT). This is intended as a temporary fix: the sub-groups will be merged back as soon as this is doable without running into our current problems. The developers have been instructed to erase all the profiles of users that have not completed the tutorial, to lighten the load. More long-term solutions are being studied.

2018:

The app has now been installed over 2000 times. A little less than 600 players have reached level 2 (Level 1 is reached automatically upon completing the tutorial). Approximately 1000 puzzles (=10 000 trials) have been played each day. Players are rather diverse and come from more than 60 different countries.

29.05.2018:

To fix the issue that was identified on 22.05 (see above), the developers released a new version of the app, where each player only sees other players if they (a)have puzzles available for that player, and/or (b) have interacted with that player, and/or (c) are live. This restriction avoids the proliferation of players that plagued the app and caused bugs. The updated version was released on the 28th for Android (morning) and iOS (evening). Android versions of the Color Game were automatically updated, while iOS versions had to be redownloaded to get the update. Today at 13:13 CET (11:13 GMT), the 2x2 groups that had been split were re-merged into the moieties that they came from. Groups A1 and A2 were re-merged into Half A, and the same happened to B1 and B2. 20.08.2018: A player made us aware that he was able to solve the same puzzle multiple times, which is not supposed to be possible at all. We have been investigating this issue with the developers since then, but the issue, which affects only a small minority of users, proves difficult to replicate on the developers' machines. On the 30th of August, this issue concerned 51 players. We are waiting for the developers to solve this issue. We decided we would not use this data for any analysis. [written 21/09] 05.09.2018: We completed our first analysis of the app's data, concerning the TREES project. We found that the proper conditions to carry on the project, split the two halves into sub-groups, and provoke the growth of distinct dialects, were not met. The project has been aborted. [written 21/09] 15.09.2018: We rolled out version 1.10 of the app for iOS and Google Play. It includes Portuguese (Brazil), Japanese, and Russian language versions of the game. 09.04.2019: We decided last week to retire the app slightly earlier than planned (the planned date was May 15th). The reasons are (a) traffic on the app has slowed down a lot, we still have several visits per day but the trend is downwards (b) the projects that required a one-year data collection period were refashioned to accommodate other scales (based on the progress of individual pairs not the group as a whole, like FRIENDS) or abandoned (like TREES). The website colorgame.net does not feature the links to the app stores since last week, and on Thursday the app will be discontinued. 10.04.2019: We consider that the data collection period is over since today 1 PM. We will be playing with the app with a group of students this afternoon but this data won't be kept.

Descriptive and exploratory analyses

The accompanying R scripts for this section can be found on this page: https://osf.io/rywsd/. This document records simple exploratory analyses on players' performance and frequentation, which were not part of preregistered studies. They are presented in the form of a research diary. No preregistration was made, but analyses were recorded as exhaustively as possible. The analyses were carried out on preliminary versions of the dataset, not on the complete and cleaned-up Color Game dataset. The code we provide should nevertheless be sufficient to replicate our analyses approximately with the final version of the dataset.

22/06/2018. Players' performance in the Color Game through time.

We looked at player performance to see whether it improved with time. It does (Fig. 8). In the next analysis, it could be interesting to tease apart three possible mechanisms for this progress.

A. Players become better decoders. In this scenario, players simply become better at solving any puzzle at all. They become better Receivers, but not necessarily better Senders. B. Messages become more intuitive. In this scenario, players simply become better at producing solvable puzzles, whoever the puzzles are destined for. They become better Senders, not necessarily better Receivers.

C. Players become better at mastering conventions. In this scenario, players become better at using the conventions that they developed for the game; since these are purely conventional, however, this skill only works with players who have also mastered these conventions, not with new players. Players do not become better at solving or creating puzzles successfully with anyone, only with experienced players.

A + B. Convention-independent progress. In this scenario, hypotheses A and B (but not C) are both true. Players become better at solving (any) puzzle, and they also become better at creating puzzles that can be successfully solved by any player.

A + B + C. All of the above. In this scenario, all the above hypotheses are true. Players become better at solving (any) puzzle, and they also become better at creating puzzles that can be successfully solved by any player. In addition, they are even more successful when playing with another experienced player. A pair's performance in the game appears to be driven by the players' experience as a pair, more than the total experience of each player (playing separately), suggesting a crucial role for pair-specific conventions.

