C Cuyamendous

K S Leung

T Durand

-Y Lee

C Oger

J.-M Galano

Synthesis and Discovery of Phytofurans: Metabolites of α-Linolenic Acid Peroxidation

Phytofurans are novel metabolites produced by non-enzymatic peroxidation of α-linolenic acid. An unprecedented Payne rearrangement-cyclization of a C2-symmetric bisepoxide permitted construction of the core 3-hydroxy-2,5-disubstituted tetrahydrofuran. LC-MS/MS investigation provided evidence for the presence of phytofurans in nuts and seeds for the first time.

Omega 3 polyunsaturated fatty acids (PUFAs) such as αlinolenic acid (ALA), eicosapentaenoic acid (EPA) and docosahexaenoic acid (DHA) are well known for their cardioprotective properties. 1 Under oxidative stress conditions these PUFAs are easily peroxidized to give oxygenated metabolites by reactive oxygen species (ROS), 2 and we recently showed that some oxygenated metabolites of DHA provide cardioprotection by exerting antiarrhythmic properties. 3 In 1998, Muller and co-workers highlighted the formation of ALA enzyme-independent cyclic oxygenated metabolites in plants and expressed them as phytoprostanes (PhytoPs) (Scheme 1). 4 Currently, PhytoPs stand as biomarkers of oxidative stress in biological systems, but also as protective defence molecules in plants and neurons. 5 The presence of PhytoPs in plant oil from nuts and seeds is also pertinent in human diet and health.

More recently, a novel non-enzymatic pathway of PUFA was uncovered leading to 3-hydroxy-2,5-disubstituted tetrahydrofuran structures, named Isofurans (IsoFs) from arachidonic acid (AA), Neurofurans (NeuroFs) from DHA and dihomo-IsoFs from adrenic acid (AdA). Our explorations in this field prompted us to investigate the occurrence of such mechanism pathways in plants and we successfully identified production of the phytofurans (PhytoFs) (named by analogy to phytoprostanes and isofurans) from ALA for the first time.

Three strategies have been reported for the synthesis of this family of natural products by the group of Taber (IsoFs), 7-9 Zanoni-Vidari (NeuroFs) 10 and our group (dihomo-IsoFs and NeuroFs). 11,12 We report herein on the development of a novel and powerful strategy for the synthesis of all members of this family of targets and their derivatives. We have applied this strategy to the synthesis of PhytoF and to our first target, ent-16-(RS)-13-epi-ST-∆ 14 -9-PhytoF (1) from the enediol family.

13

The Phytofuranic biosynthetic pathway from ALA can generate two classes of PhytoFs (alkenyl and enediol) each comprising two families, giving a potential total of 128 phytofuran isomers. 14 We made the arbitrary choice to target the enediol metabolite ent-16-(RS)-13-epi-ST-∆ 14 -9-PhytoF (1), knowing that LC-MS/MS analysis will not allow separation of the various stereoisomers in each family of PhytoFs. The synthesis of bisepoxide 3 was performed according to the work of Hoffmann et al., 16 but with slight modification of the literature procedure. It started from the known 4-chlorobut-2yn-1-ol after SOCl 2 /pyridine chlorination of commercially available but-2-yne-1,4-diol in 46% yield (Scheme 3).A cross coupling reaction between chlorinated alkyne and prop-2-yn-1-ol was performed following Ivanov's procedure, 17 yielding desired hepta-2,5-diyne-1,7-diol 4 in 81% yield. Partial reduction of diyne 4 to give the E/E-diene using freshly prepared Red-Al reagent produced desired hepta-2,5-diene-1,7-diol in 55% yield on a multigram scale. Finally, double Sharpless asymmetric epoxidation (SAE) of the bis-allylic alcohol 3 using L-(+)-diethyl tartrate afforded the desired C2symmetric (2S,2'S,3S,3'S)-bis-epoxide 3 in 76% yield and with 90% ee on a gram scale quantity and only 4 steps.

In our original plan, the epoxydiol THF C was to be obtained via two consecutive Payne epoxide rearrangement reactions (A to B) of 3 followed by cyclization (Scheme 4). However, it soon became apparent that the poor solubility of 3 in almost any organic solvent compatible with a Payne rearrangement reaction tempered this design process (a non-nucleophilic base in an aprotic media was necessary to avoid hydrolysis of the newly formed terminal epoxides from the Payne reactions). [START_REF] Hanson | Organic Reactions[END_REF] Switching to water or MeOH as the solvent resulted in complete dissolution of 3, and a screening of bases (NaOH, LiOH, KOH, MeONa) and temperature conditions (rt, 80 °C) was performed to focus on the formation of the THF derivatives. Indeed, treating 3 with KOH (5 eq.) in water at 80 °C for 2 h furnished a 4/1 mixture of inseparable tetraol THF derivatives such as 2 in 65% yield (Scheme 3).

The other alkaline bases also furnished 2 but with lower yields, and reaction at rt required more than 4 days to proceed to completion with lower yields. Intensive NMR investigations did not secure the relative configuration of the two THF compounds recovered. Gratifyingly, treatment of that mixture of THF derivatives with 2,2-dimethoxypropane and a catalytic amount of PTSA in boiling acetone gave the corresponding protected inseparable diols in 79% yield. The crystalline nature of major diastereoisomer 5 meant that it could be crystallised to give crystals suitable for X-ray diffraction, therefore allowing its configuration to be assigned. unidentified polar compounds were isolated from the reaction. Finally, a greater mechanistic understanding of the reaction sequence was obtained by treatment of 3 with MeONa in MeOH at 80 °C. This reaction afforded a mixture of THF compounds (7:3 ratio) bearing a methoxy substituent (the major compound having the MeO group in place of the nucleophilic hydroxide following scheme 4). 14 Following the discovery of a reliable and easy route to supply the functionalized THF core of PhytoFs, we continued our synthetic venture. The poor solubility and handling of 2 meant that acetonide THF 5 (together with minor unknown diastereoisomer) was used for the synthesis of the monoprotected TBS-ether 6 via a bisprotection-regioselective deprotection sequence in an overall 64% yield (Scheme 5). The primary alcohol 6 was oxidized using Dess-Martin periodinane followed by Wittig reaction with previously described BrPh 3 P(CH 2 ) 6 CO 2 Et and NaHMDS in THF at -78°C in 75% overall yield. The Z-double bond formed was then subjected to catalytic hydrogenation with palladium over charcoal to complete the first side chain of compound 7 in 87% yield. To introduce the lipidic second side chain, functionalization of the acetonide moiety was required. The best compromise was a three-step sequence developed by Rychnovsky and co-workers [START_REF] Rychnovsky | [END_REF] which permitted transformation of the acetonide 7, through an enol intermediate, into the free primary alcohol 1-methyl-1cyclopropyl hydroxyl derivative 8 in 62% yield. Interestingly, flash chromatography purification of 8 permitted removal of the minor isomer obtained from the Payne sequence.

(+ dia) (+ dia) (+ dia) f-
Oxidation of the alcohol with Dess-Martin periodinane followed by the coupling with 1-(triphenylphosphoranylidene)butan-2-one gave enone 9 in 51% over 2 steps. The cyclopropyl group of 9 was removed thereafter using N-bromosuccimide to give 10 in 85% yield. 19 Luche reduction of the enone function gave a 1:1 mixture of epimers at C16 (epimers were chosen for mass spectrometry identification), followed by TBS deprotection, before hydrolysis using LiOH to give the first synthesis of PhytoF, ent-16-(RS)-13epi-ST-∆ 14 -9-PhytoF 1. Eight milligrams of 1 was obtained using this 20-step sequence with an overall yield of 0.8%. Identification of the mass fragmentations from the MS/MS of the synthesized PhytoFs and evaluation by LC-MS/MS in plant foodstuffs will corroborate its presence without ambiguity.

For the first time, we established the presence of the natural PhytoFs in nuts and seeds, the progenitors of plants. The consumption of nuts and seeds has increased in recent years due to the popularity of the vegetarian and Mediterranean diets. ALA found in these nuts and seeds is the precursor of essential fatty acids, mainly EPA and DHA for the vegetarians, which are important components for the brain development and central nervous system, as well as in prevention of metabolic disorders. However, the conversion of dietary plant ALA to EPA and DHA in vivo is poor. Nevertheless, there are reports concerning the benefits of dietary nuts and seeds in prevention of cognitive dysfunction and cardiovascular diseases. 20,21 When all facts are considered, it is possible that the beneficial in vivo effect of ALA from plants is due to other bioactive components. PhytoPs from ALA peroxidation have emerged as potential active components in plants and humans, and several isomers have been identified previously in plant oils. 22,23 In human cells, some PhytoPs were found to display anti-inflammatory properties, and protect against oxidant-induced neuronal injuries. As depicted in Figure 1, we quantified the levels of ALA in nuts (pine nuts and walnuts) and seeds (flaxseeds and chia seeds) using the LC-MS/MS. 14 ‡ We also idenƟfied and quantified one particular family of PhytoF for the first time with the help of our synthetic ent-16-(RS)-13-epi-ST-∆ 14 -9-PhytoF 1 and the use of MS/MS. 14 It should be noted that such analyses quantify the overall sum of all possible stereoisomers of 16-(RS)-13-epi-ST-∆ 14 -9-PhytoF (because secondary MS fragmentation is likely to be similar for all isomers) which can be formed following the Please do not adjust margins Please do not adjust margins biosynthetic pathways. Surprisingly, large variations in ALA levels were observed in various samples (Figure 1A), and seeds generally had higher concentrations than the nuts. In addition, the ent-16-(RS)-13-epi-ST-∆ 14 -9-PhytoF concentration was successfully quantified in the same samples (Figure 1B). Notably, flaxseeds showed the highest ALA level but had a relatively low ent-16-(RS)-13-epi-ST-∆ 14 -9-PhytoF level compared to walnuts and chia seeds, that contained less ALA. However, pine nuts had the lowest ALA level and the lowest ent-16-(RS)-13-epi-ST-∆ 14 -9-PhytoF level, and the concentration of ent-16-(RS)-13-epi-ST-∆ 14 -9-PhytoF in walnuts and chia seeds was approximately 10-to 20-fold higher than in flaxseeds and pine nuts. 14 Our observations also indicate that the relative concentrations of the ent-16-(RS)-13-epi-ST-∆ 14 -9-PhytoF do not necessarily mirror the relative ALA concentrations in the nuts and seeds. Nuts and seeds consumed whole or in the form of an oil extract are known to have health benefits; to date, some of this bioactivity has been attributed to oxygenated ALA and potentially to phytoprostanes. However, the concentrations of ent-16-(RS)-13-epi-ST-∆ 14 -9-PhytoF found in the nuts and seeds are comparable to those of phytoprostanes reported previously 23 and so could also have a functional role in human health. Currently our group is conducting in vivo studies to validate the potency of ent-16-(RS)-13-epi-ST-∆ 14 -9-PhytoF in biological systems.

ARTICLE

In conclusion, the total synthesis of a phytofuran natural product, ent-16-(RS)-13-epi-ST-∆ 14 -9-PhytoF (1) was accomplished using a innovative synthetic strategy involving a Payne rearrangement-cyclization sequence of a C2-symmetric intermediate. We also quantified levels of ent-16-(RS)-13-epi-ST-∆ 14 -9-PhytoF in plant foodstuffs for the first time. Further studies are on-going to achieve the synthesis of the three remaining types of PhytoFs using our novel and flexible approach and to further establish their presence in dietary plants and their role in human health.

We would like to thank to University of Montpellier (UM) for the doctoral fellowship of CC and the PIVERT project N° 2014/GENESYS/WP6P4-linoprostanes. We also thank Dr. Arie Van der Lee for performing X-ray analysis.

Notes and references ‡ RelaƟve concentraƟon of ALA and PhytoF in seeds and nuts can be subject to variation depending on the source and storage conditions.

14 With regard to the stereochemistry observed for 2

 2 process (from A to C) did not occur but and that a single Payne rearrangement/ bimolecular epoxide-opening/ cyclization cascade was likely (Scheme 4). Such a conclusion also resulted from the isolation of a small amount of compound B from the reaction with KOH showing the feasibility of a double Payne reaction. Interestingly treatment of B under the same Payne conditions did not produce compound C and only starting material and

  Scheme 5. Reagents and conditions: a) TBSCl, imidazole, DMAP, CH2Cl2, RT, 88%; b) PPTS, EtOH, 0°C, 73%; c) DMP, NaHCO3, CH2Cl2, RT; d) BrPh3P(CH2)6CO2Et, NaHMDS, THF, -78°C to RT, 75% over 2 steps; e) H2, Pd/C, EtOH, RT, 87%; f); i-PrNEt, TMSOTf, CH2Cl2, 0°C to RT; g) Et2Zn, CH2I2, Et2O, RT; h) K2CO3, MeOH, RT, 62% over 3 steps; i) DMP, CH2Cl2, 0°C to RT; j) 1-(triphenylphosphoranylidene)butan-2-one, CH2Cl2, 51% over 2 steps; k) NBS, THF/H2O, RT, 85%; l) CeCl3.7H2O, NaBH4, MeOH, 0°C RT; m) TBAF, THF, RT; n) LiOH, THF/H2O, RT, 67% over 3 steps. TES = triethylsilyl, Tf = trifluoromethansulfonyl, NBS = N-bromosuccinimide, TBAF = tetrabutylammonium fluoride. TBS = tert-butyldimethylsilyl, DMAP = 4-(N,Ndimethylamino)pyridine, PPTS = pyridinium para-toluenesulfonate, DMP = Dess Martin Periodinane, NaHMDS = sodium bis(trimethylsilyl)amide.

