Jacques Lacaze
email: jacques.lacaze@ensiacet.fr

Jon Sertucha

Lena Magnusson Åberg

Microstructure of As-cast Ferritic-pearlitic Nodular Cast Irons

Keywords: nodular cast iron, as-cast microstructure, ferrite, pearlite, alloying

A review of past works on the formation of ferrite and pearlite in nodular cast iron is proposed. The effects of cooling rate after solidification and of nodule count on the formation of both constituents are stressed, though much emphasis is put on alloying elements and impurities.

Introduction

Nodular cast irons are composite materials made of graphite spheroids embedded in a Fe-rich matrix. Their mechanical properties depend both on nodule count and on the matrix constitution resulting from the transformation of high-temperature austenite formed together with graphite and sometimes to martensite in case of highly alloyed irons cast in small sections. Bainite is not observed in as-cast not be considered further here where focus is put on as-cast irons with ferritic-pearlitic structures. The ferritic-pearlitic matrix consists of halos of ferrite around the nodules and so-called bulls-eye structure illustrated in Fig. 1.

One of the most relevant advantages of nodular cast iron is their extensive application in the as-cast condition. This reduces the manufacturing cost and potential problems related to variability from heat-treatment are avoided. The transformation of austenite in the stable and metastable (mostly pearlite promoting elements). The following section is devoted to attempts to predict room temperature matrix i.e. considering

Eutectoid Transformation of Austenite

1) have austenite starting to decompose in the stable system giving ferrite halos growing symmetrically around graphite nodules. Further growth of ferrite involves transfer of carbon from the remaining austenite to graphite nodules by diffusion through the ferrite halo (this will hereafter be called the ferritic reaction). The process is thus slower and slower as low enough for nucleation and growth of pearlite in the metastable system (hereafter called the pearlitic reaction). Pearlite growth is comparatively rapid because it proceeds certainly much like in steels according to Pan et al.

2) and Venugopalan.

3) 1) could show that pearlite appears at the ferrite/austenite interface and develops as spherical colonies inside the remaining austendecomposition is most generally quickly completed.

An important number of foundries include thermal analysis for controlling melt preparation before casting using commercial standard rig and cups. Figure 2 shows an example of such a record when the output has been pursued down to low enough temperature to register the solid-state transformation of the alloy. The derivative of the temperature with respect to time (i.e. the cooling rate) is also transformations appearing as plateaus on the cooling curve give sharp changes in the derivative. Focusing on the part of the derivative record corresponding to the solid-state reactions. On such curves it is possible to measure the cooling rate of the material before the eutectoid transformation and the temperatures for the start of the ferritic and pearlitic reactions. Integration of the area below the derivative peak kinetics. The same characterization could be performed as well on differential thermal analysis (DTA) records where the cooling rate is imposed.

Formation of Ferrite

Figure 3 schematically illustrates ferrite growth described above. One particular feature of the ferritic reaction is the fact that no redistribution of substitutional elements at the ferrite/austenite interface has been reported for as cast mate--3) see appendix. If one assumes the e.g. the one drawn at 2.5 wt.% silicon in Fig. 4 graphite equilibrium at temperatures above the three-phase below this domain. It is worth noting the very low level of carbon that can dissolve in ferrite as compared to that in be rejected partly to austenite but growth of graphite will proceed mainly by diffusion of carbon to graphite through the ferrite halo. Following Lacaze et al.

the schematic of carbon distribution in Fig. 3 demonstrates that this diffusion could proceed only if

Δw C = w C α γ / w C gra / w C α γ
/ and w C gra / are the carbon content in ferrite (α) in equilibrium with austenite (γ) and graphite (gra) respectively. in the growing ferrite and in the receding austenite should be described by the metastable extrapolations (shown with dashed lines) of the equilibrium lines (solid lines). Note that Once the temperature of the alloy has reached the three nucleate but does not grow because this would involve long range redistribution of substitutional elements between ferrite and austenite. Such a redistribution needs diffusion which is very sluggish for substitutional elements at the can thus proceed only when the temperature of the alloy has reached the temperature T α given by the intersection of the extrapolation of the austenite/ferrite boundary with the on this temperature has been estimated 8) by means of thermodynamic calculations using Thermocalc 9) and data from Uhrenius: diagram. This may be related to the downward shape of the Note that the average composition of austenite w i in elements other than carbon differs from the nominal composition w i o of the cast iron because of graphite precipitation α should thus be done for corrected compositions using the simple following mass balance: where ρ gra and ρ γ are the densities of graphite and austenite respectively. Putting the appropriate values in Eq. (2) leads to w i w i o . Note also that using an average composition means not accounting for microsegregation built up during 11) growth of ferrite during the eutectoid reaction is known to be highly sensitive to cooling rate. Rehder 11) showed that at very low cooling occurs at austenite grain boundaries and proceeds with longrange redistribution of substitutional solutes between ferrite and austenite. The same observation has been reported by Brown and Hawkes 12) in case of isothermal holding in the 8) performed an analysis of data from literature which showed the above model withcooling rate is higher than about 0.02° or 1.2° .

form at austenite grain boundaries and its growth will be controlled by long range diffusion of substitutional solutes.

to plot the undercooling for the start of the ferritic reaction (ΔT α ) as measured with respect to T α i.e. ΔT α = T α where T is the temperature at which the ferritic reaction is seen to start experimentally. This is illustrated in Fig. 5 for a series of alloys containing various amounts of copper studied by Sertucha et al. It is seen that the undercooling for the start of the ferritic reaction appears to follow a power law of the cooling rate that would extrapolate to zero at zero

The undercooling for the start of the reaction shown in Fig. 5 corresponds to the so-called incubation time in the literature on time temperature transformations (TTT) and continuous cooling transformations (CCT). To get further et al. 13) performed dilatometry experiments where samples and held at a temperature below T α . During this second then a dilation. The contraction relates to carbon depletion of austenite because its solubility in austenite strongly decreases with temperature so that the associated change in the density of austenite overtakes graphite precipitation.

tion. Hazotte et al. observed however that ferrite appears very early during the second holding and that the apparent i.e. in fact to a ferrite fraction of more than 10%.

i.e. the diffusion distance for carbon from austenite to graphand more to the right of the equilibrium austenite/graphite phase boundary (graphite solvus) because there is less and less time for austenite depletion. This is illustrated schemati- cally in Fig. 6 where it is suggested that high cooling rates could lead to conditions where ferrite becomes stable only i.e. at the intersection of the thick solid line which stands for the average carbon content in austenite upon cooling with the extrapolation of the w C γ α / line. The dilatometry experiments mentioned above showed also that carbon transfer from austenite to graphite during cooling was much slower than calculated on a basis i.e. dissolution of fusion control calculations. That carbon transfer to graphite is slower than calculated for diffusion control is in fact 15) on graphitization of white cast irons where carbon transfer a difference could be due to stress effect as investigated by Hillert 16) in the case of graphitization of steels or by an interfacial reaction for carbon transfer from austenite to graphite. Simple calculations by Silva et al.

showed an increased compression towards the graphite/austenite interface that is compatible with a decreased diffusion rate sary to conclude quantitatively on the relative role of stress and interface kinetics.

Formation of Pearlite

1) and later Venugopalan 3) considered pearlite grows in cast irons as it does in steels and this is X alloys have been reported describing growth of pearlite during isothermal holding after austenitization and quenching to a temperature where pearlite could nucleate and grow. alloys described by Hillert. 20) i.e.

long range redistribution of substitutional alloying elements be observed when the carbon activity in austenite is very high. The two above cases could not apply to cast irons because of the presence of graphite precipitates that act as carbon sink and limit any increase of carbon activity in austenite. For temperature below the lower limit of the grow at low and high undercooling respectively. In both of 20) demonstrated so that its growth kinetics is controlled by carbon diffusion in austenite. It is however worth to mention the role of carbon diffusion in ferrite that Nakajima et al. 21) have been multicomponent alloys. 20) constant ortho-pearlite grows with full partitioning of substitutional solutes between cementite and ferrite while para-pearlite grows with no such partitioni.e. both cementite and ferrite inherit the austenite compartitioning of substitutional alloying elements is expected to control lamellar spacing within pearlite. The temperature for the transition between constant ortho-pearlite and para-ortho-pearlite needs being considered in the case of cast irons. To check if the assumed similarity between constant 22) analyzed literature data for the start of the pearlitic reaction during isothermal treatment of various cast irons and compared with the results of Al-Salman et al. 23) steel. A strong agreement was found in that results for the common pearlite nose with little scattering. The fact that pearlite forming during continuous cooling of cast irons inherits the composition of the parent austenite section of the metastable system as done before for the ferritic reaction in the stable system. Such an isopleth section is illustrated in Fig. 7 et al. suggested that the pearlitic reaction takes place only when the temperature of the alloy has reached the lower pearlite formation should thus be given by the intersection of the extrapolated austenite/ferrite boundary with the lower p 9) and data [START_REF] Uhrenius | Proc. Symp. Hardenability Concepts with Application to 11[END_REF] the following equation was proposed for T p (° should be due to the fact it is a carbide former owing to the above remark on the downward gamma loop in this system.

results for the detected start of the pearlitic reaction may conveniently be represented as undercooling ΔT p calculated with respect to T p expressed as: ΔT p = T p observed onset temperature of the transformation. Figure 5 shows there is a minimum undercooling necessary for pearl-°C for alloys containing copper and manganese according to the results by Sertucha et al. Because results on the ferclaimed that such an undercooling for the pearlitic reaction cooling for its formation increases much less when the cooling rate increases as compared to the ferritic reacand pearlitic reactions intersect for a cooling rate of about 180° ferrite would form above this critical cooling rate. As for undercoolings are overestimated because the transformation cant volume fraction of pearlite has appeared. Comparison between thermal analysis or DTA experiments and dilatomare not aware of such a work.

