Mame Mguenar Ndiaye

Sébastien Pillet

El-Eulmi Bendeif

Mathieu Marchivie

Guillaume Chastanet

Kamel Boukheddaden

Smail Triki

Mame Mguenar Ndiaye

Hidden hysteretic behavior of a paramagnetic iron(II) network revealed by light irradiation

Keywords: Coordination Chemistry, Iron(II), Spin-Crossover, LIESST, Photomagnetism, Hidden hysteresis, Elastic frustration

Introduction

Switching phenomenon of molecular based compounds is an open challenge for synthetic chemists worldwide. [1][2][START_REF]Spin Crossover in Transition Metal Compounds I-III[END_REF] In this context, photo-switching is under specific attention since it offers ultrafast (down to femtosecond) manipulation of physical properties. [START_REF] Nasu | Photoinduced Phase Transitions[END_REF][START_REF] Cailleau | [END_REF][6] Among a wide range of systems under investigation, materials that undergo a spin switching, whereby the central metal ion changes its spin state upon an external perturbation, have received considerable interest because of their potential applications as functional materials. [7,8] In this context, spin-crossover (SCO) complexes of 3d 4 to 3d 7 metal ions, displaying high-spin (HS) ↔ low-spin (LS) transition upon temperature, light, pressure, chemical or magnetic field perturbations, have focused a huge amount of investigations in the last 80 years because of potential involvement in optical switches or/and magneto-optical storage properties. [START_REF]Spin Crossover in Transition Metal Compounds I-III[END_REF][7][8][9][10][11] Most of the explored systems are pseudooctahedral 3d 6 iron(II) complexes that may be either paramagnetic (HS, S = 2) or diamagnetic (LS, S = 0).

The photoswitching of SCO materials is mainly based on three different effects. The first one is related to an irradiation centered on photosensitive ligands such as photo-isomerizable units. [12] This effect, named LD-LISC for Ligand-Driven Light-Induced Spin Change, has been improved to be efficient at room temperature [13] but hardly works in the solid state due to strong steric hindrance. The second way of switching by light is based on photo-thermal effects: a pulsed laser irradiation from the LS state inside the thermal hysteresis loop induces enough local and fast heating to trigger the LS to HS spin transition in few nanoseconds. [14] This effect deserves a specific optical set up for ultrafast excitations and its reversibility is still debated. [15] A third strategy for the light-induced SCO is to use light excitation centered on the metal ions, a phenomenon known as Light-Induced Excited Spin State Trapping or LIESST effect [16] and observed on a large number of compounds. One of the main issue of this effect is related to the lifetime of the photo-induced state which has focused a large amount of work to understand the physical and chemical parameters allowing its improvement. [17,18] Basically, the higher the thermal spin-crossover temperature T1/2 the faster the relaxation: a behavior that can be easily understood since the transition temperature T1/2 is proportional to the strength of the ligand field, which stabilizes (resp. destabilizes) the LS (resp. HS) state. The first important parameter that was identified to increase the lifetime of the photo-induced state was the metal-ligand bond-length distance variation upon the spin-crossover: a huge variation increases the vibronic elastic energy barrier of the metastable HS state which then slows down its thermal relaxation.

Using the T(LIESST) measurement, which aims to estimate the limiting temperature above which the photoinduced HS metastable state is erased, [19] a [T(LIESST), T1/2] database was built. [20] From this comparison of more than 50 compounds, the coordination sphere distortion was pointed as a crucial parameter to increase the lifetime of the photo-induced state. This has allowed to increase the T(LIESST) value from 60 K in the late 90's to 130 K for a pure iron(II) SCO material [21] and even 180 K in Prussian Blue analogues. [22] Abstract: The paramagnetic coordination polymer [Fe(3phOHtrz){Pt(CN)4}].2H2O was obtained by reaction of iron(II) salt with the 4-(3-hydroxyphenyl)-1,2,4-triazole (3phOH-trz) ligand and a [Pt(CN)4] 2-salt. Its structure consists in two dimensional {FePt(CN)4} layers linked by π-stacking interactions and a strong H-bonds network between water molecules and the hydroxyl group from the ligand. From the paramagnetic high spin (HS) state at 10 K, irradiation at 830 nm led to the fully low spin (LS) state, according to the reverse-LIESST process. Upon warming, this LS state undergoes a thermally induced spin transition to high spin around 105 K. This photoswitching process is reversible and the paramagnetic state can be recovered by a 510 nm light irradiation below 105 K. Moreover, a permanent irradiation (830 nm) revealed a hidden hysteresis loop of 37 K width. Photocrystallographic experiments did not evidence any structural phase transition upon excitation but underlined that the elastic frustration might be responsible for the inhibition of the spin crossover in this compound and allowed the observation of the hidden hysteresis loop. [b] Sébastien Pillet and El-Eulmi Bendeif, Université de Lorraine, CNRS, CRM2, Blvd des aiguillettes, 54506, Vandoeuvre-lès-Nancy, France. [c] Mathieu Marchivie and Guillaume Chastanet, CNRS, Université de Bordeaux, ICMCB, 87 avenue du Dr. A. Schweitzer, Pessac, F-33608, France, guillaume.chastanet@icmcb.cnrs.fr ; mathieu.marchivie@icmcb.cnrs.fr [d] Kamel Boukheddaden, UMR-CNRS 8635, Univ. Versailles, 45 Av. des Etats-Unis 78035 Versailles, France.

Supporting information for this article is available on the WWW under http://www.eurjic.org/ or from the author.

To circumvent the problem of the photo-induced state's lifetime, a new strategy has recently emerged. By taking advantage of the reversibility of the LIESST effect, [23] purely HS compounds can be converted at very low temperature into the LS state, and further undergo a LS to HS crossover while warming. In such cases, this implies that a hysteresis loop is hidden and can only be revealed by light irradiation. Since the hysteresis is due to a collective effect, the photo-induced states are stable.

There are two strategies to obtain such a hidden hysteresis loop. The first one aims at favoring the overlap between the T(LIESST) and the thermal hysteresis loop through metal dilution. [24] This has led to the observation of a light induced phase collapse on Prussian blue analogs. [25] The second strategy is based on elastic frustration, resulting from a competition between ferroelastic like (cooperativity-rigidity) and antiferroelastic like (flexibility) interactions, that can inhibit partially or completely the SCO. [26] With such strategy, the photoswitching efficiency can be improved and symmetry breaking can be promoted. [27] In this context, as a part of our and others ongoing work on the cooperative polynuclear or extended systems based on functionalized triazole ligands, [28][29][30] we have reported recently an original bidirectional photoswitching of a frustrated state. [30] Indeed, in the 2D [Fe(2-pytrz){Pt(CN)4}].3H2O network, 1, (2pytrz = 4-(2-pyridyl)-1,2,4-triazole), an intermediate HS-LS state is trapped at low temperature due to elastic strains. From this state the fully LS-LS or HS-HS states can be reached by irradiation at 830 nm and 510 nm, respectively. Moreover, a hidden hysteresis loop was evidenced by permanent light irradiation at 830 nm between the HS-LS and LS-LS states. [30] To use such strategy, materials exhibiting reverse-LIESST effect are more suitable. This makes the Hoffman phase analogs very promising. Indeed, such systems can be highly functionalized to tune the rigidity/flexibility competition [31][32][33][34] and to exhibit LIESST and reverse-LIESST effects. [29a,31] In that sense, we present here as a continuation of our work on such systems [30] a new paramagnetic Hofmann-like phase that does not exhibit any thermal spin crossover. Using the reverse-LIESST approach we have evidenced the hidden hysteresis loop from a purely HS to a purely LS state. From low temperature photo-crystallography we have correlated this observation to the evolution of the structure under light irradiation.

Results and Discussion

Synthetic strategy.

From our previous work on [Fe(2-pytrz){Pt(CN)4}].3H2O network 1, [30] we have shown that its structural topology consists of two dimensional {FePt(CN)4} sheets connected to each other by strong hydrogen bonds mediated by three non-coordinated water molecules. In this particular system, the 2D sheets are corrugated due to the presence of two different iron(II) sites and the water molecules are mainly organized around only one of these sites. From crystallographic analysis, we have shown that this iron(II) ion does not exhibit any SCO and remains trapped in the HS state. It appeared that this site was the most constrained site. However, this HS ion was converted into its LS state by irradiation with nearinfrared light through the reverse-LIESST effect, evidencing the occurrence of a hidden hysteresis loop.