6

 6

Figure 1 .

 1 Figure 1. Concentration of α-linolenic acid and ent-16-(RS)-13-epi-ST-∆ 14 -9-PhytoF in nuts and seeds. The lipid component was isolated by Soxhlet extraction and cleaned by SPE prior to LC-MS/MS measurement (refer to SI). Values of the column in graphical display is annotated as mean ± SD, n=4. Columns sharing different alphabetical superscripts in each graph are significantly different at p<0.05

Journal Name 4 |

 4 J. Name., 2012, 00, 1-3 This journal is © The Royal Society of Chemistry 20xx

Published on 07 September 2015. Downloaded by University of Montpellier on 08/09/2015 12:35:51. View Article Online

ChemComm Accepted Manuscript


F Ben Ettouil

B Pateyron

Hélène Ageorges

M El Ganaoui

P Fauchais

Study of powder dispersion at the injector exit and its influence on particles thermal and dynamic history

Keywords: powder feeder, injector, collisions, dispersion angle, velocity distribution

In plasma spraying, the particles dispersion at the injector exit controls their dynamical and thermal histories (trajectories, velocities, temperatures, melting and vaporization …) which are key parameters for the splat formation at impact and the coating construction. In this paper a model of particles conveyed within the injector is presented and their dispersions in size, velocity and direction are characterized

Introduction

At the injector exit, a particle is characterized by four parameters:

• velocity vector amplitude • velocity vector angle δ within a cone which total angle is generally below 20° (see figure 1)

• velocity vector angle in the direction defined by the angle Φ varying from 0 to 360° • mass distribution Fig. 1 Cone of particles dispersion at the injector exit For the same amplitude, the velocity vector angle plays a non negligible role as illustrated in figure 2. It presents the evolution of the surface temperature of two zirconia particles, 25 µm in diameter, both injected with the same injection velocity vector amplitude of 40 m/s, but with two different angles (δ 1 =0° et δ 2 =10°). It can be seen that the modification of only one parameter of the particle injection changes its thermal history. Similar results are obtained for its trajectory and velocity. This work presents a descriptive model of the transport of particles by a carrier gas inside the cylindrical injector. The model taking into account the size distribution of the particles, the particle collisions between themselves and with the injector wall, makes it possible to obtain the velocity and size distribution of particles at the injector exit. The adapted strategy in this simulation consists in:

-calculating the gas velocity profile, -simulating of the size distribution, -calculating of particle displacements and velocities, -detecting of particle-particle and particle-wall collisions, -calculating of displacements and velocities after collisions.

Modeling 2.1 Calculation of carrier gas velocity profile

To evaluate the viscosity of the suspension gas-powder, the formula of Einstein has been used [START_REF] Kiyoshi | Extension of Einstein's viscosity equation to that for concentrated dispersions of solutes and particles[END_REF] ) 5 . 2 1 (

s g eff ε µ µ + =
The flow mode is determined by the Reynolds number:

eff g g D u µ ρ / Re =
In the case of turbulent flow, a self-similarity of meanflow is considered [START_REF] Guo | Self-similarity of mean flow in pipe turbulence[END_REF], it is based on modified logwake law for turbulent flow [START_REF] Guo | Modified log-wake law for turbulent flow in smooth pipes, Reply to Discussion[END_REF]. The friction coefficient is written as following:

( )

5 25 . 0 3110 . 4 Re/ 1 Re 3164 . 0 + = g f
The wall shear stress is given by: ( )

4 / exp 4 ξ - = + y Y               + +       +       +       = k C Y A Y arctg k A Y arctg A A Y arctg A U 1 4 4 3 1 ln . . 3 .
with A=7, C=107, k=0.43 et k4=-0.52 The dimensionless velocity has been written as follows ( )

4 / sin . 2 4 2 ξ - + = + U u
Finally, the mean velocity at the radial position r is written as the function of the outer variable and the shear velocity:

( ) ( ) 4 / sin . 2 ) ( 4 2 * ξ - + = U u r u

Calculation of particle displacement

Particles are supposed to be spherical, they can collide between themselves and with the wall and they are especially subjected to drag force, which permits to write the momentum equation as follows:

( )

2 4 . . 2 g p g p g p D p p u u u u d C dt dv m - - - = ρ π

particle-particle collision

Particle-particle collisions are supposed to be elastic and treated with the hard sphere model. Particle parameters after collision (position, velocity, acceleration) are calculated from the conservation equations for momentum and energy.

particle-wall collision

The particle-wall collision was inspired from the collision synchronization in the domain of dynamic simulation and animation. Two approaches are possible:

-Milencovic method [START_REF] Milenkovic | Optimization based animation[END_REF]: each object is stopped on its first collision, the collision synchronisation and treatment is made at the end of the time step -Faure and Galizzi method [START_REF] Galizzi | Animation efficace de solides en contact par modèle physique[END_REF]: the rectification of positions and velocities occurs later after the collision.

Milenkovic method

Faure and Galizzi method Figure 3: Treatment of particle-wall collision In this work, the Milencovic method was adopted.

Results:

In the following are presented results from the simulation of 5000 zirconia particles (22-45 µm), conveyed within an injector 1.5 mm in diameter. The used carrier gas is argon. The size distribution is presented in figure 4. Gas and powder properties are summarized in table 1. Figure 5 shows that the spatial distribution of the velocity vectors is uniform in space. According to the surface of a hemisphere the central part corresponds to the smallest surface, thus resulting in an angle distribution close to zero in the injector axis. It is also worth to note that for the initial distribution a maximum angle of 30° was chosen for δ. Of course the initial distribution at the injector inlet should depends on the particle distribution at the connecting pipe outlet and works are in progress to calculate this distribution within the connecting pipe between the powder feeder and the injector. Figure 7 shows the δ angle distribution without taking into account particle-particle collision, the dispersion seems to be the previous as the case but with a reduced variation domain. Figure 8 shows the final axial velocity distribution, the initial one being supposed to be randomly uniform. This resulting distribution at the injector exit seems to be Gaussian and is centred on the mean gas velocity

Conclusions

This paper presents the first results of a model representing the dispersion of particles at the injector exit of a plasma torch. The model developed takes into account the size distribution, the wall-particle and particle-particle collisions. This model is aimed at determining the conditions of particles injection of in a d.c plasma jet according to the working conditions (carrier gas and powder flow rate, injector internal diameter …)

  drag coefficient (-) D : internal diameter of the injector (m) d p : particle diameter (m) f : friction coefficient (-) k : Karman constant (-) 4 k : constant (-) m p : particle mass (kg) r : radius (m) R : injector radius (m) Re : Reynolds number (-) u : velocity (m/s) * u : shear velocity (m/s) number of the wall effect (-) y = R-r distance from the wall (m) Y : similarity number combining inner and outer variables + y δ : dispersion angle (rad) Φ: azimuthal dispersion angle (rad) ρ : mass density (kg.m -3 ) µ : molecular viscosity (Pa.s) µ eff : suspension viscosity (Pa.s) ε s : solid fraction (-) υ : kinematic viscosity (m 2 /s) τω : wall shear stress (N/m2)

Fig. 2 :

 2 Fig.2: Temperature evolution on the surface of zirconia particles of 25 µm in diameter Injected with different δ angles and Φ=0

u+

  being the dimensionless velocity which is given by: The model considered is based on two composed similarity numbers U and Y, where U is the similarity number representing the pure effect of the wall and Y that combining the inner and outer variables.

Figure 4 :

 4 Figure 4: simulated particle size distribution

Figure 5 :

 5 Figure 5: Initial δ angle distribution at the injector inletFigure6shows that at the injector exit, due to collisions, some particles have a divergence angle higher than 20°, but most of them are within 20° in good agreement with the experimental results. The δ angle distribution evolves along the injector length and after 7 cm, it becomes bimodal and symmetrical. It is clear that it tends after a longer course, to a Gaussian distribution of average zero. The most of δ values are within a cone of 20° and values outside the angle are due to the particle-particle collisions.

Figure 6 :Figure 7 :

 67 Figure 6: Final δ angle distribution at the 7 cm long injector exit

Figure 8 :

 8 Figure 8: Final axial velocity distribution Figure 9 illustrates the final δ angle distribution as function of particles size. What ever may be the size class of particles, small or big, the mean value of δ is approximately zero. It seems to confirm that the

Figure 9 :

 9 Figure 9: Final δ angle distribution as function of particles size Figure 10 presents the distribution of the radial velocity as function of the particle size. The geometry of the studied system is symmetric, which explains why this distribution is almost symmetric and centred on zero.

Figure 10 :

 10 Figure 10: Final radial velocity distribution as function of particles size

Table 1 :

 1 Powder and carrier gas properties

		ρ	Flow rate	Viscosity
	Carrier gas:	1,67	4 L/min	2,09 10 -5
	Argon	kg/m3		Pa.s
	Powder :	5800	1 kg/h	-
	Zirconia	kg/m3


Professor Pierre Casal

Henri Gouin
email: henri.gouin@univ-cezanne.fr

VIBRATIONS OF LIQUID DROPS IN FILM BOILING PHENOMENA (the mathematical model )

Keywords: vibrations, liquid drop, film boiling

Flattened liquid drops poured on a very hot surface evaporate quite slowly and float on a film of their own vapour. In the cavities of a surface, an unusual type of vibrational motions occurs. Large vibrations take place and different forms of dynamic drops are possible. They form elliptic patterns with two lobes or hypotrochoid patterns with three lobes or more. The lobes are turning relatively to the hot surface. We present a model of vibrating motions of the drops. Frequencies of the vibrations are calculated regarding the number of lobes. The computations agree with experimental forms obtained in [1] by Holter and Glasscock.

INTRODUCTION

Water splashed on a moderately hot metallic flat plate spreads out, comes to the boil and then quickly evaporates. It is not the same when the metal is very hot: water remains cool, breaks up into many drops that roll, bound and are thrown on all sides. Such drops are mentioned in the literature as being in "spheroidal state". Most of the people have noticed these phenomena in their childhood when they stared at rolling drops on a very hot oven. Leidenfrost first experimented the process in 1756: a little water poured on a red hot spoon, does not damp the spoon and takes the same shape as mercury. A strong motion of vapour lying between liquid and metal supports liquid masses and causes fast vibrating motion in the liquid bulk. This is the film boiling or Leidenfrost phenomenon [START_REF] Leidenfrost | De aquae communis nonnullis qualitatibus tractatus. Duisburg (1756)[END_REF]. It is well known since before the time of Faraday and was the subject of numerous studies during the nineteenth century [START_REF] Boutigny | Sur les phénomènes que présentent les corps projetés sur des surfaces chaudes[END_REF]. One of the interests in the Leidenfrost phenomenon rises from an explosive behaviour associated with non-equilibrium of pressures. This affects all the industrial sectors where hot temperatures are used (for example in metal industry: quenching of metals [START_REF] Flament | Film boiling instability at high temperature on a vertical cylinder quenched in a subcooled liquid[END_REF], etc). This generates the unsteadiness of distillation plants in the petroleum industry and many accidents in nuclear engineering [START_REF]Thermohydraulics of two-phase systems for industrial design and nuclear engineering[END_REF]. Indeed, the first surprising effect was previously described. The liquid drops are floating on a vapour film when they are placed on hot plates. This state is usually explained as the fact that when the temperature of the wall is greater than a value depending both on the fluid and the state of the surface, the exchanges of heat are small corresponding to the formation of a thin film of vapour isolating drops from the wall. In such conditions, liquid drops maintain during a time of the order of minutes and usual descriptions present the film of vapour keeping liquid drops well below boiling. Observations of the drops give phenomena by translational and rotational motions. Photographic and stroboscopic tools have been used to have descriptions of the experiments, but the effects can be seen by naked eye. These vibrations lead to beautiful patterns that appear as solid geometric figures. Such a phenomenon is qualitatively very well described in the paper by Holter and Glasscock [START_REF] Holter | Vibrations of evaporating liquid drops[END_REF]. The different models do not give an interpretation of vibrating motions in the liquid bulk of drops on hot plates. It seems that classical treatments of vibrations (like in Lamb [START_REF] Lamb | Hydrodynamics[END_REF]) do not cover the present situation. Contrary to first impression, we determine that the present situation complicated by the gravity does not take into account heat gradients, vapour flows or more subtle effects. For the analytic approach, we consider a horizontal surface with a slight cavity invariant by rotation around its vertical symmetry axis. The drop is assumed not quite big enough to slip out of the cavity. Experiments describe such drops as flattened spheroids in the vertical plane. If the mean curvature radius of the surface is large compared to the size of the flattened drops, we assume that the motion of the liquid is approximately a motion in the horizontal plane. We consider plane motions of an incompressible liquid submitted to a convenient potential due to the effects of curvature of the surface. We investigate only rotations of drops and not random motions occurring by translation if the surface is really flat plate. Due to the film of vapour, we assume that motions are frictionless and the very small viscosity effect is neglected. The motions are not small vibrations. The form of drops in vibration are hypotrochoids [START_REF] Iyanaga | Encyclopedic Dictionary of Mathematics[END_REF] with different numbers of sides or lobes. The case of only two lobes corresponds to an elliptic form of drops. The results of computation favourably compare to experiments. It is possible to evaluate the angular velocity of vibrations. It depends on the acceleration of gravity, the curvature of the surface and the number of lobes. Times of rotation for these spheroidal drops are computed. They seem to be in accord with simple observations [START_REF] Walker | The amateur scientist[END_REF].

