Wafa Essafi
email: essafi.wafa@inrap.rnrt.tn

Wifek Raissi

Amira Abdelli

François Boue

Metastability of Large Aggregates and Viscosity, and Stability of The Pearl Necklace Conformation After Organic Solvent Treatment Of Aqueous Hydrophobic Polyelectrolyte Solutions

Keywords:

Aggregates -a phenomenon still not understood, as well as pearl necklace-like chain conformation in aqueous solutions of hydrophobic polyelectrolyte are addressed here, using treatment by an organic solvent. The second appear to be at equilibrium in water, the first appear to be metastable, and surprisingly associated with higher zero shear viscosity. The hydrophobic polyelectrolyte is poly(styrene-co-sodium styrene sulfonate), and the solution treatment is to first add to water an organic solvent, THF, which is then evaporated and replaced by the same amount of water. To investigate polyelectrolyte solutions as a function of THF treatment, we use Small Angle Neutron Scattering in the semi-dilute regime, viscosimetry in dilute and semi-dilute regime (unentangled), and osmometry in similar semidilute regime (the contribution of the counterions being dominant).

First, the structure, namely the scattering from all chains, is characterized by a maximum ("polyelectrolyte peak"). Its position, amplitude and scattered intensity at zero angle depend, at a given sulfonation rate of PSS, on the solvent quality through the added amount of organic solvent (THF). This dependence is very pronounced when the sulfonation rate is low (more hydrophobic polyelectrolyte) and is canceled when the sulfonation rate is high (more hydrophilic polyelectrolyte). Second, the viscosity of the polyelectrolyte solutions decreases with THF treatment for the hydrophobic polyelectrolytes. Third, osmometry shows no noticeable increase of the effective charge with THF treatment. It is proposed that the large scale aggregates, especially in the case of very hydrophobic polyelectrolytes, disappear irreversibly with THF treatment, while the pearl-necklace conformation of the chain remains as in its initial state.

Parallel test measurements for a fully hydrophilic polyelectrolyte, poly (sodium-2acrylamido-2-methylpropane sulfonate) co-(acrylamide): p(AMAMPS) at different sulfonation rates, show no evolution of the structure with THF treatment in the aqueous solution. The same behavior is highlighted by viscosimetry.

INTRODUCTION:

Polyelectrolytes (PELs) are polymers containing ionisable groups. Once dissolved in suitable polar solvent such as water, the ion pairs dissociate into charges linked to the polymer backbone and counter-ions dispersed in whole the solution. This combination opens a huge range of opportunities for properties, exploited in industrial as well in living worlds. For a better control of their properties, it is important to well understand the solution structure, i.e. the way polyelectrolyte chains dissolve in water: do they dissolve completely individually, or do they form aggregates? This question is very often addressed for polyelectrolyte solutions, in different situations.

Two main situations are usually considered: when water is a poor solvent for the backbone; polyelectrolytes are called hydrophobic and the process of their solvation is then a combination of electrostatic and hydrophobic interactions. Conversely, backbones of hydrophilic polyelectrolytes are under good solvent conditions in water and interactions have mainly a pure electrostatic nature.

Let us recall the general background about polyelectrolytes chain conformation and interactions in solvent, say water. We start by the dilute regime. While for the hydrophilic polyelectrolytes the single chain is theoretically described as an extended rod-like configuration of electrostatic blobs, [START_REF] De Gennes | Remarks on Polyelectrolyte Conformation[END_REF][START_REF] Dobrynin | Scaling Theory of Polyelectrolyte Solutions[END_REF][START_REF] Barrat | Advances in Chemical Physics[END_REF] for hydrophobic polyelectrolytes, the single chain is described in the framework of the pearl-necklace model. [4][START_REF] Kantor | Excess Charge in Polyampholytes[END_REF][START_REF] Dobrynin | Hydrophobic Polyelectrolytes[END_REF][START_REF] Dobrynin | Counterion Condensation and Phase Separation in Solutions of Hydrophobic Polyelectrolytes[END_REF] The balance between collapse and extension results in the formation of compact beads (the pearls) joined by narrow elongated strings, in agreement with simulations. [START_REF] Micka | Strongly Charged, Flexible Polyelectrolytes in Poor Solvents -A Molecular Dynamics Study[END_REF][START_REF] Limbach | Structure of Polyelectrolytes in Poor Solvent[END_REF][START_REF] Limbach | Single-Chain Properties of Polyelectrolytes in Poor Solvent[END_REF][START_REF] Schweins | Particle Scattering Factor of Pearl Necklace Chains[END_REF][START_REF] Liao | Counterion-Correlation-Induced Attraction and Necklace Formation in Polyelectrolyte Solutions: Theory and Simulations[END_REF] Experimental results have been reviewed in different papers, like in our previous one, [START_REF] Essafi | Weak Temperature Dependence of Structure in Hydrophobic Polyelectrolyte Aqueous Solution (PSSNa): Correlation between Scattering and Viscosity[END_REF] especially on scattering measurements. We summarize them just below.

For poly-(sodium styrene sulfonate) f -(styrene) 1-f , it was shown by SAXS that when decreasing the proportion f of hydrophilic monomers, the initially wormlike chains (for f = 1) collapse into more compact objects, further from each other. [START_REF] Williams | Electrostatic Effects in Soft Matter and Biophysics[END_REF][START_REF] Baigl | On the Pearl Size of Hydrophobic Polyelectrolytes[END_REF][START_REF] Essafi | Macro-ion Characterization. From Dilute solutions to complex fluids[END_REF] It was proposed that the low internal dielectric constant ε 17 inside the pearls leads to counter-ion/polyion undissociated pairs, explaining the reduction in osmotic pressure. [START_REF] Essafi | Anomalous Counterion Condensation in Salt-Free Hydrophobic Polyelectrolyte Solutions: Osmotic Pressure Measurements[END_REF] Another possibility is that counterions condense around the pearls as shown by using X Rays measurements 19 in the same way as they dress the chain string for PSS f = 1. [START_REF] Combet | Structure of Polyelectrolytes with Mixed Monovalent and Divalent Counterions : SAXS Measurements and Poisson-Boltzmann Analysis[END_REF] For another polymer, it was showed by using addition of miscible bad (or marginal) solvent to water, acetone, that the polyelectrolyte chain undergoes a complete coil to globule collapse transition. [START_REF] Aseyev | Contraction of a Polyelectrolyte upon Dilution. Light-Scattering Studies on a Polycation in Saltless Water-Acetone Mixtures[END_REF][START_REF] Aseyev | Neutron Scattering Studies of the Structure of a Polyelectrolyte Globule in a Water-Acetone Mixture[END_REF] A third system dealt with anionic chains neutralized by specifically interacting alkaline earth cations, also showing a pearlstring-like conformation. [START_REF] Schweins | Calcium Induced Shrinking of NaPA Chains: A SANS Investigation of Single Chain Behavior[END_REF][START_REF] Goerigk | The distribution of Sr 2+ Counterions Around Polyacrylate Chains Analyzed by Anomalous Small-Angle X-ray Scattering[END_REF] In the semi-dilute regime where many chains are interpenetrated with each others, in the case of hydrophilic polyelectrolyte, the chain is a random walk of correlation blobs, [START_REF] De Gennes | Remarks on Polyelectrolyte Conformation[END_REF][START_REF] Dobrynin | Scaling Theory of Polyelectrolyte Solutions[END_REF] each blob is an extended configuration of electrostatic blobs, and the correlation length ξ i.e. the mesh size of the isotropic transient network of overlapping chains scales as c p -1/2 where c p is the polyelectrolyte concentration. Experimentally, the evolution of ξ with c p was verified for highly charged hydrophilic polyelectrolyte. [START_REF] Essafi | Structural Evidence of Charge Renormalization in Semi-Dilute Solutions of Highly Charged Polyelectrolytes[END_REF][START_REF] Nierlich | Characteristic lengths and the structure of salt free polyelectrolyte solutions. A Small Angle Neutron Scattering Study[END_REF][27] In the case of hydrophobic polyelectrolyte two regimes have been predicted by Dobrynin et al [START_REF] Dobrynin | Hydrophobic Polyelectrolytes[END_REF] : the string controlled regime and the bead-controlled regime. The string controlled regime exists as long as the pearl size is much smaller than the correlation length ξ; a classical polyelectrolyte behaviour is expected where ξ scales as c p -1/2 . When the pearl size becomes of the order of ξ, a bead-controlled semi-dilute regime is expected with ξ scaling as c p -1/3 and the system behaves as a solution of charged beads of constant size. Experimentally, two regimes with two scaling exponents being -1/2 and -1/7 have been observed for a solvophobic polyelectrolyte in nonaqueous solvent. [START_REF] Waigh | Semidilute and Concentrated Solutions of a Solvophobic Polyelectrolyte in Nonaqueous Solvents[END_REF] For the model hydrophobic PSS in water, the total structure function was measured by Small Angle Scattering studies (SAXS and SANS) and it was found that ξ scales as c p -α where α decreases from 0.5 to less than 0.4 when decreasing the sulfonation rate of the chain f . [START_REF] Essafi | Macro-ion Characterization. From Dilute solutions to complex fluids[END_REF][START_REF] Spiteri | Conformation et Corrélations Spatiales Dans Les Solutions de Polyélectrolytes: Etude Par Diffusion de Neutrons aux Petits Angles[END_REF] Later, this was also observed by Baigl et al. through SAXS and Atomic Force Microscopy studies. [START_REF] Baigl | Correlation Length of Hydrophobic Polyelectrolyte Solutions[END_REF][START_REF] Qu | Dependence of Structural Forces in Polyelectrolyte Solutions on Charge Density: A Combined AFM/SAXS Study[END_REF] Finally, the free counterion concentration in aqueous hydrophobic polyelectrolyte solutions has been studied experimentally and it was found that the effective charge is strongly reduced, [START_REF] Essafi | Anomalous Counterion Condensation in Salt-Free Hydrophobic Polyelectrolyte Solutions: Osmotic Pressure Measurements[END_REF] compared to the hydrophilic case. [START_REF] Oosawa | Polyelectrolytes[END_REF][START_REF] Manning | Limiting Laws and Counterion Condensation in Polyelectrolyte Solutions I. Colligative Properties. and Limiting Laws and Counterion Condensation in Polyelectrolyte Solutions II. Self--Diffusion of the Small Ions[END_REF] Condensation has been considered theoretically and recently by including the idea that the pearls can be penetrable to the counterions. [START_REF] Chepelianskii | On the effective charge of hydrophobic polyelectrolytes[END_REF] Apart from that in the scattering, the presence of low q upturn is most of the time observed for all kinds of polyelectrolyte solutions. It is the signature of large scale aggregates, but their origin remains unanswered: static versus dynamic, hydrophobic versus electrostatic ? To tackle the hydrophobic origin while keeping electrostatic interactions, one route is to monitor the solvent quality (its ability to dissolve the hydrophobic domains and the polyelectrolyte backbone), which we achieved here by adding an organic solvent to water. Indeed, it was shown that the addition of small amount of THF reduces the important low q upturn found with hydrophobic polyelectrolyte solutions: the large aggregates are dissolved by THF addition. [START_REF] Essafi | Behavior of Hydrophobic Polyelectrolyte Solution in Mixed Aqueous/Organic Solvents Revealed by Neutron Scattering and Viscosimetry[END_REF] Now, we would like to take advantage of another possibility offered by THF plus water mixtures: it is possible, after adding THF, to evaporate it and substitute it by water in order to keep the same concentration. We will call this "THF treatment". This can enable us to check whether the solution in water is under thermodynamic equilibrium at two scales: at the scale of the pearls and at the scale of the aggregates: do these two kind of objects reform after removing THF back to their initial shape or not ? So, the aim of the present work is to investigate the effect of THF treatment on the structure of aqueous solutions of sulfonated PSSNa at intermediate sulfonation rates, thus increasing the hydrophobicity of the highly charged polyelectrolyte:

-either by Small Angle Neutron Scattering (SANS) in the semi-dilute regime and thus we determine the total structure function, -or by viscosimetry measurements in the dilute and semi-dilute unentangled regimes, allowing the determination of the intrinsic viscosity of the different polyelectrolytes as a function of THF treatment.

For comparison, we will study also the behavior of p(AMAMPS), a polyelectrolyte in good solvent in water, at the same intermediate sulfonation rates.

II-MATERIAL

II-1-Polymer synthesis and characterization

The hydrophobic polyelectrolyte used in this study is a copolymer of styrene and sodium styrene sulfonate (PSSNa) poly-(sodium styrene sulfonate) f -co-(styrene) 1-f whose chemical structure is shown on figure 1. It was prepared by post-sulfonation of polystyrene (M w = 280,000 g/mole of batch n° 16311DB of Sigma-Aldrich Reference 182427) based on the Makowski procedure, [START_REF] Essafi | Macro-ion Characterization. From Dilute solutions to complex fluids[END_REF][START_REF] Makowski | Flexible polymeric compositions comprising a normally plastic polymer sulfonated to about 0.2 to about 10 mole % sulfonate[END_REF] which enables partial sulfonation and leads to a well-defined polyelectrolyte. [START_REF] Baigl | Preparation and Characterization of Hydrosoluble, Partially Charged Poly(styrenesulfonate)s of Various Controlled Charge Fractions and Chain Lengths[END_REF] We used also a commercial fully charged PSSNa f = 1 (M w = 666,000 g/mol, its Polydispersity Index M w /M n < 1.2) purchased from Polymer Standard Service (Mainz, Germany), for viscosimetry measurements. The sulfonation rate f of the polyelectrolytes was varied between 0.3 (the limit for solubility in water) and 1 (fully charged). The sulfonation rate f is thus always above the Manning condensation limit for the chemical rate, equal to a/l B ~ 0.3 for PSS in water (a, length of one unit, l B ~ e 2 /(εkT) ~ 7 Å, Bjerrum length in water).

The hydrophilic polyelectrolyte also studied in this paper is poly-(sodium-2acrylamido-2-methylpropane sulfonate) f -co-(acrylamide) (1-f) , whose chemical structure is also shown on figure 1. The average molar mass of the monomer is 71 +158*f. It was synthesized by radical copolymerisation of acrylamide with 2-acrylamido-2-methyl propane sulfonic acid, a method reported formerly, [START_REF] Mccormick | Water-soluble copolymers. IV. Random copolymers of acrylamide with sulfonated comonomers[END_REF] which was slightly modified by adjusting the ratio of the two monomers to obtain, after neutralisation, a fraction of ionisable unit (AMPS), f, between 0.3 and 1. Thus these chains are also highly charged polyelectrolytes. Note that both polyelectrolytes (AMAMPS and PSS) are salts of strong acid, bearing SO 3 -anions as side groups when ionised, with Na + counterions. Both backbones are equally flexible (bare persistence length of order 1 nm). So the two polyelectrolytes used in this study differ mainly by the solvation characteristics of their backbone (hydrophobic in the case of PSS, hydrophilic in the case of AMAMPS).

II-2. Preparation of solutions

For the SANS measurements, solutions were prepared by dissolving the dry polyelectrolyte in D 2 O (used as delivered from Eurisotop) or in the mixture D 2 O/deuterated THF noted (THF d ) and letting at rest for two days under stirring. The treatment of the solutions consists in evaporating subsequently THF d , under nitrogen flow and substituting it exactly by the same volume of D 2 O. So, the polyelectrolyte solutions are in pure D 2 O at the same concentration than initially in the mixture D 2 O/THF d .

For viscosity or cryoscopy measurements, the dry polyelectrolyte was dissolved in deionised H 2 O (a resistivity of 18 MΩ.cm) or in a mixture of H 2 O/THF. After two days under stirring, the solutions undergo the same treatment as previously.

NH

All solution concentrations are expressed in mol/L of the corresponding monomer.

II-3-SANS measurements

Small Angle Neutron Scattering (SANS) measurements were performed on the PACE spectrometer at the Orphée reactor of LLB, CEA -Saclay, France (www-llb.cea.fr).

A range of scattering vector q = (4π/λ).sin(θ/2) between 0.003 and 0.36 Å -1 was covered by using the following three settings: D = 1m -λ= 6 Å, D = 4.7 m -λ = 13.2 Å and D = 4.7 m -λ = 6 Å. The scattered intensity was recorded on a multidetector with 30 concentric, 1 cm wide rings. The response of each ring was normalized to the (flat) incoherent scattering of light water. Samples were contained in 2 mm thick quartz cells.

All measurements were done at room temperature. The cells were capped and sealed in order to prevent D 2 O evaporation.

The recorded intensity was corrected for sample thickness, transmission, incoherent and background scattering (the solvent contribution). Using direct beam measurements of Cotton method, [START_REF] Cotton | Comment faire une calibration absolue des mesures de DNPA[END_REF] the intensities were obtained in absolute units of cross-section; they were then divided by the contrast factor. The SANS data are plotted as S(q) versus q.

II-4-Viscosity measurements

Viscometric measurements of the polyelectrolyte solutions were performed at 25 ± 0.01°C using a SCHOTT viscometer (AVS 470) with a micro Ubbelohde tube of 0.53 mm diameter requiring a sample volume of about 4 ml. Each solution was kept about 5 minutes in the bath prior measurements, for temperature equilibrium.

II-5-Osmometry measurements

The osmometry measurements were performed by mean of a KNAUER vapor pressure osmometer K-7000 which has been designed to exactly measure the total osmolality of fluids. Its principle is based on vapor pressure reduction in solutions corresponding to Raoults law. Two identical thermistors are located in a cell where the gas phase is saturated with solvent vapor. If both thermistors carry drops of pure solvent, the difference in potential between the two is zero. During measurement, one of the solvent droplets is replaced with a droplet of solution (with reduced vapor pressure). This difference is compensated as follows: some vapor of the pure solvent that saturates the gas phase condenses on the droplet of solution. The heat resulting from condensation increases the temperature of the solution droplet, what causes an increase of its vapour pressure, until it equals that of the pure solvent droplet. This increase of the solution temperature is measured with an accuracy of ± 1*10 -3 °C and the number of particles dissolved in the solution, is proportional to this increase. Thus, the osmolality can be determined.

III-RESULTS

The effect of the THF treatment on the hydrophobic polyelectyrolyte solutions was followed mainly by means of two techniques : SANS and viscosimetry.

In what follows, we note "% THF" the percentage by volume of THF used initially in the solvent mixture before evaporation.

III-1-Small Angle Neutron Scattering study in semi-dilute regime

III-1-a-Effect on a hydrophilic polyelectrolyte

Figure 2 shows the evolution of the SANS profiles without and with treatment from different solvent mixtures of D 2 O/THF d , for the fully charged p(AMPS) (f = 1) in totally aqueous solutions in the semi-dilute regime. The polyelectrolyte concentration is kept equal to 0.34 mol/L, for all the samples. It emerges that the scattering function is independent of the treatment applied to the hydrophilic polyelectrolyte solution, in shape and in intensity. Namely, the position of the peak is unchanged as well as its intensity and the scattered intensity at zero angle (figure 2) is constant whatever the treatment applied to the polyelectrolyte solution.

This indicates that the solvent quality is not improved nor abated as treatment is applied or not to the hydrophilic polyelectrolyte solution.

III-1-b-Effect on a hydrophobic polyelectrolyte

Figure 3 shows the evolution of the scattering profiles for PSSNa f = 0.88 at a concentration of 0.34 mol/L, before and after treatment from different solvent mixtures D 2 O/THF d , at different angles. We see that all the profiles can be considered as identical, and so that the scattering function is independent of the treatment applied to this highly charged hydrophobic polyelectrolyte solution, in shape and in intensity. Also, the scattering at the zero q limit (figure 3) is independent of the treatment. So, in the semi-dilute regime, for aqueous solutions of this hydrophobic polyelectrolyte with still a high sulfonation rate, the solvent quality is not improved with treatment. We remain in the situation of the hydrophilic polyelectrolyte solution.