We wanted to check that the players' progress in the Color Game was due to the establishment of shared conventions-that players didn't simply improve by playing a lot, independently of establishing conventions with other players. To make sure of this, we considered two variables that could affect the performance of a pair of players. One is the pair's experience as a pair: the number of trials that the two players composing the pair played together. The other is the pair's players' total experience: the number of trials played by either player of the pair, together or separately.

To avoid the confounding effects introduced by the fact that players gain new keyboard symbols with experience, we only studied all the pairs of players where one of the players had, at some point, reached the level at which all 35 symbols are available. Looking at the whole dataset from 26/04 to 22/06, this means 4074 pairs, made up of 1412 distinct players. Fig. 9 shows the result. The full figure was made with plotly and has scroll-over features not accessible here (but see R code). It clearly shows that joint experience is correlated with a performance increase. This could be due to a tendency for players to keep playing with the partners they achieved high scores with. But this seems unlikely: the pairs with very high joint experience perform at extremely high levels (80% success rates being common) which pairs with less joint experience seldom reach. The experience of players outside the pairs, in contrast, does not seem to matter as much, even though the variance for this variable is much greater. This is shown in Fig. 10, which plots performance against a pair of players' general experience, gained together or separately. Most performance scores cluster around round values (0.1, 0.2, etc.), because many pairs only play 10 or 20 trials together. The odd shape of the data in the 10K -20K trials range is probably due to the fact that a few highly experienced players engaged a lot of players with very little experience. This was also a good occasion to see what share of the game's activity could be attributed to what number of players, and identify possible problems with the data. We mapped the frequentation of the app: the number of players, number of trials, when they play and how often, how they perform, and their favourite time of day for playing. Players are asked to fill in a consent form allowing their player data, collected through the app and entirely anonymised, to be used by researchers for scientific purposes. The data collected during the period of time where the app will be in use is to be made publicly available to other researchers in the future, after an embargo period. Any player wishing to have their data removed from the dataset may ask us to do so (at colorgame@shh.mpg.de) and their request will be immediately fulfilled. (Since the app's anonymised IDs do not allow us to identify individual players, the players will need to provide us with a way to identify them: for instance, the exact date and time of their first connection, or the exact date and time of their last five puzzles played.)

This study has been approved by the ethical committee attached to the president of the Max Planck Society (advice n° 2017_05). The advice is attached to this document, followed by our official request for authorisation."

Fig. 3 .

 3 Fig. 3. The game's colour space. Each colour is given its associated Hex code (as used by the app).

Fig 4 .

 4 Fig 4. How colour arrays were built. Top row: The composition of two colour arrays, one marked by white dots, the other by black dots, is shown relative to the colour space. Bottom row: Six contiguous colour arrays (out of 32), including the white-dot and blackdot ones.

  2.2.3. LANGUAGE (https://osf.io/a8bge/).(Authors:Müller, Winters, Morisseau, Noveck, Morin.)

Fig. 6 .

 6 Fig. 6. The Color Game's parameters on May 4 th , 2018. (Screen capture of the app's back end.)

Fig. 8 .

 8 Fig. 8. The average number of correct trials for a set of 10 went from 4 to 7 in the last five weeks. The error bars represent simple variance (not standard deviations). n = 17832 sets of 10 trials.

  07-20-pairXP.R for the data; a few days later we did an updated version, 2018-07-25-pairXP.R. -Figure-2018-07-20.R for the figure.

Fig. 9 .

 9 Fig. 9. Player pairs' performances as a function of their past performance playing together as a pair. Each dot stands for a pair of players (n = 4074 pairs, from 1412 players). Y-axis: the pair's chance of scoring a hit (0 to 1, chance level = 0.25). X-axis: The number of trials played by the pair as a pair. The colour and size of each dot represents the pair's general experience: The number of trials played by either player, separately or jointly.

Fig. 10 .

 10 Fig. 10. Player pairs' performances as a function of their past performance playing together as a pair. Each dot stands for a pair of players (n = 4074 pairs, from 1412 players). Y-axis: the pair's chance of scoring a hit (0 to 1, chance level = 0.25). X-axis: The number of trials played by the pair's players, separately or together. The colour and size of each dot represents the pair's joint experience: the number of trials played by either player, separately or jointly.