A CLASS OF PLANE MOTIONS FOR INCOMPRESSIBLE PERFECT FLUIDS

Motions of a fluid

The motion of a continuous flow can be represented by a surjective differential mapping

ζ → χ = Φ(ζ) , (1)
where ζ = (t, ξ) belongs to W, an open set in the time-space occupied by the fluid between time t 1 and time t 2 . The position in the reference space D o is denoted by χ; its position at time t in D t is denoted by ξ [START_REF] Serrin | Mathematical principle of classical fluid mechanics[END_REF]. We assume that distinct points of the continuous fluid remain distinct throughout the entire motion. At t fixed, transformation (1) possesses an inverse.

In order to describe a particular case of plane motions of a fluid let us introduce relatively to fixed Cartesian plane systems of coordinates, the inverse mapping

Z → z = ϕ(t, Z). (2)
The complex number Z = X + iY is such that χ = (X, Y ) ∈ R 2 represents Lagrangian coordinates in the reference space D o . The complex number z = x + iy with ξ = (x, y) ∈ R 2 represents Eulerian coordinates in the Euclidean plane D t . Let us notice that (X, Y ) is not necessarily a spatial position of a particle. Then, (X, Y ) are material coordinates.

In polar form Z = re iθ . Here we call (r, θ) the reference position. Now, we write (r, θ) ∈ D o and in polar form, transformation (2) is written

(r, θ) ∈ D o → z = ϕ t (r, θ) ∈ D t . (3)
Let us consider a particular transformation (2) of the form

z = Z + f (Z) e iωt , ( 4
)
where ω is a real constant and the function f = p + iq is an analytic function of Z with p is the real part and q is the imaginary part of f . Components of the velocity V are ( ẋ, ẏ) and ż = iωf (Z) e iωt .

Components of the acceleration are (ẍ, ÿ) and z = -ω 2 f (Z) e iωt . Let us denote F (Z) a primitive function of f (Z). At t fixed, relation ( 4) yields

e -iωt f (Z)dz = e -iωt f (Z)dZ + (p + iq)(dp -idq) or e -iωt f (Z)dz = e -iωt dF (Z) + 1 2 d(p 2 + q 2 ) + i(qdp -pdq) and R[e -iωt f (Z)dz] = dΦ with Φ = R[e -iωt f (Z)] + 1 2 f (Z)f (Z) . (5)
(R denotes the real part of a complex number).

Consequences of transformation (4) are:

(a) ẋdyẏdxi( ẋdx + ẏdy) = -i żdz = -ω e -iωt f (Z)dz and consequently ẋdy

-ẏdx = -ω dΦ . Then div V = 0 (6)
and -ω Φ is the stream function.

(b) ẍdx + ÿdy + i(ẍdyÿdx) = zdz = -ω 2 e -iωt f (Z)dz and consequently ẍdx + ÿdy = -ω 2 dΦ .

Then, Γ = -ω 2 grad Φ .

Equation [START_REF] Iyanaga | Encyclopedic Dictionary of Mathematics[END_REF] shows that the acceleration Γ is the derivative of a potential.

Pressure of these plane motions

In the case of plane motion of an incompressible perfect fluid, the equation of motion yields

Γ = - 1 ρ grad p -grad W, ( 8
)
where ρ is the constant volumic mass, p is the pressure and W is the extraneous force potential.

From equation [START_REF] Iyanaga | Encyclopedic Dictionary of Mathematics[END_REF] we get

p ρ = ω 2 Φ -W. (9)
Relation ( 9) yields the value of the pressure field as a function of Φ and W .

PARTICULAR CLASS OF PLANE MOTIONS

For a reference set D o such that

D o ≡ Z = r e iθ with r ∈ [0, r o ] , θ ∈ [0, 2π] . (10)
Let us consider two particular cases of motions defined by a function f such that:

(a) f (Z) = Z n a n-1 , n ∈ N, n > 1, a ∈ R + * (b) f (Z) = λ Z, λ ∈ ]0, 1[ 3.1 Case (a) Z ∈ D o -→ z = Z + Z n a n-1 e iωt ∈ D t (11)
is the representation of the motion.

In parametric representation, relation (11) yields:

x = r cos θ + r n a n-1 cos(ωt -nθ) y = r sin θ + r n a n-1 sin(ωt -nθ) (12)
with (r, θ) ∈ D o .

The trajectory of a particle with reference position (r, θ) is a circle (C r,θ ). In the system coordinates, the centre of (C r,θ ) is (r cos θ, r sin θ) and the radius is R = r n /a n-1 . The particle associated (r, θ) moves on the circle (C r,θ ) with the angular velocity ω.

r o a n-1 < 1 n . ( 13
)
Consequently, E t (r o ) is the free boundary of the fluid and the length a verifies

a > r o n-1 √ n .
One verifies that ϕ t (r, θ) = ϕ o (r, θ -Ωt)e iΩt with Ω = ω n + 1 .

The hypotrochoids E t (r) where r ∈ [0, r o ] and specially the free boundary E t (r o ) of the fluid, are turning relatively to the hot plate with the angular velocity Ω = ω n + 1 .

Case (b)

Z ∈ D o → z = Z + λZe iωt ∈ D t (14
) is the representation of the motion. In parametric representation, relation (14) yields:

x = r cos θ λr cos(ωt -θ) y = r sin θ + λr sin(ωt -θ) (15)
with (r, θ) ∈ D o . Contrary to case (a), no limitation is imposed on r o . At time t, the particles whose reference positions in D o are on the circle centered on O and radius r move in the physical space on an elliptic curve E t (r). Ellipses E t (r) have semi-axes of length r(1 + λ) and r(1 -λ). They are turning relatively to the hot plate with the angular velocity Ω = ω/2.

MOTIONS OF A PERFECT FLUID IN THE BOTTOM OF A LARGE CAVITY.

With a convenient Euclidean frame Oxyz, the equation of the hot surface is

z = f (r) with r 2 = x 2 + y 2 . ( 16
) Function f is a C 2 -function and f (0) = f ′ (0) = 0. The vertical direction is Oz. Equation (16) yields z = f ′′ (0) 2 r 2 + r 3 ε(r) with lim r→0 ε(r) = 0 . (17)
For a surface whose meridian curve is of small curvature, we use the equivalent approximation

z = f ′′ (0) 2 r 2 . ( 18
)
The potential of gravity forces is

W = k(x 2 + y 2 ) with k = g f ′′ (0) 2 . (19)
Relation ( 18) is used in the following computation: the mean curvature of the surface at (0, 0, 0) is R = g 2k and

k = g 2R . ( 20
)
Let us consider the two cases (a) and (b):

In case (a)

f (Z) = Z n a n-1 =⇒ F (Z) = Z n+1 (n + 1) a n-1 .
We deduce the stream function,

Φ = r 2n 2a 2n-2 + r n+1 (n + 1)a n-1 cos[ωt -(n + 1)θ] .
For the potential (19) we deduce from relation (12) the expression

W = k r 2 + r 2n a 2n-2 + 2 r n+1 a n-1 cos (ωt -(n + 1)θ) .
Equation (10) yields

p ρ = ω 2 [ r 2n 2a 2n-2 + r n+1 (n + 1)a n-1 cos(ωt -(n + 1)θ)] -k[r 2 + r 2n a 2n-2 + 2 r n+1 a n-1 cos(ωt -(n + 1)θ)] + C te .
In the case ω 2 = 2(n+1)k and for a pressure p o on the free boundary associated with r = r o , we obtain

p ρ = p o ρ + ω 2 2(n + 1) (r 2 o -r 2 ) - n 2(n + 1) ω 2 r 2n o -r 2n a 2n-2 . ( 21
)
In the case (b)

F (Z) = λ 2 Z 2 , Φ = λ 2 2 r 2 + λ 2 r 2 cos(ωt -2θ) and W = kr 2 [1 + 2λ cos(ωt -2θ) + λ 2 ] .
Equation (12) yields

p ρ = r 2 ω 2 [ λ 2 2 + λ 2 cos(ωt -2θ)] -kr 2 [1 + 2λ cos(ωt -2θ) + λ 2 ] + C te .
In the case ω 2 = 4k and for a pressure p o on the free boundary, we obtain

p ρ = p o ρ + ω 2 4 (1 -λ 2 )(r 2 o -r 2 ).
In the two cases, equation (20) yields ω 2 = (n + 1)g/R and fixes the angular rotation of the drop

Ω 2 = g (n + 1)R . ( 22
)
Consequently, following the different values of n, different modes are possible. The period of the rotation of the vibrating drop is given by

T = 2π (n + 1)R g . (23)
In fact, after an interval of time T n+1 = T /(n + 1), the drop is identical to its previous position and ν n+1 = 1/T n+1 is the apparent frequency of changes.

ν n+1 = 1 2π (n + 1)g R . (24)

COMPUTATION AND GRAPHS OF THE MOTION

Experiments were performed with a very simple apparatus in [START_REF] Holter | Vibrations of evaporating liquid drops[END_REF]. No data were performed, but it seems possible to investigate the above frequencies. Some apparent frequencies of the drop vibrations are calculated in Table 1.

We have drawn different cases of vibrating drops from two lobes (basic mode) to eight lobes. Drops are turning around their center of symmetry and we have drawn different positions of a drop with two and three lobes (Fig. 1).

In the case of three-lobed modes, the trajectory of a particle is represented as a circle for a particular drop. The graphs have the same forms than the ones given in [START_REF] Holter | Vibrations of evaporating liquid drops[END_REF]. Only the case of two lobes present some discrepancy with the interpretation given in [START_REF] Holter | Vibrations of evaporating liquid drops[END_REF]. In [START_REF] Holter | Vibrations of evaporating liquid drops[END_REF], the middle of the drop may be, in some cases, squeezed by comparison with the shape given here. Depending on the value of a and n, the drop can be like vibrating polygons of any type (Fig. 2). The two orientations are similar by computation; we notice also it is possible that the drop reverses the direction of rotation of the motion.

  At time t, particles whose reference positions in D o are on the circle with center O and radius r move in the physical space D t on a hypotrochoid curve E t (r). Curves E t (r) are obtained as circular disks of radius ρ = r/n rolling internally inside a fixed circle of radius r + ρ. Two different points of D o correspond to two different points of D t and the hypotrochoid has no double point. Due to det set D o of material variables verify

Fig. 1 .

 1 Fig. 1. In the case of two-lobed mode or three-lobed mode, the drops are turning around their centre of symmetry. The trajectory of a particle is a circle represented in the case of a three-lobed mode.

Fig. 2 .

 2 Fig. 2. Two possible forms of drop are presented in cases of three-lobed to eight-lobed modes. In many cases the vibrating drops are approximately polygons which may have any of three to n sides.

 1Some apparent frequencies ν n+1 (in Hertz) of drops as function of the mean curvature radius of the surface (in cm) and the number n+1 of lobes of the drop (g=981 cm/s 2 ).


Ai-Hua Fan

AND Ling-Min Liao
email: lingmin.liao@u-picardie.fr

Jacques Peyri Ère
email: jacques.peyriere@math.u-psud.fr

Jacques Peyri

GENERIC POINTS IN SYSTEMS OF SPECIFICATION AND BANACH VALUED BIRKHOFF ERGODIC AVERAGE

Keywords: 1991 Mathematics Subject Classification. Primary: 37B40, 37B45; Secondary: 37A05 Generic points, Systems of specification, Topological entropy, Variational principle

We prove that systems satisfying the specification property are saturated in the sense that the topological entropy of the set of generic points of any invariant measure is equal to the measure-theoretic entropy of the measure. We study Banach valued Birkhoff ergodic averages and obtain a variational principle for its topological entropy spectrum. As application, we examine a particular example concerning with the set of real numbers for which the frequencies of occurrences in their dyadic expansions of infinitely many words are prescribed. This relies on our explicit determination of a maximal entropy measure.