3) described pearlite growth in nodular cast irons with a model ing pearlite growth was the same for all cast irons except those containing molybdenum. This agrees with detailed experimental studies by Lalich and Loper 19) and Pan et al. 2) which have been analyzed in terms of growth kinetics by Lacaze. [START_REF] Lacaze | [END_REF] 0.5 wt.% As and 0.3 wt.% Sn all showed similar pearlite growth rates. Lacaze and Sertucha as well for alloys with copper up to 0.9 wt.% and tin up to 0.11 wt.%. It may thus be concluded that pearlite growth in cast iron is controlled by silicon partitioning between fer- containing molybdenum and certainly also chromium which are known to affect the growth rate of pearlite in cast irons and CCT curves by Röhrig and Fairhurst. 25) The decrease in pearlite growth kinetics observed when these elements are added is in agreement with the fact that they are known to strongly affect carbon diffusion in austenite which has been stated to control constant ortho-pearlite growth.

could have been expected to affect pearlite growth rate as they should strongly partition. It does not appear this particertainly be worth of further investigation.

Growth Competition of Ferritic and Pearlitic Reactions

window in which ferrite could grow without the risk of pearlite appearing for usual cast irons. From Eq. (5) it would be concluded that manganese is a are very strong and mid pearlite promoter respectively.

Copper and nickel appear as mid ferrite promoters while they are generally reported to be pearlite promoters. It may thus be concluded that considering the temperature difference for the start of ferritic and pearlitic reactions could be for analyzing the effect of these usual alloying elements on as-cast materials which are continuously cooled during the eutectoid transformation.

Effects of Pearlite Promoters

The way copper and manganese act on the eutectoid explanations as reviewed by Pan et al. 2) In a later work on the effects of alloying on the eutectoid transformation in [START_REF] Venugopalan | Proc. Int. Symp. Fundamentals and Applications of Int. J. Metalcast[END_REF] summarized the role of austenite have different effect on carbon diffusion: manganese would would form a barrier around the nodules that retards carbon transfer to graphite as suggested by Lalich and Loper. 19) As stable eutectoid temperature which then certainly leads to promoting effect because it is not expected that additions of ganese and nickel (excluding austenitic cast irons) in cast in ferrite. austenite boundary with its extrapolation below the ferrite/ graphite line (dashed line) is shown for the three Fe-C-2.5Si-X alloys. Note that the temperature of the intersection between these two boundaries is the T α temperature because chromium or molybdenum was found to change very little the T α the above additions did leave the ferrite/graphite boundary unchanged.

of the driving force for carbon diffusion from austenite to

Δw C the amount of ferrite. At given undercooling ΔT α decreasing Δw C ing effect. decrease T α -Table 1). It rite Curie temperature. The Curie temperature of pure iron ° and Si. Sertucha et al.

°C

From Fig. 8 it would be expected that low addition of and Loper 19) have effectively reported that small addition et al. 5) noticed that copper favors slightly the ferritic reaction when added at similar low level. The known pearlite promoting effect of copper appears at higher content as illustrated in Fig. 9 at given cooling conditions. Sertucha et al. then suggested that when the alloy composition is such that the ferritic reacis abruptly slowed down because of the drop in the carbon - 26) have shown that the relative amount of ferrite and pearlite is sensitive to change in nodule counts or cooling rate only when the nodule count is low. When required to suppress ferrite growth.

The analysis of literature data performed by Lacaze et Fig. 8.

ΔwC for an alloy with 2.5 wt.% Si. al. showed that the same kind of behavior as seen for copper in Fig. 9 For the thermal conditions prevailing in thermal analysis obtained for Sn eq equal to or higher than 0.13 wt.%. How-28) it could also be concluded from literature data that addition of copper i.e. a fully pearlitic matrix. 1) 29) compared the role of thin layer at the surface of graphite that would be a barrier to 29) mentions that such a diffusion barrier could not be evidenced at -1) this layer may have formed in the solid state by rejection of tin and antimony from graphite. This further suggests that there should be a strong interaction between carbon atoms on the basal planes of graphite and tin or antimony that limits carbon transfer from the matrix to graphite and thus leads to their pearlite promoter effect. present a similar gamma loop indicates that tin is a ferrite T α and T p et al. 2) on an experimental basis. Looking at the slope of the gamma proposed that a term + Sn found that ΔT p extrapolates to zero at zero cooling rate for cast irons alloyed with more than 0.05 wt.% Sn. This suggests that Sn acts in strongly decreasing the undercooling for cementite precipitation and thus for pearlite formation. The pearlite growth may have the same reason. There is however Cho et al. 30) have reported that small additions (up to 0.3 observed by Hsu et al. 31) et al. 32) for et al. 33) for Ni and Yu and Loper for [START_REF] Venugopalan | AFS Trans. 98 Metalovedenie I Termicheskaya Obrabotka Metallov[END_REF] related the pearlite promoter effect of nickel to its effect on the eutectoid temperature. These authors discussed also the complex effect of molybdenum in the same line as those followed by considering molybdenum decreases carbon 36) have effectively reported long ago such an effect of molybdenum. That molybdenum decreases carbon diffusion in ferrite and in of this element to cast irons does shift the ferritic and pearlitic reactions to longer times in TTT and CCT diagrams as could be seen in the compendium of Rohrig and Fairhurst. 25)

Impurities and Trace Elements Effects

Björkegren has reviewed some of the relations available at the beginning of the 1980's to relate composition and ferrite or pearlite fraction for given casting conditions. The still most extensive work available is the one published by Thielemann [START_REF] Thielemann | Giessereitechnik 16 Giessereiforschung 22 English)[END_REF] who proposed the following relation for the fraction of ferrite in the matrix (%): 8) to data from other authors than Thielemann and found a satisfactory description of the transition from ferritic-pearlitic to fully accounting for minor elements such as magnesium or for impurities such as phosphorus was important: in any series data from various sources remained high which may be due because cooling rate differed from one study to another.

f exp (
Another source of scattering is the nodule count which may greatly vary even when melt preparation and casting parameters are expected to be constant. Such a variability has been observed by Sertucha et al. who reported that the nodule count (N A ) varied from 190 mm to 590 mm at given casting and inoculation conditions. The composition ranges of the four series of alloys that were investigated are given in Table 2 for the main elements and Table 3 be observed a two-fold relation between N A and sulfur as seen in Fig. 10. Half of the data showed no sensitivity to the sulfur content while the other half showed a clear increase of N A was observed for both fully ferritic and partly pearlitic casts. Looking systematically at the change in composition of the noticed that copper and titanium contents presented also a two-fold correlation with the sulfur content. This suggests that very complex chemical reactions take place during melt preparation that would preclude any possibility of predicting without an appropriate description of these reactions which is far from being available.

Microsegregation microsegregation

used experimental characterization of the microsegregations to calculate the local T α and T p temperatures and found these temperatures do not change much for 80% of the matrix. Further these results illustrated clearly that the undercooling of the ferritic reaction increases all along the transformaconstant for the pearlitic reaction. This is in line with the above description of these reactions. build-up of microsegregation of other solutes depends on it as calculated by Lacaze. Scheil's model for substitutional solutes but assuming rapid solid-state diffusion of carbon may conveniently represent of nodular cast iron. Such a calculation was performed using the Scheil's module of Thermocalc 9) base selecting an alloy in the middle range of those studied at 3.6 wt.% to give a eutectic composition and 0.1 wt.% Cr was also added. In Fig. 11 plotted versus solid fraction which is "equivalent" to the distance from the surface of a nodule (solid fraction equal to zero) to half distance between two neighboring nodules (solid fraction equal to one). Note the logarithmic scale of the Y axis in Fig. 11. It is seen that the predicted microsegre- gations agree very well with those measured by Rundman and with information from literature he reported.

Figure 12 illustrates the local evolutions of T α and T p for the above alloy and compares them to those for the unalloyed cast iron (Fe-3.6 wt.% C-2.5 wt.% Si). Note that the correction given by Eq. (2) was not applied here. As tures for the start of the stable and metastable transforma-Curie temperature of ferrite. The second effect is to affect the temperature difference between these two reactions which are seen to intersect at some point away from the graphite nodules in alloyed iron when they remain well separated in the unalloyed one. This is seen to be due to the fact that T p remains constant and even increases because of the marked segregation of molybdenum and chromium which affect positively T p . Both of these effects will favor the formation of pearlite instead of ferrite.

While austenite was considered as chemically homogethis is not the case and the graph in Fig. 12 demonstrates how this affects the local reference temperatures. Accordreaction has been observed to start at higher undercooling than in fully pearlitic matrix as reported by Samuel and Viswanathan and Lacaze and Sertucha. This is certainly to be associated with microsegregation.

Conclusion

Control of the eutectoid transformation of austenite to get a fully ferritic or fully pearlitic matrix is achieved by alloying or by change in the cooling rate (which depends in all practicality on the casting section size). The most common count may also have a positive effect on ferrite fraction by favoring carbon diffusion from austenite to graphite nodules but it has been stated this is more effective at low nodule count nodular cast irons will be sensitive to cooling rate while very high cooling rates would be necessary to avoid some ferrite forming at high nodule count.