  14/08/2018. Another look at pairs' performances.

Fig. 12 .

 12 Fig. 12. The previous figure, with the joint experience log-transformed.

Fig. 13 .

 13 Fig. 13. Player pairs' performances as a function of their past performance playing together as a pair. Each dot stands for a pair of players (n = 5506 pairs, from 1528 players). Y-axis: the pair's chance of scoring a hit (0 to 1, chance level = 0.25). X-axis: The number of trials played by the pair's players, separately or together. The colour and size of each dot represents the pair's joint experience: The number of trials played by either player, separately or jointly.

Fig. 14 ,

 14 15, and 17 cover the time span from May 15 th to September 5 th , 2018 on the xaxis. Fig.16shows player affluence by time of day.

Fig. 14 .

 14 Fig. 14. Number of trials per day.

Fig. 15 .

 15 Fig. 15. Average performance rates per day. The performance increase seen on 22/06/2018 (see that entry) did not continue.

Fig. 16 .

 16 Fig. 16. Number of trials played on the app (y-axis) for each minute of the day (xaxis). Time is given as Universal Time. Central European Time is two hours later (20:00 = 22:00).

Fig. 17 .

 17 Fig. 17. Players visits to the Color Game. Each row represents one player, with the earliest visitors shown at the bottom. Each dot represents one visit on one day, by one player.

Table 1 : Color Game levels.

 1

			Number of
			symbols
	0	0	15
	1	250	17
	2	500	19
	3	750	21
	4	1250	23
	5	1500	27
	6	2000	31
	7	2500	33
	8	3000	35
	9	5000	35
	10	7500	35
	11	15000	35
	12	40000	35
	13	80000	35

Acknowledgements

The Color Game app was designed by the Minds and Traditions Research Group within the Department for Linguistic and Cultural Evolution of the Max Planck Institute for the Science of Human History in Jena, in collaboration with Etter studios (Zurich). The basic principle of the color game was an original idea of Olivier Morin and Tiffany Morisseau in 2015. The game was then developed and investigated experimentally by Olivier Morin, Thomas Müller, James Winters, Lidiia Romanova at the Minds and Traditions Research Group of the Max Planck Institute for the Science of Human History. The blueprint for the app was conceived by Olivier Morin, James Winters, Thomas Müller, Tiffany Morisseau, Lidiia Romanova, and Christian Etter. We are grateful to Helene Kreysa and Simon Greenhill for their helpful comments. The blueprint was implemented by Etter studios in Zurich under the supervision of Christian Etter. The technical supervision for server maintenance and data storage in Jena was assured by Thomas Baumann and his team. The website colorgame.net is administered by Syahrul Anwar at the institute. The app's symbols were designed by Thomas Müller, Lisa Jeschke, Liliana Lovallo, Barbara Pavlek, Piers Kelly, James Winters, Olivier Morin, Lidiia Romanova, Olena Tykhostup. The game's graphic design is the joint work of Michelle O'Reilly, Olivier Morin and James Winters, in collaboration with Etter studios. Translations were the work of Thomas Müller, Olivier Morin, Lisa Jeschke, Charlélie Goldschmidt, Ezequiel Koile, and Tao Li -with special thanks to Messrs. Koile and Li. We are grateful for the help of our Beta-testers. In no particular order: Teresa Blasco, Alina Shron, Pierrick Bourrat, Réka Finta, Abel Gerschenfeld, Charlélie Goldschmidt, Liem Binh Luong Nguyen, Olivier Mascaro, Pierre-Yves Le Coq, Barbara Pomiechowska, András Szanto, Radu Umbres, Laëtitia de Villepin, Rosie Ivády, Chris Winters, Katie Finch, Matt Spike, Stefan Becker, Suzan Huijgen, Maria Brackin, Seán Roberts, George Dunford, Claudia & Stefan Geßlein, Thomas Borchert, Sarina Parschick, Mathias Löbl, Anneke Hamann, Lisa Kerscher, Jana Koplanski, Julia Belger, Judith Lidzba, Elisabeth Wienß, Susanna Sabin, Ezequiel Koile, Sarah Wand, George Taylor, Ian Gillespy, Raul Azevedo, Camille Kerdanet, Cécile Eymond, Johan Morisseau, Maximilien Quesnel, Jérôme Redouté, Laurent Jacquod, Ira Noveck, Katell Lavéant, Katrina & Barbara Pavlek. The game's promotion is assured by the Grafikerorg agency (Jena). We are grateful to Aditya Kumar Lankapalli, Eirini Skourtanioti, Hiba Babiker, Cosimo Posth, and Xintian Li, who acted in the advert. Inside the Max Planck Institute for the Science of Human History, we are especially grateful for the help of Lidiia Romanova, Lisa Jeschke, Anne Gibson, Petra Mader, Christian Nagel & Barbara Pavlek. The work leading to this paper has received funding from the "Frontiers in Cognition" EUR grant, ANR-17-EURE-0017 EUR.