Introduction

By dynamical system (X, T ), we mean a continuous transformation T : X → X on a compact metric space X with metric d. We shall adopt the notion of topological entropy introduced by Bowen ( [START_REF] Bowen | Equilibrium states and the ergodic theory of Anosov diffeomorphisms[END_REF], recalled in the section 2), denoted by h top , to describe the sizes of sets in X. We denote by M inv the set of all T -invariant probability Borel measures on X and by M erg its subset of all ergodic measures. The measure-theoretic entropy of µ in M inv is denoted by h µ .

Let us first recall some notions like generic points, saturated property and the specification property which are quite known in dynamical systems nowadays.

For µ ∈ M inv , the set G µ of µ-generic points is defined by

G µ :=    x ∈ X : 1 n n-1 j=0 δ T j x w * -→ µ    ,
where w * -→ stands for the weak star convergence of the measures. A dynamical system (X, T ) is said to be saturated if for any µ ∈ M inv , we have h top (G µ ) = h µ .

Bowen ([8]) proved that on any dynamical system, we have h top (G µ ) ≤ h µ for any µ ∈ M inv . So, saturatedness means that G µ is of optimal topological entropy. One of our main results is to prove that systems of specification share this saturatedness.

A dynamical system (X, T ) is said to satisfy the specification property if for any ǫ > 0 there exists an integer m(ǫ) ≥ 1 having the property that for any integer k ≥ 2, for any k points x 1 , . . . , x k in X, and for any integers

a 1 ≤ b 1 < a 2 ≤ b 2 < • • • < a k ≤ b k with a i -b i-1 ≥ m(ǫ) (∀2 ≤ i ≤ k), there exists a point y ∈ X such that d(T ai+n y, T n x i ) < ǫ (∀ 0 ≤ n ≤ b i -a i , ∀1 ≤ i ≤ k).
The specification property was introduced by Bowen ( [START_REF] Bowen | Topological entropy for noncompact sets[END_REF]) who required that y is periodic. But the present day tradition doesn't require this. The specification property implies the topological mixing. Blokh ([7]) proved that these two properties are equivalent for continuous interval transformations. Mixing subshifts of finite type satisfy the specification property. In general, a subshift satisfies the specification if for any admissible words u and v there exists a word w with |w| ≤ k (some constant k) such that uwv is admissible. For β-shifts defined by T β x = βx( mod 1), there is only a countable number of β's such that the β shifts admit Markov partition (i.e. subshifts of finite type), but an uncountable number of β's such that the β-shifts satisfy the specification property ( [START_REF] Schmeling | Symbolic dynamics for β-shifts and self-normal numbers[END_REF]).

Our first result is stated as follows.

Theorem 1.1. If the dynamical system (X, T ) satisfies the specification property, then it is saturated.

As application, we study Banach-valued Birkhoff averages for saturated systems. Let B be a real Banach space and B * its dual space, their duality being denoted by •, • . We consider B * as a locally convex topological space with the weak star topology σ(B * , B). For any B * -valued continuous function Φ : X → B * , we consider its Birkhoff ergodic averages

A n Φ(x) = 1 n n-1 j=0 Φ(T j x) (n ≥ 1).
We would like to know the asymptotic behavior of A n Φ(x) in the σ(B * , B)-topology for different points x ∈ X.

Let us state the problem we are studying as follows.

X Φ (α; E) = x ∈ X : lim sup n→∞ A n Φ(x) E ≤ α .
The set X Φ (α; B) will be simply denoted by X Φ (α). This is the set of points

x ∈ X such that lim n→∞ A n Φ(x) = α in σ(B * , B)-topology. If E denotes the convex cone of E which consists of all aw ′ + bw ′′ with a ≥ 0, b ≥ 0 and w ′ ∈ E, w ′′ ∈ E, then X Φ (α, E) = X Φ (α, E). So we may always assume that E is a convex cone. If E is symmetric in the sense that E = -E, then we have X Φ (α, E) = x ∈ X : lim n→∞ A n Φ(x) E = α .
By entropy spectrum we mean the function

E E Φ (α) := h top (X Φ (α; E)).
Invariant measures will be involved in the study of the entropy spectrum E E Φ (α). We set

M Φ (α; E) = µ ∈ M inv : Φdµ E ≤ α
where Φdµ denotes the vector-valued integral in Pettis' sense (see [START_REF] Rudin | Functional analysis[END_REF]) and the inequality " E ≤" means Φ, w dµ ≤ α, w for all w ∈ E.

For saturated systems, we prove the following variational principle.

Theorem 1.2. Suppose that the dynamical system (X, T ) is saturated. Then

(a) If M Φ (α; E) = ∅, we have X Φ (α, E) = ∅. (b) If M Φ (α; E) = ∅, we have h top (X Φ (α; E)) = sup µ∈MΦ(α;E) h µ . (1.1)
When B is a finite dimensional Euclidean space R d and E = R d , the variational principle (1.1) with E = R d was proved in [START_REF] Fan | On the distribution of long-term time averages on symbolic space[END_REF][START_REF] Fan | Recurrence, dimensions and entropies[END_REF] for subshifts of finite type, then for conformal repellers ( [START_REF] Feng | Ergodic limits on the conformal repellers[END_REF]) and later generalized to systems with specification property [START_REF] Takens | On the variational principle for the topological entropy of certain non-compact sets[END_REF]. There are other works assuming that Φ is regular (Hölder for example). See [START_REF] Besicovitch | On the sum of digits of real numbers represented in the dyadic system[END_REF][START_REF] Eggleston | The fractional dimension of a set defined by decimal properties[END_REF] for classical discussions, [START_REF] Barreira | Higher dimensional multifractal analysis[END_REF][START_REF] Barreira | Distribution of frequencies of digits via multifractal analysis[END_REF][START_REF] Olivier | Dimension de Billingsley d'ensembles satures[END_REF][START_REF] Olivier | Multifractal analysis in symbolic dynamics and distribution of pointwise dimension for g-measures[END_REF][START_REF] Pesin | The multifractal analysis of Birkhoff averages and large deviation. Global analysis of dynamical systems[END_REF][START_REF] Tempelman | Multifractal analysis of ergodic averages: a generalization of Eggleston's theorem[END_REF] for recent developments on Birkhoff averages, and [START_REF] Barral | Comparing multifractal formalisms: the neighboring boxes condition[END_REF][START_REF] Ben Nasr | The validity of the multifractal formalism: results and examples[END_REF][START_REF] Brown | On the multifractal analysis of measures[END_REF][START_REF] Collet | The dimension spectrum of some dynamical systems[END_REF][START_REF] Nasr | Analyse multifractale de mesures[END_REF] for the multifractal analysis of measures.

The study of infinite dimensional Birkhoff averages is a new subject. We point out that [START_REF] Peyrière | A vectorial multifractal formalism[END_REF] provides another point of view, i.e. the thermodynamical point of view which was first introduced by physicists.

The above variational principle (1.1), when E = B, is easy to generalize to the following setting. Let Ψ be a continuous function defined on the closed convex hull of the image Φ(X) of Φ into a topological space Y . For given Φ, Ψ, and β ∈ Y , we set

X Ψ Φ (β) = x ∈ X : lim n→∞ Ψ A n Φ(x) = β and M Ψ Φ (β) = µ ∈ M inv : Ψ Φ dµ = β .
We also set

X Ψ Φ (β) = x ∈ X : Ψ lim n→∞ A n Φ(x) = β = α: Ψ(α)=β X Φ (α). It is clear that X Ψ Φ (β) is a subset of X Ψ Φ (β). Theorem 1.3. Suppose that the dynamical system (X, T ) is saturated. Then (1) if M Ψ Φ (β) = ∅, we have X Ψ Φ (β) = ∅, (2) if M Ψ Φ (β) = ∅, we have h top X Ψ Φ (β) = h top X Ψ Φ (β) = sup µ∈M Ψ Φ (β) h µ . (1.2)
This generalized variational principle (1.2) allows us to study generalized ergodic limits like

lim n→∞ n-1 j=0 Φ(T j x) n-1 j=0 g(T j x) , (1.3)
where g : X → R + is a continuous positive function. It suffices to apply (1.2) to Φ replaced by (Φ, g) and Ψ defined by Ψ(x, y) = x/y, with x ∈ B * and y ∈ R + .

It also allows us to study the set of points x ∈ X for which the limits

A ∞ Φ(x) = lim n→∞ A n Φ(x) verify the equation Ψ A ∞ Φ(x) = β.
(1.4)

The present infinite dimensional version of variational principle would have many interesting applications. We will just illustrate the usefulness of the variational principle by the following study of frequencies of blocks in the dyadic development of real numbers. It can be reviewed as an infinitely multi-recurrence problem.

Let us state the question to which we can answer. All but a countable number of real numbers t ∈ [0, 1] can be uniquely developed as follows

t = ∞ n=1 t n 2 n (t n ∈ {0, 1}).
Let k ≥ 1. We write 0 k for the block of k consecutive zeroes and we define the 0 n -frequency of t as the limit (if it exists)

f (t, k) = lim n→∞ 1 n ♯{1 ≤ j ≤ n : t j t j+1 • • • t j+k-1 = 0 k }. Let (a 1 , a 2 , • • • ) be
1 = a 0 ≥ a 1 ≥ a 2 ≥ . . . ; a i -2a i+1 + a i+2 ≥ 0 (i ≥ 0). (1.5)
If the condition (1.5) is fulfilled, we have

h top (S(a 1 , a 2 , • • • )) = -h(1 -a 1 ) + ∞ j=0 h(a j -2a j+1 + a j+2 ) (1.6)
where h(x) = -x log x.

Furthermore, it is proved that there is a unique maximal entropy measure, which is completely determined (see Lemma 5.6).

The paper is organized as follows. In the section 2, we give some preliminaries. In the section 3, we prove Theorem 1.1. In the section 4, we prove the theorems 1.2 and 1.3 and examine the case where B = ℓ 1 (Z). In the section 5, we apply the variational principle (1.1) to the study of the recurrence into an infinite number of cylinders of the symbolic dynamics. Especially, we study the set of orbits whose recurrences into infinitely many cylinders are prescribed. This relies on the explicit determination of a maximal entropy measure which, by definition, maximizes the supremum in (1.1).

Preliminary

Before proving the main theorems, we wish to recall the notions of topological entropy, the Bowen lemma and two propositions about the measure-theoretic entropy.

Recall that X is a compact metric space with its metric d and that T : X → X is a continuous transformation on X. For any integer n ≥ 1 we define the Bowen metric d n on X by

d n (x, y) = max 0≤j<n d(T j x, T j y).
For any ǫ > 0, we will denote by B n (x, ǫ) the open d n -ball centered at x of radius ǫ.

Topological entropy and Bowen lemma.

Let Z ⊂ X be a subset of X. Let ǫ > 0. We say that a collection (at most countable)

R = {B ni (x i , ǫ)} covers Z if Z ⊂ i B ni (x i , ǫ). For such a collection R, we put n(R) = min i n i . Let s ≥ 0. Define H s n (Z, ǫ) = inf R i exp(-sn i ),
where the infimum is taken over all covers R of Z with n(R) ≥ n. The quantity H s n (Z, ǫ) is non-decreasing as a function of n, so the following limit exists

H s (Z, ǫ) = lim n→∞ H s n (Z, ǫ).
For the quantity H s (Z, ǫ) considered as a function of s, there exists a critical value, which we denote by h top (Z, ǫ), such that

H s (Z, ǫ) = +∞, s < h top (Z, ǫ) 0, s > h top (Z, ǫ).
One can prove that the following limit exists

h top (Z) = lim ǫ→0 h top (Z, ǫ).
The quantity h top (Z) is called the topological entropy of Z ( [START_REF] Bowen | Equilibrium states and the ergodic theory of Anosov diffeomorphisms[END_REF]). For x ∈ X, we denote by V (x) the set of all weak limits of the sequence of probability measures n -1 n-1 j=0 δ T j x . It is clear that V (x) = ∅ and V (x) ⊂ M inv for any x. The following Bowen lemma is one of the key lemmas for proving the variational principle. Lemma 2.1 (Bowen [START_REF] Bowen | Topological entropy for noncompact sets[END_REF]). For t ≥ 0, consider the set

B (t) = {x ∈ X : ∃µ ∈ V (x) satisfying h µ ≤ t} . Then h top (B (t) ) ≤ t.
Let µ ∈ M inv be an invariant measure. A point x ∈ X such that V (x) = {µ} is said to be generic for µ. Recall our definition of G µ , we know that G µ is the set of all generic points for µ. Bowen proved that h top (G µ ) ≤ h µ for any invariant measure. This assertion can be deduced by using Lemma 2.1. In fact, the reason is that x ∈ G µ implies µ ∈ V (x). Bowen also proved that the inequality becomes equality when µ is ergodic. However, in general, we do not have the equality h top (G µ ) = h µ (saturatedness) and it is even possible that G µ = ∅. Cajar [START_REF] Cajar | Billingsley Dimension in Probability Spaces[END_REF] proved that full symbolic spaces are saturated. Concerning the µ-measure of G µ , it is well known that µ(G µ ) = 1 or 0 according to whether µ is ergodic or not (see [START_REF] Denker | Ergodic Theory on Compact Spaces[END_REF]).