Following this concept, we propose that favoring and strengthening the hydrogen bonding network would possibly end up with a fully constrained network and a purely HS material that can be erased using the reverse-LIESST effect. For that purpose, we have designed a new triazole-based ligand possessing a hydroxyl group to promote hydrogen bonds with water molecules. This approach has been previously used to strengthen the hydrogen bonding network using saltrz ligand ((E)-2-(((4H-1,2,4triazol-4-yl)imino)methyl)phenol). [29a] This ligand connects a phenol ring to a triazole moiety by an imine function. In order to have a shorter connection between the phenol and the triazole and to enhance the rigidity of the ligand, the 4-(3-hydroxyphenyl)-1,2,4-triazole (3phOH-trz, scheme 1)) was synthesized as described in the experimental section. Small colorless square crystals of the [Fe(3phOH-trz){Pt(CN)4}].2H2O (2) were obtained by slow diffusion. Except the bands attributed to the [Pt(CN)4] 2-anion (Figure S1 and S4), the IR spectrum of the title compound is similar to that measured for the free 3phOH-trz ligand. The blue shift of the characteristic intense C=N stretching band νCN observed at 2169 cm -1 , which was located at 2133 and 2122 cm -1 in K2[Pt(CN)4].xH2O, suggests the μ4-bridging coordination mode of the [Pt(CN)4] 2-moiety. Moreover, the νCN bands of the free triazole unit, observed at 1598 and 1202 cm -1 , are shifted to 1599 and 1211 cm -1 in 2, indicating its coordination to the Fe(II).

Structural investigations.

The crystal structure of 2 was solved and refined firstly at 296 K and 110 K then, according to the photomagnetic properties, complementary photo-crystallography was performed at 10 K. The light induced structural behaviour will be detailed in the following paragraphs.

Compound 2 crystallizes in the monoclinic P21/c space group. Crystallographic data and the structural parameters of the FeN6 octahedron are listed in Tables 1 and2, respectively. The crystal structure is built from one Fe(II) site and one half [Pt(CN)4] 2-anion both located on an inversion center at (½, 0, ½) and (0, ½, ½), respectively. One monodentate 3phOH-trz ligand and one water molecule, both located on general positions, complete the crystal structure (Figure 1). 8)° at 296 K and 110 K, respectively (Figure 3). In addition to the direct interactions through the NC-Pt-CN bridges, the elastic connection between the iron centers within the 2D planes is assisted by a direct H-bond interaction between a C-H from the triazole moiety and the OH group of an adjacent ligand and by indirect significant H-bonding via the water molecules between two triazole within the a direction (Figure 3 and table 3).

The 2D layers are stacked and interdigitated along the c direction, connected to each other by interlayer π-stacking interactions between the triazole moiety and the phenyl groups of 3phOH-trz ligands of successive layers along the b direction (Figure 4). Additionally, H-bonds between the hydroxyl group and 2) 62( 2) 72( 2) 57( 2)

1 1-X,1-Y,1-Z ;
a  is the sum of the deviation from 90° of the 12 cis-angles of the FeN6 octahedron [35] ; b  is the sum of the deviation from 60° of the 24 trigonal angles of the projection of the FeN6 octahedron onto its trigonal faces. [35] It is worth to note that the longest Fe-Fe distance ( 7.6149(2) Å at 293 K and 7.6230(2) Å at 110 K) of the Fe4 rectangles correspond to the intra-layer H-bonding interactions direction, whereas the shortest ones ( 7.3105(2) Å at 293 K and 7.2775(2) Å at 110 K) to the inter-layer π-stacking interactions (figures 2, 3 and 4). Due to the trans position of the hydroxyl group with respect to the free N atom of the triazole on the 3phOH-trz ligand, the water molecules are connected on both side of the ligand by H-bonding interactions leading to a symmetrical distribution of the solvent molecules around the ligands. These two latter observations constitute the main differences in the local crystal structure compared to compound 1, [Fe(2-pytrz){Pt(CN)4}].3H2O [30] . This parent compound displays two different Fe(II) centers leading to two different FeN6 environments for which only one of them exhibits a thermal SCO behaviour, while in compound 2, the crystal structure is built from only one iron(II) center. Moreover, 1 shows a different interactions network where water molecules were all located on the same side of the ligand, leading to an asymmetric distribution of the solvent molecules around the ligand. Additionally, no direct or indirect interactions were found between the ligands within the same layer, conversely to what is observed in 2. These observations explain (i) why the corrugation of the 2D network is much less pronounced in this case as the symmetrical distribution of the water molecules prevent the tilt of one iron center and (ii) why the pseudo-squares observed in the parent compound are much more distorted in 2 to reach a rectangular geometry as the presence of significant interactions between ligands in the same layer impose a longer distance within the a direction (Tables SI1 andSI2). These observations lead us to predict a paramagnetic behavior or a complete one-step SCO transition.

The coordination sphere around the Fe atom remains almost unchanged at 296 K and 110 K and adopts a slightly distorted octahedral geometry involving an FeN4 equatorial plane arising from four nitrogen atoms of the [Pt(CN)4]2-anions and two N atoms in axial positions from the 3phOH-trz ligands. The mean Fe-N bond length -2.169 Å and 2.159 Å at 296 K and 110 K respectivelyrange close to those expected for a iron(II) cation in the HS state. The octahedron distortion is small for a HS iron(II) at both temperatures as is usually observed for "Hoffman like" spin crossover compounds (table 2). [33] Nevertheless the present octahedron distortion is more pronounced than other 2D SCO "Hoffman like" compounds [34] . Interestingly, compared to the parent compound 1, the iron site distortion is closer to the Fe2 site that did not undergo the thermal spin transition and higher than those observed for the others hydrated triazole-based 2D SCO Hofmann-like networks. [29] . Paez-Espejo et al. [26] introduced an elastic model to explain the conditions of occurrence of multistep transitions in 2D spin crossover solids, as a function of a frustration parameter ξ. For the present compound (2), the frustration parameter ξ can be evaluated slightly higher than that calculated for 1 (ξ ~ 0.2 for 2 and ~ 0.1 for 1) or identical if the model chosen in the case of compound 2 is rectangular (ξ ~ 0.1). For such ξ values one-step transitions are expected in both cases. However, it was already suggested that the additional square distortion (square distortion Sd of the pseudo-squares as defined in legend of table 4) should play a significant role on the spin transition leading to a two-step transition for the parent compound . [30] It possibly precludes here the SCO of the compound 2 as the "pseudo-square" around the iron site is strongly distorted. Using the same parameter Sd [30] the square distortion of compound 2 appears high at 296 K and increases at 110 K. This value is significantly higher than the

D-H┈A d(H-A) d(D-A) D-H-A d(H-A) d(D-A) D-H-A d(H-A) d(D-A) D-H-A d(H-A) d(D-A) D-H-A

C2-H2┈O1 [29d] derivative that does not undergo any SCO on the whole studied temperature range. This higher distortion of compound 2 could result in stronger elastic frustration that may quench the spin transition in the present case. It seems that square distortion greater than 4-5 %, also linked to higher iron octahedron deformation, could preclude the spin transition on the corresponding site. It is worth noting that, within this family, when several iron sites are present, the less distorted site undergoes the SCO firstly when cooling down, leading to an increase of the square distortion of the second site. When this distortion becomes too high, the SCO may be quenched on the second site as it is observed for compound 1.

Magnetic and photomagnetic properties.

The magnetic properties of 2 were measured on a polycrystalline sample from 300 K to 10 K. Figure 5 reports the thermal evolution of χMT where χM is the magnetic molar susceptibility and T the temperature. At 300 K, the χMT value of 3.96 cm 3 K mol -1 is consistent with a HS iron(II) ion with a g value of 2.3. Upon cooling, the χMT product remains almost constant down to 50 K. Below this temperature a faster decrease is observed, probably due to the zero-field splitting of the paramagnetic iron(II) ion. [START_REF] Kahn | Molecular Magnetism[END_REF] This behavior is completely reversible upon warming. The absence of SCO behavior for 1 fulfils to our goal that was to inhibit it by elastic strains. Compare to the compound from Sciortino et al. [29a] that reported a 4-steps SCO using a phenol substituted triazole ligand, the absence of any SCO in 2 could be explained by a more rigid ligand used in our case.

The reverse-LIESST effect was then attempted on the sample using a 830 nm irradiation. Figures 5 and6 report the experiment performed at 10 K. A decrease of χMT is recorded upon irradiation down to 0.04 cm 3 K mol -1 after one hour indicating a fully efficient reverse-LIESST in a relatively short time. From the generated photo-induced LS state, two experiments can then be performed.

The first one that can be performed is the photocycling of the sample. From the LS state, an irradiation at 510 nm induces the complete population of the HS state in less than one hour (Figure 6). The photoswitching properties of this material are then reversible, reaching at each step the fully saturated LS and HS states. In addition, the photoswitching time appears equivalent for the LIESST and the reverse-LIESST processes, which is unusual. Finally, the HS populated by a 510 nm irradiation exhibits the same paramagnetic behavior than the pristine compound.