Code:

-PairExperience.R

We wrote up a much more elegant version of the code used for the preceding entry. We did a re-run of this code on the updated dataset. Results in Fig. 111213.

Code:

-CGFrequentation-2018-10-02.R.

We plotted the peak traffic times of the day with a circular plot (Fig. 18).

Creative Commons Licence

The Color Game dataset and the present paper are distributed with a Creative Commons Attribution 4.0 International Public License.

Ethics and data privacy

The following information was displayed on the colorgame.net website, as part of a downloadable pdf document that also contained the official ethical approval for the project.

"The Color Game app stores smartphone identifiers securely and confidentially. It also records, in fully anonymised format, the players' actions within the game. This fully anonymised data will be shared with the scientific community, but no personal data will be. Participants are free to opt out of our research and ask for their data to be erased. This study has been approved by the ethical committee attached to the president of the Max Planck Society.

The app identifies its players' phones (via UDID on iOS, UUID on Android) to make sure that players do not have to restart the game at the basic level every time they open the app. The app also checks its players' IP addresses (but does not record it). These data, along with all the data used by the Color Game app, are exclusively stored on the secure servers of the Max Planck Society. The app does not require or record the following information: names, email addresses, geolocalisation tags.

Inside the game, players identify their contacts by randomly generated aliases that change for every player pair (so that A is known to be by a certain name, but known to C by a different name), making it impossible for contacts to identify one another outside the game. In addition, players are identified on the scoreboard by a pseudonym that is randomly generated by the app (although players are free to reject pseudonyms that they dislike).

The researchers of the Max Planck Society access a fully anonymised subset of the data generated by the app, which they store and exploit for research purposes, and may share with other researchers. It is a record of the players' moves in the game in anonymised format. For instance, the colours picked by Receivers and the symbols sent by Senders are recorded for every puzzle. The time at which puzzles are played is also recorded. The IP addresses, UDID/UUID, aliases, and pseudonyms will not be part of that dataset kept for research purposes.


Ugo Boscain
email: ugo.boscain@sorbonne-universite.fr

Eugenio Bellini
email: eugenio.bellini@outlook.it

Surfaces of genus g ≥ 1 in 3D contact sub-Riemannian manifolds

We consider surfaces embedded in a 3D contact sub-Riemannian manifold and the problem of the finiteness of the induced distance (i.e., the infimum of the length of horizontal curves that belong to the surface).

Recently it has been proved that for a surface having the topology of a sphere embedded in a tight co-orientable structure, the distance is always finite. In this paper we study closed surfaces of genus larger than 1, proving that such surfaces can be embedded in such a way that the induced distance is finite or infinite. We then study the structural stability of the finiteness/not-finiteness of the distance.