Two propositions about the measure-theoretic entropy.

We denote by C(X) the set of continuous functions on X, by M = M(X) the set of all Borel probability measures.

In the sequel, we fix a sequence (p i ) i≥1 such that p i > 0 for all i ≥ 1 and

∞ i=1 p i = 1 (for example, p i = 2 -i will do). Suppose that s n = (s n,i ) i≥1 (n = 1, 2, • • • ) is a sequence of elements in ℓ ∞ . It is obvious that s n converges to α = (α i ) i≥1 ∈ ℓ ∞ in the weak star topology (i.e. each coordinate converges) is equivalent to lim n→∞ ∞ i=1 p i |s n,i -α i | = 0.
We also fix a sequence of continuous functions

{Φ 1 , Φ 2 , . . .} which is dense in the unit ball of C(X). Write Φ = (Φ 1 , Φ 2 , . . .). It is evident that Φ : X → ℓ ∞ is continuous when ℓ ∞ is equipped with its weak star topology. Fix an invariant measure µ ∈ M inv . Let α = (α 1 , α 2 , • • • α i , • • • ) where α i = Φ i dµ.
The set of generic points G µ can be described as follows

G µ = x ∈ X : lim n→∞ ∞ i=1 p i |A n Φ i -α i | = 0 = X Φ (α).
(

It is well known that the weak topology of M is compatible with the topology induced by the metric

d(µ, ν) = ∞ i=1 p i Φ i dµ -Φ i dν (2.2)
where both (p i ) i≥1 and {Φ i } i≥1 are chosen as above.

The following two results will be useful for us.

Proposition 2.2 (Young [START_REF] Young | Large deviations in dynamical systems[END_REF]). For any µ ∈ M inv and any numbers 0 < δ < 1 and 0 < θ < 1, there exist an invariant measure ν which is a finite convex combination of ergodic measures, i.e.

ν = r k=1 λ k ν k , where λ k > 0, r k=1 λ k = 1, ν k ∈ M erg , r ∈ N + such that d(µ, ν) < δ, h ν ≥ h µ -θ.
This is a consequence of the following result due to Jacobs (see [START_REF] Walters | An introduction to ergodic theory[END_REF], p. 186). Let µ ∈ M inv be an invariant measure which has the ergodic decomposition µ = Merg τ dπ(τ ) where π is a Borel probability measure on M erg . Then we have

h µ = Merg h τ dπ(τ ).
Proposition 2.3 (Katok [START_REF] Katok | Lyapunov exponents, entropy and periodic orbits for diffeomorphisms[END_REF]). Let µ ∈ M erg be an ergodic invariant measure. For ǫ > 0 and δ > 0, let r n (ǫ, δ, µ) denote the minimum number of ǫ-balls in the Bowen metric d n whose union has µ-measure more than or equal to 1δ. Then for each δ > 0 we have

h µ = lim ǫ→0 lim sup n→∞ 1 n log r n (ǫ, δ, µ) = lim ǫ→0 lim inf n→∞ 1 n log r n (ǫ, δ, µ).
In [START_REF] Katok | Lyapunov exponents, entropy and periodic orbits for diffeomorphisms[END_REF], it was assumed that T : X → X is a homeomorphism. But the proof in [START_REF] Katok | Lyapunov exponents, entropy and periodic orbits for diffeomorphisms[END_REF] works for the transformations we are studying.

Systems with specification property are saturated

In this section, we prove Theorem 1.1 which says that every system satisfying the specification property is saturated. Because of Bowen's lemma (Lemma 2.1), we have only to show h top (G µ ) ≥ h µ . The idea of the proof appeared in [START_REF] Fan | On the distribution of long-term time averages on symbolic space[END_REF][START_REF] Fan | Recurrence, dimensions and entropies[END_REF] and was developed in [START_REF] Takens | On the variational principle for the topological entropy of certain non-compact sets[END_REF]. It consists of constructing the so-called dynamical Moran sets which approximate the set of generic points G µ .

Dynamical Moran sets and their entropies.

Fix ǫ > 0. Let {m k } k≥1 be the sequence of integers defined by m k = m(2 -k ǫ) which is the constant appeared in the definition of the specification property (k = 1, 2, . . .). Let {W k } k≥1 be a sequence of finite sets in X and {n k } k≥1 be a sequence of positive integers. Assume that

d n k (x, y) ≥ 5ǫ (∀x, y ∈ W k x = y). (3.1)
Let {N k } k≥1 be another sequence of positive integers with N 1 = 1. Using these data, we are going to construct a compact set of Cantor type, called a dynamical Moran set, which will be denoted by

F = F (ǫ, {W k }, {n k }, {N k }).
We will give an estimate for its topological entropy. Denote

M k = #W k . Fix k ≥ 1. For any N k points x 1 , • • • , x N k in W k i.e. (x 1 , • • • , x N k ) ∈ W N k k , we choose a point y(x 1 , • • • , x N k ) ∈ X,
which does exist by the specification property, such that

d n k (x s , T as y) < ǫ 2 k (s = 1, . . . , N k ) (3.2)
where

a s = (s -1)(n k + m k ).
Both (3.1) and (3.2) imply that for two distinct points (

x 1 , • • • , x N k ) and (x 1 , • • • , xN k ) in W N k k we have d t k (y(x 1 , • • • , x N k ), y(x 1 , • • • , xN k )) > 4ǫ (3.3)
where

t k = a N k + n k , i.e. t k = (N k -1)m k + N k n k . In fact, let y = y(x 1 , • • • , x N k ) and ȳ = y(x 1 , • • • , xN k ). Suppose x s = xs for some s ∈ {1, • • • , N k }. Then d t k (y, ȳ) ≥ d n k (T as y, T as ȳ) ≥ d n k (x s , xs ) -d n k (x s , T as y) -d n k (x s , T as ȳ) > 5ǫ -ǫ/2 -ǫ/2 = 4ǫ. Let D 1 = W 1 , D k = y(x 1 , • • • , x N k ) : (x 1 , • • • , x N k ) ∈ W N k k (∀k ≥ 2).
Now define recursively L k and ℓ k as follows. Let

L 1 = D 1 , ℓ 1 = n 1 .
For any x ∈ L k and any y ∈ D k+1 (k ≥ 1), by the specification property, we can find a point z(x, y) ∈ X such that

d ℓ k (z(x, y), x) < ǫ 2 k+1 , d t k+1 (T ℓ k +m k+1 z(x, y), y) < ǫ 2 k+1 .
We will choose one and only one such z(x, y) and call it the descend from x ∈ L k through y ∈ D k+1 . Let

L k+1 = {z(x, y) : x ∈ L k , y ∈ D k+1 } , ℓ k+1 = ℓ k + m k+1 + t k+1 = N 1 n 1 + k+1 i=2 N i (m i + n i ).
Observe that for any x ∈ L k and for all y, ȳ ∈ D k+1 with y = ȳ, we have

d ℓ k (z(x, y), z(x, ȳ)) < ǫ 2 k (k ≥ 1), (3.4)
and for any x, x ∈ L k and y, ȳ ∈ D k+1 with (x, y) = (x, ȳ), we have

d ℓ k+1 (z(x, y), z(x, ȳ)) > 3ǫ (k ≥ 1). (3.5)
The fact (3.4) is obvious. To prove (3.5), first remark that d ℓ1 (z, z) ≥ 5ǫ > 4ǫ for any z, z ∈ L 1 with z = z, and that for any x, x ∈ L k and y, ȳ ∈ D k+1 with (x, y) = (x, ȳ) we have

d ℓ k+1 (z(x, y), z(x, ȳ)) ≥ d ℓ k (x, x) -d ℓ k (z(x, y), x) -d ℓ k (z(x, ȳ), x)
and

d ℓ k+1 (z(x, y), z(x, ȳ)) ≥ d t k+1 (y, ȳ) -d t k+1 (T ℓ k +m k+1 z(x, y), y) -d t k+1 (T ℓ k +m k+1 z(x, ȳ), ȳ).
Now using the above two inequalities, we prove (3.5) by induction. For any x, x ∈ L 1 and y, ȳ ∈ D 2 with either x = x or y = ȳ, we have

d ℓ2 (z(x, y), z(x, ȳ)) > 4ǫ - ǫ 2 2 - ǫ 2 2 = 4ǫ - ǫ 2 .
Suppose we have obtained that

d ℓ k (z(x, y), z(x, ȳ)) > 4ǫ - ǫ 2 - ǫ 2 2 -• • • - ǫ 2 k-1 .
Then for any x, x ∈ L k and y, ȳ ∈ D k+1 with (x, y) = (x, ȳ) we have

d ℓ k+1 (z(x, y), z(x, ȳ)) > 4ǫ - ǫ 2 - ǫ 2 2 -• • • - ǫ 2 k-1 - ǫ 2 k+1 - ǫ 2 k+1 = 4ǫ - ǫ 2 - ǫ 2 2 -• • • - ǫ 2 k-1 - ǫ 2 k > 3ǫ Now define our dynamical Moran set F = F (ǫ, {W k }, {n k }, {N k }) = ∞ k=1 F k , where F k = x∈L k B ℓ k (x, ǫ2 -(k-1) )
(B(x, r) denoting the closed ball of center x and radius r). The set F is Cantor-like because for any distinct points x ′ , x ′′ ∈ L k , by (3.5) we have

B ℓ k (x ′ , ǫ2 -(k-1) ) B ℓ k (x ′′ , ǫ2 -(k-1) ) = ∅
and if z ∈ L k+1 descends from x ∈ L k , by (3.4) we have

B ℓ k+1 (z, ǫ2 -k ) ⊆ B ℓ k (x, ǫ2 -(k-1)
).

Proposition 3.1 (Entropy of F ). For any integer n ≥ 1, let k = k(n) ≥ 1 and 0 ≤ p = p(n) < N k+1 be the unique integers such that

ℓ k + p(m k+1 + n k+1 ) < n ≤ ℓ k + (p + 1)(m k+1 + n k+1 ).
We have

h top (F ) ≥ lim inf n→∞ 1 n (N 1 log M 1 + • • • + N k log M k + p log M k+1 ).
Proof. For every k ≥ 1, consider the discrete measure σ k concentrated on

F k σ k = 1 #L k x∈L k δ x .
It can be proved that σ k converges in the week star topology to a probability measure σ concentrated on F . Moreover, for sufficiently large n and every point

x ∈ X such that B n (x, ǫ/2) ∩ F = ∅, we have

σ(B n (x, ǫ/2)) ≤ 1 #(L k )M p k+1 = 1 M N1 1 • • • M N k k M p k+1 .
(see [START_REF] Takens | On the variational principle for the topological entropy of certain non-compact sets[END_REF]). Then we apply the mass distribution principle to estimate the entropy.

Box-counting of G µ .