Impurities or trace elements appear to have a marked effect on graphite shape and on room temperature prop-ling cast iron melt chemistry. Attempts made in the past such as the extensive work by Thielemann [START_REF] Thielemann | Giessereitechnik 16 Giessereiforschung 22 English)[END_REF] should be updated as impurities and their level are not the same in the present days as they were at that time. Supporting such a work with the development of an appropriate databank for thermodynamic calculations of inclusion formation during chromium are present would be of great interest.

Fig. 1 .

 1 Fig. 1. Bull-eyes structure of nodular cast irons showing halos of ferrite (white contrast) around graphite nodules (dark congrey contrast).

Fig. 2 .Fig. 3 .

 23 Fig. 2. Cooling curve and its derivative obtained by casting a standard thermal analysis cup with a nodular cast iron et al. Tα and Ttrans are respectively the experimental temperatures for the start of the ferritic and pearlitic

Fig. 4 .

 4 Fig. 4.Si in the stable eutectoid range. ΔwC shows the difference between the carbon content at the ferrite/austenite interw C α γ /

  ............ (2)

Fig. 5 .

 5 Fig. 5. Evolution with cooling rate of the start of the ferritic and

  Fig. 6.possible evolution of the average carbon content in austenite (very thick lines) during cooling at low and high cooling rate.

  rite and cementite even when (low) additions of the above experimental data by Al Salman et al.23) the growth rate of pearlite could then be expressed as function of ΔT p as the pearlite nose.

Fig. 7 .

 7 Fig. 7.Si in the metastable eutectoid range.

  % Ni on the ferrite/austenite boundary in an isopleth 9)

Fig. 9 .

 9 Fig. 9. Effect of copper addition on the ferrite fraction after cooling at different rates after austenitization or casting in a thermal analysis cup (60°C/min). The alloys contained After Sertucha et al.

Fig. 10 .

 10 Fig. 10. Evolution of the nodule count NA with the sulfur content of fully ferritic (large circles) and partly pearlitic (small et al.

Fig. 11 .

 11 Fig. 11. Build-up of microsegregation of some substitutional sol-

Fig. 12 .

 12 Fig. 12. Local reference temperatures for the ferritic and pearlitic 11. Both the case of an unalloyed cast iron and of an alloyed one are shown.

Table 1 .

 1

2

D C

DC . TCurie is the Curie temperature of ferrite.

  exists for tin with a critical amount at about 0.05 wt.%. Available results concerning manganese present a more regular effect though it could not be excluded that a drop exists also for this element. Considering data obtained by Pan et al. 2) (Sn eq

	Sn	eq	=	0 075 .	⋅	w	Mn	+	0 125 .	⋅	w	Cu	+	w	Sn	............ (6)

Table 3 .

 3 Composition ranges in other elements (wt.%) of the four series.

	Series	B	Ti	Al	Bi	N	Nb	Zr	Cr	Sn		V	Sb
	ferritic series	0 0.032	0 0.028	0	0 0.005	0.0032	0 0.012 0.003 0
	Ni series		<0.01						0.02 0.05	<0.005 <0.01 0.01
	Si series			0.006					0.029		<0.01 0.013
	Pearlitic series	0	0 0.36	0 0.02		0 0.0086	0	0 0.005	0 0.35	0 0.025	0 0.98	0 0.5	0 0.005

Table 2 .

 2 A (mm ) and ferrite fraction f α (%) for the four series.

	Series	C	Si		P	S	Ni	Cu		NA	f α
	ferritic series	3.36	1.51	0.09 0.39		0.02	0	0.01 0.15	0.021 0.059	190 590	100
	Ni series		1.88	0.09 0.13	0.012	0.003 0.005		0.01 0.02	0.052 0.058	191
	Si series	3.02 2.20 0.05 3.81 0.11	0.016 0.032	0.005 0.009	0.030 <0.020 0.052	0.033 0.060	136 323	89 100
	Pearlitic series	2.32	1.66	0.12 1.06	0.012	0.008 0.021	0.026 0.060	1.18		95	0

Appendix

Redistribution of substitutional elements between ferrite and austenite or between pearlite and austenite during austenite decomposition needs long range diffusion of these elements in austenite in front of the growing interface. The thickness of the corresponding diffusion spike would scale V the growth rate of the interface. redistribution of substitutional solutes. The same should apply to the pearlitic reaction which proceeds at even larger growth rates than the ferritic reac-Stefanescu seems questionable and suggests there were strong chemical heterogeneities in their alloys due to the precipitation of eutectic carbides that were then totally or partly dissolved during austenitization.


Sheila Sandon

AN INTEGER VALUED BI-INVARIANT METRIC ON THE GROUP OF CONTACTOMORPHISMS OF R 2n × S 1

In his article [Vit92] on generating functions Viterbo constructed a bi-invariant metric on the group of compactly supported Hamiltonian symplectomorphisms of R 2n . Using the set-up of [S09] we extend the Viterbo metric to the group Cont c 0 (R 2n × S 1 ) of compactly supported contactomorphisms of R 2n × S 1 isotopic to the identity. We also prove that Cont c 0 (R 2n × S 1 ) is unbounded with respect to this metric.

Introduction

Gromov's non-squeezing theorem [START_REF] Gromov | Pseudoholomorphic curves in symplectic manifolds[END_REF] marked the beginning of the study of symplectic capacities, which are global symplectic invariants that measure the size of symplectic manifolds. Some years later Hofer [START_REF] Hofer | On the topological properties of symplectic maps[END_REF] discovered a way of measuring the size (or energy) of Hamiltonian symplectomorphisms, by looking at the total variation of their generating Hamiltonian. This notion gave rise in fact to the definition of a bi-invariant metric (the Hofer metric) on the group of Hamiltonian symplectomorphisms [START_REF] Hofer | On the topological properties of symplectic maps[END_REF][START_REF] Polterovich | Symplectic displacement energy for Lagrangian submanifolds[END_REF][START_REF] Lalonde | The geometry of symplectic energy[END_REF]. The Hofer metric is deeply related to symplectic capacities. It was proved by Hofer that for any domain U of R 2n the Hofer-Zehnder capacity c HZ (U) is a lower bound for the energy of any compactly supported Hamiltonian symplectomorphism φ such that φ(U) ∩ U = ∅. Lalonde and McDuff [START_REF] Lalonde | The geometry of symplectic energy[END_REF] gave moreover a direct geometric construction relating non-degeneracy of the Hofer metric to Gromov's non-squeezing theorem.

Using the theory of generating functions, Viterbo constructed in [START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF] a new symplectic capacity for domains of R 2n and a new bi-invariant metric d V on the group Ham c (R 2n ) of compactly supported Hamiltonian symplectomorphisms 1 , and proved an energy-capacity inequality relating these two notions. Moreover he defined a partial order ≤ V on Ham c (R 2n ), giving Ham c (R 2n ), d V the structure of a partially ordered metric space 2 .

The generalization of the constructions in [START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF] to the contact case [START_REF] Sandon | A partial order on the group of contactomorphisms of R 2n+1 via generating functions[END_REF][START_REF] Sandon | Contact Homology, Capacity and Non-Squeezing in R 2n × S 1 via Generating Functions[END_REF] has been motivated by the theory of orderability of contact manifolds introduced by Eliashberg and Polterovich [START_REF] Eliashberg | Partially ordered groups and geometry of contact transformations[END_REF], and by the related contact rigidity phenomena studied in [START_REF] Eliashberg | Geometry of contact transformations and domains: orderability vs squeezing[END_REF]. Bhupal [START_REF] Sandon | A partial order on the group of contactomorphisms of R 2n+1 via generating functions[END_REF] extended the Viterbo partial order to the group Cont c 0 (R 2n+1 ) of compactly supported and isotopic to the identity contactomorphisms of R 2n+1 , thereby proving orderability of R 2n+1 . In [S09] we obtained a new proof of the contact non-squeezing theorem of Eliashberg, Kim and Polterovich [START_REF] Eliashberg | Geometry of contact transformations and domains: orderability vs squeezing[END_REF] by extending the Viterbo capacity to the contact manifold R 2n × S 1 .

In introducing the notion of orderability Eliashberg and Polterovich were in fact motivated by the question of finding some geometric structure on the group of contactomorphisms. They showed in [START_REF] Eliashberg | Partially ordered groups and geometry of contact transformations[END_REF] that the concept of relative growth, which is available in any partially ordered group, can be applied to the contactomorphism group of an orderable contact manifold (M, ξ) in order to 1 It was proved by Bialy and Polterovich [START_REF] Bialy | Geodesics of Hofer's metric on the group of Hamiltonian diffeomorphisms[END_REF] that the Viterbo and the Hofer metric coincide on a neighborhood of the identity in Ham c (R 2n ), but Sorrentino and Viterbo [START_REF] Sorrentino | Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms[END_REF] recently found examples showing that the two metrics are different in general.

2 Recall that a partially ordered metric space is a metric space (Z, d) endowed with a partial order ≤ such that

for every a, b, c in Z with a ≤ b ≤ c it holds d(a, b) ≤ d(a, c).
associate to it a metric space Z(M, ξ), δ . This can be done by defining, in terms of the relative growth, a pseudo-distance δ on the group of those contactomorphisms of (M, ξ) that are generated by a positive Hamiltonian, and then by considering the quotient of this group by the equivalence classes of elements which are at zero distance from each other.

In the present article we show that in the case of the contact manifold R 2n × S 1 it is possible to define a bi-invariant metric directly on the group of all (not necessarily positive) compactly supported contactomorphisms isotopic to the identity. This metric is a generalization of the Viterbo metric and can be easily constructed by using the set-up developed in [S09]. However a crucial difference is that, in contrast with the symplectic case, our metric only takes values in Z. We refer to the introduction of [S09] for an explanation, in terms of generating functions, of the special role played by the integers in the study of rigidity phenomena for the contact manifold R 2n × S 1 .