Introduction

Consider a three dimensional contact sub-Riemannian manifold (M, D, g), where M is a smooth manifold of dimension three, D is a contact distribution (i.e., such that rank D = 2 and D + [D, D] = T M ) and g is a sub-Riemannian metric [START_REF] Agrachev | A comprehensive introduction to sub-Riemannian geometry[END_REF][START_REF] Montgomery | A tour of subriemannian geometries, their geodesics and applications[END_REF]. Let S be a surface embedded in M . The intersection D ∩ T S defines a field of directions with singularities on S, where the singularities are the points q ∈ S such that D q = T q S. Such points are called characteristic points. In the following the set of characteristic points is denoted with Σ(S). The set of integral curves of D ∩ T S on S ∖ Σ(S) is called the characteristic foliation of S, and it is denoted with DS. For an example of characteristic foliation of a surface in the Heisenberg group see Figure 1. Each leaf is a one-dimensional Riemannian manifold, the metric being the restriction of g. A lipschitz curve γ ∶ [0, T ] → S is called admissible if γ(t) ∈ D γ(t) for almost every t ∈ [0, T ]. We can use the metric g to measure the length of admissible curves on S L(γ) ∶= T 0 g( γ(t), γ(t)) dt, and we define the induced distance between two points q 0 , q 1 ∈ S as

d S (q 0 , q 1 ) = inf{ L(γ) ∶ γ admissible, γ(0) = q 0 , γ(T ) = q 1 }, (1.1)
with the understanding that the induced distance is infinite whenever there is no admissible curve connecting q 0 to q 1 . In this paper we are interested in studying the finiteness of d S . We say that the induced distance is finite if it is finite for every pair of points, in that case the couple (S, d S ) defines a metric space and in particular a length space. Notice moreover that this distance does not coincide with the restriction of the sub-Riemannian distance on S, moreover d S is never continuous with respect to the topology of the surface [D B22]. If the sub-Riemannian manifold admits a global orthonormal frame (F 1 , F 2 ), or equivalently if D is a trivial vector bundle over M , then the problem of finding the curve realizing the distance between two points q 0 and q 1 can be written as the optimal control problem

γ(t) = u 1 (t)F 1 (γ(t)) + u 2 (t)F 2 (γ(t)), γ(0) = q 0 , γ(T ) = q 1 , T 0 u 2 1 (t) + u 2 (t) 2 dt → min, with the state constraint γ(t) ∈ S.
In the study of contact distributions on 3-manifolds from a topological perspective, embedded surfaces play a central role [START_REF] Bennequin | Entrelacements et équations de Pfaff[END_REF][START_REF] Giroux | Convexité en topologie de contact[END_REF][START_REF] Giroux | Structures de contact en dimension trois et bifurcations des feuilletages de surfaces[END_REF]. Recently related topics have been studied when the contact distribution is endowed with a Riemannian metric (surfaces embedded in 3D contact sub-Riemannian manifolds). See for instance [DGN12; DGN07] for Carnot groups, [D B22] for generic structures, [BTV17; BTV20; Vel20] for Gauss-Bonnet theorems and [START_REF] Barilari | Stochastic processes on surfaces in threedimensional contact sub-Riemannian manifolds[END_REF] for stochastic evolution equations. In particular in [D B22] the authors have proved that the induced distance is always finite for spheres embedded in a cooriented tight 3D-contact sub-Riemannian manifolds. Recall that a 3D-contact sub-Riemannian manifold (M, D, g) is co-orientable if the distribution D can be globally expressed as the kernel of a 1-form ω. This form is called the contact form, and in the context of sub-Riemannian geometry is normalized in such a way that

dω D = Vol g , (1.2)
where Vol g is the area form induced on D by the metric g. The contact condition D + [D, D] = T M can be expressed in term of the contact form ω as ω ∧ dω ≠ 0, thus a co-orientable contact manifold M is necessarily orientable. Recall moreover that a contact distribution is called overtwisted (see 4.5 of [START_REF] Geiges | An introduction to contact topology[END_REF]) if admits an overtwisted disk, i.e., an embedding of a disk with horizontal boundary whose characteristic foliation has a unique singular point: an elliptic point in the interior of the disk, see Figure 2.

∶ z = 1 2 (π 2 -r 2 ), r ≤ π, θ ∈ [0, 2π) } in the contact structure (R 3 , D = span{∂ r , cos r r ∂ θ -sin r∂ z }), expressed in cylindrical coordinates (r, θ, z).
A contact distribution is called tight if it is not overtwisted. Since every overtwisted contact structure contains a sphere S having closed orbits in its characteristic foliation, which necessarily implies that d S is not finite, the result obtained in [D B22] can be stated as a metric characterization of tightness.