However, if the charge density is decreased, and therefore the hydrophobicity increases, the behaviour becomes different: indeed, for PSSNa f = 0.50 (figure 4a) the peak position q* remains constant with treatment but its height decreases, and also the scattered intensity at small angles decreases. We note that no further structural evolution occurs over time, since the scattering profile recorded after 12 hours is not different from the one just at the end of treatment (figure 4b).

Finally, this trend for change is reinforced when the sulfonation rate decreases furthermore: for f = 0.38 (figure 5), the structure variation with THF treatment is significant:

-Obviously, the peak position q* increases slightly and its height decreases (the peak width Δq/q remaining constant). Moreover, the scattered intensity at large angles overlap and are independent of the treatment (figure 5).

-also, at the right of the peak, we notice for the non-treated solution a shoulder (see insert of figure 5). This shoulder transforms into a peak in q 2 I(q) representation as shown in the insert of figure 5. As earlier commented, [START_REF] Essafi | Macro-ion Characterization. From Dilute solutions to complex fluids[END_REF][START_REF] Spiteri | Conformation et Corrélations Spatiales Dans Les Solutions de Polyélectrolytes: Etude Par Diffusion de Neutrons aux Petits Angles[END_REF] this is due to the scattering from compact small objects )

q (Å -1
)

, which are the pearls. As a function of treatment, the shoulder is reduced but it still exists, as unambiguous shown in insert of figure 5.

-for the low q region, changes are noticeable for f = 0.38. The upturn at q→0 decreases considerably with treatment and as a consequence on the right of the foot of the upturn, the plateau part widens progressively and also the plateau height lowers.

-at the lowest q, the scattered intensity decreases with treatment. These effects grow in strength with increasing THF from 5 to 25%. However, there is saturation when passing from a treatment of 25% THF d to 50% THF d . The two curves are perfectly superimposed. Finally, as in the case of PSSNa f = 0.50, no structural evolution occurs over time, after 12 hours (figure 6a).

For f = 0.38, we have also some measurements at a lower concentration, 0.17 mol/L (figure 6b) and at higher concentration, 0.50 mol/L (figure 6c). We observe the same effects.

III-2-Viscometric Study in dilute and semi-dilute unentangled regime III-2-a-Behaviour of an Hydrophilic polyelectrolyte

The variation of the reduced viscosity versus the concentration of AMAMPS f = 0.35 with treatment, is presented in figure 7a. All the curves show a polyelectrolyte behaviour, namely a typical upturn of the reduced viscosity η red at low concentrations. This is classically explained as follows: when the polyelectrolyte concentration decreases, the screening -due mostly to the free counter-ions -is reduced: the range of the intramolecular repulsion -the Debye length -is increased. Thus the polyelectrolyte chain becomes more extended and consequently η red increases. [START_REF] Fuoss | II. Poly-4-Vinylpyridonium Chloride and Poly-4-vinyl-N-n-Butylpyridonium Bromide[END_REF][START_REF] Dragan | Viscometric Study of Poly(sodium 2-acrylamido-2methylpropanesulfonate) and Two Random Copolymers[END_REF] We also see on figure 7b that THF treatment does not affect the reduced viscosity of the fully charged of p(AMPS) f = 1 polyelectrolyte solution. This agrees with the idea of a conformation unaffected by THF treatment in the corresponding concentration range for hydrophilic PEL.

III-2-a-Behaviour of an Hydrophobic polyelectrolyte

The effect of THF treatment on the viscosity of the PSSNa hydrophobic polyelectrolyte aqueous solutions at 25°C, is shown on figure 8.

For the two extreme sulfonation rates, the PSSNa presents like for the hydrophilic polyelectrolyte AMAMPS, the typical polyelectrolyte behaviour: the reduced viscosity increases as the concentration decreases. The explanation is the same as given for AMAMPS.

When treatment is applied, the evolution is different for the two f's. For f = 1 (highly charged PSSNa), figure 8a shows that the reduced viscosity is independent of the THF treatment, like for AMAMPS: PSSNa f = 1 behaves as an hydrophilic polyelectrolyte. However, the effect of THF treatment is different for the second f: indeed, for f = 0.38, the reduced viscosity decreases with THF treatment. Moreover, the reduced viscosity of PSSNa f = 0.38, as a function of polyelectrolyte concentration, treated from a solvent mixture : 90% H 2 O/10% THF overlaps that obtained from a solvent mixture 75% H 2 O/25% THF. So, beyond a certain amount of THF in the solvent mixture, a further addition of THF is useless.

III-3-Osmometry measurements

Assuming that the osmolality c osmol of the polyelectrolyte solution measured directly by the osmometer, is proportional to the concentration of free counterions of the polyelectrolyte solution, so the ratio c osmol /c p is proportional to the effective charge of the polyelectrolyte. To convert c osmol /c p in term of effective charge, we need first a reference which is the p(AMPS) f = 1, for which the effective charge is equal to 0.36. [START_REF] Essafi | Anomalous Counterion Condensation in Salt-Free Hydrophobic Polyelectrolyte Solutions: Osmotic Pressure Measurements[END_REF] On the other hand, the contribution of the polyelectrolyte chain and the residual salt of the solvent to the osmolarity were subtracted by measuring the osmolarity of the corresponding "neutral" polyelectrolyte p(AM).

We use the following empirical formula to determine the effective charge:

36 . 0 ) c / c ( ) c / c ( ) c / c ( ) c / c ( f ) AM ( P p osmol 1 f ) AMPS ( P p osmol ) AM ( P p osmol PSS p osmol eff * - - = = (Eq.1)
The osmolarity of all the polyelectrolyte solutions was measured at a concentration of 0.17 mol/l, corresponding to a regime where the osmolarity of the polyelectrolyte solution provides only from the free counterions and hence the contribution of the polymer is negligible. [START_REF] Essafi | Anomalous Counterion Condensation in Salt-Free Hydrophobic Polyelectrolyte Solutions: Osmotic Pressure Measurements[END_REF] Figure 9 shows the evolution of the effective charge f eff of PSSNa f = 0.38 as a function of treatment, obtained by vapour pressure osmometry at a concentration of 0.17 mol/L. It emerges first that the effective charge for the polyelectrolyte solution without treatment determined by vapour pressure osmometer is very close to that determined by osmotic-pressure measurements [START_REF] Essafi | Anomalous Counterion Condensation in Salt-Free Hydrophobic Polyelectrolyte Solutions: Osmotic Pressure Measurements[END_REF] and which is found to be around 0.07, highly reduced to the value of 0.36 found for the hydrophilic p(AMPS) f = 1. On the other hand, the most striking result is that the effective charge remains constant with treatment. So, the THF treatment does not affect the effective charge of the single polyelectrolyte chain.

In summary of the Results section, when the polyelectrolyte has a clear cut hydrophobic character in water, viscosimetry as well as scattering suggest an evolution towards a limit structure, as a function of THF treatment indicating an irreversible disappearance of the large scale aggregates in separated chains.

IV-DISCUSSION

IV-1-Small Angle Neutron Scattering study in semi-dilute regime

IV-1-a-Effect of THF treatment on hydrophilic polyelectrolyte

Concerning the hydrophilic polyelectrolyte, p(AMPS) f = 1, it emerges firstly from data that the scattering is independent of THF d treatment. The curves for 0%, 10% and 25% THF d , just overlap. This means that the polyelectrolyte chain network of the p(AMPS) has its mesh size and degree of order unchanged, and that the effective charge remains constant with THF treatment. Finally, the constancy of the scattered intensity at the lowest q value available (3.10 -3 Å -1 ) also suggests that the effective charge is constant, if we assume that S(q→0) which is related to the osmotic compressibility is ~ 1/f eff . Knowing also that for most of polyelectrolyte solutions, measurements at lower q are perturbed by a low q upturn generally observed. Let us briefly detail below. Intermediate and large q behaviour. The analytical expression for the dependence of the position of the maximum in the scattering intensity q* is predicted by the theoretical models of the semi-dilute polyelectrolyte solutions derived from the isotropic phase model (de Gennes et al 1 ) and by Dobrynin et al 2 for polyelectrolytes in good solvents, therefore far from theta temperature Θ. The chain is a random walk of correlation blobs of size ξ, each of which is an extended configuration of electrostatic blobs of g e monomers inside the blob diameter D. Therefore ξ is close to a geometrical distance between rod-like strands of chains and depends on their effective linear density of segments (g e /D), which we will define as equal to B/a, defining a as the segment length and B as in Ref. 2:

(Eq.2)

Thus we consider now the case of good solvent (T>> Θ): g e and D are determined by the fact that the total electrostatic energy between charges inside the electrostatic blob is ~ k B T, due to thermal agitation, with no influence of the distance to theta and using again f eff = a/l B , q* writes now:

(Eq.3)

Following equation 3, the THF treatment should not affect the position q* since the dielectric constant and temperature are constant, which is in agreement with our experimental results.

Concerning the variation of S(q*), it can be expressed by 1 :

(Eq.4)

(g is the number of monomer inside the blob ξ).

We do not observe experimentally any variation of S(q*) with THF treatment, which is in agreement with the theoretical predictions.

Low q behaviour. According to the Dobrynin model 2 :

(Eq.5)

(where is the polymer volume fraction, V molecular is the molecular volume of the PSS -;

V molecular = V Molar /N Avogadro and V Molar is the PSS -molar volume ).

This should lead to a flat scattering at q tending to zero. However on top of this predicted scattering, scattering from aggregates are known to be responsible for the low q upturn. Since the low q scattering does not vary neither in the upturn nor in the flat part (0.01 Å -1 < q < 0.02

2 / 1 p a c B ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∝ ξ ( ) ( ) 7 / 1 7 / 1 B 2 / 1 p 7 / 2 eff 7 / 1 B 2 / 1 p T l a c f a l a c B 2 * q ε ξ π ∝ ∝ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∝ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∝ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∝ - - Θ >> T ( ) 7 / 3 3 3 p T * q c g 2 S *) q ( S - - ε = ≈ ξ ≈ ≈ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ξ π = Θ >> T molecular eff p p p V . f c kTc ) 0 q ( S Φ = Π ∂ ∂ = → p Φ Å -1
), we can propose that both f eff and the aggregate shape are not changed by THF treatment. Consistent with this, is the fact that the origin of the aggregates does not arise from hydrophobic attractive interactions, hence they have most probably an electrostatic origin.