Recall that α = (α i ) i≥1 ∈ ℓ ∞ and Φ = (Φ i ) is a dense sequence in the unit ball of C(X). For δ > 0 and n ≥ 1, define

X Φ (α, δ, n) = x ∈ X : ∞ i=1 p i |A n Φ i (x) -α i | < δ . For ǫ > 0, let N (α, δ, n, ǫ) denote the minimal number of balls B n (x, ǫ) to cover the set X Φ (α, δ, n). Define Λ Φ (α) := lim ǫ→0 lim δ→0 lim sup n→∞ 1 n log N (α, δ, n, ǫ) (3.6)
By the same argument in [START_REF] Fan | On the distribution of long-term time averages on symbolic space[END_REF] (p. 884-885), we can prove the existence of the limits, and the following equality:

Λ Φ (α) = lim ǫ→0 lim δ→0 lim inf n→∞ 1 n log N (α, δ, n, ǫ). Proposition 3.2. Λ Φ (α) ≥ h µ .
Proof. It suffices to prove Λ Φ (α) ≥ h µ -4θ for any θ > 0. For each i ≥ 1, define the variation of Φ i by

var(Φ i , ǫ) = sup d(x,y)<ǫ |Φ i (x) -Φ i (y)|.
By the compactness of X and the continuity of Φ i , lim ǫ→0 var(Φ i , ǫ) → 0. So

lim ǫ→0 +∞ i=1 p i var(Φ i , ǫ) → 0.
This, together with (3.6), allows us to choose ǫ > 0 and δ > 0 such that

+∞ i=1 p i var(Φ i , ǫ) < δ < θ (3.7)
and lim sup

n→∞ 1 n log N (α, 5δ, n, ǫ) < Λ Φ (α) + θ. (3.8)
For the measure µ, take an invariant measure ν = r k=1 λ k ν k having the properties stated in Proposition 2.2. For 1 ≤ k ≤ r and N ≥ 1, set

Y k (N ) = x ∈ X : ∞ i=1 p i A n Φ i (x) -Φ i dν k < δ (∀n ≥ N ) .
Since ν k is ergodic, by the Birkhoff theorem, we have

lim n→∞ ∞ i=0 p i A n Φ i (x) -Φ i dν k = 0 ν k -a.e. (3.9)
Then by the Egorov theorem, there exists a set with ν k -measure greater than 1θ on which the above limit (3.9) is uniform. Therefore, if N is sufficiently large, we have

ν k (Y k (N )) > 1 -θ (∀ k = 1, • • • , r).
(3.10) Apply the second equality in Proposition 2.3 to the triple (ν k , 4ǫ, θ) in place of (µ, ǫ, δ). When ǫ > 0 is small enough, we can find an integer

N k = N k (ν k , 4ǫ, θ) ≥ 1 such that r n (4ǫ, θ, ν k ) ≥ exp(n(h ν k -θ)) (∀n ≥ N k ). This implies that if n ≥ N k , then the minimal number of balls B n (x, 4ǫ) to cover Y k (N ) is greater than or equal to exp(n(h ν k -θ)). Consequently, if we use C(n, 4ǫ) to denote a maximal (n, 4ǫ)-separated set in Y k (N ), then #C(n, 4ǫ) ≥ exp(n(h ν k -θ)) (∀n ≥ N k ). (3.11)
Choose a sufficiently large integer N 0 such that

n k := [λ k n] ≥ max(N 1 , • • • , N k , N ) (∀k = 1, . . . , r; ∀n ≥ N 0 ) ([•]
denoting the integral part). By the specification property, for each r points x 1 ∈ C(n 1 , 4ǫ), . . . , x r ∈ C(n r , 4ǫ), there exist an integer m(ǫ) depending on ǫ and a point y = y(x 1 , . . . , x r ) ∈ X such that

d n k (T a k y, x k ) < ǫ (1 ≤ k ≤ r) (3.12)
where

a 1 = 0, a k = (k -1)m + k-1 s=1 n s (k ≥ 2).
Write n = a r + n r , i.e. n = (r -1)m + r s=1 n s .

We claim that for all such y = y(x 1 , . . . , x r ), we have

y = y(x 1 , . . . , x r ) ∈ X Φ (α, 5δ, n) (3.13)
when n is sufficiently large, and that for two distinct points (x 1 , . . . , x r ) and (x ′ 1 , . . . , x ′ r ) in C(n 1 , 4ǫ) × • • • × C(n r , 4ǫ), the points y = y(x 1 , . . . , x r ) and y ′ = y(x ′ 1 , . . . , x ′ r ) satisfy d n(y, y ′ ) > 2ǫ.

(3.14) If we admit (3.13) and (3.14), we can conclude. In fact, the balls B n(y, ǫ) are disjoint owing to (3.14) and hence there are #C(n 1 , 4ǫ) × . . . × #C(n r , 4ǫ) such balls. Therefore, because of (3.13), the minimal number of (n, ǫ)-balls needed to cover X Φ (α, 5δ, n) is greater than the number of such points y's. That is to say

N (α, 5δ, n, ǫ) ≥ #C(n 1 , 4ǫ) × . . . × #C(n r , 4ǫ)
Then by (3.11), we get

N (α, 5δ, n, ǫ) ≥ exp r k=1 [λ k n](h ν k -θ).

By noticing that

[λ k n] n → λ k as n → ∞ and r k=1 λ k = 1, we get lim inf n→∞ 1 n log N (α, 5δ, n, ǫ) ≥ h µ -3θ
This, together with (3.8), implies Λ Φ (α) ≥ h µ -4θ. Now return to prove (3.13) and (3.14). The proof of (3.14) is simple: suppose (3.13). Recall that α i = Φ i dν and ν = r k=1 λ k ν k . We have

x k = x ′ k for some 1 ≤ k ≤ r. By (3.12), d n(y, y ′ ) ≥ d n k (T a k y, T a k y ′ ) ≥ d n k (x k , x ′ k ) -2ǫ > 4ǫ -2ǫ = 2ǫ. Now prove
|A nΦ i (y) -α i | ≤ A nΦ i (y) - r k=1 λ k Φ i dν k + Φ i dν -Φ i dµ . Since d(µ, ν) < δ i.e. ∞ i=1 p i | Φ i dµ -Φ i dν| < δ, we have only to show that ∞ i=1 p i A nΦ i (y) - r k=1 λ k Φ i dν k < 4δ. ( 3.15)
Write

A nΦ i (y) = 1 n r k=1 [λ k n]-1 j=0 Φ i (T a k +j y) + 1 n r k=2 a k -1 j=a k -m Φ i (T j y) = r k=1 [λ k n] n A [λ k n] Φ i (T a k y) + 1 n r k=2 a k -1 j=a k -m Φ i (T j y).
Then

A nΦ i (y) - r k=1 λ k Φ i dν k ≤ I 1 (i) + I 2 (i) + I 3 (i) + I 4 (i)
with

I 1 (i) = r k=1 [λ k n] n A [λ k n] Φ i (T a k y) -A [λ k n] Φ i (x k ) I 2 (i) = r k=1 [λ k n] n A [λ k n] Φ i (x k ) -Φ i dν k I 3 (i) = r k=1 [λ k n] n -λ k |Φ i |dν k I 4 (i) = 1 n r k=2 a k -1 j=a k -m |Φ i (T j y)|.
Since [λ k n] ≤ λ k n and x k satisfies (3.12), by (3.7) we get

∞ i=1 p i I 1 (i) ≤ ∞ i=1 p i r k=1 λ k var(Φ i , ǫ) = ∞ i=1 p i var(Φ i , ǫ) < δ. Since x k ∈ Y k (N ) and [λ k n] ≥ N , we have ∞ i=1 p i I 2 (i) ≤ ∞ i=1 p i r k=1 λ k A [λ k n] Φ i (x k ) -Φ i dν k = r k=1 λ k ∞ i=1 p i A [λ k n] Φ i (x k ) -Φ i dν k ≤ δ r k=1 λ k = δ. Since Φ i ≤ 1, we have ∞ i=1 p i I 3 (i) ≤ r k=1 [λ k n] n -λ k < δ ∞ i=1 p i I 4 (i) ≤ (r -1)m(ǫ) n ∞ i=1 p i = (r -1)m(ǫ) n < δ
when n is sufficiently large because n → ∞ and [λ k n] n → λ k . By combining all these estimates, we obtain (3.15).

Saturatedness of systems with specification.

In this subsection we will finish our proof of Theorem 1.1, which says that systems satisfying the specification property are saturated. It remains to prove h top (G µ ) ≥ Λ Φ (α). In fact, by Proposition 3.2, we will have h top (G µ ) ≥ h µ . On the other hand, it was known to Bowen [START_REF] Bowen | Topological entropy for noncompact sets[END_REF]

that h µ ≥ h top (G µ ). So, we will get h top (G µ ) = h µ . Proposition 3.3. h top (G µ ) ≥ Λ Φ (α)
Proof. It suffices to prove h top (G µ ) ≥ Λ Φ (α)θ for any θ > 0. To this end, we will construct dynamical Moran subsets of G µ = X Φ (α), which approach X Φ (α). The construction is based on separated sets of X Φ (α, δ, n).

Let m k = m(2 -k ǫ) be the constants in the definition of specification. By the definition of Λ Φ (α) (see (3.6)), when ǫ > 0 is small enough there exist a sequence of positive numbers {δ k } decreasing to zero and a sequence of integers {n k } increasing to the infinity such that

n k ≥ 2 m k (3.16)
and that for any k ≥ 1 we can find a (n k , 5ǫ)-separated set

W k of X Φ (α, δ k , n k ) with M k := ♯W k ≥ exp(n k (Λ Φ (α) -θ)). (3.17)
Choose a sequence of integers {N k } such that

N 1 = 1 (3.18) N k ≥ 2 n k+1 +m k+1 , k ≥ 2 (3.19) N k+1 ≥ 2 N1n1+N2(n2+m2)•••+N k (n k +m k ) , k ≥ 1 (3.
lim inf n→∞ 1 n (N 1 log M 1 + • • • + N k log M k + p log M k+1 ) ≥ Λ Φ (α) -θ.
By Proposition 3.1, we have

h top (F ) ≥ Λ Φ (α) -θ.
Thus we have only to prove F ⊆ X Φ (α). Or equivalently

lim n→∞ 1 n ∞ i=0 p i |S n Φ i (x) -nα i | = 0 (x ∈ F ). (3.21)
Let us use the same notations as in the last subsection including ℓ k , t k , D k and L k etc.

Fix n ≥ 1. Let k ≥ 1 and 0 ≤ p < N k+1 be the integers, which depend on n, such that

ℓ k + p(m k+1 + n k+1 ) < n ≤ ℓ k + (p + 1)(m k+1 + n k+1 ) Write q = n -ℓ k + p(m k+1 + n k+1 ) , b s = (s -1)(m k+1 + n k+1 ). Decompose the interval [0, n) (⊂ N) into small intervals [0, n) = [0, ℓ k ) [ℓ k , ℓ k + p(m k+1 + n k+1 )) [ℓ k + p(m k+1 + n k+1 ), n)
and decompose still [ℓ k , ℓ k + p(m k+1 + n k+1 )) into intervals alternatively of lengths n k+1 and m k+1 . Then cut the sum 0≤j<n Φ i (T j x) into sums taken over small intervals. Thus we get

|S n Φ i (x) -nα i | ≤ J 1 (i) + J 2 (i) + J 3 (i) + J 4 (i)
where

J 1 (i) = |S ℓ k Φ i (x) -ℓ k α i | J 2 (i) = p s=1 S m k+1 Φ i (T ℓ k +bs x) -m k+1 α i J 3 (i) = p s=1 S n k+1 Φ i (T ℓ k +bs+m k+1 x) -n k+1 α i J 4 (i) = S q Φ i (T ℓ k +p(m k+1 +n k+1 ) x) -qα i Since Φ i ≤ 1 (hence |α i | ≤ 1), we have J 2 (i) ≤ 2pm k+1 , J 4 (i) ≤ 2q ≤ 2(m k+1 + n k+1 ).
By (3.19), we have

lim n→∞ 1 n ∞ i=0 p i J 2 (i) = 0, lim n→∞ 1 n ∞ i=0 p i J 4 (i) = 0. (3.22)
Now let us deal with J 1 (i) and J 3 (i). We claim that for any x ∈ F there exists an x ∈ L k such that

d ℓ k (x, x) < ǫ 2 k-1 , (3.23)
and that for all 1 ≤ s ≤ p, there exists a point x s ∈ W k+1 such that

d n k+1 (x s , T us x) < ǫ 2 k-1 (3.24)
where u s = ℓ k + b s + m k+1 . In fact, by the construction of F , there exists a point z ∈ L k+1 such that

d ℓ k+1 (z, x) ≤ ǫ 2 k . (3.25)
Assume that z descends from some x ∈ L k through y ∈ D k+1 . Then

d ℓ k (x, z) < ǫ 2 k+1 (3.26) and d t k+1 (y, T ℓ k +m k+1 z) < ǫ 2 k+1 . ( 3
d ℓ k (x, x) ≤ ǫ 2 k+1 + ǫ 2 k < ǫ 2 k-1 . Thus (3.
d n k+1 (x s , T us x) ≤ d n k+1 (x s , T bs y) + d n k+1 (T bs y, T us z) + d n k+1 (T us z, T us x) ≤ d n k+1 (x s , T bs y) + d t k+1 (y, T ℓ k +m k+1 z) + d ℓ k+1 (z, x) < ǫ 2 k+1 + ǫ 2 k+1 + ǫ 2 k = ǫ 2 k-1 .
It is now easy to deal with J 3 (i), which is obviously bounded by

J 3 (i) ≤ p s=1 S n k+1 Φ i (T us x) -S n k+1 Φ i (x s ) + p s=1 S n k+1 Φ i (x s ) -n k+1 α i .
Using (3.24), we obtain

S n k+1 Φ i (x s ) -S n k+1 Φ i (T us x) ≤ n k+1 var(Φ i , ǫ2 -(k-1) ).
On the other hand, since x s ∈ W k+1 ⊆ X Φ (α, δ k+1 , n k+1 ), we have, by definition,

∞ i=1 p i S n k+1 Φ i (x s ) -n k+1 α i ≤ n k+1 δ k+1 .
Then, combining the last three estimates, and using the facts ∞ j=1 p j = 1 and pn k+1 ≤ n, we get

1 n ∞ i=1 p i J 3 (i) ≤ ∞ i=1 p i var(Φ i , ǫ2 -(k-1) ) + δ k+1 .
Since k can be arbitrarily large, we finally get

lim n→∞ 1 n ∞ i=1 p i J 3 (i) = 0. ( 3.29)
Now it remains to prove

lim n→∞ 1 n ∞ i=1 p i J 1 (i) = 0. (3.30) Observe that J 1 (i) ≤ |S ℓ k Φ i (x) -S ℓ k Φ i (x)| + |S ℓ k Φ i (x) -ℓ k α i | . By (3.23), we have |S ℓ k Φ i (x) -S ℓ k Φ i (x)| ≤ ℓ k var(Φ i , ǫ2 -(k-1) ). Then J 1 (i) ≤ ℓ k var(Φ i , ǫ2 -(k-1) ) + R k,i where R k,i = max z∈L k |S ℓ k Φ i (z) -ℓ k α i | .
Since var(Φ i , ǫ2 -(k-1) ) tends to zero as k → ∞, the desired claim (3.30) is reduced to

lim n→∞ 1 n ∞ i=1 p i R k,i = 0. (3.31)
We need two lemmas to estimate R k,i .