Theorem 1. ([D B22]) Let (M, D, g) be a co-oriented 3D-contact sub-Riemannian manifold, then (M, D) is tight if and only if every embedded surface homemorphic to a sphere has finite induced distance.

In this paper we prove that the characterization given by Theorem 1 works only with spheres. In particular we obtain the following result.

Corollary 1. Let (M, D, g) be a co-orientable 3D-contact sub-Riemannian structure. Let S be a closed orientable surface of genus g ≥ 1. Then there exist an embedding of S in M for which the induced distance is finite, and one for which the induced distance is not finite.

The latter result is a corollary of the following theorem, stating that generically the finiteness (not finitess) of the distance is a structurally stable property.

Theorem 2. Let (M, D, g) be a co-orientable 3D-contact sub-Riemannian manifold, let S be a closed orientable surface of genus g ≥ 1, and let Emb(S, M ) be the space of embeddings of S in M endowed with the C ∞ -topology. Then there exist two disjoint non empty open subsets, U f , U ∞ ⊂ Emb(S, M ), having dense union, such that the induced distance is finite on U f and not finite in U ∞ .

The proof of this theorem is largely based on the theory of Morse-Smale vector fields, which are structurally stable. In Section 2 we recall the basic facts concerning these vector fields. We prove that the only obstruction to the finiteness of the induced distance on Morse-Smale characteristic foliations is the existence of closed orbits. We then recall the elimination lemma, a result from contact topology which we use to break up the closed orbits of a Morse-Smale foliation by means of an arbitrarily small C 0 perturbation. These facts together ensure the existence of the set U f of Theorem 2. In Section 3 we prove the existence of the set U ∞ and we conclude the proof of Theorem 2.

Morse-Smale vector fields

The characteristic foliation of an oriented surface S embedded in a co-oriented contact manifold (M, ω) is driven by a global vector field: there exists a smooth vector field X on S vanishing on Σ(S), satisfying

T q S ∩ D q = span R X q , ∀ q ∈ S ∖ Σ(S).
(2.1)

A vector field with the properties described above is called a characteristic vector field of S. Characteristic vector fields are not unique, and can be obtained one from the other via multiplication by a positive function. In particular the choice of a characteristic vector field corresponds to the choice of an area form Ω on the orientable surface S, compatible with the orientation of the latter. Indeed, once the area form Ω is chosen, the corresponding characteristic vector field is the unique solution to the equation

i X Ω = ω S .
(2.2)

The contact condition ω ∧ dω > 0 implies that the divergence div Ω X(q) is non zero for every q ∈ Σ(S):

d ○ i X Ω q = div Ω X(q)Ω q ≠ 0. (2.3)
The sign of a critical point q ∈ Σ(S) is defined as sign(q) ∶= sign(div Ω X(q)).

(2.4)

For a generic surface S, the characteristic vector field is Morse-Smale: a type of vector field with particularly simple dynamical features, recalled in the definition below.

Definition 1. Let S be a closed orientable surface, a vector field X on S is called Morse-Smale if

• X has finitely many critical points and closed orbits, all of which are non degenerate,

• the α-limit of every trajectory is either a critical point or a closed orbit, and the same holds for ω-limits,

• there are no saddle connections.

In its celebrated stability theorems, M.M. Peixoto showed that such vector fields are generic and structurally stable on closed orientable surfaces ([Pei62],[M C59]). These results are summarized in the following theorem.

Theorem 3. Let X be a Morse-Smale vector field on a closed orientable surface S. The dynamic of any vector field X ′ sufficiently C 1 -close to X is topologically conjugated to the dynamic of X. The homeomorphism realizing the topological equivalence can be chosen C 0 -close to the identity. Morse-Smale vector fields of class C r , with 1 ≤ r ≤ ∞, on a closed orientable surface form an open and dense subset of the set o vector fields on S endowed with the C r -topology.

Actually, Morse-Smale vector fields are generic among the characteristic vector fields of closed orientable surfaces in contact 3-manifolds, as it follows from the following proposition. Proposition 1. [Proposition 4.6.11 of [START_REF] Geiges | An introduction to contact topology[END_REF]] Let S be a closed orientable surface embedded in a co-oriented contact manifold. There exists a surface S ′ , isotopic and C ∞ close to S, having a Morse-Smale characteristic foliation.