IV-1-b-Effect of THF treatment on hydrophobic polyelectrolyte

This evolution of the scattered intensity is more pronounced as f decreases.

• for f = 0.88 and in the q ~ q*, the change with THF treatment is limited to the low qs : more precisely the scattering minimum (corresponding to the flat part region, q around 0.02 Å -1 ) is slightly lowered. Following what explained just above, this can be assigned to (i) an increase of the effective charge, or (ii) to less scattering from large aggregates. If (i) was right, the peak position should moves to higher q's, which is not the case. Hence this decrease of the intensity is better related to a lower scattering from aggregates.

• for f = 0.50 and in the q ~ q*, the intensity of the peak decreases with THF treatment (with a saturation for 25% and beyond). This is surprising because the peak height is predicted by Eq.4 to scale as . Since q* and c p are constant, S(q*) should remain constant with treatment. At the intermediate q range (q < 0.03 Å -1 ), the flat part of the scattered intensity decreases with THF treatment. This decrease of the flat part should be ascribed to a less scattering large scale aggregates. Concerning the kinetics evolution of the scattered intensity, the lack of evolution after 12 hours (figure 4b) means that the initial hydrophobic aggregates disappear after treatment, irreversibly: they were not at the equilibrium state. • for f = 0.38 (figure 5), changes are strong :

-in the range q ~ q*, the intensity of the peak decreases significantly with THF treatment (with saturation from 25% THF d ) and the peak is displaced to higher q. Note that for hydrophobic polyelectrolyte where T < Θ, q* is expressed by [START_REF] Dobrynin | Scaling Theory of Polyelectrolyte Solutions[END_REF][START_REF] Dobrynin | Hydrophobic Polyelectrolytes[END_REF] :

2 / 1 p 4 / 1 eff 4 / 1 c ) f ( * q - τ ∝ for T < Θ (Eq.6)
where τ = ( ) Θ -Θ T . So, an increase of q* is attributed to an increase of the concentration or of the effective charge (τ remains constant since temperature is kept constant and the used solvent is usually water). From osmometry measurements, we deduce that the effective charge remains constant with THF treatment; hence the displacement of the peak should be attributed to an increase of the PEL concentration. This should only come from the dissolved aggregates; the concentration change would however be strong: the shift from q* = 0.0413 Å -1 to q* = 0.0481 Å -1 corresponds to an increase from 0.34 mol/L to 0.58 mol/L, according to the variation law of q* as a function of c p for PSSNa f = 0.38, before treatment.

-in the low q region, the scattered intensity at q→0 measured just at the foot of the upturn q ≈ 0.020641 Å -1 , is decreased by THF treatment in two ways:

-the upturn at q → 0 is reduced significantly: this can be interpreted as the reduction of large hydrophobic aggregates.

the value of the 'flat part' is lowered down and also this region widens. Since we assume from osmometry measurements a constant osmotic compressibility, it

3 p 3 p * q c c *) q ( S - ≈ ξ ≈
can be related to less scattering objects and so to a decrease of aggregates scattering. The saturation effect above 25% THF d treatment suggests that the aggregates that are likely to dissolve have disappeared entirely, at least with this kind of treatment. Note that despite the decay of the scattered intensity in the whole low q region, it remains however higher than for the highly charged PSSNa f = 0.88. This result shows that the effective charge of PSSNa after treatment is lower than that of PSSNa f = 0.88, which is expectable, and that even after treatment the large scale inhomogeneities for f = 0.38 are still scattering more than the well-known large scale inhomogeneities characteristics of pure polyelectrolyte solutions, attributed to electrostatics.

• Concerning the behaviour at the right of the peak (q ~ 0.05 -0.4 Å -1 ) including the domain of existence of the "second shoulder", [START_REF] Essafi | Macro-ion Characterization. From Dilute solutions to complex fluids[END_REF] enhanced in the q 2 S(q) representation (insert of figure 5), the abscissa of the shoulder does not vary with THF treatment; hence the pearl size remains constant with THF treatment. Finally, we can compare the variation observed here with the one observed with added THF. For 10% added THF (see figure SI1 or A1 in Supporting Information or in Appendix of Ref 35), the shift of q* is shifted towards 0.05 Å -1 , so only slightly more than for THF treatment: part only of the effect is reversible. But when amount of added THF reaches 25%, q* becomes 0.065 Å -1 : in this case most of the effect of THF is reversible. So, the conformational change of the polyelectrolyte chain in presence of THF addition is largely reversible after THF treatment.

IV-2-Viscometric study in dilute and semi-dilute unentangled regimes

IV-2-a-Hydrophilic polyelectrolyte

The viscosity of polyelectrolyte solutions in semi-dilute unentangled regime can be described by the evolution of the 1/η red = f(c p 1/2 ). We will use this representation, which gives often a linear variation in practice, in agreement with the so-called "Fuoss law" [START_REF] Fuoss | II. Poly-4-Vinylpyridonium Chloride and Poly-4-vinyl-N-n-Butylpyridonium Bromide[END_REF] :

η red = (Eq.7)
Though of empirical origin, Eq.7 is recovered by the theory of Dobrynin and al for hydrophilic polyelectrolytes. [START_REF] Dobrynin | Scaling Theory of Polyelectrolyte Solutions[END_REF] It is frequently used to determine the intrinsic viscosity values [η] for polyelectrolytes in aqueous solution, from the intercept of the inverse of the Fuoss representation. For the p(AMPS) f = 1, as well as for f = 0.35 with the different solvent compositions, plotting 1/η red versus , gives linear evolutions for all the solvent compositions (Figure 10).

Concerning THF treatment, we note that all curves overlap. The intercept A remains constant with THF treatment as shown on figure 10.

Let us compare with the evolution under THF treatment of SANS, which also show no change: we conclude that viscosimetry and SANS appear complementary methods and show a fair correlation (though the same concentration ranges are most of the time not even overlapping: maximum C pmax for η is 0.06 mol/L to compare with 0.34 mol/L for S(q)) -see Figure 7a; 7b and 8a, except for PSSNa f = 0.38, where C pmax is 0.17 mol/L -see Figure 8b.

IV-2-b-Hydrophobic polyelectrolyte

For the hydrophobic polyelectrolyte, the same agreement with Fuoss law is observed for all PSS sulfonation rates and all THF treatments (figures 11a and 11b). Firstly, for f = 1, there is no effect of THF treatment (figure 8a), the same behaviour as found for the hydrophilic polyelectrolyte AMAMPS. Secondly, for f = 0.38, the viscosity decreases considerably with THF treatment. In other words, the hydrodynamic volume of the PSSNa decreases with THF treatment. Fuoss law 40 fits well the experimental results here also (figure 11b) and, accordingly, the extracted intrinsic viscosity decreases with THF treatment (see figure 11c). Note that opposite to what expected if the chain was expanding (as observed when simply adding THF). What is more revealing with this representation is that the slope of the Fuoss representation remains constant with THF treatment. According to the theories of Witten [START_REF] Witten | Structure and Viscosity of Interpenetrating Polyelectrolyte Chains[END_REF] and Rabin, [START_REF] Rabin | Anomalous Viscosity of Polyelectrolyte Solutions[END_REF] this slope is related to the effective charge f eff of the polyelectrolyte; we deduce that f eff remains constant with THF treatment. This behavior is in agreement with osmometry measurements. To conclude this paragraph, the decrease of the intrinsic viscosity and thus the hydrodynamic volume of the chain can be related only to the dissolution of the hydrophobic large scales aggregates.

Under this assumption, we thus find agreement between SANS and viscosimetry measurements.

To end this discussion it is interesting to compare with the case of solutions with added THF, without evaporation : in presence of THF the aggregates vanish, but the chains expand, which is the dominant effect for viscosity and results in an increase. [START_REF] Essafi | Behavior of Hydrophobic Polyelectrolyte Solution in Mixed Aqueous/Organic Solvents Revealed by Neutron Scattering and Viscosimetry[END_REF] This is schematized in Figure 12 a and b. After evaporation (figure 12 c), the chains recover their initial contracted conformation, so that their contribution to viscosity returns to the value before treatment. But the aggregates contribution is reduced by the treatment, so that the viscosity decreases even more.

V-CONCLUSION

To summarize our results, the effect of THF treatment is negligible for hydrophilic PELs, AMAMPS or PSSNa f ≈ 0.9-1, hardly visible for PSSNa f = 0.50, and noticeable for low sulfonation rate (PSSNa f = 0.38). These results are observed both on the scattering, related to the spatial structure of the solution, in the semi-dilute regime of concentration, and on the viscosity, measured from the dilute to the semi-dilute concentration range (sometimes comprising the value used for scattering).

From this point of view, there is thus a good agreement between SANS and viscometry. But the sign of the variation of the two quantities requires more discussion. This can be enlightened by the complementary osmometry measurements, which are consistent with the latter, though they actually require even more subtlety in the discussion. We will focus on the hydrophobic case, f = 0.38. The variation of SANS results comprises two different behaviours: • (i) at low q, the signal from aggregates decreases clearly : the simplest explanation is some aggregates have dissolved in presence of THF and do not reform when back in water. Their dissolution is irreversible. • (ii) at intermediate and large q, the peak is shifted to large q. This could be explained by a better dissolution of aggregates into separate polyelectrolyte chains, while the conformation of the polyelectrolyte chains remains constant with the treatment. This behaviour corroborates that the polyelectrolyte is at equilibrium state in water. the conformation of the polyelectrolyte chain remaining constant with THF treatment is also correlated to a constancy of the effective charge f eff with THF treatment, as also revealed by the osmometry measurements. • the viscosity varies but in Fuoss representation, the slope which is attributed to f eff , is the same as well, confirming the result of the constancy of the effective charge. So, we proposed that the shift of q* could be due to an increase of concentration due to the dissolution of the aggregates.