Lemma 3.4. For any y ∈ D k+1 , we have

∞ i=1 p i S t k+1 Φ i (y) -t k+1 α i ≤ ∞ i=1 p i N k+1 n k+1 var(Φ i , ǫ2 -(k+1) ) + 2(N k+1 -1)m k+1 + N k+1 n k+1 δ k+1 .
Proof. For any s = 1, . . . , N k+1 , there exists x s ∈ W k+1 such that

d n k+1 (x s , T bs y) < ǫ 2 k+1 (3.32)
where b s = (s -1)(m k+1 + n k+1 ). Write

S t k+1 Φ i (y) = N k+1 s=1 S n k+1 Φ i (T bs y) + N k+1 -1 s=1 S m k+1 Φ i (T bs+n k+1 y). Then S t k+1 Φ i (y) -t k+1 α i ≤ N k+1 s=1 S n k+1 Φ i (T bs y) -n k+1 α i + N k+1 -1 s=1 S m k+1 Φ i (T bs+n k+1 y) -m k+1 α i . Since x s ∈ W k+1 ⊆ X Φ (α, δ k+1 , n k+1 ), by (3.32), we have ∞ i=1 p i S n k+1 Φ i (T bs y) -n k+1 α i ≤ ∞ i=1 p i S n k+1 Φ i (T bs y) -S n k+1 Φ i (x s ) + ∞ i=1 p i S n k+1 Φ i (x s ) -n k+1 α i ≤ ∞ i=1 p i n k+1 var(Φ i , ǫ2 -(k+1) ) + n k+1 δ k+1 .
On the other hand,

S m k+1 Φ i (T bs+n k+1 y) -m k+1 α i ≤ 2m k+1 .
Now it is easy to conclude.

Lemma 3.5.

∞ i=1 p i R k,i ≤ 2 ∞ i=1 p i k j=1 ℓ j var(Φ i , ǫ2 -j ) + 2 k j=1 N j m j + k j=1 ℓ j δ j .
Proof. We prove it by induction on k. When k = 1, we have

L 1 = D 1 = W 1 ⊆ X Φ (α, δ 1 , n 1 ) and then ∞ i=1 p i R 1,i ≤ n 1 δ 1 = ℓ 1 δ 1 .
Suppose the lemma holds for k. For any z ∈ L k+1 there exist x ∈ L k and y ∈ D k+1 , such that

d ℓ k (x, z) < ǫ 2 k+1 , d t k+1 (y, T ℓ k +m k+1 z) < ǫ 2 k+1 . Write S ℓ k+1 Φ i (z) = S ℓ k Φ i (z) + S m k+1 Φ i (T ℓ k z) + S t k+1 Φ i (T ℓ k +m k+1 z). Then S ℓ k+1 Φ i (z) -ℓ k+1 α i is bounded by |S ℓ k Φ i (z) -ℓ k α i | + S m k+1 Φ i (T ℓ k z) -m k+1 α i + S t k+1 Φ i (T ℓ k +m k+1 z) -t k+1 α i . Notice that |S ℓ k Φ i (z) -ℓ k α i | ≤ |S ℓ k Φ i (z) -S ℓ k Φ i (x)| + |S ℓ k Φ i (x) -ℓ k α i | ≤ ℓ k var(Φ i , ǫ2 -(k+1 ) + R k,i , S m k+1 Φ i (T ℓ k z) -m k+1 α i ≤ 2m k+1 and S t k+1 Φ i (T ℓ k +m k+1 z) -t k+1 α i ≤ S t k+1 Φ i (T ℓ k +m k+1 z) -S t k+1 Φ i (y) + S t k+1 Φ i (y) -t k+1 α i ≤ t k+1 var(Φ i , ǫ2 -(k+1) ) + S t k+1 Φ i (y) -t k+1 α i . By Lemma 3.4, we have ∞ i=1 p i R k+1,i ≤ ∞ i=1 p i R k,i + ∞ i=1 p i (ℓ k + t k+1 + N k+1 n k+1 )var(Φ i , ǫ/2 k+1 ) +2N k+1 m k+1 + N k+1 n k+1 δ k+1 .
Then according to the induction hypothesis the Lemma holds for k + 1, because

ℓ k + t k+1 ≤ ℓ k+1 , N k+1 n k+1 ≤ ℓ k+1 .
Let us finish the proof of Theorem 1.1 by showing (3.31). Since n j ≥ 2 mj , we have

N j m j ℓ j ≤ N j m j N j (n j + m j ) = m j n j + m j → 0 (j → ∞).
Then the estimate in Lemma 3.5 can be written as

∞ i=1 p i R k,i ≤ k j=1 ℓ j c j
where c j → 0 (j → ∞). By (3.20), we have

ℓ k ≥ 2 ℓ k-1 . It follows that 1 ℓ k ∞ i=1 p i R k,i ≤ c k + 1 ℓ k k-1 i=1 c j ℓ j .
This implies (3.31).

Variational principle

In this section, we prove variational principles for saturated systems (Theorem 1.2 and Theorem 1.3). The proof of Theorem 1.3 is similar to that of Theorem 1.2.

Proof of Theorem 1.2 (a). It suffices to prove that if there exists a point x ∈

X Φ (α; E), then M Φ (α; E) = ∅. That x ∈ X Φ (α; E) means lim sup n→∞ A n Φ(x), w ≤ α, w (∀w ∈ E). (4.1)
Let µ be a weak limit of n -1 n-1 j=0 δ T j x . That is to say, there exists a sequence n m such that

lim m→∞ 1 n m nm-1 j=0 f (T j x) = f dµ (4.2)
for all scalar continuous functions f . We deduce from (4.1) and (4.2) that for all w ∈ E we have

Φ, w dµ = lim m→∞ A nm Φ(x), w ≤ lim sup n→∞ A n Φ(x), w ≤ α, w .
So µ ∈ M Φ (α; E). Proof of Theorem 1.2 (b). Let t = sup µ∈MΦ(α;E) h µ . What we have just proved above may be stated as follows: if x ∈ X Φ (α; E), then

V (x) ⊂ M Φ (α; E).
It follows that h µ ≤ t for any µ ∈ V (x). Thus

X Φ (α; E) ⊂ {x ∈ X : ∀ µ ∈ V (x) satisfying h µ ≤ t} ⊂ {x ∈ X : ∃ µ ∈ V (x) satisfying h µ ≤ t} .
Then, due to Lemma 2.1, we get h top (X Φ (α; E)) ≤ t. Now we prove the converse inequality. For any µ ∈ M Φ (α; E), consider G µ the set of generic points. We have

G µ ⊂ X Φ (α; B) ⊂ X Φ (α; E).
The second inclusion is obvious and the first one is a consequence of the fact that x ∈ G µ implies lim n→∞ A n Φ(x) = Φdµ = α in the σ(B * , B)-topology. Thus

h top (X Φ (α; E)) ≥ h top (G µ ).
Since µ is an arbitrary invariant measure in M Φ (α; E), we can finish the proof because the system (X, T ) is saturated (i.e. h top (G µ ) = h µ ).

It is useful to point out the following facts appearing in the proof:

(i) If x ∈ X Φ (α, E), then V (x) ⊂ M Φ (α, E). (ii) We have µ∈MΦ(α,E) G µ ⊂ X Φ (α, E) ⊂ µ∈MΦ(α,E) G µ with G µ = {x ∈ X : V (x) ∋ µ}.
It is worth to notice the fact that all the G µ are disjoint.

It is clear that M Φ (α, E) is a compact convex subset of the space M inv of Borel probability invariant measures. If h µ , as a function of µ, is upper semi-continuous on M inv , then the supremum in the variational principle is attained by some invariant measure, called the maximal entropy measure in M Φ (α, E). Usually, the structure of M Φ (α, E) is complicated. But it is sometimes possible to calculate the maximal entropy.

Proof of Theorem 1.3. Let x ∈ X Ψ Φ (β) and let µ be a weak limit of n -1 n-1 j=0 δ T j x . Then there exists a subsequence of integers {n m } such that A nm Φ(x) tends to Φdµ in the weak star topology as m → ∞ because we have an expression similar to (4.2) with f = Φ, w (w ∈ B being arbitrary). Hence

Ψ Φdµ = lim m→∞ Ψ(A nm Φ(x)) = lim n→∞ Ψ(A n Φ(x)) = β.
Thus we have proved that µ ∈ M Ψ Φ (β). That is to say

V (x) ⊂ M Ψ Φ (β) (∀x ∈ X Ψ Φ (β)
). It follows that (a) holds and that due to Lemma 2.1 we have

h top (X Ψ Φ (β)) ≤ sup µ∈M Ψ Φ (β) h µ .
The converse inequality is a consequence of the variational principle (1.1) and the relationship

X Ψ Φ (β) ⊃ X Ψ Φ (β) = α:Ψ(α)=β X Φ (α).
In fact,

h top (X Ψ Φ (β)) ≥ h top ( X Ψ Φ (β)) ≥ sup α:Ψ(α)=β h top (X Φ (α)) = sup α:Ψ(α)=β sup µ∈MΦ(α) h µ = sup µ∈M Ψ Φ (β) h µ .
4.2. ℓ ∞ (Z)-valued ergodic average. Let us consider the special case where B = ℓ 1 (Z). Then B * = ℓ ∞ (Z). Any ℓ ∞ (Z)-valued function Φ can be written as

Φ(x) = Φ n (x) n∈Z , with sup n |Φ n (x)| < ∞.
Recall that ℓ ∞ (Z) is equipped with the σ(ℓ ∞ , ℓ 1 )-topology. A ℓ ∞ (Z)-valued function Φ is continuous if and only if all coordinate functions Φ n : X → R are continuous, because for any w = (w n ) n∈Z ∈ ℓ 1 we have

Φ(x), w = n∈Z w n Φ n (x).
Let us give an application of the variational principle in this setting. Let I be a finite or infinite subset of positive integers. Let {Φ i } i∈I be a family of real continuous functions defined on X. We suppose that sup i∈I Φ i C(X) < ∞. For two given sequences of real numbers a = {a i } i∈I and b = {b i } i∈I , we denote by S(a, b) the set of points x ∈ X such that

a i ≤ lim inf n→∞ A n Φ i (x) ≤ lim sup n→∞ A n Φ i (x) ≤ b i (∀i ∈ I).
Corollary 4.1. Suppose that the system (X, T ) is saturated. The topological entropy of S(a, b) defined above is equal to the supremum of the measure-theoretical entropies h µ for those invariant measures µ such that

a i ≤ Φ i dµ ≤ b i (∀i ∈ I).
Proof. For n ∈ Z, let e n be the n th element of the canonical basis of ℓ 1 (Z). Let Φ be a function whose nth coordinate and -n th coordinate are respectively equal to Φ n and -Φ n for each n ∈ I (other coordinates may be taken to be zero). Take the set E ⊂ ℓ 1 , which consists of e i and e -i for i ∈ I. Take α ∈ ℓ ∞ such that α -i = -a i and α i = b i for i ∈ I. Now we can directly apply the variational principle by noticing that

Φ, e i = Φ i , Φ, e -i = -Φ i (i ∈ I).
The result contained in this corollary is new, even when I is finite. If I is finite and if a i = b i for i ∈ I, the preceding corollary allows one to recover the results in [START_REF] Fan | Recurrence, dimensions and entropies[END_REF] and [START_REF] Takens | On the variational principle for the topological entropy of certain non-compact sets[END_REF].

The validity of the variational principle is to some extent equivalent to the fact that the system (X, T ) is saturated. Theorem 4.2. Let (X, T ) be a compact dynamical system. The system is saturated if and only if the variational principle (Theorem 1.2 (b)) holds for all real Banach spaces B.