One clearly necessary condition for a surface to have finite induced distance is the absence of closed orbits. For a surface with a Morse-Smale characteristic foliation this condition is also sufficient.

Lemma 1. Let S be a closed orientable surface embedded in a co-oriented 3Dcontact sub-Riemannian manifold M , having characteristic foliation driven by a Morse-Smale vector field X. Then the induced distance d S is finite if and only if X does not have closed orbits.

Proof. As we have already said, if X has closed orbits, then the induced distance is not finite. Assume now that X does not have closed orbits. The Morse-Smale property ensures that any leaf of the characteristic foliation is admissible, and hence has finite length: the finiteness of these lengths is proved in [D B22] Proposition 1.3, and it is a consequence of the fact that the integral curves of X contained in the stable manifold of a non-degenerate critical point q converge sub-exponentially to q. Denoting with A x,y the set of admissible curves joining x to y, we only need to show that A x,y ≠ ∅ for any x, y ∈ S. Let S ′ be the complement of the hyperbolic points (saddles) in S, which is open and connected. By definition of Morse-Smale vector field, every saddle is connected to some elliptic point, thus it is sufficient to show that A x,y ≠ ∅ for any x, y ∈ S ′ . For every x, y ∈ S we define the equivalence relation Given any orientable surface S embedded in a 3D-contact manifold, with an arbitrarily small C ∞ perturbation, we can achieve an isotopic surface with characteristic foliation of Morse-Smale type. The new surface may well present closed orbits, even if the original surface S did not have any. The elimination lemma, a result due to Giroux which we are about to state, describes a procedure which allows us to destroy closed orbits preserving the Morse-Smale property, by means of an arbitrarily C 0 -small perturbation of the surface, see Figure 3 and4. Lemma 2. [Lemma 4.6.26 of [START_REF] Geiges | An introduction to contact topology[END_REF]] Let (M, D) be a three dimensional coorientable contact manifold and S and embedded closed oriented surface with characteristic foliation of Morse-Smale type. Assume that there exist two singular point of the same sign, one elliptic point q e and one hyperbolic point q h , connected by a a separatrix γ of the latter. Let U be an arbitrarily small neighbourhood of γ. Then there exists an isotopy ϕ t ∶ S → M , t ∈ [0, 1], with the following properties:

x ∼ y ⇐⇒ A x,y ≠ ∅, ( 2
• ϕ 0 ∶ S → M is the inclusion of S in M ,
• ϕ t can be chosen arbitrarily C 0 -close to ϕ 0 ,

• The isotopy is fixed on γ and outside U ,

• ϕ 1 (S) has a characteristic foliation of Morse-Smale type,

• the characteristic foliation of ϕ 1 (S) has no singularities in ϕ 1 (U ). The elimination lemma allows us to reduce the number of singularities in the characteristic foliation, without destroying the Morse-Smale property. Of course, this procedure can be reversed: we can use the elimination procedure to increase the number of singular points. In particular by means of an arbitrarily small C 0 perturbation of the surface S, it is possible to break any closed orbit introducing a pair of singular points along it, both positive if the orbit is repelling and negative if the orbit is attracting. We summarize this immediate consequence of the lemma in the following corollary.

Figure 4: Elimination of the critical point q e , q h depicted in Figure 3 . Putting together Proposition 1, Lemma 1 and Corollary 2, we deduce the following fact.

Proposition 2. Let (M, D, g) be a contact sub-Riemannian manifold, let g be a positive integer and let Emb(S, M ) be the space of embeddings of the genus g orientable surface in M , endowed with the C 0 topology. The set of surfaces having finite induced distance is dense in Emb(S, M ).

Proof of Theorem 2

It is left to show that the subset of Emb(S, M ) of embedded surfaces with infinite induced distance is non empty, for any contact 3-manifold M . The result is a consequence of the following lemma, paired with Darboux theorem. Lemma 3. For every positive integer g there exist a closed surface of genus g embedded in the three dimensional Heisenberg group, having infinite induced distance.

Proof. The contact structure of the 3-dimensional Heisenberg group can be written as (R 3 , D

= span{∂ x + y 2 ∂ z , ∂ y -x 2 ∂ z }).
Let us denote the characteristic foliation of a surface S embedded in (R 3 , D) by DS. Consider first the case g = 1.