Our results can be summarized in the following theorem.

Theorem 1.1. The group Cont c 0 (R 2n × S 1 ) admits an unbounded integer valued bi-invariant metric d. This metric is compatible with the Bhupal partial order ≤ B in the sense that ≤ B turns Cont c 0 (R 2n × S 1 ), d into a partially ordered metric space.

The proof of Theorem 1.1 is based on results of [S09].

This article is organized as follows. In the first two sections we recall the set-up of [S09]. In particular, in Section 2 we give some preliminaries on generating functions and in Section 3 we discuss the generalization to Cont c 0 (R 2n × S 1 ) of the invariants c + and c -that were constructed by Viterbo in [START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF]. In Section 4 we define the metric d on Cont c 0 (R 2n × S 1 ), and discuss an energy-capacity inequality relating it to the contact capacity for domains of R 2n × S 1 that was constructed in [S09]. In Section 5 we recall the definition of the Bhupal partial order ≤ B on Cont c 0 (R 2n × S 1 ) and show that d and ≤ B are compatible. We also show how this implies that the energy (i.e. the distance to the identity) does not decrease along contact isotopies that are generated by a non-negative Hamiltonian. In the last section we prove that d is unbounded.

Acknowledgements. I thank my supervisor Miguel Abreu for his support and mathematical guidance, and Leonid Polterovich for feedback on preliminary versions of this article. My research was supported by an FCT graduate fellowship, program POCTI-Research Units Pluriannual Funding Program through the Center for Mathematical Analysis Geometry and Dynamical Systems and Portugal/Spain cooperation grant FCT/CSIC-14/CSIC/08.

Preliminaries on generating functions

We start by presenting some preliminaries on generating functions, referring to [S09] and the bibliography therein for more details and background information.

Let B be a closed manifold, and consider a function S : E → R defined on the total space of a fiber bundle p : E -→ B. We will assume that dS : E -→ T * E is transverse to N E := { (e, η) ∈ T * E | η = 0 on ker dp (e) }, so that the set Σ S of fiber critical points is a submanifold of E, of dimension equal to the dimension of B. To any e in Σ S we associate an element v * (e) of T * p(e) B by defining v * (e) (X) := dS ( X) for X ∈ T p(e) B, where X is any vector in T e E with p * ( X) = X. Then i S : Σ S -→ T * B, e → p(e), v * (e) is an exact Lagrangian immersion, with i * S λ can = d (S |Σ S ). Its lift to J 1 B = T * B × R , ker(dz -λ can ) is the Legendrian immersion j S : Σ S → J 1 B, e → p(e), v * (e), S(e) . The function S : E → R is called a generating function for L S := i S (Σ S ) ⊂ T * B and for its lift L S := j S (Σ S ) ⊂ J 1 B. A fundamental property of S is that its critical points correspond under i S to intersections of L S with the 0-section 0 B , and under j S to intersections of L S with the 0-wall 0 B × R. Note also that if e is a critical point of S then its critical value is given by the R-coordinate of the point j S (e).

A generating function S : E -→ R is said to be quadratic at infinity if p : E -→ B is a vector bundle and if there exists a non-degenerate quadratic form

Q ∞ : E -→ R such that dS -∂ v Q ∞ : E -→ E * is
bounded, where ∂ v denotes the fiber derivative. Existence of generating functions quadratic at infinity for all Legendrian submanifolds of J 1 B contact isotopic to the 0-section was proved by Chaperon [START_REF] Chaperon | On generating families[END_REF] and Théret [START_REF] Théret | Utilisation des fonctions génératrices en géométrie symplectique globale[END_REF], and independently by Chekanov [START_REF] Chekanov | Critical points of quasi-functions and generating families of Legendrian manifolds[END_REF]. Their theorem is a generalization of the analogous result for Lagrangian submanifolds of T * B Hamiltonian isotopic to the 0-section, that was proved by Sikorav [Sik86, Sik87] using ideas of [START_REF] Laudenbach | Persistance d'intersection avec la section nulle au cours d'une isotopie hamiltonienne dans un fibre cotangent[END_REF] and [START_REF] Chaperon | Une idée du type "géodésiques brisées"pour les systémes hamiltoniens[END_REF]. A second fundamental result is the uniqueness theorem for generating functions quadratic at infinity, that is due to Viterbo [START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF] and Théret [START_REF] Théret | Utilisation des fonctions génératrices en géométrie symplectique globale[END_REF][START_REF] Théret | A complete proof of Viterbo's uniqueness theorem on generating functions[END_REF].

Relying on the Uniqueness Theorem, Viterbo [START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF] applied Morse theoretical methods to generating functions in order to define invariants for Lagrangian submanifolds of T * B Hamiltonian isotopic to the 0-section. As observed by Bhupal [Bh01], Viterbo's invariants can also be defined in the more general class L of Legendrian submanifolds of J 1 B contact isotopic to the 0-section. The construction goes as follows. Let L be an element of L with generating function S : E → R.

Denote by E a , for a ∈ R ∪ ∞, the sublevel set of S at a, and by E -∞ the set E -a for a > 0 big. We consider the inclusion i a : (E a , E -∞ ) → (E, E -∞ ), and the induced map on cohomology

i * a : H * (B) ≡ H * (E, E -∞ ) -→ H * (E a , E -∞ ) where H * (B) is identified with H * (E, E -∞ ) via the Thom isomorphism. For any u = 0 in H * (B) we define c(u, L) = c(u, S) = inf { a ∈ R | i * a (u) = 0 }. Note that c(u, L) is a critical value of S.
(i) c v ∪ w, L 1 + L 2 ≥ c(v, L 1 ) + c(w, L 2 )
where L 1 + L 2 is defined to be the set3 of points (q, p, z) in J 1 B such that p = p 1 + p 2 and

z = z 1 + z 2 with (q, p 1 , z 1 ) ∈ L 1 and (q, p 2 , z 2 ) ∈ L 2 . (ii) c(µ, L) = -c(1, L),
where L denotes the image of L under the map

J 1 B → J 1 B, (q, p, z) → (q, -p, -z). (iii) Assume that L ∩ 0 B = ∅. Then c(µ, L) = c(1, L) if and only if L is the 0-section. In this case we have c(µ, L) = c(1, L) = 0.
The proof of this lemma is purely algebraic topological, and does not require any argument of symplectic or contact topology. It was originally given by Viterbo in the setting of Lagrangian submanifolds of T * B Hamiltonian isotopic to the 0-section, but its extension to the contact case is immediate (see [START_REF] Sandon | A partial order on the group of contactomorphisms of R 2n+1 via generating functions[END_REF] or [S09]). In the symplectic case, the symplectic character of c is given by the fact that

(1) c u, Ψ(L) = c u, L -Ψ -1 (0 B )
for every Hamiltonian symplectomorphism Ψ of T * B (see [START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF] or [S09]). The analogue of this result does not hold in the contact case. However Bhupal proved that the following weaker statement is still true.

Lemma 2.2 ([Bh01]

). For any contactomorphism Ψ of J 1 B contact isotopic to the identity,

0 = u ∈ H * (B) and L ∈ L it holds c u, Ψ(L) = 0 ⇔ c u, L -Ψ -1 (0 B ) = 0.
The idea of the proof is to study the bifurcation diagram of a 1-parameter family S t of generating functions of

Ψ -1 t Ψ(L) -Ψ -1 t (0 B )
, where Ψ t is a contact isotopy connecting Ψ to the identity, and to show that there cannot be a 1-parameter family c t of critical values of S t crossing the critical value 0. The key reason why the critical value 0 plays a special role is that critical points with critical value 0 correspond to intersections of the generated Legendrian submanifold with the 0-section. As we observed in [S09], if Ψ is 1-periodic in the R-coordinate of J 1 B = T * B × R then the argument of Bhupal can also be applied if we replace 0 by any other integer, to get the following result. We denote by • (respectively • ) the smallest (respectively largest) integer that is greater or equal (respectively smaller or equal) to the given number.

Lemma 2.3 ([S09]

). Let Ψ be a contactomorphism of J 1 B which is 1-periodic in the R-coordinate of J 1 B = T * B × R, and isotopic to the identity through 1-periodic contactomorphisms. Then for every u = 0 in H * (B) and

L ∈ L it holds c u, Ψ(L) = c u, L -Ψ -1 (0 B ) and c u, Ψ(L) = c u, L -Ψ -1 (0 B ) .

The invariants c + and c -

The invariants for Lagrangian submanifolds of T * B discussed in the previous section where applied by Viterbo [START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF] to the special case of a compactly supported Hamiltonian symplectomorphism φ of R 2n , ω = dx ∧ dy , by regarding its compactified graph as a Lagrangian submanifold of T * S 2n . Viterbo obtained in this way two invariants c + (φ) and c -(φ), defined by using respectively the orientation and unit cohomology classes of S 2n . The invariants c + and c -were generalized in [START_REF] Sandon | A partial order on the group of contactomorphisms of R 2n+1 via generating functions[END_REF] and [S09] respectively to the case of compactly supported contactomorphisms of R 2n+1 and R 2n × S 1 . We will review in this section the construction and properties of c + and c -, discussing directly the contact case 4 .

Let φ be a contactomorphism of R 2n+1 , ξ 0 = ker (dz -ydx) , with φ * (dz -ydx) = e g (dz -ydx).