The second experiment to be performed is to check the magnetic behavior of the photo-induced LS state. After red light irradiation, the light was switched off and the temperature was increased at 0.4 K/mn up to 120 K. Figure 5 clearly shows an abrupt LS to HS spin transition with a T1/2 of 105 K. In a certain way, this opens a 105 K large hysteresis since upon cooling the system remains HS. By analogy to other examples, [23,30,[START_REF] Hinek | [END_REF] this curve should reflect the spin crossover behavior of the LS state. This behavior is only accessible by light irradiation since the HS state exhibits an extremely long lifetime. This can be evidenced by a comparison of the time dependence of χMT at 50 K with and without irradiation (Figure SI5). During seven hours in the dark, the system retains its HS state, without any evidence of signal decrease while in two hours of irradiation at 830 nm, the fully LS state is populated. This observation can be compared to the pressure-induced SCO phenomenon that has been observed in the [Fe II Pd-(CN)4(thiome)2]•2H2O [29d] that goes from HS to a complete 2-steps SCO under pressure. The long-lived HS state lets us to suspect the presence of a hidden hysteresis loop whose upper branch would correspond to the measured LS → HS transition curve. To probe the existence of this hysteresis loop we have performed the LITH measurements. [19a,38] This approach was used to reveal the hidden hysteresis in 1. [30] Upon constant light irradiation at 830 nm, the sample is heated from the photoexcited LS state at 10 K to the HS state at 110 K and then cooled down at 0.4 K/mn. It results in a non-equilibrium competition between the light-assisted population of the LS state and the thermal cooperative population of the HS state through the thermally induced spin crossover.

= |d(Pt•••Pt a ) -d(Pt•••Pt b )|/d(Pt•••Pt b ).
The LITH experiment of 2 is depicted in figure 7. It clearly shows the presence of a thermal hysteresis loop of 55 K width. Since such hysteresis is the result of an equilibrium between two antagonist pathways, it is very sensitive to the measurement timescale (temperature sweeping rate). [38,39] To have access to the true equilibrium (or the quasi-static) curve, several photostationary points were recorded at several temperatures along the cooling and warming branches. Under light irradiation, the temperature was set at the desired value, and the temporal evolution of mT under light was recorded. This procedure led to reach the photo-stationary states, from which an envelope of the quasi-static LITH loop (black curves) of 37 K width can be drawn.

The photostationary points at 100 K (Figure SI6) are illustrative of the presence of a bistable region under light irradiation: from the pure LS state χMT reaches a value of 2.18 cm 3 K mol -1 while it remains constant at 3.64 cm 3 K mol -1 from the cooling branch. This bistable region spreads itself over 37 K.

Regarding the quasi-static LITH, the warming branch under irradiation in figure 7 is close to the warming branch without irradiation (see Figure 5). This weak difference strengthens the argument that the heating branch of the reverse-LITH curve describes the thermally-induced spin transition of a hidden stable LS state towards the HS phase around 100 K. On cooling from the HS state, near-infrared irradiation favored the occurrence of the reversible HSLS transition around 55 K. This closed thermal hysteresis loop is related to the transition of iron center, while in the dark the system stays trapped in the HS state. This reveals the presence of a hidden thermal hysteresis which was not accessible by classical thermal ways.

Recently, a 2D paramagnetic {[Fe(bbtr)3](BF4)2} sample was evidenced to exhibit a hidden hysteresis revealed by reverse-LIESST effect as well as a light-induced bistability using selective wavelengths. [40] The hidden hysteresis curve was suspected from the presence of an instability curve, typical of hysteretic systems. The experimental envelop of this hysteresis was, however, not recorded, and a crystallographic phase transition was evidenced. With our current study, we produce a new example with bidirectional selective photoswitching in the class of rare materials exhibiting this behavior, and overall we give chemical tools to promote such effects: elastic frustration is a parameter on which chemist can play to tune the properties of the resulting materials.

Photocrystallographic investigation.

In order to relate the frustration to a possible structural phase transition, photo-crystallographic experiments have been performed to explore the structural properties of 1 at 10K before and after irradiation at 850 nm. In both cases no crystallographic phase transition was observed and 1 crystallizes in the same system and space group than at room temperature, the crystal structure can then be described in the same way. At 10K after irradiation the unit cell volume decreases by 90 Å 3 , which represents ca. 7.4 % of the unit cell volume before irradiation suggesting a HS to LS transition of the Fe cation. The FeN6 geometry is significantly affected by the light irradiation. The mean Fe-N distances decrease from 2.157(2) Å before irradiation to 1.956(2) Å after with a more regular octahedral geometry (table 2). It evidences a full HS to LS transition by the reverse-LIESST process. At 10K in the HS state the distortion of the "pseudo-squares" geometry (Sd = 5.1%) is higher than those at 110 K and 296 K (table 4), which confirms that the distortion increases when the temperature is cooled down. The difference between the nature of the interactions within the a and b directions (strong intra-layer H-bonding in the a direction and essentially interlayer π-stacking in the b direction) may explain this anisotropic structural behavior. The intermolecular interactions are significantly affected by the light induced HS to LS transition at 10K whereas their modifications due to the temperature variation (296K to 10K in the HS state) are comparatively small, especially concerning the H-bond intra-layer network (table 3). These modifications, in addition to the change of the spin state, lead to a more distorted pseudo-squares at 10K in the LS state (Sd = 8.1%), that should increase the elastic frustration of the lattice.

Conclusions

A new Hoffman-type two-dimensional network [Fe(3phOHtrz){Pt(CN)4}].2H2O was synthesized specifically to promote a strong hydrogen bonds network that we previously suspected to be responsible of the inhibition of the SCO behaviour. This effect was here confirmed, leading to a completely paramagnetic compound that remains HS down to 10 K. This compound consists of iron centers covalently linked by [Pt(CN)4] 2-anions giving a 2D coordination polymer within slightly corrugated layers formed by a succession of Fe4 perfect rectangles. Compared to the parent [Fe(2pytrz){Pt(CN)4}].3H2O compound 1, the elastic frustration is higher in 2, leading to a complete inhibition of the spin crossover. From the HS state, the reverse-LIESST process was applied to populate the fully LS state at 10 K. Upon near infrared irradiation no particular structural changes occurred (except the usual unit cell volume change) and the network remained always constituted of one iron(II) crystallographic site. This LS state can be converted back to the HS state through a green light irradiation (510 nm) or upon warming. Indeed, heating in the dark the photo-induced LS state revealed a thermal LS to HS transition at 105 K. When the near infrared irradiation (830 nm) is applied while the sample is thermally cycled, a hidden hysteresis loop of 37 K width is revealed with an upper transition temperature equal to that of the dark process, suggesting that the LS state might be the true thermodynamically stable state of the system. This study evidences that the constraints brought by the water molecules network leads to a strong elastic frustration that inhibits the spin crossover behaviour. This complete structure-property relationship study gives important insights on the role of elastic frustration on the level of inhibition of the spin crossover. This gives important trends to target wavelength selective photoswitching between two stable states, which is highly favourable compared to the current LIESST process that involves a metastable state with a given lifetime.

Experimental Section

Methods. The starting materials 1,2-diformylhydrazine, aminophenol and iron(II) perchlorate hydrate were purchased from Sigma Aldrich and used without further purification. Solvents were used and purified by standard procedures. IR spectra were obtained with a Vertex 70 BRUKER spectrometer equipped with a platinum ATR accessory. NMR spectra were recorded on a Bruker Advance 400 MHz, J values are given in Hz. [41] The reaction was carried out under nitrogen atmosphere. 1,2-diformylhydrazine (3.0g, 34 mmol.) and aminophenol (3.75g, 34 mmol.) with a small amount of hydroquinone were heated at 170°C for 30 mn. Then, the mixture was cooled down to 70°C, 20 mL of freshly distilled ethanol was further added and the mixture was stirred for 5 mn. It was then cooled down to 20°C and a volume of 100 mL of diethylic ether was added. The mixture was then kept at -20°C overnight. The crude product was collected after filtration as a dark brown powder, washed with ether and dried under vacuum. (yield 2.93g, 18 mmol, 53 %). 1 H NMR ((CD3)2SO), 400 MHz) δ: 6.84 (dd, 1H, 3 JH-H = 2.4 Hz, [START_REF]Spin Crossover in Transition Metal Compounds I-III[END_REF] JH-H = 2Hz), 7.04 (m, 2H), 7.17 (t, 1H, [START_REF]Spin Crossover in Transition Metal Compounds I-III[END_REF] JH-H = 8 Hz), 9.05 (s, 2H), 10.05 (broad, 1H) ppm. 13 C NMR ((CD3)2OD, 75 MHz) δ: 110. 06, 112.55, 113.63, 132.02, 136.20, 143.15, 160.36. (cm -1 ): 3106(w), 3064(w), 2965(w), 2861(w), 2838(w), 2727(w), 2662(w), 2627(w), 1667(m), 1598(s), 1529(s), 1493(s), 1389(m), 1287(s), 1248(m), 1203(s), 1164(m), 1091(m), 1030(w), 995(w), 860(s), 831(w), 783(s), 685(s), 645(w), 607(m), 553(m), 504(m), 467(w), 458(m).