The second interesting point to discuss is the sign of the variation of the viscosity: it decreases. The opposite is expected from a more extended chain, as was actually observed in our former papers. [START_REF] Essafi | Weak Temperature Dependence of Structure in Hydrophobic Polyelectrolyte Aqueous Solution (PSSNa): Correlation between Scattering and Viscosity[END_REF][START_REF] Essafi | Behavior of Hydrophobic Polyelectrolyte Solution in Mixed Aqueous/Organic Solvents Revealed by Neutron Scattering and Viscosimetry[END_REF] Hence we are driven to this explanation: this decrease can be due to the dissolution of some aggregates (figure 12). Such possibility points a striking role of aggregates or other types of large scale concentration fluctuations in the viscosity of polyelectrolyte solutions.

Figure 2 : 1 )Figure 3 : 1 )

 2131 Figure 2: Evolution of the SANS profiles of totally heavy water solutions of p(AMPS) f = 1 with treatment from initially different solvent mixtures D 2 O/THF d . The polyelectrolyte concentration c p = 0.34 mol/L.

Figure 4b : 1 )Figure 5 :

 4b15 Figure 4b: kinetic of evolution of SANS profiles of PSSNa f = 0.50 in D 2 O, with treatment (25% THF d ). The polyelectrolyte concentration c p = 0.34 mol/L.

Figure 6a :

 6a Figure 6a: kinetic of evolution of SANS profiles of PSSNa f = 0.38 in D 2 O, with treatment (50% THF d ). The polyelectrolyte concentration c p = 0.34 mol/L.

Figure 6b :

 6b Figure 6b: Evolution of the SANS profiles of PSSNa f = 0.38 in D 2 O without and with treatment from initially different solvent mixtures D 2 O/THF d . The polyelectrolyte concentration c p = 0.17 mol/L.

1 )Figure 6c : 1 )Figure 7a :

 16c17a Figure 6c: Evolution of the SANS profiles of PSSNa f = 0.38 in D 2 O without and with treatment from initially different solvent mixtures D 2 O/THF d . The polyelectrolyte concentration c p = 0.50 mol/L.

Figure 7b :Figure 8a :

 7b8a Figure 7b: Evolution of the reduced viscosity versus the polyelectrolyte concentration for p(AMPS) f = 1, as a function of treatment at 25°C.

Figure 8b :

 8b Figure 8b: Evolution of the reduced viscosity versus the polyelectrolyte concentration for PSSNa f = 0.38, as a function of treatment at 25°C.

Figure 9 :

 9 Figure 9: Evolution of the effective charge f eff as a function of THF treatment, for PSSNa f = 0.38, obtained by vapour pressure osmometry at a concentration of 0.17 mol/L.

Figure 10a :

 10a Figure 10a: Fit of the viscometry data AMAMPS f = 0.35 at 25°C as a function of treatment with the Fuoss Equation.

Figure 10b :

 10b Figure 10b: Fit of the viscometry data p(AMPS) f = 1 at 25°C as a function of treatment with the Fuoss Equation.

Figure 11a :

 11a Figure 11a: Fit of the viscometry data PSSNa f = 1 at 25°C as a function of treatment with the Fuoss Equation.

Figure 11b :

 11b Figure 11b: Fit of the viscometry data PSSNa f = 0.38 at 25°C as a function of treatment with the Fuoss Equation.

Figure 11

 11 Figure 11 c: The evolution of the extrapolated intrinsic viscosity for PSS f = 0.38 with treatment.

Figure 12 :

 12 Figure 12: Schematization of the hydrophobic polyelectrolyte aqueous solution (a), after THF addition (b) and after THF treatment (c).

ACKNOWLEDGMENTS

We thank Souha BEN MAHMOUD and Ons MHIRI for their helpful in the preparation of some polyelectrolyte solutions.

SUPPORTING INFORMATION AVAILABLE

The Evolution of the SANS profiles with the percentage of added THF to aqueous solutions of PSSNa f = 0.38. The polyelectrolyte concentration is cp = 0.34 mol/L. The insert represents the Kratky plot of the intensity scattered by the same solutions. (reprint from Figure 5 in ref. 35). This information is available free of charge via the Internet at http://pubs.acs.org

For Table of Contents Use Only

Aggregation and viscosity in hydrophobic polyelectrolyte aqueous solutions are metastable while pearl necklace is not changed by organic solvent treatment

Wafa ESSAFI, Wifek RAISSI, Amira ABDELLI and François BOUE


Tom Ás

Ibarluc Ía

Julien Melleray

DYNAMICAL SIMPLICES AND MINIMAL HOMEOMORPHISMS

We give a characterization of sets K of probability measures on a Cantor space X with the property that there exists a minimal homeomorphism g of X such that the set of g-invariant probability measures on X coincides with K. This extends theorems of Akin (corresponding to the case when K is a singleton) and Dahl (when K is finite-dimensional). Our argument is elementary and different from both Akin's and Dahl's.

INTRODUCTION

The study of minimal homeomorphisms (those for which all orbits are dense) on a Cantor space is a suprisingly rich and active domain of research. In a foundational series of papers (see [HPS], [START_REF] Giordano | Topological orbit equivalence and C * -crossed products[END_REF] and [START_REF]Full groups of Cantor minimal systems[END_REF]), Giordano, Herman, Putnam and Skau have pursued the analysis of minimal actions of Z (and later Z d ), and developed a deep theory. In particular, it is proved in [START_REF] Giordano | Topological orbit equivalence and C * -crossed products[END_REF] that the partition of a Cantor space X induced by the orbits of a minimal homeomorphism g is completely determined, up to a homeomorphism of X, by the collection of all g-invariant measures.

Gaining a better understanding of sets of invariant measures then becomes a natural concern, and that is our object of study here: given a Cantor space X, and a simplex K of probability measures, when does there exist a minimal homeomorphism g of X such that K is exactly the simplex of all g-invariant measures? Downarowicz [D2] proved that any abstract Choquet simplex can be realized in this way; here we are not given K as an abstract simplex, but already as a simplex of measures, so the problem has a different flavour.

A theorem of Glasner-Weiss [GW] imposes a necessary condition: if g is a minimal homeomorphism, K is the simplex of all g-invariant measures, and A, B are clopen subsets of X such that µ(A) < µ(B) for all µ ∈ K, then there exists a clopen subset C ⊆ B such that µ(C) = µ (A) for all µ ∈ K. This is already a strong, nontrivial assumption when K is a singleton; in that case the Glasner-Weiss condition is essentially sufficient, as was proved by Akin. [A]). Assume that µ is a probability measure on a Cantor space X which is atomless, has full support, and is good, that is, for any clopen sets A, B, if µ(A) < µ(B) then there exists a clopen C ⊆ B such that µ(C) = µ (A).

Theorem (Akin

Then there exists a minimal homeomorphism g of X such that the unique g-invariant measure is µ.

Following Akin, we say that a simplex is good if it satisfies the necessary condition established by Glasner and Weiss. Akin's theorem suggests that, modulo some simple additional necessary conditions, any good simplex of measures could be the simplex of all invariant measures for some minimal homeomorphism. This idea is further reinforced by an unpublished result of Dahl, which generalizes Akin's theorem.

Theorem (Dahl [D1]). Let K be a Choquet simplex made up of atomless probability measures with full support on a Cantor space X. Assume that K is good and has finitely many extreme points, which are mutually singular. Then there exists a minimal homeomorphism whose set of invariant probability measures coincides with K.

Dahl actually obtains a more general result. To formulate it, we recall her notation: given a simplex K of probability measures on a Cantor space X, let Aff (K) denote the set of all continuous affine functions on K, and G(K) ⊆ Aff (K) be the set of all functions µ → X f dµ, where f belongs to C(X, Z). Theorem (Dahl [D1]). Let K be a Choquet simplex made up of atomless probability measures with full support on X. Assume K is good and the extreme points of K are mutually singular. If G(K) is dense in Aff (K), then there exists a minimal homeomorphism whose set of invariant measures is exactly K.

As pointed out in the third section of [D1], it follows from Theorem 4.4 in [E] that G(K) being dense in Aff (K) is necessary for G(K) to be a so-called simple dimension group, which is in turn necessary for the existence of g as above. The other conditions are also necessary, so Dahl could have formulated her theorem as an equivalence.

Using Lyapunov's theorem, Dahl proves that any finite-dimensional Choquet simplex of probability measures on X with mutually singular extreme points is such that G(K) is uniformly dense in Aff (K), thus deducing the theorem we stated previously from the one we just quoted. Dahl's proof of her second theorem above uses some high-powered machinery following the Giordano-Herman-Putnam-Skau approach to topological dynamics via dimension groups, K-theory, Bratteli diagrams and Bratteli-Vershik maps. By contrast, Akin's proof is elementary and rather explicit, though somewhat long.

Here, we take an approach which is different from both Akin's and Dahl's: we build a minimal homeomorphism preserving a prescribed set of probability measures by constructing inductively a sequence of partitions which will turn out to be Kakutani-Rokhlin partitions for that homeomorphism (we recall the definition of Kakutani-Rokhlin partitions and other basic notions of topological dynamics in the next section). While pursuing this approach, we unearthed a necessary condition for K to be the simplex of invariant measures for some minimal homeomorphism. This led us to the following definition.

Definition. We say that a nonempty set K of probability measures on a Cantor space X is a dynamical simplex if it satisfies the following conditions:

• K is compact and convex.

• All elements of K are atomless and have full support.

• K is good.

• K is approximately divisible, i.e. for any clopen A, any integer n and any ε > 0, there exists a clopen B ⊆ A such that nµ(B) ∈ [µ(A)ε, µ (A)] for all µ ∈ K.

Note that, assuming that K is good, the assumption that B ⊆ A in the last item above is redundant; we nevertheless include it because this is how approximate divisibility is used in our arguments.

We borrow the terminology "dynamical simplex" from Dahl, but our definition is different. Using Lyapunov's theorem as in [D1], it is easy to see that the condition of approximate divisibility is redundant when K is finite dimensional. The main result of this paper is the following.

Theorem 1.1. Given a simplex K of probability measures on a Cantor space X, there exists a minimal homeomorphism g whose set of invariant measures is K if, and only if, K is a dynamical simplex.