Following Bhupal [START_REF] Sandon | A partial order on the group of contactomorphisms of R 2n+1 via generating functions[END_REF], we define a Legendrian embedding Γ φ : R 2n+1 -→ J 1 R 2n+1 to be the composition τ • gr φ , where gr φ : R 2n+1 -→ R 2(2n+1)+1 is the Legendrian embedding q → (q, φ(q), g(q)) and τ : R 2(2n+1)+1 -→ J 1 R 2n+1 the contact embedding (x, y, z, X, Y, Z, θ) → x, Y, z, Y -e θ y, x -X, e θ -1, xY -XY + Z -z . Here we consider the product contact structure e θ (dz -ydx) -(dZ -Y dX) on R 2(2n+1)+1 . More explicitely, for φ = (φ 1 , φ 2 , φ 3 ) we have that Γ φ : R 2n+1 -→ J 1 R 2n+1 is given by

Γ φ (x, y, z) = (x, φ 2 , z, φ 2 -e g y, x -φ 1 , e g -1, xφ 2 -φ 1 φ 2 + φ 3 -z).
Note that Γ φ can also be written as Γ φ = Ψ φ (0-section) where Ψ φ is the local contactomorphism of J 1 R 2n+1 defined by the diagram

R 2(2n+1)+1 φ / / τ R 2(2n+1)+1 τ J 1 R 2n+1 Ψ φ / / J 1 R 2n+1
with φ the contactomorphism (p, P, θ) → (p, φ(P ), g(P ) + θ). This shows in particular that Γ φ is contact isotopic to the 0-section. Notice also that the diagram above behaves well with respect to composition: for all contactomorphisms φ and ψ we have namely that

Ψ φ • Ψ ψ = Ψ φψ (in particular Γ φ • ψ = Ψ φ (Γ ψ )) and Ψ -1 φ = Ψ φ -1 .
If φ is compactly supported then we can see Γ φ as a Legendrian submanifold of J 1 S 2n+1 , thus we can associate to it a generating function. The same is true if φ is a contactomorphism of R 2n+1 which is 1-periodic in the z-coordinate and compactly supported in the (x, y)-plane (i.e. a compactly supported contactomorphism of R 2n × S 1 ) because then we can see Γ φ as a Legendrian submanifold of J 1 S 2n × S 1 . We denote respectively by Cont c 0 (R 2n+1 ) and Cont c 0 (R 2n × S 1 ) the groups of compactly supported contactomorphisms of R 2n+1 and R 2n × S 1 that are isotopic to the identity. In the following we will always regard compactly supported contactomorphisms of

R 2n × S 1 as 1-periodic contactomorphisms of R 2n+1 . For φ in Cont c 0 (R 2n+1 ) or in Cont c 0 (R 2n × S 1 ) we define c + (φ) := c(µ, Γ φ ) c -(φ) := c(1, Γ φ )
where µ and 1 are respectively the orientation and the unit cohomology class either of S 2n+1 or of S 2n × S 1 . Note that c + (φ) and c -(φ) are critical values for any generating function of Γ φ . Moreover, exactly as in the symplectic case, they satisfy the following properties.

Lemma 3.1. For all φ, ψ in Cont c 0 (R 2n+1 ) or Cont c 0 (R 2n × S 1 ) it holds:

(i) c + (φ) ≥ 0 and c -(φ) ≤ 0. (ii) c + (φ) = c -(φ) = 0 if and only if φ is the identity.
Proof. Point (ii) is an immediate consequence of Lemma 2.1(iii). As for (i), it can be seen as follows [START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF]. We will prove that c(1, Γ φ ) ≤ 0 and c(1, Γ φ ) ≤ 0 for any φ, so that c -(φ) ≤ 0 and, using Lemma 2.1(ii), c

+ (φ) = c(µ, Γ φ ) = -c(1, Γ φ ) ≥ 0. Since c(1, Γ φ ) = inf { a ∈ R | i *
a (1) = 0 }, we need to prove that i * 0 (1) = 0. Let S : E → R be a g.f.q.i. for Γ φ (respectively Γ φ ) and take a point P in B, where B denotes either S 2n+1 or S 2n × S 1 , outside the support of φ. Consider the commutative diagram

H * (E 0 , E -∞ ) / / H * (E 0 P , E -∞ P ) H * (B) / / (i0) * O O H * ({P }) ∼ = O O
where the horizontal maps are induced by the inclusions {P } → B and E P → E. Since P is outside the support of φ we have that Γ φ and Γ φ coincide with the 0-section on a neighborhood of P , and so S |E P : E P → R is a quadratic form. It follows that the vertical map on the right hand side is an isomorphism. Since the horizontal map on the bottom sends 1 to 1, we see that i * 0 (1) = 0 as we wanted.

In the symplectic case the relation (1), together with the properties in Lemma 2.1, implies that for every φ, ψ in Ham c (R 2n ) we have c [START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF] or [S09]). In the contact 1-periodic case we only get the following weaker statement, using Lemma 2.3.

-(φ) = -c + (φ -1 ), c + (φψ) ≤ c + (φ) + c + (ψ) and c -(φψ) ≥ c -(φ) + c -(ψ) (see

Lemma 3.2 ([S09]

). For all φ, ψ in Cont c 0 (R 2n × S 1 ) it holds:

(i) c -(φ) = -c + (φ -1 ) . (ii) c + (φψ) ≤ c + (φ) + c + (ψ) and c -(φψ) ≥ c -(φ) + c -(ψ) .
Proof.

(i) Note first that c(u, Γ φ -1 ) = c(u, Γ φ ) for all u (apply Lemma 2.3 to L = 0 B and Ψ = Ψ φ -1 ). Using this and Lemma 2.1(ii) we have

c -(φ) = c(1, Γ φ ) = -c(µ, Γ φ ) = -c(µ, Γ φ -1 ) = -c + (φ -1 ) . (ii) We have c + (ψ) = c(µ, Γ ψ ) = c µ, Ψ φ -1 (Γ φψ ) thus by Lemma 2.3 it holds c + (ψ) = c µ, Γ φψ -Ψ φ (0 B ) . But, by Lemma 2.1(i)-(ii) c µ, Γ φψ -Ψ φ (0 B ) ≥ c µ, Γ φψ + c 1, Γ φ = c + (φψ) -c + (φ). Thus c + (ψ) ≥ c + (φψ) -c + (φ) ≥ c + (φψ) -c + (φ)
as we wanted. The statement about c -follows now from (i).

Similarly, in the case of R 2n+1 we can use Lemma 2.2 to show that c -(φ) = 0 if and only if c + (φ -1 ) = 0 and that if c ± (φ) = c ± (ψ) = 0 then c ± (φψ) = 0 (see [START_REF] Sandon | A partial order on the group of contactomorphisms of R 2n+1 via generating functions[END_REF]).

A fundamental property of c + and c -in the symplectic case is that they are invariant by conjugation, i.e. c ± (φ) = c ± (ψφψ -1 ) for all φ, ψ in Ham c (R 2n ) (see [START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF] or [S09]). This is a consequence of the fact that the set of critical values of the generating function of a Hamiltonian symplectomorphism φ of R 2n coincides with the action spectrum of φ, which is invariant by conjugation: if q is a fixed point of φ then ψ(q) is a fixed point of ψφψ -1 with the same symplectic action. This crucial fact does not hold in the contact case. Given a contactomorphism φ = (φ 1 , φ 2 , φ 3 ) of R 2n+1 with φ * (dz -ydx) = e g (dz -ydx), the critical points of a generating function of φ coincide with the translated points of φ, i.e. the points q = (x, y, z) such that φ 1 (q) = x, φ 2 (q) = y and g(q) = 0. Moreover, the critical value is given by the contact action of the corresponding translated point, i.e. the value φ 3 (q) -z (see [START_REF] Sandon | A partial order on the group of contactomorphisms of R 2n+1 via generating functions[END_REF] or [S09]). Note that the contact action is not invariant by conjugation. In fact, not even the property of being a translated point is invariant by conjugation: if q is a translated point of φ then in general ψ(q) is not a translated point of ψφψ -1 . However this is the case if the contact action is 0, because translated points with contact action 0 are fixed points of φ. This fact has been used by Bhupal to prove that, for all φ, ψ in Cont c 0 (R 2n+1 ), c ± (φ) = 0 if and only if c ± (ψφψ -1 ) = 0. As for Lemma 2.2, the idea of the proof is to study the bifurcation diagram of a 1-parameter family S t of generating functions for ψ t φψ -1 t , where ψ t is a contact isotopy connecting ψ to the identity, and to show that there can be no path c t of critical values for S t crossing the critical value 0. As observed in [S09], in the 1-periodic case the same argument can also be applied if we replace 0 by any other integer, to show that the integer part of c + and c -is invariant by conjugation.

Lemma 3.3 ([S09]

). For all φ, ψ in Cont c 0 (R 2n × S 1 ) it holds that c ± (φ) = c ± (ψφψ -1 ) and c ± (φ) = c ± (ψφψ -1 ) .