There exists an embedded torus in (R 3 , D) such that at least one leaf of DT is an embedded closed curve. Indeed given 0 < r < R, consider the torus T R,r given by the embedding T 2 ∋ (θ 1 , θ 2 ) ↦ ((R + r cos θ 1 ) cos θ 2 , (R + r cos θ 1 ) sin θ 2 , r sin θ 1 ).

(3.1)

The characteristic foliation on T R,r is driven by the vector field

X = (R + r cos θ 1 ) 2 2 ∂ θ1 - r cos θ 1 2 ∂ θ2 . (3.2)
Thanks to Proposition 1 we know that with a C ∞ small perturbation of T r,R we can obtain a torus T having a Morse-Smale characteristic foliation, which can be assumed to be devoid of singular points, since the field X in (3.2) is nowhere vanishing. A Morse-Smale characteristic foliation without singular points must contain closed orbits. This follows from the second property of Morse-Smale vector fields listed in Definition 1. The lemma is proved for g = 1. Let now Σ g-1 be any embedded surface of genus g -1, not intersecting the torus T mentioned above. Let ∈ DT be a closed leaf and let B ⊂ T be a ball not intersecting , ∩ B = ∅. We define Σ g ∶= T #Σ g-1 , where the connected sum is made deleting B from T , deleting a small ball from Σ g-1 , and gluing together (smoothly) the resulting boundary circles, see Figure 5. Since B ∩ = ∅, DΣ g still has the closed curve among its leaves, thus Σ g has infinite induced distance. We conclude the proof of Theorem 2.

Figure 1 :

 1 Figure 1: Characteristic foliation of the Euclidean unitary sphere centered at the origin, embedded in the Heisenberg group (R 3 , D = span{∂ x + y 2 ∂ z , ∂ y -x 2 ∂ z }).

Figure 2 :

 2 Figure 2: The image depicts the overtwisted disk {(r, θ, z) ∶ z = 1 2 (π 2r 2 ), r ≤ π, θ ∈ [0, 2π) } in the contact structure (R 3 , D = span{∂ r , cos r r ∂ θ -sin r∂ z }), expressed in cylindrical coordinates (r, θ, z).

  .5) and we denote with [x] the equivalence class of x. By definition of Morse-Smale vector field, for every x ∈ S ′ the limitsω(x) = lim t→∞ e tX (x), α(x) = lim t→∞ e -tX (x),(2.6) exist and are critical points of X. Moreover at least one of the two limits must be an elliptic point, because there are no trajectories joining hyperbolic points. Therefore for every x ∈ S ′ the set [x] ∩ S ′ contains an open neighbourhood of x, which is the stable (unstable) manifold of some elliptic point. It follows that the equivalence classes are open, thus, S ′ being connected, there is only one equivalence class.

Figure 3 :

 3 Figure3: An elliptic point q e and a saddle point q h in elimination position, as described by Lemma 2.

Corollary 2 .

 2 Let S be a closed orientable surface embedded in an co-oriented contact three manifold (M, D), having characteristic foliation of Morse-Smale type. There exists a surface S ′ arbitrarily C 0 -close to S, having a characteristic foliation of Morse-Smale type without closed orbits.

Figure 5 :

 5 Figure 5: Connected sum between the torus T containing the closed leaf , on the left, and the surface Σ g-1 , on the right.

Proof. We define U ∞ , U f ⊂ Emb(S, M ) to be set of embeddings of S having a characteristic foliation of Morse-Smale type with and without closed orbits respectively. They are both open because of structural stability of Morse-Smale vector fields. Moreover, it follows from Proposition 1 that the union U f ∪ U ∞ is dense in Emb(S, M ). Finally Lemma 3 ensures that U ∞ ≠ ∅, while Proposition 2 proves that U f ≠ ∅.

Aknowledgements

This work has been partly supported by the ANR-DFG projects "CoRoMo" ANR-22-CE92-0077-01 and received financial support from the CNRS through the MITI interdisciplinary programs 80 Prime. The authors would like thank Andrei A. Agrachev for very helpful discussions.