Our construction produces a minimal homeomorphism g whose set of invariant measures is K and such that the topological full group [[g]] is dense in the closure of the full group [g] (in the terminology of [BK], g is saturated). When one starts off by assuming that K is the simplex of T-invariant measures for some minimal homeomorphism T, the existence of such a homeomorphism follows from a combination of theorems of Giordano-Putnam-Skau and Glasner-Weiss, see [START_REF] Bezuglyi | Topologies on full groups and normalizers of Cantor minimal systems[END_REF]Theorem 1.6]. Here we provide an elementary proof of that fact, which seems interesting on its own.

For finite dimensional simplices, our theorem generalizes Dahl's result, showing that the assumption that extreme points are mutually singular is actually a consequence of her other hypotheses. It would be interesting to gain a better understanding of the relationship between her conditions and ours (see Remark 3 at the end of the paper).

We would like to point out that the ideas of our construction are different from both Akin's and Dahl's, and relatively elementary; in particular our argument completely bypasses the use of dimension groups, Bratteli diagrams, etc. It is our hope that such ideas could be used to give elementary dynamical proofs of some other theorems of topological dynamics.

Acknowledgment. The authors are grateful to G. Aubrun, I. Farah and B. Weiss for interesting comments and discussions. We would also like to thank an anonymous referee for spotting some inacurracies and making valuable suggestions.

BACKGROUND AND NOTATIONS

Throughout, X is a Cantor space; we fix some compatible distance on X, and whenever we mention the diameter of a set if will be with respect to this distance. We denote by Prob(X) the compact space of all probability measures on X, endowed with its usual topology (which comes from seeing Prob(X) as a subset of the dual of C(X), endowed with the weak- * topology).

The group Homeo(X) of all homeomorphisms of X is a Polish group when endowed with the topology of uniform convergence on X. Since X is the Cantor space, one can also describe this topology using Stone duality: a homeomorphism of X corresponds to an automorphism of the boolean algebra of clopen sets of X, Clop(X); identifying Homeo(X) with automorphisms of this algebra yields that a basis of neighbourhoods of identity is given by sets of the form {g ∈ Homeo(X) : ∀A ∈ A g(A) = A}, where A runs over all clopen partitions of X (note that by compactness all clopen partitions are finite). It is readily checked that the two topologies we just described coincide on Homeo(X).

Definition 2.1. Given g ∈ Homeo(X), its topological full group [[g]] is the group of all homeomorphisms h of X such that there exists a clopen partition A 1 , . . . , A n and integers n i with the property that for all x ∈ A i one has h(x) = g n i (x).

The full group [g] is the group of all homeomorphisms h such that for all x there exists n satisfying h(x) = g n (x).

By definition, the topological full group is countable (there are only countably many clopen sets) and contained in the full group. The reason these groups are relevant to our concerns is the following, which follows easily from Proposition 2.6 of [GW].

Theorem 2.2 (Glasner-Weiss). Let g be a minimal homeomorphism. The closure of [g] in Homeo(X) consists of all homeomorphisms which preserve all g-invariant probability measures on X.

Proof. Let H denote the group of all homeomorphisms which preserve each ginvariant measure. By definition, H is closed and [g] ⊆ H, so that [g] ⊆ H. Towards proving the converse inclusion, pick h ∈ H and an open neighborhood

O = {k ∈ H : ∀A ∈ A k(A) = h(A)} of h, where A is a clopen partition of X.
For any A ∈ A there exists, by Proposition 2.6 of [GW], some It is not always true that [[g]] is dense in [g]; when that happens we say that g is saturated (this follows terminology introduced in [BK]).

k A ∈ [g] such that k A (A) = h(A). Then, the map k defined by setting k(x) = k A (x) whenever x ∈ A is a homeomorphism (because A is a clopen partition,
We next recall the definition of a Kakutani-Rokhlin partition associated to a minimal homeomorphism.

Definition 2.3. A Kakutani-Rokhlin partition T associated to a minimal homeomorphism g is a clopen partition of X of the form A 0 ⊔ . . . A k , where each A i is further subdivided in B i,0 , . . . , B i,j i (possibly j i = 0) and for all i and all r ∈ {0, . . . ,

j i -1}, g(B i,r ) = B i,r+1 .
The union of all B i,0 is called the base of the partition, and the union of all B i,j i is its top. Each A i is called a column of the partition, and the definition ensures that g must map the top of the partition onto its base.

To obtain such a partition, one can first choose a clopen base B; then subdivide it into B 1 , . . . , B N , with B i made up of all x ∈ B such that i = min{j > 0 : g j (x) ∈ B}; and set B i,j = g j (B i ) for all j ∈ {0, . . . , i -1}.

Below we represent a Kakutani-Rokhlin partition; the arrows correspond to the action of the homeomorphism on the partition, which is prescribed on all atoms except those contained in the top (all we know there is that the top is mapped onto the base). The base is colored in blue and the top in red; note that on the picture the base and top do not intersect. They are allowed to, but will not intersect as soon as we take a small enough base.

It is a standard, important fact in topological dynamics that, given a minimal homeomorphism g, one can produce a sequence of Kakutani-Rokhlin partitions for g whose atoms generate the algebra of clopen sets, and whose top and base have vanishing diameter. Such a sequence naturally defines a basis of neighborhoods of g in Homeo(X).

FIGURE 1. A Kakutani-Rokhlin partition

Below, to obtain a minimal homeomorphism with prescribed set of invariant measures, we will define a sequence of partitions which will turn out to be Kakutani-Rokhlin partitions for that homeomorphism.

We recall the definition of a dynamical simplex given in the introduction.

Definition 2.4. We say that a nonempty set K of probability measures on X is a dynamical simplex if it satisfies the following conditions:

• K is compact and convex.

• All elements of K are atomless and have full support.

• K is good, i.e. for any two clopen sets A, B such that µ(A) < µ(B) for all µ ∈ K, there exists a clopen subset Given a set K of probability measures, and two clopen sets A, B, we use the notation A ∼ K B to denote the fact that µ(A) = µ(B) for all µ ∈ K. Note that, modulo goodness, approximate divisibility may be stated equivalently by saying that there exist clopen subsets B 1 , . . . , B n such that B 1 , . . . , B n are disjoint, contained in A, B i ∼ K B j for all i, j, and A \ B i has measure less than ε for all µ ∈ K.

C ⊆ B such that µ(C) = µ(A) for all µ ∈ K. • K is approximately divisible, i.
We should point out again that we borrow the term "dynamical simplex" from Dahl [D1], and that our definition is, at least formally, different from Dahl's : the definition given in [D1] includes the assumption that K is a Choquet simplex and extreme points of K are mutually singular, and does not mention approximate divisibility. We briefly discuss the relations between our conditions and Dahl's in Remark 3 at the end of the paper.

We note that, when K has finitely many extreme points, the assumption of approximate divisibility is redundant, as follows from the proposition below.

Proposition 2.5. Assume that K is a compact subset of Prob(X), and all the measures in K are atomless and have full support. Then the following properties hold.

(1) For any nonempty clopen set A, inf{µ(A) : µ ∈ K} > 0.

(2) For any ε > 0, there exists δ > 0 such that, for any clopen set A of diameter less than δ, one has µ(A) ≤ ε for all µ ∈ K.

(3) If K has finitely many extreme points, then K is approximately divisible.

Proof. The first two items are well-known when K is the simplex of all invariant measures for a minimal homeomorphism, and the proofs are simple and similar to that case. We give them for the reader's convenience.

For the first item, assume that there exists a sequence (µ n ) of elements of K and a nonempty clopen set A such that µ n (A) converges to 0. Then by compactness of K we find some µ ∈ K such that µ(A) = 0, contradicting the fact that µ has full support (it is perhaps worth recalling that a sequence µ n of elements of Prob(X) converges to µ ∈ Prob(X) exactly if µ n (A) converges to µ(A) for all clopen set A).

The second item requires a bit more work; we follow the argument of [BM, Proposition 2.3] and assume for a contradiction that there exists a sequence of clopen subsets (A n ) of vanishing diameter and ε > 0 such that µ n (A n ) ≥ ε for all n. Up to passing to a subsequence, we may assume that (A n ) converges to a singleton {x} for the Hausdorff distance on compact subsets of X, and that µ n converges to µ ∈ K. Let O be any clopen neighborhood of x; we will have A n ⊆ O for all large enough n, so that µ n (O) ≥ ε for all large n. As above, this implies that µ(O) ≥ ε for all n, so that (taking the intersection over all clopen neighborhoods of x) µ({x}) ≥ ε, contradicting the fact that µ is atomless.

To see why the third point holds, we argue in a way similar to [D1]. Fix ε > 0 and let µ 1 , . . . , µ n denote the extreme points of K. Lyapunov's theorem on vector measures tells us that

{(µ 1 (B), . . . , µ n (B)) : B a Borel subset of A} is convex. In particular, there exists a Borel subset B of A such that µ i (B) = 1 n µ i (A) for i ∈ {1, . . . , n}. Using the regularity of µ 1 , . . . , µ n we obtain a clopen subset C ⊂ A such that µ i (C) ∈ [ µ i (A)-ε n , µ i (A)
n ] for all i, which is what we wanted.

Remark 1. If we had assumed that the extreme points of K were mutually singular, we would not have needed Lyapunov's theorem to conclude that K is approximately divisible when K has finitely many extreme points; but we do not need to make this assumption. We also do not include the assumption that K is a Choquet simplex in our definition of a dynamical simplex, as we do not need it in the arguments. Both these assumptions are clearly necessary for K to be the simplex of all invariant measures of an homeomorphism, hence follow from the others, given the main result of the paper. We do not know if, in general, approximate divisibility is a consequence of the other assumptions (to which one could add the fact that K is a Choquet simplex with mutually singular extreme points, if necessary) as is the case when K is finitedimensional. The proof above does not seem to adapt: Lyapunov's theorem does extend to more general situations, but this extension (known as Knowles' theorem, see [START_REF] Diestel | Vector measures[END_REF]IX.1.4]) requires the existence of a finite control measure ν, which will exist only when K has finitely many extreme points. More precisely, one would like to apply the Extension Theorem [START_REF] Diestel | Vector measures[END_REF]I.5.2] to the vector-valued measure F : Clop(X) → C(K), F(A)(µ) = µ(A), but then, assuming K has countably many mutually singular extreme points, it is not difficult to see that the second item of the theorem fails. Nevertheless, one can certainly prove that approximate divisibility is redundant in some infinite-dimensional situations, for instance when the extreme boundary of K has only one non-isolated point (this was remarked during a conversation with I. Farah).