Proof. It remains to prove that the variational principle implies the saturation of the system. Take a countable set {f i } i∈N which is dense in the unit ball of C(X) (C(X) being the space of all real valued continuous functions on X). Consider the function

Φ = (f 1 , f 2 , • • • , f n , • • • ) which takes values in B = ℓ ∞ (N). For any invariant measure µ ∈ M inv , define α = f 1 dµ, f 2 dµ, • • • ∈ ℓ ∞ (N).
It is clear that M Φ (α) = {µ}. Then the variational principle implies h top (X Φ (α)) = h µ . This finishes the proof because X Φ (α) is nothing but G µ .

An example: recurrence in an infinite number of cylinders

We have got a general variational principle. In order to apply this principle, one of the main questions is to get information about the convex set M Φ (α, E) and the maximal entropy measures contained in it and to compute the maximal entropy. Let us consider the symbolic dynamical system ({0, 1} N , T ), T being the shift. The structure of the space M inv is relatively simple. To illustrate the main result, we shall consider a special problem of recurrence into a countable number of cylinders. Let W = {w i } i∈I with I ⊂ N be a finite or infinite set of words. Let α = {a i } i∈I be a sequence of non-negative numbers. We are interested in the following recurrence set R(a; W) = {x ∈ X : R(x, [w i ]) = a i for all i ∈ I} whose topological entropy will be computed by the variational principle which takes a simpler form. Remark that the shift transformation on the symbolic space is expansive. Thus the entropy function h µ is upper semi-continuous( [START_REF] Walters | An introduction to ergodic theory[END_REF], p. 184). Hence we can obtain the supremum in Corollary 5.1.

Recall that any Borel probability measure on X is uniquely determined by its values on cylinders. In other words, a function µ defined on all cylinders can be extended to be a Borel probability measure if and only if The entropy h µ of any invariant measure µ ∈ M inv can be expressed as follows

h µ = lim n→∞ x1,••• ,xn -µ([x 1 , • • • , x n ]) log µ([x 1 , • • • , x n ]) µ([x 1 , • • • , x n-1 ]
) .

The sum in the above expression which we will denote by h

(n) µ is nothing but a conditional entropy of µ and it is also the entropy of an (n -1)-Markov measure µ n , which tends towards µ as n goes to ∞.

A Markov measure of order k is an invariant measure ν ∈ M inv having the following Markov property: for all n > k and all (x 1 ,

• • • , x n ) ∈ {0, 1} n ν([x 1 , • • • , x n ]) ν([x 1 , • • • , x n-1 ]) = ν([x n-k , • • • , x n ]) ν([x n-k , • • • , x n-1 ])
.

A Markov measure of order k is uniquely determined by its values on the (k + 1)cylinders. The preceding approximating Markov measure µ n has the same values as µ on n-cylinders.

To apply the above corollary, we have to maximize the entropy h µ among all invariant measures µ with constraints µ([w i ]) = a i for i ∈ I. The entropy h µ is a function of an infinite number of variables µ([w]). So we have to maximize a function of an infinite number of variables. However, in some cases it suffices to reduce the problem to maximize the conditional entropy which is a function of a finite number of variables.

Denote by |w| the length of the word w. Let W n := {w ∈ W : |w| ≤ n} and M(a, W n ) := {µ ∈ M inv : µ([w i ]) = a i , w i ∈ W n }. Let µ * be a maximal entropy measure over M(a, W) and µ * n be the (n -1)-Markov measure which converges to µ * . Let µ (n) be a maximal entropy measure over M(a, W n ). Then

h µ * = lim n→∞ h µ * n ≤ lim inf n→∞ h µ (n) ≤ lim sup n→∞ h µ (n) ≤ h µ * . Hence lim n→∞ h µ (n) = h µ * = max µ∈M(a,W) h µ .
However, for any measure µ ∈ M(a, W n ), we have

h µ = h µn = h (n)
µ , where µ n is the (n -1)-Markov measure which converges to µ ( [START_REF] Fan | Recurrence, dimensions and entropies[END_REF]). Thus, µ (n) is the maximal point of the conditional entropy function h Proposition 5.2. The maximal entropy over M(a, W) can be approximated by the maximal entropies over M(a, W n )'s.

  Fix a subset E ⊂ B. For a sequence {ξ n } ⊂ B * and a point ξ ∈ B * , we denote by lim sup n→∞ ξ n E ≤ ξ the fact lim sup n→∞ ξ n , w ≤ ξ, w for all w ∈ E. The meaning of " E =" is obvious. It is clear that lim sup n→∞ ξ n B ≤ ξ, or equivalently lim sup n→∞ ξ n B = ξ, means ξ n converges to ξ in the weak star topology σ(B * , B). Let α ∈ B * and E ⊂ B. The object of our study is the set

20 )

 20 Consider the dynamical Moran set F = F (ǫ, {W k }, {n k }, {N k }) as we constructed in the last subsection. From (3.16)-(3.20), we get

. 27 )

 27 On the other hand, according to the definition of D k+1 , there exists an x s ∈ W k+1 such that d n k+1 (x s , T bs y) < ǫ 2 k+1 . (3.28) Now by the trigonometric inequality, the fact d ℓ k (z, x) ≤ d ℓ k+1 (z, x) and (3.25) and (3.26) we get

  23) is proved. By (3.25),(3.27) and (3.28), we can similarly prove (3.26):

4. 1 .

 1 Proofs of Theorems 1.2 and 1.3.

5. 1 .

 1 Symbolic space. Let X = {0, 1} N and T be the shift transformation. As usual, an n-cylinder in X determined by a word w= x 1 x 2 • • • x n is denoted by [w] or [x 1 , • • • , x n ].For any word w, define the recurrence to the cylinder [w] of x by R(x, [w]) = lim n→∞ ] (T j x) if the limit exists.

Corollary 5 . 1 .

 51 We have h top (R(a; W)) = max µ∈M(a,W) h µ where M(a, W) = {µ ∈ M inv : µ([w i ]) = a i for all i ∈ I} .

  x1,••• ,xn µ([x 1 , • • • , x n ]) = 1 and ǫ∈{0,1} µ([x 1 , • • • , x n-1 , ǫ]) = µ([x 1 , • • • , x n-1 ]).Such a probability measure µ is invariant if and only if

  ǫ, x 2 , • • • , x n ]) = µ([x 2 , • • • , x n ]).

  we have proved the following proposition.

  a sequence of non-negative numbers. We denote by S(a1 , a 2 , • • • ) the set of all numbers t ∈ [0, 1] such that f (t, k) = a k for all k ≥ 1.As a consequence of the variational principle ( 1.1), we prove Theorem 1.4. The set S(a 1 , a 2 , • • • ) is non-empty if and only if the following condition is satisfied

We thank the referee for telling us that Pfister and Sullivan [25] also obtained the same result as our Theorem 1.1 but with a different method.

Example: Frequency of dyadic digital blocks.

Let us consider a special example:

where 0 k means the word with 0 repeated k times.

Theorem 5.3. Let W = {[0 n ]} n≥1 and a = {a n } n≥1 ⊂ R + . We have (a) R(a; {[0 n ]} n≥1 ) = ∅ if and only if 1 = a 0 ≥ a 1 ≥ a 2 ≥ . . . ; a i -2a i+1 + a i+2 ≥ 0 (i ≥ 0).

(5.1) (b) If the above condition is satisfied, then

where h(x) = -x log x.

The proof of the above theorem is decomposed into several lemmas which actually allow us to find the unique invariant measure of maximal entropy and to compute its entropy.

Let µ be an invariant measure. The consistence and the invariance of the measure imply that we may partition all (n + 2)-cylinders into groups of the form

) of (n + 1)-cylinders. More precisely, if we write p w = µ([w]), then for any word w of length n, we have

Lemma 5.4. Suppose µ ∈ M(a, W). If w = 0 n with n ≥ 0, we have

(5.4)

(5.6)

Proof. The relation (5.4) is a consequence of the consistence p 00 n 1 + p 00 n 0 = p 00 n and the facts p 00 n = a n+1 and p 00 n 0 = a n+2 ; the relation (5.5) is a consequence of the invariance p 10 n 0 + p 00 n 0 = p 0 n 0 and the same facts; to obtain the relation (5.6) we need both the invariance and the consistence: p 10 n 1 + p 00 n 1 = p 0 n 1 = p 0 np 0 n 0 . Then by (5.4) we get

Let a, b, c be three positive numbers such that a + b ≥ c. Consider the function

defined on R +4 , where h(x) = -x log x.

Lemma 5.5. Under the condition

the function F defined by (5.7) attains its maximum at

Proof. From the condition we may solve t, u, v as functions of w:

So, maximizing F (t, u, v, w) under the condition becomes maximizing the function

which is strictly concave in its domain. Since h ′ (x) = -1log x, we have

. The corresponding t, u, v are as announced in (5.8) Lemma 5.6. Suppose that {a n } n≥0 is a sequence of real numbers such that

(5.9)

There exists an invariant measure µ such that if w is not a block of 0's, we have

(5.10)

The above recursion relations (5.10) together with (5.3-5.6) completely determine the measure µ, which is the unique maximal entropy measure among those invariant measures ν such that ν([0 n ]) = a n for n ≥ 1.

Proof. For any µinM(a, W), we must have µ([0]) = a 1 and µ([1]) = 1a 1 . By Proposition 5.2, we are led to find the measure µ (n+2) which maximizes h (n+2) µ for each n ≥ 0. Let µ be an arbitrary invariant measure in M(a, W n+2 ), n ≥ 0. We identify µ with the sequence p w = µ([w]) indexed by finite words. By Lemma 5.4, we have

So, we have

(5.11)

Let us now consider the conditional entropy h (n+2) µ

for n ≥ 1, which is a function of µ-measures of (n + 2)-cylinders, whose general form is [ǫwǫ ′ ] with w a word of length n and ǫ, ǫ ′ ∈ {0, 1}. If w = 0 n , by Lemma 5.4, the measures of the four cylinders [ǫ0 n ǫ ′ ] with ǫ, ǫ ′ ∈ {0, 1} are determined by {a k } k≥1 . If w = 0 n , the four quantities p ǫwǫ ′ are linked to each other by p 0w0 + p 0w1 = p 0w , p 1w0 + p 1w1 = p 1w , p 0w0 + p 1w0 = p w0 through measures of (n + 1)-cylinders: a := p w0 , b := p 1w , c := p 0w . Consider the four measures p ǫwǫ ′ as variables, there is only one free variable and three others are linked to it. Thus to any word w = 0 n of length n is associated a free variable.

In fact, h

is the sum of all terms

If w = 0 n , the corresponding term (5.12) is a constant depending on the sequence {a n } (see Lemma 5.4). If w = 0 n , there is a free variable in the term (5.12). So, maximizing h

is equivalent to maximizing all above terms, or equivalently to maximizing -

(5.13) Applying Lemma 5.5 to the above function in (5.13) provides us with the maximal point µ (n+2) described by (5.10) with |w| = n. It is easy to check that the family {p w } defined by (5.3-5.6) and (5.10) for all words w, verifies the consistency and the invariance conditions. The measure determined by the family is the unique measure of maximal entropy and µ (n+2) is its (n + 1)-Markov measure.

Lemma 5.7. The entropy of the invariant measure µ of maximal entropy determined in the above lemma is equal to

where h(x) = -x log x.

Proof. Recall that for any invariant measure µ we have

p wǫ log p wǫ p w .

Thus we may write

For n ≥ 0, write

where ′ means that the sum is taken over w = 0 n (0 0 meaning the empty word so that I 1 (0) = 0). When n = 0, we get h (2) µ = h(a 2 ) + 2h(a 1a 2 )h(a 1 )h(1a 1 ) + h(1 -2a 1 + a 2 ). (5.16) This coincides with (5.11).

Suppose n ≥ 1. By the recursion relation (5.10), we have

where

From (5.15) and (5.17) we get

(5.18)

On the one hand, by using the invariance and the consistence we can simplify I 3 as follows

On the other hand, we have

+a n+1 log a n+1 + (a na n+1 ) log(a na n+1 ).

(5.20)

By combining (5.18), (5.19) and (5.20) we get

+h(a n -2a n+1 + a n+2 ).

Finally using (5.14) we get

h(a j -2a j+1 + a j+2 ).

Remark that the measure of maximal entropy in R(a, W) is not necessarily ergodic. Here is an example. If a n = a (∀n ≥ 1) is constant, then there is a unique invariant measure in M(a, W), which is aδ0 + (1a)δ1. In this case, R(a, W) is not empty but of zero entropy. Notice that R(a, W) contains no point in the support of the unique invariant measure. If a = 1 2 , R( 1 2 , W) contains the following point 01001100011100001111... (The terms in the two sequences {0 k } and {1 k } are alternatively appended.)