[Fe(3phOH-trz)2{Pt(CN)4}].2H2O. The ligand L1 (20 mg, 0.10 mmol.) was solubilized in 10 mL of dry ethanol. A solution of potassium tetracyanoplatinate (26 mg, 0.06 mmol.) in 5 mL of water was slowly poured into this solution. The mixture was stirred at r.t. for 15 mn. 2 mL of this solution was then transferred into a thin glass tube. Next, a freshly prepared aqueous solution (6 mL) of Fe(ClO4)2, 6H2O (15 mg, 0.01 mmol.) was slowly layered on the top of the tube for slow diffusion. After several days, thin colorless plates appeared at the interface and were collected in an amount sufficient for the structural and physical studies. Anal. Calcd for C20H18FeN10O4Pt: C, 33.7; H, 2.5; N, 19.6 %. Found: C, 33.5; H, 2.6; N, 19.4 %. IR (ATR), (cm -1 ): 3620(w), 3186(w), 3115(w), 2992(w), 2169(s), 1599(s), 1531(s), 1494(s), 1366(w), 1278(s), 1241(w), 1211(s), 1167(w), 1096(m), 1043(m), 1006(w), 981(w), 899(w), 869(s), 858(s), 778(s), 669(s), 600(w), 533(w), 500(w), 478(w), 454(m).

Magnetic and Photomagnetic Studies.

Magnetic susceptibility data were collected using a Quantum Design MPMS 5 SQUID magnetometer under an applied field of 1 T. All measurements were performed on homogenous polycrystalline samples and in the temperature range 10 to 290 K. Diamagnetic corrections for the sample holder and the material (using Pascal constants) were applied. Photomagnetic measurements were performed with a set of photo-diodes coupled through an optical fiber to the cavity of a MPMS-55 Quantum Design SQUID magnetometer operating at 2 T. The intensity of irradiation was set to prevent heating of the sample. The crystalline sample was prepared in a thin layer (0.66 mg) to promote full penetration of the irradiated light. Our previously published standardized method for obtaining LIESST data was followed. [19,20] The sample was first slow cooled to 10 K, ensuring that potential trapping of HS species at low temperatures did not occur. Irradiation was carried out at a set wavelength and the power at the sample surface was adjusted to 5 mW cm -2 . Once photo-saturation was reached, irradiation was ceased and the temperature increased at a rate of 0.4 K.min -1 to ~100 K and the magnetization measured every 1 K to determine the T(LIESST) value given by the extreme of the δ χMT /δT versus T curve for the relaxation. The T(LIESST) value describes the limiting temperature above which the light-induced magnetic high-spin information is erased in a SQUID cavity. In the absence of irradiation, the magnetism was also measured over the temperature range 10 -290 K to follow the thermal spin transition and to obtain a low temperature baseline.

Crystallographic and photo-crystallographic investigations:

Single crystal X-ray studies were performed at 296 K and 110 K on Xcalibur 2 κ-CCD diffractometer using Mo Kα radiation (λ = 0.71073 Å). The corresponding structures were solved by direct methods with the SHELXS program and refined on F 2 by weighted full matrix least-squares methods using the SHELXL program. [42] . All non-hydrogen atoms were refined anisotropically, hydrogen atoms were located in difference Fourier maps and treated using a riding model. Crystallographic data and refinement details are provided in Table 1.

10K single crystal diffraction data were collected on a Microfocus Supernova diffractometer equipped with a two dimensional ATLAS detector, using Mo Kα radiation, and a Helijet He open flow cryosystem. In situ photo-excitations were carried out using a 850 nm diode laser (duration 10 min), until the photo-stationary state was reached to populate the LS state by reverse-LIESST. The single crystal sample was rotated continuously during excitation to ensure a homogeneous and complete excitation. The corresponding structures were solved by direct methods with the SHELXS program and refined on F 2 by weighted full matrix leastsquares methods using the SHELXL program. All non-hydrogen atoms were refined anisotropically, hydrogen atoms were located in difference Fourier maps and treated using a riding model.

CCDC

1574585-1574588 contain the supplementary crystallographic data. These data can be obtained free of charge from The Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/data_requ-est/cif.

Scheme 1 :

 1 Scheme 1: synthesis of the 4-(3-hydroxyphenyl)-1,2,4-triazole.

Figure 1 .

 1 Figure 1. Thermal ellipsoid drawing (30% probability ellipsoids) of the asymmetric unit of 2 at 296 K showing the labeling scheme.

Figure 3 .

 3 Figure 3. View along the [0, 1, 0] direction at ambient temperature showing the FeN4 equatorial plane slightly tilted with respect to the [Pt(CN)4] 2-plane (P1-P2 angle = 7.46(8)°). i 1+x,y,z ; ii -1+x,y,z ; iii 2-x,2y,1-z ; iv 1-x,2-y,1-z ; v -x,2-y,1-z

Figure 2 .

 2 Figure 2. View of the 2D arrangement of 2 at 296 K: projection along the c direction showing the Fe4 pseudosquares.

Figure 4 .

 4 Figure 4. 3D crystal packing of 2 at ambient temperature along the [0, 1, 0] direction showing the intermolecular interactions and the quasi inexistent corrugation. i -1+x,y,z ; ii 1-x,-½+y,½-z

Figure 6 :Figure 5 :

 65 Figure 6: Photocycling of the sample 2 upon alternative irradiation at 830 nm () and 510 nm (∆).

Figure 7 :

 7 Figure 7: Thermal dependence of χMT under permanent light irradiation at 830 nm (). The up and down red triangles stand for the isothermal kinetics of the HS fraction under light recorded at different temperatures in order to drawn the envelope of the quasi-static LITH loop (black curve) made of photostationnary states.

  synthesized according to the procedure previously published byElokhina et al.

Table 2 .

 2 Selected bond lengths and distortion parameters of the coordination sphere of compound 2.

	Atoms		distance / Å
		296 K HS	110 K HS	10 K HS	10 K LS*
	Fe1-N1	2.197(2)	2.191(2)	2.187(2)	1.989(2)
	Fe1-N4	2.158(3)	2.137(2)	2.142(2)	1.936(2)
	Fe1-N51	2.152(3)	2.148(2)	2.140(2)	1.944(2)
	<Fe1-N>	2.169(3)	2.159(2)	2.157(2)	1.956(2)
			Distortion / °

a Σ (°) 24(1) 26(

1

) 27(

1

) 22(1) b Θ (°) 59(

Table 1 .

 1 Crystallographic data for compound 2

		HS	HS	HS	LS
	T / K	296	110	10	10
	Color	colorless	colorless	colorless	pink
	a Chemical formula	C20H18FeN10O4Pt	C20H18FeN10O4Pt	C20H18FeN10O4Pt	C20H18FeN10O4Pt
	Formula weight	713.38	713.38	713.38	713.38
	Crystal system	Monoclinic	Monoclinic	Monoclinic	Monoclinic
	Space group	P21/c	P21/c	P21/c	P21/c
	a/Å	7.6149(2)	7.6230(2)	7.6326(3)	7.4562(3)
	b/Å	7.3105(2)	7.2775(2)	7.2627(3)	6.8994(3)
	c/Å	22.3832(7)	22.2303(9)	22.1577(9)	22.1699(9)
	β/°	98.137(3)	98.048(3)	98.071(4)	99.107(4)
	Volume/Å 3	1233.5(1)	1221.1(1)	1216.1(1)	1126.1(1)
	Z	2	2	2	2
	ρcalc g/cm 3	1.921	1.946	1.954	2.110

Table 3 .

 3 Intermolecular interactions of compound 2 (Å, °)

	T / spin state	293 K / HS	110 K / HS	10 K / HS	10 K / LS
			H-bonds interaction

stacking interactions Planes angle a distance b shift c angle a distance b shift c angle a distance b shift c angle a distance b shift c

	value obtained for 1 (table 4) and is always greater than those
	observed for others hydrated triazole based 2D "Hoffman like"
	compounds except in case of the [Fe II Pd-(CN)4(thiome)2]•2H2O
	(thiome	=	4-[(E)-2-(5-methyl-2-thienyl)
	vinyl]-1,2,4-triazole)
	2	2.32 3.250(5) 177	2.29	3.219(4) 176	2.28 3.207(3) 176	2.24 3.172(3) 177
	C1 H1┈OW1 1	2.26 3.180(4) 170	2.25	3.176(3) 171	2.26 3.180(3) 172	2.22 3.105(3) 159
	OW1-HW1A┈N2 2.01 2.778(4) 157	1.93	2.761(3) 165	1.94 2.756(3) 160	1.92 2.719(3) 163
	O1-H1A┈OW1 3	1.95 2.745(4) 172	1.91	2.723(4) 171	1.91 2.721(3) 172	1.87 2.687(3) 171
	π-plane1 / plane2 4 5.261 3.973 1.710 5.426	3.955	1.712	5.324	3.936	1.701	4.321	3.813	1.615
	plane1 / plane2 3	5.261	3.660 1.081	5.426	3.622	1.045	5.324	3.614	1.033	4.321	3.579	1.313

Table 4 .