Remark 3.4. Every Hamiltonian symplectomorphism ϕ of R 2n can be lifted to a contactomorphism ϕ of R 2n+1 or R 2n × S 1 by defining ϕ(x, y, z) = ϕ 1 (x, y), ϕ 2 (x, y), z + F (x, y) where ϕ = (ϕ 1 , ϕ 2 ) and F is the compactly supported function satisfying ϕ * (ydx) -ydx = dF . It can be proved (see [S09])

that c + ( ϕ) = c + (ϕ) and c -( ϕ) = c -(ϕ). 4. The bi-invariant metric d on Cont c 0 (R 2n × S 1 )
In [START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF] Viterbo used the invariants c + and c -to construct a bi-invariant partial order ≤ V and a bi-invariant metric d V on Ham c (R 2n ), and a symplectic capacity for domains in R 2n . Bhupal showed in [START_REF] Sandon | A partial order on the group of contactomorphisms of R 2n+1 via generating functions[END_REF] that the weaker properties of c + and c -that are still satisfied in the case of Cont c 0 (R 2n+1 ) are in fact enough to extend the Viterbo partial order to that group. Note that, by Lemma 3.2 and Lemma 3.3, these same properties are also satisfied by elements of Cont c 0 (R 2n ×S 1 ) so that Bhupal's contruction can be applied to the case of R 2n × S 1 as well (see [S09]). However, we will now show that Lemma 3.2 and Lemma 3.3, that are only available in the 1-periodic case, allow us to extend also the Viterbo metric to Cont c 0 (R 2n × S 1 ).

Recall that the Viterbo metric on Ham c (R 2n ) is defined by d V (φ, ψ) := c + (φψ -1 ) -c -(φψ -1 ).

Similarly, our metric d on Cont c 0 (R 2n × S 1 ) is defined by d (φ, ψ) := c + (φψ -1 ) -c -(φψ -1 ) . Proof. Positivity and symmetry follow from Lemma 3.1(i) and Lemma 3.2(i) respectively. Using Lemma 3.2(ii) we have

d(φ, ϕ) = c + (φϕ -1 ) -c -(φϕ -1 ) = c + (φψ -1 ψϕ -1 ) -c -(φψ -1 ψϕ -1 ) ≤ c + (φψ -1 ) + c + (ψϕ -1 ) -c -(φψ -1 ) -c -(ψϕ -1 ) = d (φ, ψ) + d (ψ, ϕ)
proving the triangle inequality. By Lemma 3.1(ii) we have c + (id) = c -(id) = 0, thus d (φ, φ) = 0. Suppose now that d (φ, ψ) = 0. Then, because of Lemma 3.1(i), we must have c + (φψ -1 ) = c -(φψ -1 ) = 0 and so φ = ψ by Lemma 3.1(ii). This proves non-degeneracy. As for bi-invariance, we have

d (φϕ, ψϕ) = c + (φϕϕ -1 ψ -1 ) -c -(φϕϕ -1 ψ -1 ) = c + (φψ -1 ) -c -(φψ -1 ) = d (φ, ψ)
and, by Lemma 3.3,

d (ϕφ, ϕψ) = c + (ϕφψ -1 ϕ -1 ) -c -(ϕφψ -1 ϕ -1 ) = c + (φψ -1 ) -c -(φψ -1 ) = d (φ, ψ).
The energy of an element φ of Cont c 0 (R 2n × S 1 ) is defined to be its distance to the identity, i.e.

E(φ)

:= c + (φ) -c -(φ) .
Given an open and bounded domain V of R 2n × S 1 , its displacement energy is defined by

E(V) := inf { E(ψ) | ψ(V) ∩ V = ∅ }.
This definition can be extended to arbitrary domains of R 2n ×S 1 by setting

E(U) = sup { E(V) | V ⊂ U, V bounded } if U is open, and E(A) = inf { E(U) | U open, A ⊂ U }

Relation with the Bhupal partial order

Recall from [START_REF] Sandon | A partial order on the group of contactomorphisms of R 2n+1 via generating functions[END_REF] and [S09] that, similarly to the symplectic case, the Bhupal partial order ≤ B on Cont c 0 (R 2n+1 ) and Cont c 0 (R 2n × S 1 ) is defined by

φ 1 ≤ B φ 2 if c + (φ 1 φ -1 2 ) = 0.
We will now show that the metric d and the partial order ≤ B are compatible.

Proposition 5.1. Cont c 0 (R 2n × S 1 ), d, ≤ B is a partially ordered metric space.

φ in Cont c 0 (R 2n × S 1 ) which is the time-1 flow of a Hamiltonian supported in V we have that E(φ) = c + (φ) + c + (φ -1 ) ≤ 2 c(V). If φ is generated by a Hamiltonian supported in V then so is φ n as well, thus lim n→∞ E(φ n ) n = 0 for all φ. Hence Cont c 0 (R 2n × S 1 ) is not stably unbounded. Similarly, Ham c (R 2n ) is unbounded but not stably unbounded with respect to the Viterbo metric5 . We conclude by discussing the case of the contact manifold S 1 , ker(dz) , that can be seen as R 2n × S 1 , ker(dz -ydx) for n = 0. It was proved in [START_REF] Burago | Conjugation-invariant norms on groups of geometric origin, in Groups of Diffeomorphisms: In Honor of Shigeyuki Morita on the Occasion of His 60th Birthday[END_REF] that the diffeomorphism group of S 1 does not admit any non-trivial (up to equivalence) bi-invariant metric. However, the construction of our metric d on R 2n × S 1 does not contradict this result. First of all notice that, in the case n = 0, d is defined on the group of 1-periodic contactomorphisms of R. While in higher dimension compactly supported contactomorphisms of R 2n × S 1 can be seen as 1-periodic contactomorphisms of R 2n+1 , there is no canonical way to do that if n = 0 (we have namely that the group of 1-periodic contactomorphisms of R is the universal cover of the contactomorphism group of S 1 ). Moreover, as we will now explain, our construction does not give a metric even on the group of 1-periodic contactomorphisms of R. Note that for a diffeomorphism φ of R it holds that φ * dz = φ dz, thus φ is a contactomorphism if and only if φ > 0 i.e. if and only if φ is orientation preserving. Let φ be an orientation preserving 1-periodic diffeomorphism of R. We can then associate to φ a Legendrian submanifold Γ φ of J 1 R by defining Γ φ = τ • gr φ , where τ : R 3 → J 1 R is the contact embedding τ (z, Z, θ) = z, e θ -1, Z -z .

Thus Γ φ : R → J 1 R is given by Γ φ (q) = q, φ (q) -1, φ(q) -q . We have that Γ φ is the 1-jet of the function S : R → R, S(q) = φ(q) -q thus in other words S is a generating function for φ. Notice that c + (φ) = max(S) is not necessarily non-negative, and c -(φ) = min(S) not necessarily non-positive. Hence if we define d as in Section 4 we do not get a metric in this case, because positivity fails. Notice that the proof of non-negativity of c + (φ) and non-positivity of c -(φ) in Lemma 3.1(i) used in a crucial way the possibility of choosing a point outside the support of φ. This cannot be done in general in the S 1 case.

  Lemma 2.1 ([START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF]). Let µ ∈ H n (B) denote the orientation class of B, and 0 B the 0-section in J 1 B. The map H * (B) × L -→ R, (u, L) -→ c(u, L) satisfies the following properties:

  Proposition 4.1. d is a bi-invariant metric on Cont c 0 (R 2n × S 1 ), i.e. (i) (positivity) d (φ, ψ) ≥ 0 for all φ, ψ. (ii) (non-degeneracy) d (φ, ψ) = 0 if and only if φ = ψ. (iii) (symmetry) d (φ, ψ) = d (ψ, φ). (iv) (triangle inequality) d (φ, ϕ) ≤ d (φ, ψ) + d (ψ, ϕ) (v) (bi-invariance) d (φϕ, ψϕ) = d (ϕφ, ϕψ) = d (φ, ψ).

  for an arbitrary domain A. In[S09] we extended the Viterbo capacity to domains of R 2n × S 1 by definingc(V) = sup { c + (φ) | φ ∈ Cont (V) }where Cont (V) denotes the set of time-1 maps of contact Hamiltonians supported in V. The energy-capacity inequality c(V) ≤ E(V) follows from [S09, 3.6.1].

Note that L 1 + L 2 is not necessarily a smooth submanifold. However it is generated by the function S 1 S 2 :E 1 ⊕ E 2 -→ R which is defined by S 1 S 2 (x; ξ 1 , ξ 2 ) = S 1 (x; ξ 1 ) + S 2 (x; ξ 2 ), where S 1 : E 1 -→ R and S 2 : E 2 -→ R are generating functions for L 1 and L 2 respectively. For a cohomology class u in B, by c(u, L 1 + L 2 ) we mean in fact c(u, S 1 S 2 ). If L 1 + L 2 is a smooth submanifold contact isotopic to the 0-section, then this is consistent with the definition given above.

The same also holds for Ham c (R 2n ) with respect to the Hofer metric[START_REF] Sikorav | Systemes Hamiltoniens et topologie symplectique[END_REF].

Proof. Suppose that φ 1 ≤ B φ 2 ≤ B φ 3 . Then, since c + (φ 1 φ -1 2 ) = c + (φ 2 φ -1 3 ) = c + (φ 1 φ -1 3 ) = 0, using Lemma 3.2(i) and (ii) we get

Consider now the relation on Cont

1 can be written as the time-1 flow of a non-negative Hamiltonian. This relation is clearly reflexive and transitive. The deep fact that is also anti-symmetric (hence a partial order) follows from antisymmetry of ≤ B and the implication

In the language of Eliashberg and Polterovich [START_REF] Eliashberg | Partially ordered groups and geometry of contact transformations[END_REF], antisymmetry of proves that R 2n ×S 1 and R 2n+1 are orderable contact manifolds. The implication (2) can easily be proved using the fact that c + and c -are monotone with respect to : if [START_REF] Sandon | A partial order on the group of contactomorphisms of R 2n+1 via generating functions[END_REF] or [S09]). Note that an analogous relation is also defined in Ham c (R 2n ) by setting

1 can be written as the time-1 flow of a non-negative Hamiltonian. Then φ 1 ≤ V φ 2 if φ 1 φ 2 , proving that is a partial order (see [START_REF] Viterbo | Symplectic topology as the geometry of generating functions[END_REF]).