Proposition 2.6. Assume that g is a minimal homeomorphism of X, and that K is the simplex of all g-invariant probability measures. Then K is a dynamical simplex.

Proof. Clearly g-invariant measures always form a compact convex subset of Prob(X), and when g is minimal any g-invariant measure must be atomless and have full support. The fact that the simplex K is good follows from the theorem of Glasner and Weiss recalled in the introduction, so we only need to explain why K is approximately divisible.

Start from a nonempty clopen set A, and consider the map g A defined by g A (x) = g n (x), where n = min{i > 0 : g i (x) ∈ A}. Then g A is a homeomorphism of A, and is minimal. The restriction of any µ ∈ K defines a g A -invariant measure on A, which we still denote by µ. Pick N ≥ n such that n/N < ε. Since g A is aperiodic, we can find a clopen set U such that U, g A U, . . . , g N A U are disjoint. In particular, µ(U) is less than µ(A)/N for all µ ∈ K.

Now, consider a Kakutani-Rokhlin partition of A associated to g A , with base U. Let C 0 , . . . , C M denote the columns of this partition; we have

C i = C i,0 ⊔ C i,1 . . . ⊔ C i,n i , with n i ≥ N ≥ n. Denote n i + 1 = k i n + p, p
∈ {0, . . . , n -1}. For i ∈ {0, . . . , M} and j ∈ {0, . . . , n -1}, let B i,j denote the union of the levels of C i of height < k i n and equal to j modulo n, and let B j be the union of all B i,j . Then we have B j ∼ K B l for all j, l, and the complement of their union is of measure less than nµ(U) ≤ εµ (A) for all µ ∈ K.

The following picture is supposed to illustrate the procedure we just described: below, n = 3; the domains in blue, red and green are ∼ K equivalent, and the measure of the remainder is less than 3 times the measure of the base, for all µ ∈ K. Proposition 2.7. Assume K is a good simplex of probability measures on a Cantor space X with full support. Let G = {g ∈ Homeo(X) : ∀µ ∈ K g * µ = µ}. If U, V are clopen sets with U ∼ K V, then there is g ∈ G such that gU = V.

Proof. We construct a K-preserving automorphism g of the algebra Clop(X) by a standard back-and-forth argument. Let {A n }, {B n } be two enumerations of Clop(X), with A 0 = U and B 0 = V. Let A 0 be the partition of X into A 0 and its complement. We set gA 0 = B 0 , g(X \ A 0 ) = X \ B 0 . Now assume inductively that we have defined g on the atoms of a finite clopen partition A n such that: (i) the sets {A i } i<n are contained in the algebra generated by A n , (ii) the image

gA n = {gC : C ∈ A n } is a partition of X, (iii) µ(gC) = µ(C) for all C ∈ A n , µ ∈ K,
and (iv) the sets {B i } i<n are contained in the algebra generated by gA n .

Let C 1 , . . . , C m be the atoms of the partition A n . We take

C 0 i = C i ∩ A n , C 1 i = C i \ A n . Since K is good, we can find D 0 i ⊆ gC i such that µ(D 0 i ) = µ(C 0 i ) for all µ ∈ K; we set D 1 i = gC i \ D 0 i . Now we take D j,0 i = D j i ∩ B n , D j,1 i = D j i \ B n . Again, since K is good, we can find a clopen partition A n+1 = {C j,k i } j,k<2 i<m such that C j,k i ⊆ C j i and µ(C j,k i ) = µ(D j,k i ) for each i, j
, k and all µ ∈ K. Then we extend the definition of g to A n+1 by setting gC j,k i = D j,k i . The construction ensures that properties (i)-(iv) are preserved.

At the end we get a K-preserving automorphism of Clop(X) sending U to V. By Stone duality, this induces an homeomorphism g as required.

CONSTRUCTION OF A SATURATED ELEMENT

In this section and the next, we fix a dynamical simplex of measures K on a Cantor space X, and we let G denote the group of all homeomorphisms g of X such that g * µ = µ for all µ ∈ K.

Definition 3.1. We say that g ∈ G is K-saturated if for any clopen sets U, V such that U ∼ K V there exists h ∈ [[g]] with h(U) = V. Proposition 3.2. Assume that g is K-saturated. Then [[g]] is dense in G and g is mini- mal.
Proof. It is easy to see that [[g]] being dense in G is equivalent to g being Ksaturated once once has Theorem 2.2 in hand. To see that a K-saturated element is minimal, pick any nonempty clopen set U. Given any x ∈ X, a sufficiently small clopen neighborhood V of x will be such that sup K µ(V) < inf K µ(U), thus there exists g ∈ G such that gU ⊇ V ∋ x. Hence X = g∈G gU.

Now, for all h ∈ G there exists k ∈ [[g]] such that kU = gU, so that X = k∈[[g]] kU. Since for all k ∈ [[g]] we have kU ⊆ i∈Z g i U, we obtain that X = i∈Z g i U. This means that X is the unique nonempty open g-invariant set, which is the same as saying that g is minimal.

Next, we introduce partitions which resemble Kakutani-Rokhlin partitions. Essentially, we are trying to build a homeomorphism from a sequence of partitions, rather than the other way around.

Definition 3.3. A KR-partition T is a clopen partition of X of the form A 0 ⊔ . . . A k ,
where each A i is further subdivided into B i,0 , . . . , B i,j i (possibly j i = 0) and for all i and all r, s ∈ {0, . . . ,

j i } B i,r ∼ K B i,s .
The union of all B i,0 is called the base of the partition, and the union of all B i,j i is its top. Each A i is called a column of the partition.

To each KR-partition, one can associate the algebra A T whose atoms are all B i,r with r < j i , and the top of the partition; and the partial automorphism of Clop(X) with domain A which maps each B i,r to B i,r+1 for all i and r < j i , and maps the top of the partition to its base. Note that the ordering of atoms within each column matters.

We say that a KR-partition S refines a KR-partition T if the base (respectively, top) of S is contained in the base (respectively, top) of T , and S-towers are obtained by cutting and stacking towers of T on top of each other (see Figure 3). More precisely: for each column C = (D j ) 0≤j≤J of S there exist columns

A i k = (B i k ,j ) 0≤j≤j i k of T (0 ≤ k ≤ K)
and clopen subsets S k j ⊆ B i k ,j such that J = ∑ 0≤k≤K (j i k + 1) and such that, for each k, we have D j+∑ l<k (j i l +1) = S k j for every 0 ≤ j ≤ j i k .

Note that if S refines T then the algebra and partial automorphism associated to S refine those that are associated to T . Proposition 3.4. Given a KR-partition T , and ε > 0, there exists a KR-partition S which refines T and which is such that the base and top of S both have diameter less than ε.

Proof. We let again A 0 , . . . , A k denote the columns of T , B i,0 , B i,j i denote respectively the base and top of the i-th column, and B be the base of T .

We begin by describing how to deal with a very favorable particular case, where there exists an integer n ≥ 2 such that µ(B 0,0 ) = µ(B 1,0 ) = 1 n µ(B) and both B 0,0 and B 1,j 1 have diameter less than ε. Let R = B \ (B 0,0 ∪ B 1,0 ). Then we have µ(R) = (n -2)µ(B 0,0 ) for all µ ∈ K, so, by goodness, as long as n > 2, we can find a copy C of B 0,0 inside R. Now, the bases of A 2 , . . . , A k induce a partition of C, C = k j=2 (C ∩ B j,0 ). Furthermore, by goodness (or homogeneity), we can find an equivalent cutting of B 0,0 into pieces P j ∼ K (C ∩ B j,0 ). We pass on this cutting of B 0,0 throughout the column A 0 , and similarly we pass on the cutting of C throughout the columns A 2 , . . . , A k . Finally, we stack each new column starting with C ∩ B j,0 on the top of the new column based on P j ; this gives us a refinement T ′ of T .

As long as n ′ = n -1 > 2, we repeat this process, only that now we find copies C j ⊆ R ′ = R \ C of each P j . As before, we cut them with the bases of A 2 , . . . , A k , then imitate this cutting on the corresponding P j and pass it on throughout the columns; then we stack the new columns based on subsets of R ′ on top of the corresponding new columns based on subsets of B 0,0 . Once this process has been repeated n -2 times, we apply it, lastly, on A 1 . What we obtain is a refinement S of T whose base is B 0,0 and whose top is B 1,j 1 .

The picture below illustrates the procedure we just described. The column containing the new base is colored in blue, and the one containing the top is red; we keep track of what happens to them in the picture (in a very simple case for readability).

FIGURE 3. Cutting and stacking in the favorable case

That is the last picture that we will include in this article, as the next arguments are a bit harder to illustrate; nevertheless, we invite the reader to draw her own pictures, since we feel that the ideas become more transparent in this way.

To deal with the general case, we use the fact that K is good and approximately divisible to reduce to this favorable case, modulo a small error (which would not appear if K were exactly divisible). First, by cutting B 0,0 (and throughout A 0 ) if necessary, we can ensure that B 0,0 has small diameter. Moreover, by picking a subset of B 0,j 0 of small diameter and cutting again, we can ensure that both the top and base of the first column of our partition have small diameter. Cutting yet again, we make sure that the union B 0,j 0 ∪ B 1,j 1 has small diameter. Cutting and using goodness once more, we ensure µ(B 0,j 0 ) = µ(B 1,j 1 ) for all µ ∈ K.