 4 Fe•••Fe and Pt•••Pt distances (Å) and squares distortion (Sd, %) in the pseudosquares within the 2D layers

		293 / HS	110K / HS 10K / HS	10K / LS
			distance / Å
	Fe…Fe a	7.6149(2)	7.6230(2)	7.6326(3)	7.4562(3)
	Fe…Fe b	7.3105(2)	7.2775(2)	7.2627(3)	6.8994(3)
	Pt…Pt a	7.6149(2)	7.6230(2)	7.6326(3)	7.4562(3)
	Pt…Pt b	7.3105(2)	7.2775(2)	7.2627(3)	6.8994(3)
	Sd c	0.042	0.047	0.051	0.081

a within the (100) crystallographic direction, b within the (010) crystallographic direction, c Sd is defined as the difference of both sides of the pseudosquare around Fe divided by the shortest side of the pseudo squares. This definition gives Sd

1+X,+Y,+Z;

-1+X,+Y,+Z;

1-X,1/2+Y,1/2-Z ;

1-X,-1/2+Y,1/2-Z ; a Angle between planes ; b Centroid-Centroid distance ; c Shift distance ; Plane1=C3C4C5C6C7C8 ; Plane2=N1C1N3C2N2.

Acknowledgments

We thank the CNRS, the Universities of Brest, Paris-Saclay Lorraine, Versailles and Bordeaux, the "Agence Nationale de la Recherche" (ANR project BISTA-MAT: ANR-12-BS07-0030-01) and "Région Nouvelle Aquitaine" for the funding of this work.


Aris Daniilidis

AND Olivier Ley

Stéphane Éphane Sabourau

ASYMPTOTIC BEHAVIOUR OF SELF-CONTRACTED PLANAR CURVES AND GRADIENT ORBITS OF CONVEX FUNCTIONS

Keywords: Mathematics Subject Classification. Primary 37C10 ; Secondary 34D20, 37B35, 37N40, 52A10 Planar dynamical system, gradient trajectory, convex function, convex foliation, Lojasiewicz inequality

published or not. The documents may come

Introduction

This work is mainly devoted to the study of the length of bounded trajectories of the gradient flow of convex (or quasiconvex) functions in the plane. The motivation for this study comes from a well-known result due to S. Lojasiewicz (see [START_REF] Lojasiewicz | Sur la géométrie semi-et sous-analytique[END_REF]), asserting that if f : R n → R is a real-analytic function and x ∈ f -1 (0) is a critical point of f , then there exist two constants ρ ∈ [1/2, 1) and C > 0 such that

||∇f (x)|| ≥ C |f (x)| ρ (1.1)
for all x belonging to a neighborhood U of x. An immediate by-product is the finite length of the orbits of the gradient flow of f lying in U . The proof is straightforward using (1.1) and is illustrated below: let γ : [0, +∞) → U be a gradient trajectory of f , that is, γ(t) = -∇f (γ(t)). Then,

- 1 1 -ρ d dt f (γ(t)) 1-ρ = -γ(t), ∇f (γ(t) f (γ(t)) -ρ = || γ(t)|| 2 f (γ(t)) -ρ ≥ C || γ(t)||, yielding length(γ) = +∞ 0 || γ(t)||dt < +∞. (1.
2)

The aforementioned inequality (1.1) has been extended by K. Kurdyka in [START_REF] Kurdyka | On gradients of functions definable in o-minimal structures[END_REF] for C 1 functions belonging to an arbitrary o-minimal structure (we refer to [START_REF] Van Den Dries | Geometries categories and o-minimal structures[END_REF] for the relevant definition), in a way that allows us again to deduce the finiteness of the lengths of the gradient orbits in this more general context. In [START_REF] Bolte | The Lojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems[END_REF] and [START_REF] Bolte | Clarke subgradients of stratifiable functions[END_REF], a further extension has been realized to encompass (nonsmooth) functions and orbits of the corresponding subgradient systems.

It should be noted that in the above cases the functions enjoy an important structural property (o-minimality) and that, for general functions, there is no hope to prove a result like (1.2). A classical example of J. Palis and W. De Melo ( [16, page 14]) asserts that the bounded trajectories of the gradient flow of an arbitrary C ∞ function need not converge (in particular, they are of infinite length). In the aforementioned example the critical set of the function is not reduced to a singleton: in Section 7.1, we provide another example of a smooth function having a unique critical point towards which all corresponding orbits converge, but again are of infinite length.

The case when f is a convex coercive function is particularly interesting in view of its potential impact in numerical optimization (see [START_REF] Absil | Convergence of the Iterates of Descent Methods for Analytic Cost Functions[END_REF], [START_REF] Bolte | The Lojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems[END_REF], [START_REF] Bolte | Characterizations of Lojasiewicz inequalities and applications, 57p[END_REF], for example). But convex functions are far from being analytic and they do not satisfy neither the Lojasiewicz inequality nor its generalized form established by Kurdyka, unless a growth condition is assumed (see [START_REF] Bolte | Characterizations of Lojasiewicz inequalities and applications, 57p[END_REF] for a sufficient condition and a counter-example). Nevertheless, their rigid structure makes it natural to think that the orbits of their gradient flow are of finite length. It is rather surprising that the answer of this question is not yet known in the literature except in some particular cases.

Let us mention that in the framework of Hilbert spaces, this has been stated as an open problem by H. Brézis [START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF]Open problems,p. 167]. In infinite dimension, R. Bruck [START_REF] Bruck | Asymptotic convergence of nonlinear contraction semigroups in Hilbert space[END_REF] proved that the (sub)gradient orbits of convex coercive functions are converging towards a global minimizer of f but this convergence holds only with respect to the weak topology. Indeed, B. Baillon [START_REF] Baillon | Un exemple concernant le comportement asymptotique de la solution du problème du/dt + ∂ϕ(u) ∋ 0[END_REF] constructed a counterexample of a lower semicontinuous convex function f in a Hilbert space whose gradient orbits do not converge for the norm topology. A straightforward consequence is that these orbits have infinite length. Concurrently, there are some cases where a convex coercive function f : H → R is known to have (sub)gradient orbits of finite length. This is true when the set of minimizers of f has nonempty interior in the Hilbert space H (see H. Brézis [START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF]), or whenever f satisfies a growth condition. For a detailed discussion and the proofs of these facts, we refer to [START_REF] Bolte | Characterizations of Lojasiewicz inequalities and applications, 57p[END_REF].

The aforementioned results do not cover the simplest case of a convex smooth function defined in the plane and having a unique minimum. One of the main results of this work is to prove the following: Theorem 1.1 (Convex gradient system). Let f : R 2 → R be a smooth convex function with a unique minimum. Then, the trajectories γ of the gradient system

γ(t) = -∇f (γ(t))
have a finite length.

The proof of this result does not use the whole convexity of f but, instead, rather relies on the convexity of its level-sets. More precisely, the conclusion of Theorem 1.1 remains also true for the orbits of the gradient flow of a quasiconvex function (see Corollary 6.3).

Actually, both results will follow as consequences of a much more general result (Theorem 1.3) about bounded self-contracted planar curves, which allows us to provide a unified framework for this study.

Definition 1.2 (Self-contracted curve). A curve γ : I → R n defined on an interval I of [0, +∞) is called self-contracted, if for every t 1 ≤ t 2 ≤ t 3 , with t i ∈ I, we have dist(γ(t 1 ), γ(t 3 )) ≥ dist(γ(t 2 ), γ(t 3 )). (1.3) In other words, for every [a, b] ⊂ I, the map t ∈ [a, b] → dist(γ(t), γ(b)) is nonincreasing.
We prove the following.

Theorem 1.3 (Main result). Every bounded continuous self-contracted planar curve γ is of finite length. More precisely, length(γ) ≤ (8π + 2) D(γ) where D(γ) is the distance between the endpoints of γ.

Let us finally mention that, even if gradient orbits of convex functions have finite length in the plane, they do not enjoy all the properties of the gradient orbits of real-analytic functions. Indeed, on the one hand, the so-called Thom conjecture for the gradient orbits of real-analytic functions holds true: if x ∞ denotes the limit of the orbit γ(t), then the secants (γ(t)-x ∞ )/||γ(t)x ∞ || converge towards a fixed direction of the unit sphere (see K. Kurdyka, T. Mostowski and A. Parusinski [START_REF] Kurdyka | Proof of the gradient conjecture of R. Thom[END_REF]). On the other hand, as we show in Section 7.2, an analogous result fails in the convex case. Indeed, the orbits of a convex gradient flow may turn around their limit infinitely many times.

Our techniques only work in the two-dimensional case. We do not know whether Theorem 1.1 and Theorem 1.3 hold in greater dimension.

The article is organized as follows. In Section 2, we present basic properties of self-contracted curves. In Section 3, we decompose each polygonal approximation of a bounded self-contracted curve in an annulus centered at its endpoint into horizontal and vertical segments. We establish upper bounds on the total length of the vertical segments in Section 3 and on the total length of the horizontal segments in Section 4. The proof of the main result is presented in Section 5. In Section 6, we show that the orbits of various dynamical systems are self-contracted curves. Two (counter)-examples are presented in Section 7.