Notice that (2) and Proposition 5.1 immediately imply that Cont c 0 (R 2n ×S 1 ), d, is a also partially ordered metric space, so that in particular the energy does not decrease along non-negative contact isotopies.

Unboundedness of d

We will now show that the diameter of Cont c 0 (R 2n × S 1 ) with respect to our metric is infinite, or in other words that d is unbounded. As we will see, this fact follows immediately from the analogous result in the symplectic case and Remark 3.4.

Unboundedness of the Viterbo metric on Ham c (R 2n ) is well known. It can be seen for instance by considering the sequence of Hamiltonian symplectomorphisms of R 2n supported in B 2n (R) = { π n i=1 x 2 i + y 2 i < R } that was constructed by Traynor in [START_REF] Traynor | Symplectic Homology via generating functions[END_REF] to calculate the symplectic homology of B 2n (R), and by noticing that the energy of the elements of this sequence tends to R (that can be chosen to be arbitrarily big). Traynor's sequence

Since the H ρi are positive, by monotonicity of c -we have that c -(φ ρi ) = 0 and thus E(φ ρi ) = c + (φ ρi ). Moreover, it was proved by Traynor [Tr94] that c + (φ ρi ) tends to R for i → ∞. If we now lift the sequence φ ρ1 ≤ φ ρ2 ≤ φ ρ3 ≤ • • • to R 2n × S 1 as explained in Remark 3.4, we get a sequence of contactomorphisms whose energy tends to the integer part of R, which can be chosen arbitrarily big. It follows thus that Cont c 0 (R 2n ×S 1 ) is also not bounded.

The following terminology is taken from [START_REF] Burago | Conjugation-invariant norms on groups of geometric origin, in Groups of Diffeomorphisms: In Honor of Shigeyuki Morita on the Occasion of His 60th Birthday[END_REF]. Two norms on a group G are said to be equivalent if their ratio is bounded away from 0 and ∞. In particular, a norm ν on G is equivalent to the trivial one (i.e. the norm that is everywhere 1 except at the identity) if and only if it is bounded and not fine. A norm ν on G is called fine if 0 is a limit point of ν(G). Since the metric d on Cont c 0 (R 2n × S 1 ) takes values in Z, it is not fine. However, being unbounded, it is not equivalent to the trivial one. An unbounded norm ν on a group G is called stably unbounded if lim n→∞ ν(f n ) n = 0 for some f in G. Note that, by definition of the capacity c, for every


Karolina Golec

Florence Zara

Stéphane Nicolle

Jean-François Palierne

Guillaume Damiand

New Mass Spring System formulation to model the behavior of soft tissues

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Computer-based medicine has been largely developed in recent years for helping surgery, diagnosis and treatment [1]. Unfortunately, from a mechanical standpoint, modeling complex materials such as soft tissues is still a challenge requiring accuracy and possibility of user interaction. Among computational models (such as Finite Element Method [2], or Position Based Dynamics [3]), we chose the Mass-Spring System (MSS) as model. It offers several advantages, like adaptability and fast computation, while allowing topological modifications (like cutting or piercing) without pre-computations. In this paper we present a new MSS formulation for soft tissue simulations. The validation of the mechanical response of the model is based on work of Nicolle et al. [4] which reported the shear behavior of three vital internal organs (kidney, liver and spleen). Our method is based on a topological-physical model called TopoSim [START_REF] Golec | rd International Conference on Computer Graphics WSCG[END_REF][START_REF] Flechon | st International Conference on Computer Graphics WSCG[END_REF] which allows to easily implement different physical models and performs fast and accurate modeling of different kinds of tissues. Thanks to this model, we are able to easily adapt the simulation to desired requirements and to perform topological modifications during simulations if necessary.

Method

The environment damping and the tissue's parameters (like Young's modulus, Poisson's ratio, density) are directly connected to the parameters of the MSS. With the formulation proposed in [START_REF] Baudet | th International Conference on Computer Graphics WSCG[END_REF] we compute the stiffness (k) of springs. Then, we propose to change the usual springs' force formulation to be able to reproduce the analytical solution reported by Nicolle [4]. Our non-linear formulation of the force of springs (equation 1) allows us to reconstruct the viscous part (the term in first brackets) artificially. We derive it from both: viscous and elastic terms. In the formula c represents the spring damping, n, p are constants and l0 and l are respectively initial and the current spring length.

F= ( c ⋅ dl dt +k ⋅(l -l 0 ) 1 -n ) (1+α|l -l 0 | p -( 1 -n ) ) α= 1 n⋅l 0 p-1 (1)

Results

We compare our results obtained for shearing solicitations with the analytical model proposed by Nicolle. We studied the force response of the top layer of the tissue, exactly as performed for the real tissue experiments. We executed the simulation on liver tissue at 0.0151 s-1 strain rate and present results obtained in Fig. 1.

Discussion

In our simulation experiments we successfully reproduce the non-linear viscoelastic behavior of soft tissues. However, we are working on a generic formulation, which will enforce the correct response in an easy and natural way for different deformation speeds, under other mechanical solicitations. We are working on simulations for other tissues (kidney, liver).

Figure 1 :

 1 Figure 1: MSS simulation and the analytical curve: comparison (log-log scale)

Acknowledgements

This work was supported by the LABEX PRIMES (ANR-11-LABX-0063) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).


Ahlem Louati

Jean Rhys

Wide Sargasso

Celebrating the Female Cultural Other in Jean Rhys's Wide Sargasso Sea

Keywords: celebration, Jean Rhys, Charlotte Brontë, fragmentation, hybridity, excessive sexuality

is a postmodern parodic rewriting of Charlotte Brontë"s canonical novel Jane Eyre 2 (1847) in which the postcolonial woman novelist writes the absented side of the silenced Bertha Mason. This paper seeks to study Rhys"s text as a coming-out party for Bertha, who leaves the confines of voicelessness and savageness to celebrate, first, her splitting-subjectivity that endows her with much leeway not only to challenge her readerly representation as the "madwoman in the attic" in the original work but also to embrace her capacity to rebirth herself as Antoinette Cosway who espouses the multiplicity of her identities and the plurality of her selves. Second, her fragmented narrative structure enables her to upset the conventional concepts of objective truth and conclusive meaning, opening the floor to the free play of signs and the freefloating articulation of different realities. Third, her hybridity is no longer perceived as a dysfunctional space; it turns out to be a liberating site which enables her to subvert the conventional concept of "purity," stressing the fact that cultural identity is a matter of being and becoming rather than of being. Fourth, Bertha"s excessive passion and uncontained sexuality are openly articulated instead of being repressed.

A. Antoinette as a Splitting-Subject:

In her endeavour to redeem Antoinette/Bertha from her fixed identity as the Creole "horrible lunatic" woman from the West Indies, Jean Rhys recuperates new forms of being for her heroine who embraces the multiplicity of her selves and the fluidity of her evolving subjectivities. She takes her reader on a journey back to the Caribbean of the 1830s where she invites the marginalized Bertha Mason for a coming-out party to re-represent herself as Antoinette Cowsay, the counterpart of Bertha Mason in JE. Quickly after her mother"s marriage to Mr. Mason, Rhys"s female protagonist becomes Antoinette Cosway Mason.

Antoinette"s nomadic identity that resists closure and finality is further addressed in the novel when she gets married to Edward Rochester. According to the English Law, she is given the last name of her husband to become Antoinette Cosway Mason Rochester. Within this accretion of names, Antoinette is endowed with different identities that display her capacity to embrace different and evolving subjectivities. Indeed, her embrace of her English stepfather"s as well as Rochester"s surnames, without dropping one of them, can better translate her transgressive endeavour to masquerade the British identity. Actually, while identifying herself with the colonial subject, she upsets the imperialist ideology that fixes her within the confines of the cultural Other, opting for a cross-cultural identity. Her self-splitting female protagonist breaks down the conventional boundaries of fixity, dwelling in the realm of openness that celebrates the futurity of her identity and the flexibility of her perpetual state of becoming.

B. Fragmented Narrative Structure:

Rhys merrily overlooks the nineteenth-century chronological order, opting for a disjointed and disrupted narrative structure that fulminates against the seams of cohesion. She projects Antoinette"s fragmented identity on her inconclusive narrative, making her jump into the future and regales her readers with her coming acts. In so doing, she overturns the hierarchical supremacy of realism in the Brontean text that conjures an alternative reality about Bertha"s fatal demise. Shuttling between her past life in Jamaica and her presence in her husband"s house arrest, her heroine subverts the traditional spatio-temporal conventions blurring the barriers of two chronotopic zones. Her narrative is ripping off chicks of reality, cutting the logical cords between events and celebrating the process of shuffling back and forth. The eruption of her homeland memory that intrudes into the waking world of the heroine in the attic of Thornfield Hall showcases incompatible fragments of life which break away the logical sequence of events and disturbs its coherence. As she gazes to her red dress lying on the floor, she recalls her past life in Jamaica, where she used to enjoy the natural wilderness of the island, and notably Goulibri"s untamed garden which is compared to the Garden of Eden: "Our garden was large and beautiful as that garden in the Bible-the tree of life grew there. But it had gone wild" (Rhys, the emphasis is mine 4). Actually, the lush landscapes of the island represent the maternal space where Antoinette feels immune, fortified and safe. Her nostalgic return to Coulibri which is accompanied by a remembrance of "the orchids and the stephanotis and the jasmine and the trees of life," articulates Rhys"s tendency to articulate a gendered chronotope (123). In other words, her heroine"s non-sequential perception of time and space is well represented in her travelling from the symbolic masculine sphere England, which epitomizes the paternal law and colonial world that are related to linearity and rationality to the pre-Oedipal feminine Sargasso Sea that represents the semiotic or maternal space for her.