Next, pick some integer n such that 1 n < µ(B 0,0 ) for all µ ∈ K. Using the fact that K is good and approximately divisible, we find clopen sets

C 0 ∼ K C 1 ∼ K . . . ∼ K C n-1 contained in B, pairwise disjoint, such that C 0 ⊆ B 0,0 , C 1 ⊆ B 1,0 and E := B \ n-1 i=0 C i ⊆ B 0,0 .
We cut B 0,0 into two pieces, one of which is the error E, inducing a further KR-partition, one column of which has base E. We set apart this column A E , that is, we consider Y = X \ A E . Then, our current KR-partition of X induces a KR-partition of Y; cutting one last time we obtain columns based on C 0 and on C 1 , which we set to be, respectively, the first and the second column of that partition. We have thus obtained a KR-partition (of Y) which satisfies the assumptions of the favorable case described above. Applying the stacking procedure given for that case, we obtain a new KR-partition of Y whose base is contained in B 0,0 and whose top is contained in B 1,j 1 . Finally, considering this partition together with the column A E , we get a KR-partition of X whose base (contained in B 0,0 ) and whose top (contained in B 0,j 0 ∪ B 1,j 1 ) have both small diameter.

We say that a partition is compatible with a clopen set U if U is a union of atoms of the partition. Proposition 3.5. Given a KR-partition T , and two clopen subsets U ∼ K V, one can find a KR-partition S refining T , compatible with U and V, and such that in each column there are as many atoms contained in U as atoms contained in V.

Note that then, if g is any element of G which extends the partial automorphism associated to S, there exists an h ∈ [[g]] such that h(U) = V (because one can map U to V while only permuting atoms within each column of S).

Proof. First, note that by goodness there is a KR-partition S refining T and compatible with U, V: consider the algebra generated by T and U, V, then pull back the associated partion of atoms of T to the base of T (via an automorphism g i,s ∈ G mapping B i,s to B i,0 ), and push it back up (using g -1 i,s ). We obtain a new KRpartition S, refining T , compatible with U and V (each column has been subdivided into smaller columns, and no stacking has taken place).

Now, for all such KR-partitions, we can associate to any column C the numbers

u C = ♯{atoms of C contained in U}, v C = ♯{atoms of C contained in V},
and n C = u Cv C . Our aim is to find a KR-partition with u C = v C for all columns.

We distinguish the columns of the following types: C + , the set of columns with n C > 0, and C -, those where n C < 0. We let n S denote the maximum value of |n C | among all columns C, and finally let n denote the smallest possible n S among all S refining T and compatible with U, V.

We suppose for a contradiction that n = 0, and we pick S with n S = n. Without loss of generality, we assume that n = n C for some C ∈ C + , and we consider the set D of all columns C such that n C = n. Let B be the union of all the bases of columns in D, and B ′ the union of the bases of elements of C -. Then we observe that either µ(B) = µ(B ′ ) for all µ ∈ K, or µ(B) < µ(B ′ ) for all µ ∈ K (otherwise µ(U) and µ(V) would not be equal). Thus one can build a new KR-partition by cutting and stacking on top of each column of D some element of C -(just map arbitrarily the union of the tops of elements of D into the union of the bases of C -, then refine accordingly). This has the effect of producing a new KR-partition such that every column in C + satisfies n C < n. Doing the same (if necessary) with C , we obtain a contradiction to the minimality of n.

The previous two propositions provide us with the tools to construct the Ksaturated homeomorphism we were looking for. Proposition 3.6. There exists a K-saturated element in G.

Proof. Fix an enumeration (U n , V n ) of all pairs of clopen sets (U, V) such that U ∼ K V. Using Propositions 3.4 and 3.5, we build a sequence of KR-partitions S n with the following properties:

(1) For all n, S n+1 refines S n .

(2) For all n, S n is compatible with U n , V n , and in each column of S n there are as many atoms contained in U n as atoms contained in V n . (3) The diameters of the base and top of S n converge to 0. Then, there exists a unique g ∈ Homeo(X) which extends the partial automorphisms associated to S n . The construction ensures that g ∈ G, and that g is K-saturated.

Remark 2. The set of all K-saturated homeomorphisms, as well as the set of all minimal homeomorphisms in G, are G δ subsets of G. The argument above proves that the closure of the set of K-saturated elements contains all minimal elements of G; thus, in G, a generic minimal homeomorphism if K-saturated. It would be interesting to determine precisely the closure of the set of all minimal homeomorphisms. It is tempting to believe that it corresponds to the set of all g ∈ G such that, for any clopen A different from ∅ and X, one has g(A) = A (see [START_REF] Bezuglyi | Topologies on the group of homeomorphisms of a Cantor set[END_REF]Theorem 5.9] for the analogous result in Homeo(X)).

So far, we have managed to build a K-saturated, hence minimal, element which preserves all measures in a given dynamical simplex K. It is a priori possible that this element preserves measures not belonging to K; saturation prevents this from happening. That was the original motivation for trying to build a K-saturated element of G rather than merely a minimal homeomorphism belonging to G. To deal with this issue, we have one remaining task: proving that the set of G-invariant probability measures, which by definition contains K, in fact coincides with K.

SATURATED ELEMENTS CANNOT PRESERVE UNWANTED MEASURES

We will denote the simplex of G-invariant probability measures by K G . Given g ∈ G, the simplex of g-invariant probability measures will be denoted K g . Proposition 4.1. Let g be a K-saturated element. Then K g = K G .

Proof. Clearly, K G ⊆ K g . For any µ ∈ K g and any h ∈ [[g]] we have h * µ = µ. Since {h : h * µ = µ} is closed in Homeo(X), and the closure of [[g]] is G since g is K-saturated, we obtain as desired that h * µ = µ for all h ∈ G.

The last remaining piece of our puzzle is thus the following proposition. Proposition 4.2. We have K G ⊂ K.

Proof. We proceed by contradiction, and assume that ν ∈ K is such that g

* ν = ν for all g ∈ G. By Proposition 2.7, if U, V are clopen sets with U ∼ K V, then ν(U) = ν(V).
Note first that, if µ(A) < 1 n for all µ ∈ K, then there exist disjoint A 1 , . . . , A n such that A i ∼ K A for all i ∈ {1, . . . , n}; since by the remark in the previous paragraph ν(A 1 ) = . . . = ν(A n ) = ν(A), we also have ν(A) < 1 n . This observation will be used twice below.

Using the Hahn-Banach theorem (see for instance Theorem 3.4(b) of [R] for the version we use here, valid in locally convex Hausdorff topological vector spaces; note that we endow the dual of C(X) with the weak* topology, so that its topological dual naturally identifies with C(X), see Theorem 3.10 of [R]), we know that there exists a continuous function f :

X → R such that ∀µ ∈ K X f dµ < X f dν.
Replacing f by f + max{| f (x)| : x ∈ X}, and using the fact that all our measures are probability measures, we may assume that f (x) ≥ 0 for all x. Using uniform continuity of f , we may further assume that f only takes finitely many values, which are all non-negative rational numbers; multiplying by a large enough integer, we finally reduce to the case when f takes finitely many integer values. Hence we have finitely many clopen sets (A i ) 1≤i≤N and positive integers n i such that

∀µ ∈ K N ∑ i=1 n i µ(A i ) < N ∑ i=1 n i ν(A i ). Pick integers p, q > 1 such that ∀µ ∈ K N ∑ i=1 n i µ(A i ) < p q < N ∑ i=1 n i ν(A i ).
Using the fact that K is approximately divisible, we may find for all i some clopen sets B i,1 ∼ K B i,2 ∼ K . . . ∼ K B i,p contained in A i such that µ(A i \ B i,j ) is arbitrarily small for all µ ∈ K, hence also ν(A i \ B i,j ) is arbitrarily small. Thus,

ν(A i ) p -ν(B i,1
) can be made arbitrarily small, so we can ensure that ∀µ ∈ K N ∑ i=1 n i µ(B i,1 ) < 1 q < N ∑ i=1 n i ν(B i,1 ).

We can then build a set B which is a disjoint union of n 1 copies of B 1,1 (that is, n 1 clopen sets which are ∼ K -equivalent to B 1,1 ), n 2 copies of B 2,1 , etc; we have ∀µ ∈ K µ(B) < 1 q and ν(B) > 1 q .

This contradicts the observation made at the beginning of the proof.

Corollary 4.3. Assume that g is a K-saturated element of G. Then K = K g .

Proof. We have K ⊂ K g by definition of G, and the converse inclusion follows from the previous propositions.

We have finally proved Theorem 1.1.

Remark 3. Using the idea of the proof of the previous proposition, one can check that goodness and approximate divisibility imply that any affine function on K of the form µ → X f dµ, where f ∈ C(X, [0, 1]), can be approximated arbitrarily well by an affine function of the form µ → X χ B , where χ B is the characteristic function of a clopen set. This implies, in the terminology of Dahl, that G(K) is dense in Aff (K).

It seems clear that G(K) being dense in Aff (K) and approximate divisibility are related conditions; it appears to follow from Dahl's arguments in [D1] that, whenever K is a Choquet simplex and G(K) is dense in Aff(K), K must be approximately divisible.

We conclude by discussing a further line of enquiry suggested by our work here. As mentioned at the end of the introduction, our argument gives an elementary proof of the fact that, given a minimal homeomorphism g, there exists a saturated homeomorphism h preserving the same measures as g. It follows from a theorem of Krieger [K], the proof of which is relatively short and elementary, that any two saturated homeomorphisms preserving the same measures are orbit equivalent. It then becomes interesting to try and find a dynamical proof of the fact that, if g is a minimal homeomorphism, S is a saturated minimal homeomorphism, and K g = K S , then g and S are orbit-equivalent : combining such a proof, the main result of this paper, and Krieger's theorem, one would obtain a new dynamical proof of the Giordano-Putnam-Skau relating orbit equivalence and invariant measures.

  and h as well as each k A are homeomorphisms) and belongs to O ∩ [g].

  e. for any clopen set A, any integer n and any ε > 0, there exists a clopen B ⊆ A such that nµ(B) ∈ [µ(A)ε, µ(A)] for all µ ∈ K.

FIGURE 2 .

 2 FIGURE 2. A partition in three ∼ K pieces plus a rest of small measure

Research partially supported by Agence Nationale de la Recherche projects GruPoLoCo (ANR-11-JS01-0008) and ValCoMo (ANR-13-BS01-0006).

1