Notations. Throughout the manuscript, we shall deal with the finite-dimensional Euclidean space R n equipped with the canonical scalar product •, • . We denote by • (respectively, dist(•, •)) the corresponding norm (respectively, distance). Therefore, the distance between two points x and y of R 2 will be denoted by xy , dist(x, y) or sometimes |xy|. We also denote by dist (x, S) the distance of a given point x ∈ R n to a set S ⊂ R n , by B(x, r) the closed ball with center x ∈ R n and radius r > 0 and by S(x, r) its boundary, that is, the sphere of the same center and the same radius. For 0 < r < R, we denote by

U (r, R) := {x ∈ R n | r < x ≤ R}
(1.4) the annulus centered at the origin O with outer radius R and inner radius r and by ∆R = Rr its width. Let [p, q] := {p + t(qp) | t ∈ [0, 1]} be the closed segment with endpoints p, q ∈ R n . A subset S of R n is called convex, if [p, q] ⊂ S for every p, q ∈ S.

Self-contracted curves

Throughout this paper, we shall deal with curves γ : I → R n , defined on an interval I of R. We recall that the length of a curve γ : I → R n is defined as

length(γ) = sup k i=1 dist(γ(t i ), γ(t i+1 ))
where the supremum is taken over all the finite subdivisions {t i } k+1 i=1 of I. We shall need the following definition. Definition 2.1 (Convergence of a curve). A curve γ : I → R n is said to converge to a point x 0 ∈ R n if γ(t) converges to x 0 when t goes to t + := sup I. A curve γ : I → R n is said to be bounded, if its image γ(I) is a bounded subset of R n .

We start with an elementary property of self-contracted curves. Proposition 2.2 (Existence of left/right limits). Let γ : I → R n be a bounded self-contracted curve and (a, b) ⊂ I. Then, γ has a limit in R n when t ∈ (a, b) tends to an endpoint of (a, b). In particular, every self-contracted curve can be extended by continuity to the endpoints of I (possibly equal to ±∞).

Proof. Since γ lies in a compact set of R n , there exists an increasing sequence {t i } in (a, b) with t i → b such that γ(t i ) converges to some point of R n , noted γ(b) + . Fix any i, j ∈ N * and let t i < t < t i+j . By (1.3), we have dist(γ(t), γ(t i+j )) ≤ dist(γ(t i ), γ(t i+j )).

Letting j go to infinity, we derive dist(γ(t), γ(b)

+ ) ≤ dist(γ(t i ), γ(b) + ) which gives γ(t) → γ(b) + .
Further, using the triangle inequality and the inequality (1.3), we have dist(γ(t 1 ), γ(t 2 )) ≤ dist(γ(t 1 ), γ(t 3 )) + dist(γ(t 3 ), γ(t 2 )) ≤ 2 dist(γ(t 1 ), γ(t 3 )).

Using this inequality, we can show, as previously, that γ(t) converges as (a, b) ∋ t → a.

The last part of the assertion is straightforward.

The following result is a straightforward consequence of Proposition 2.2.

Corollary 2.3 (Convergence of bounded self-contracted curves). Every bounded self-contracted curve γ : (0, +∞) → R n converges to some point x 0 ∈ R 2 as t → +∞. Moreover, the function t → dist(x 0 , γ(t)) is nonincreasing.

In the sequel, we shall assume that every self-contracted curve γ : I → R n is (defined and) continuous at the endpoints of I. (ii) A self-contracted curve might not be (left/right) continuous. A simple example is provided by the following planar self-contracted curve:

γ(t) =    (t, 1) if t ∈ (-∞, 0) (0, 0) if t = 0 (t, -1) if t ∈ (0, +∞) (iii) If t ∈ (a, b) → γ(t) is a self-contracted curve, then the curve t ∈ (a, b) → γ(a + b -t) is not necessarily self-contracted.
(iv) Corollary 2.3 reveals that the trajectories of a general gradient system

γ(t) = -∇f (γ(t)), γ(0) = x 0 ∈ R n
might fail to be self-contracted curves. Indeed in [16, page 14] an example of a C ∞ function f : R 2 → R is given, for which all trajectories of its gradient system are bounded but fail to converge.

(v) In Section 6, we show that whenever f is (quasi)convex, the gradient trajectories are selfcontracted curves. Thus, bounded self-contracted curves might fail to converge for the strong topology in a Hilbert space (see Baillon's example in [START_REF] Baillon | Un exemple concernant le comportement asymptotique de la solution du problème du/dt + ∂ϕ(u) ∋ 0[END_REF]).

From now on, we restrict ourselves to the two-dimensional case, and study the asymptotic behaviour of self-contracted planar curves.

Horizontal and Vertical directions

In this section, we introduce a binary-type division of planar segments into horizontal and vertical ones. We shall apply this decomposition for segments issued from polygonal line approximations of a bounded self-contracted curve. In this section, we derive an upper bound on the total length of the vertical segments, while in the next section we shall do the same for the total length of the horizontal ones. Combining both results we shall thus obtain an upper bound estimation on the total length of a bounded self-contracted curve, establishing Theorem 1.3.

Fix 0 < r < R and let U (r, R) be the annulus defined in (1.4). Let σ be a segment of U (r, R), not reduced to a point. Denote by p and q the endpoints of σ and by m its midpoint. Switching p and q is necessary, we can assume that q is closer to the origin O than p, that is dist(O, q) ≤ dist(O, p). Let Omq := θ be the angle between the vectors --→ mO and -→ mq, cf.

Fig. 1. Note that θ ∈ [-π 2 , π 2
] (by convention, inverse-clockwise angles are positive).

Lemma 3.1 (Segment length estimate). Let σ be a segment of U (r, R) with endpoints p and q such that θ = ± π 2 . Then

length(σ) ≤ 2 cos θ |dist(O, p) -dist(O, q)|.
Proof. Let p be the orthogonal projection of p to the line Om. Using elementary trigonometry in the right-angled triangle ppm, we derive

dist(m, p) = 1 2 cos θ • length(σ). Hence, dist(O, p) -dist(O, m) ≥ 1 2 cos θ • length(σ). Since dist(O, q) ≤ dist(O, m), the conclusion follows. O r R q θ m p Figure 1. Fix α ∈ (0, π
2 ), let λ ∈ (0, 1) be such that sin α < λ < 1, and set r := λR. Denote by

A := U (λR, R)
the corresponding annulus of (1.4), with width equal to ∆R = (1λ)R. We now introduce a crucial definition in the proof of our main result.

Definition 3.2 (Classification of the segments). Let α ∈ (0, π 2 ), λ ∈ (0, 1) and A as above. A nontrivial segment σ of A is said to be

• vertical, if θ lies in (-π 2 + α, π 2 -α) ; • horizontal, pointing in the positive direction, if θ lies in [-π 2 , -π 2 + α] ; • horizontal, pointing in the negative direction, if θ lies in [ π 2 -α, π 2
]. For instance, the segment [p, q] in Fig. 1 points in the negative direction. Definition 3.3 (Polygonal approximation). Let γ : I → R 2 be a continuous self-contracted planar curve converging to the origin O. We consider polygonal approximations {σ i } k+1 i=1 of γ in the annulus A, as follows: Let t 1 < t 2 < • • • < t k+1 be a sequence of points of I with γ(t i ) = γ(t i+1 ) such that the restriction of γ to [t 1 , t k+1 ] lies in A. Refining the subdivision if necessary, we can further assume that for every i ∈ {1, . . . , k} the segment σ i with endpoints p i = γ(t i ) and q i = p i+1 = γ(t i+1 ) lies in A and that the length of σ i is within any desired precision η > 0 of the length of γ |[t i ,t i+1 ] (this precision η > 0 will be defined at the beginning of Section 4 and will only depend on α, λ and R). Since the function t → dist(O, γ(t)) is nonincreasing (cf. Corollary 2.3), we can further assume that, if σ i := [p i , q i ] is vertical, then q i is the closest point of σ i to the origin. We denote by m i := 1 2 (p i + q i ) the midpoint of σ i . Remark 3.4. It is worth noticing that the polygonal approximation of a self-contracted curve introduced above is no more a self-contracted curve in general. Nevertheless, one still has

dist(p i , p l ) ≥ dist(p j , p l ) when 1 ≤ i ≤ j ≤ l ≤ k + 1,
which is the property we will use.

The total length of the vertical segments satisfies the following upper bound.

Lemma 3.5 (Total vertical length upper bound). If {σ i } k+1

i=1 is a polygonal approximation of γ in the annulus A, cf. Definition 3.3, then

i∈V length(σ i ) ≤ 2 cos α ∆R
where the sum is taken over all indices i ∈ V ⊂ {1, . . . , k + 1} corresponding to the vertical segments.

Proof. Let θ i denote the angle between

--→ m i O and --→ m i q i . Since i ∈ V, it follows that |θ i | < α, whence (cos θ i ) -1 < (cos α) -1 . From Lemma 3.1 (segment length estimation), we obtain i∈V length(σ i ) < 2 cos α i∈V dist(O, p i ) -dist(O, q i ). Since dist(O, p i ) > dist(O, q i ), for all i ∈ {1, . . . , k} we deduce i∈V dist(O, p i ) -dist(O, q i ) ≤ k i=1 dist(O, p i ) -dist(O, q i ).
Now, since q i = p i+1 , the right-hand term is equal to dist(O, p 1 )dist(O, q k ), which is less or equal to the width ∆R of A. The proof is complete.