In addition to her use of flashback techniques, Antoinette"s different dreams also unsettle the chronological flow of events, foreshadow the coming actions in the plot and allow the reader to have access to the character"s subjective consciousness. For example, Antoinette"s third dream liberates her f from the conventional spatio-temporal parameters through her violation of the bounds between reality and dream, creating a magical realist scene. While blurring the boundaries between her present life in Thornfield with her memories of Coulibri, Antoinette defies the realistic conception of objective truth, creating her own reality and world. She refuses to acknowledge the world of Thornfield Hall that imposes on her a fixed identity and defines her as the marginalised Other. In the final section of the novel, Antoinette "doubts the identity of her residence" insisting that the mansion is not in England (Müller 74). She believes that England, which she previously depicts as "quite unreal, and like a dream" is an imaginary place, a "cardboard world" made of paper. She insists that she has never reached England and that she is still sailing the wide Sargasso Sea.

The indeterminacy and uncertainty that Antoinette creates overthrow the conventions of clarity and finality. Her refusal to accept England as an objective and external reality testifies her subversive capacity to create her own reality and to define herself in terms of a space of in-betweenness also stands for fluidity and flux which are in accordance with her hybrid and fluid identity. The novel"s title seems to signify Antoinette"s unstable world of plurality and difference, one that is set against Rochester"s world that denotes fixedness and stability reflected in the word "land" in England. Thinking that she and Rochester have lost their way in the Sargasso Sea casts doubt on the authority and power of Rochester as well as the reliability of Bronte"s objective reality.

II. Hybridity: A. Hybrid Cultural Identity:

Born to a British ex-slave owner father and White French Creole mother, Antoinette finds herself straddling the boundaries of two different cultural stools; she is Caribbean but not black, white but not English. As a White Creole, Antoinette enjoys the multiplicity of her identities stemming from her cultural hybridity; she subverts cultural hierarchisation and overturns the realm of imperialist and the colonialist system of binarism based on the dichotomous opposites of "either/or," "us/them," "self/Other." In other words, it calls into question Englishness as the purest cultural and national identity and troubles it as a monolithic and monologic signifier for whiteness. Her hybridity is a liberating force that opens up an "inbetween space" which Homi Bhabha calls the "Third Space of enunciation" that works to release the tensions arising from the encounter between the colonised and the coloniser and to celebrate the politics of inclusion rather than exclusion ("Cultural Diversity and Cultural Differences" 209).

Antoinette"s joyful celebration of her Creole identity and thus of her hybridity is better seen in the closing scene of the novel. To explain more, her condonation of Rochester"s calls "Bertha! Bertha" that define her as an English woman and her leap into fire where she sees the figure of Tia, her black Caribbean alter ego, should not be read as a movement towards her personal cancellation in which she seeks a complete identification with the black girl.

Instead, it is an instance of rebirth in which Antoinette acknowledges her belonging to the Caribbean world and, thus, asserts her hybrid identity, embracing her in-betweeness and celebrating both parts, black and white, of her national identity. In a self-conscious act, Antoinette resurrects herself by thinking of identity as a production which is never complete or finished. Her embrace of her hybridity becomes a site of resistance, a liberating force that enables her to experience herself as a subject in process who is engaged in a continuous becoming and being.

B. Hybrid Language:

Rhys"s novel in English "serves to interrupt pure narratives of nation," creating thus, a variety of dialects and an array of speech styles that ordinary people use in their use of language. Her novel turns out to be a heteroglot writing that encompasses the very presence of heteroglossia that Bakhtin defines as: "The internal stratification of any single national language into social dialects" ("Discourse in the Novel" 484). This incorporates the presence of many marginalized voices and styles that erupt to subvert "the refined and serious language of high culture" and challenge the uniformity and universality of the English language while ceremonially enthrone Caribbean language (Yaneva 42). In other words, the text incorporates both the Standard English used by Rochester and Mr. Mason and the English, French, and Patois used by the Caribbean islanders such as Creoles and Blacks. The novel incorporates many Creole"s songs like Christophine"s lullaby "Ma belle ka di maman li" meaning "my beautiful girl said to her mother," that represents the mother"s voice. Other childish words are also used by Christophine when she talks to Antoinette such as "doudou," meaning "little darling," "doudou ché" meaning "dear little darling," and others as "béké" meaning a white person Sumillera 29). Rhys"s embodiment of these terms in a text written in English is not only directed to decentralize the universality of her Rochester"s mother tongue, but also to display the linguistic diversities and complexities of the West Indian society, rejecting Brontë"s imperialist ideology that presents her Creole heroine as a voiceless beast that "growled like some strange wild animal" (Brontë 321). She goes beyond that to display "that Creoles do have linguistic vitality.

III. Celebration of her Sexuality:

Although both women writers address the issue of female sexuality as a way to subvert the patriarchal society, they approach it differently. In JE, Brontë violates the muteness that enveloped the nineteenth-century Victorian connubial intimacy by providing her readers a glimpse of Bertha Mason, the ethically perverted wife of her Byronic hero. She reproduces patriarchal and racist binary thought. She represents the proper female behaviour by encountering two antithetical figures of women: "The angel in the house" symbolized by Jane and "the madwoman in the attic" represented by Bertha. In other words, while Brontë defines Bertha"s sexuality within the confines of the evil and monstrous and thus, lapses in the reproduction of the patriarchal sexual politics that force women to be passive, Rhys gives voice to the female passion negated in JE. She re-examines the appropriate English norms of female sexual politics. Antoinette represents herself as a woman who is sexually lustful and full of wolfish of excess, subverting the traditional conceptualization of the female body on which the conventional moral etiquettes enacting normality and abnormality are written.

While acutely embracing the physically and sexually immoderate pleasures and desires, Antoinette/Bertha violates the officially established cultural order which is used as a tool of gender domination and control in JE. Thus, unlike the Brontean narrative which is motivated by Jane"s journey of maturity as she learns the necessity of controlling her passion, the Rhysian text celebrates Antoinette"s celebration of her body"s excesses and signals the heroine"s subversive energy to resist the patriarchal desire to control, contain and master the female body.

Rampant with Antoinette"s obsessive and excessive expression of her anarchic sexual intercourse, WSS becomes a festive merriment of the female body"s corporeality that unseats the sexual/textual politics of the Brontean text. While challenging the cultural ideologies written on her heroine"s body, Rhys presents WSS as a text written through the body as it blends the sexual and the textual instead of arranging them in opposite poles. Conscious of her textualized sexuality in the original text, the Creole heroine strives to challenge the panoptical bodily statement of her husband who inscribes her excessive sexual proclivity within a discourse of savage grotesqueness and bestial madness. In so doing, she offers new textual and sexual poetics which castrate male textual/sexual politics while liberating and redeeming herself from the regulatory system that incarcerates her within the ideological discourse of either passivity and purity or absence and lack. Rhys"s text becomes a body that heeds to voice the repressed sexual desires of the grotesque Antoinette/Bertha who challenges the stimulus/response approach which is used to indoctrinate women how to behave properly, turning them to "docile bodies" disciplined to obey structures of cultural order, to use Michel Foucault"s own words. Antoinette Cosway turns out to be a projection of patriarchal fear of female sexuality. On the one hand, she threatens the English conceptions of morality and normality. Her transgressive and excessive sexual appetites drive her to perversion and promiscuity as Rhys writes: "She thirsts for anyone-not for me ... She"ll loosen her black hair, and laugh and coax and flatter" (Rhys, emphasis is mine 107). Antoinette has a sexual affairs and notably with her cousin Sandi. She jeopardizes her husband"s sense of superiority: she is no longer a sexual object of a male gaze as she redefines her subject position while disturbingly inverts and subverts the traditional sexual politics based on the binary opposite of active male/passive female. Antoinette"s active sexuality overpowers her husband who is captivated by her physical beauty that looks alien and strange, as Rhys writes: "[he is] bewitch [ed] with her. She is in [his] blood and [his] bones. By night and by day" (59). The patriarchal Rochester finds himself sexually dependent on his wife who performs the active role of the dominator. Antoinette"s provocative sexuality and unrestrained passion horrify the English patterns of proper femininity.

Conclusion

WSS releases the Creole Bertha from the shroud of vampirism while endowing her with an alternative identity. By offering her a plausible past life, the Caribbean female novelist opens the ground for her heroine who shares with her the same origin and background to celebrate her fragmented identity that defies the unified and fixed identity of Jane Eyre as well as her cohesive and chronological narrative. While breaking off the spatiotemporal barriers, the heroine inserts fragments of her past that trouble the objective reality/truth of Rochester"s account of her story of madness, opening the floor to a liberating and celebratory space that embraces the mushrooming of multiple "truths." This process of celebration is also manifested in her cuddle of her cultural in-betweeness as well as her hybrid creole language that enables her to upset the conventional concept of purity. This latter is also challenged while making Antoinette/Bertha mock the pure and chaste Jane. The Rhysian female protagonist comes to liberate, articulate and celebrate her excessive sexual desires, subverting thus her female precursor"s complicity with the patriarchal sexual politics.