Length estimate for horizontal directions

In this section, we keep the notations and the definitions from the previous section. In particular,

α ∈ (0, π 2 ), sin α < λ < 1, A := U (λR, R) (4.1)
and {σ i } k+1 i=1 is a polygonal approximation of γ in the annulus A, cf. Definition 3.3. We establish an upper bound on the total length of the horizontal segments issued from the polygonal approximation of γ.

Let x ∈ A. The distance from the origin to the half-line L x passing through x and making an angle α > 0 with -→ xO is equal to sin α • dist(O, x). Thus, the half-line L x intersects the circle S(0, λR) of radius λR centered at the origin at two points. These two points are noted π(x) and π ′ (x), with π(x) closer to x than π ′ (x). Furthermore, the half-line L x intersects A along two segments ∆ x and ∆ ′

x , where the endpoints of ∆ x agree with x and π(x), and one of the endpoints of ∆ ′

x agrees with π ′ (x). Note that min x∈A π(x)π ′ (x) > δ > 0. The half-line L x extends to a line which bounds a (closed) half-plane H x containing the origin of the plane. Denote by H c

x the (closed) half-plane with the same boundary as H x , not containing the origin (see Fig. 2 for an illustration of these notations). The mappings x → π(x) and x → π ′ (x) from A to S(0, λR) are clearly continuous, thus also uniformly continuous. Therefore, there exists η > 0 such that for every pair of points x, y ∈ A which are η-close from each other (i.e. dist(x, y) < η), we have

dist(x, π(y)) < dist(x, π ′ (y)), dist(π(x), π(y)) < dist(π(x), π ′ (x)), and dist(π(x), π(y)) < dist(π(x), π ′ (y)). (4.2)
We shall further need the following technical lemmas.

O x ∆ x π(x) π ′ (x) ∆ ′ x H c x L x H x y ∆ y S(0, R) S(0, λR) Figure 2.
Lemma 4.1 (Essential disjointness of ∆ x , ∆ y ). Let x and y be two distinct points of A such that dist(O, x) > dist(O, y). If y ∈ H x , then the segment ∆ y lies in H x . Furthermore, ∆ y does not intersect ∆ x , except possibly at y.

Proof. Suppose first that y lies in ∆ x . One easily checks that the angle Ozπ(x) increases when the point z moves from x to π(x) along ∆ x . In particular, the angle Oyπ(x) is greater than α. Therefore, the segment ∆ y lies in H x and meets ∆ x only at y.

Suppose now that y / ∈ ∆ x and (towards a contradiction) that ∆ y intersects ∆ x at z = y. Let y ′ denote the intersection point of ∆ x with the circle of radius |Oy| centered at the origin. Then, the image of ∆ y by the rotation around the origin taking y to y ′ does not lie in H x (the image of z should lie in H c

x ). On the other hand, this image agrees with ∆ y ′ since the rotation sends the ray Oy to Oy ′ . From the previous discussion, we conclude that the image of ∆ y is contained in H x . Hence a contradiction.

Finally, suppose that y / ∈ ∆ x , ∆ y ∩ ∆ x = ∅ and ∆ y intersects ∆ ′ x at z. If z = π ′ (x), then obviously ∆ y lies in H x . Suppose now that z = π ′ (x) (thus z = π(y)). Since the angle Ozπ(y) is positive while the angle Ozπ ′ (x) is negative, we obtain that π ′ (x)zπ(y) is positive which is not possible. The proof is complete. Lemma 4.2 (Injectivity of π). Let σ := [p, q] be an horizontal segment of A, with midpoint m, pointing in the positive direction. Assume dist(O, q) ≤ dist(O, p). Then, (1) the restriction of π to σ is injective;

(2) if the length of σ is at most η, then the circular arc π([p, m]) lies in H c m . Proof. Let x, y ∈ [p, q] with dist(p, x) < dist(p, y). Since the horizontal segment σ points in the positive direction, the angle xOy is positive and y lies in H x . From Lemma 4.1 (Essential disjointness of ∆ x , ∆ y ), π(x) and π(y) are distinct (the case y = π(x) is impossible since it would yield that the angle π(x)Ox = yOx is positive, a contradiction). Hence the first part of the lemma follows.

Let x ∈ (m, p]. From above, the midpoint m of σ lies in H x and the segments ∆ x and ∆ m do not intersect each other from Lemma 4.1. By definition of η, in view of (4.2) the points x and π(x) are closer to π(m) than to π ′ (m). Thus, the segment ∆ x , which does not intersect ∆ m , does not intersect ∆ ′ m either. That is, ∆ x lies either in H m or in H c m . Since the horizontal segment σ is pointing towards the positive direction, the point x belongs to H c m . Therefore, the same holds true for the other endpoint π(x) of ∆ x . It follows that the circular arc π([p, m]) with endpoints π(p) and π(m) is contained in H c m . Lemma 4.3 (Length estimate for horizontal segments). Let σ := [p, q] be an horizontal segment of A with midpoint m, pointing in the positive direction. Assume dist(O, q) ≤ dist(O, p). Then, When a point x, starting at m ′ , moves along C, the angle Oxp increases from less than π 2 to π. Thus, there exists a unique point m ′′ of C where the angle Om ′′ p is equal to π 2 + α. By definition of an horizontal segment pointing in the positive direction, the angle Omp lies between π 2 and π 2 + α. Therefore, the point m lies in C between m ′ and m ′′ , cf. Fig. 3. The angles Om ′′ p and Om ′′ π(m ′′ ) are equal to π 2 + α and α. Therefore, the ray pm ′′ makes a right angle at m ′′ with the line D ′′ passing through m ′′ and π(m ′′ ). Thus, the line D ′′ is tangent to C at m ′′ . This implies that the points O, m and m ′′ lie in the same half-plane delimited by the line D ′ passing through m ′ and parallel to D ′′ . Since the ray Om ′ makes an angle less than α with D ′ at m ′ , the angle between D ′ and L m ′ (the half-line passing through m ′ and making an angle α > 0 with Proof. Switching the indices i and j if necessary, we can assume that i < j. From Lemma 4.2, the arc π([p i , m i ]) is contained in H c m i . To prove the desired result, it is enough to show that π([p j , m j ]) lies in the complement of H c m i (i.e. the interior of H m i ). From the definition of a self-contracted curve, the points p j and q j are closer to q i than to p i (see Remark 3.4). Thus, p j and q j , and so their barycenter m j , lie in the half-plane delimited by the perpendicular bisector of σ i . (Notice that this half-plane also contains the origin O, in view of Corollary 2.3.) The half-line of this bisector with endpoint m i which makes an acute angle with the ray m i O is noted D m i . Since the horizontal segment σ i points in the positive direction, its half-bisector D m i makes an angle less or equal to α with m i O. Thus, L m i lies in the half-plane delimited by the perpendicular bisector of σ i not containing the origin. Now, since the function t → dist(O, γ(t)) is nonincreasing, the points p j and q j , and so their barycenter m j , belong to the disk of radius |Oq i | < |Om i | centered at the origin. Therefore, the points p j and m j lie in H m i , and dist(O, m i ) > max{dist(O, p j ), dist(O, m j )}. From Lemma 4.1 (Essential disjointness of ∆ x , ∆ y ), the segments ∆ p j and ∆ m j lie in H m i and do not intersect its boundary. Therefore, their endpoints π(p j ) and π(m j ) also lie in the interior H m i .

length(σ) ≤ 2 λ length(π([p, m])).
--→ m ′ O) is positive, cf.
The total length of the horizontal segments satisfies the following upper bound. Proof. From Lemma 4.3 (Length estimate for horizontal segments), the sum of the lengths of the horizontal segments σ i pointing in the positive direction satisfies

O m i p i q i D m i L m i S(0, λR)
i∈H + length(σ i ) ≤ 2 λ i∈H + length(π([p i , m i ])),
where H + is the set of indices corresponding to the horizontal segments pointing in the positive direction. Since the arcs π([p i , m i ]) of the circle S(0, λR) are disjoint, cf. Lemma 4.4, we have

i∈H + length(π([p i , m i ])) ≤ 2πλR.
Analogous arguments lead to a similar bound for the sum of the lengths of the horizontal segments σ i pointing in the negative direction. Recalling that ∆R = (1λ)R the result follows.

Proof of the main result

In order to prove our main theorem (cf. Theorem 1.3), we shall first need the following result. Proof. Consider a decomposition of γ into segments σ i as in Definition 3.3 (refining a subdivision does not decrease the sum). From Lemma 3.5 (Total vertical length upper bound) and Lemma 4.5 (Total horizontal length upper bound), the length L of the polygonal line

σ 1 σ 2 • • • σ k satisfies L = i∈H length(σ i ) + i∈V length(σ i ) ≤ 8π 1 -λ + 2 cos α ∆R.
By taking the supremum of L over all such decompositions respecting the annulus A, we derive the same upper bound for the length of γ ∩ A. Finally, by letting α and λ go to zero, we obtain length(γ ∩ A) ≤ (8π + 2) ∆R. Now, we can derive our main result.

Proof of Theorem 1.3. From Corollary 2.3, the bounded continuous self-contracted curve γ converges to a point. Using a translation if necessary, we can assume that this point agrees with the origin O of R 2 . Let t -= inf I. Denote by γ(t -) the limit of γ(t) when t goes to t -, (cf. Proposition 2.2 (Existence of left/right limits)). Set R 0 = dist(O, γ(t -)). For i ∈ N, let A i be the planar annulus centered at the origin with outer radius R i and inner radius R i+1 , where R i+1 = λR i , with λ ∈ (0, 1) given in (4.1). From Proposition 5.1 (Length estimate in the annulus), we have

length(γ ∩ A i ) ≤ (8π + 2) ∆R i (5.1)
where ∆R i is the width of A i . Since λ < 1, the sequence R i goes to zero and the sum of the width of the disjoint annulus A i is equal to R 0 . Thus, taking the sum of the above inequalities (5.1) for i ∈ N we obtain the desired bound

length(γ) ≤ (8π + 2) dist(O, γ(t -)).
The proof is complete.

Gradient and subgradient systems, and convex foliations

In this section, we apply Theorem 1.3 to derive length estimates, first for orbits of dynamical systems of gradient or subgradient type, then for orbits orthogonal to a convex foliation. The key fact is to observe that in some interesting particular cases (for instance, f convex or quasiconvex) these curves are self-contracted. Recall however that this is not the case for gradient dynamical systems defined by a general C ∞ function, as already observed in Remark 2.4 (iv).

6.1. Gradient dynamical system -quasiconvex case. Let f : R n → R be a C k function (k ≥ 1), x 0 ∈ R n and consider the Gradient Dynamical System γ(t) = -∇f (γ(t)), t > 0 γ(0) = x 0 ∈ R n . (6.1)
It follows directly from the standard theory of Ordinary Differential Equations (see [START_REF] Hale | Ordinary Differential Equations[END_REF], for example) that the system (6.1) admits a solution (trajectory) γ : I -→ R n , where I ⊂ [0, +∞), which is a curve of class C k-1 . Note that the case k = 1 corresponds to mere continuity of γ, while for k > 1 (or more generally, if f is assumed C 1,1 , that is, ∇f is Lipschitz continuous), the trajectory γ is unique. In the sequel, we shall always consider maximal solutions of (6.1), that is, for which I = [0, T ), where T > 0 is the maximal time such that γ is defined in [0, T ). We shall refer to them as orbits of the gradient flow of f . We will also need the following definition.

Definition 6.1 (Convex, quasiconvex and coercive functions). A function

f : R n → R is called convex (respectively, quasiconvex ) if its epigraph epi f := {(x, y) ∈ R n+1 | f (x) ≤ y} is a convex subset of R n × R (respectively, if for every y ∈ R the sublevel set {x ∈ R n | f (x) ≤ y} is a convex subset of R n ).
A function f is called coercive (or proper ), if its level sets are bounded, or equivalently, if lim

x →+∞ f (x) = +∞. (6.2)
It is straightforward to see that whenever f is coercive the corresponding flow orbits are bounded curves and therefore I = [0, +∞) (the trajectories are defined for all t ≥ 0). Notice in particular that the function t -→ f (γ(t)), t ∈ [0, +∞) is a natural Lyapunov function for the orbits of the flow, i.e. it is nonincreasing along the trajectories. Moreover, unless γ meets a critical point (i.e. ∇f (γ(t * )) = 0 for some t * ∈ [0, +∞)), the function defined in (6.1) is decreasing and γ is injective.

Let us finally recall (e.g. [9, Theorem 2.1]) that a (differentiable) function f : R n → R is quasiconvex if and only if for every x, y ∈ R n the following holds:

∇f (x), y -x > 0 ⇒ f (y) ≥ f (x). (6.3)
We are now ready to establish the following result.

Proposition 6.2 (Quasiconvex orbits are self-contracted curves). The orbits of the gradient flow of a quasiconvex C 1,1 function are self-contracted curves.

Proof. Let γ : I -→ R n be an orbit of the gradient flow of f . Let 0 ≤ t ≤ t 1 be in I and consider the function

g(t) = 1 2 ||γ(t) -γ(t 1 )|| 2 , t ∈ I.
In view of (6.1), we easily deduce that g ′ (t) = ∇f (γ(t)), γ(t 1 )γ(t) .

If g ′ (t) > 0 for some t ∈ [0, t 1 ), then the quasiconvexity of f would imply that f (γ(t 1 )) ≥ f (γ(t)) (see (6.3)), which in view of (6.1) would yield that γ(t ′ ) = γ(t 1 ) for all t ′ ∈ [t, t 1 ] and ∇f (γ(t)) = 0, a contradiction. Thus, g is nonincreasing in the interval [0, t 1 ], which proves the assertion.

The following corollary is a straightforward consequence of the previous proposition and Theorem 1.3 (Main result). Corollary 6.3 (Orbits of a gradient quasiconvex flow). Let f : R 2 → R be a coercive C 1,1 quasiconvex function. Then, for every x 0 ∈ R 2 , the orbit of the gradient flow (6.1) converges and has finite length.

Subgradient dynamical systems -convex case.

A convex function is a particular case of a quasiconvex function. Therefore, Corollary 6.3 implies that the orbits of the gradient flow of C 1,1 convex functions are of finite length. It is well-known ( [START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF]) that in the case of a (nonsmooth) convex function f : R n → R (or more generally, for a semiconvex function [START_REF] Degiovanni | Evolution equations with lack of convexity[END_REF]), the gradient system (6.1) can be generalized to the following differential inclusion, called Subgradient Dynamical System γ(t) ∈ -∂f (γ(t)) a.e. t ∈ [0, +∞),

γ(0) = x 0 ∈ R n , (6.4)
where ∂f is the set of the subgradients (subdifferential) of f. If f : R n → R is convex, then this latter set is defined as

∂f (x) = {p ∈ R n | f (y) ≥ f (x) + p, y -x , ∀y ∈ R n } for all x ∈ R n .
The above formula defines always a nonempty convex compact subset of R n , and reduces to {∇f (x)} whenever f is differentiable at x, cf. [START_REF] Clarke | Optimization and non-smooth analysis[END_REF]. It is also known that (6.4) has a unique absolutely continuous solution γ : [0, +∞) → R n , that is, the derivative γ(t) = d dt γ(t) exists almost everywhere and for every 0

≤ t 1 ≤ t 2 , γ(t 2 ) = γ(t 1 ) + t 2 t 1 γ(t) dt .
The analogous of Proposition 6.2 holds true. Proposition 6.4 (Subgradient convex flow orbits are self-contracted curves). Let f : R n → R be a convex continuous function. Then, for every x 0 ∈ R n , the trajectory γ of the subgradient system (6.4) is a self-contracted curve.

Proof. We give a sketch of proof for the reader convenience (we refer to [START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF] for details). It is easy to prove that t ∈ [0, +∞) → f (γ(t)) is convex and that for almost all t ≥ 0 we have

d dt f (γ(t)) = -|| γ(t)|| 2 ≤ 0.
Therefore t → f (γ(t)) is nonincreasing and γ(t) ∈ {f ≤ f (x 0 )} is bounded. Moreover, for all t 1 > 0 and for almost all t ∈ [0,

t 1 ] 1 2 d dt ||γ(t) -γ(t 1 )|| 2 = γ(t), γ(t) -γ(t 1 ) ≤ f (γ(t 1 )) -f (γ(t)) ≤ 0.
This implies that t ∈ [0, t 1 ) → ||γ(t)-γ(t 1 )|| 2 is nonincreasing yielding that γ a self-contracted curve.

When n = 2, we have the following generalization of Theorem 1.1.

Corollary 6.5 (Orbits of a subgradient convex flow). Let f : R 2 → R be a convex continuous function which admits a minimum. Then, for every x 0 ∈ R 2 , the orbit of the gradient flow (6.4) converges and has finite length.

6.3.

Trajectories orthogonal to a convex foliation. In this section we consider orbits that are "orthogonal" to a convex foliation. Let us introduce the definition of the latter. (For any subset C ⊂ R 2 , int C denotes the interior of C and ∂C its boundary.) Let {C α } α∈[0,A] (where A > 0) be a family of subsets of R 2 such that

(i) For all α ∈ [0, A], C α is convex compact (ii) If α > α ′ , then C α ⊂ int C α ′ , (iii) For every x ∈ C 0 \ int C A , there exists a unique α ∈ [0, A] such that x ∈ ∂C α . (6.5)
We shall refer to the above as a foliation made up of convex surfaces. We shall now define a notion of "orthogonality" for an orbit γ with respect to this foliation. To this end, let T ∈ (0, +∞] and γ : [0, T ) → R 2 be an absolutely continuous curve. We say that the curve γ is "orthogonal" to the foliation defined in (6.5) if the following conditions hold:

(i) for every t ∈ [0, T ), there exists α ∈ [0, A] such that γ(t) ∈ ∂C α , (ii) for almost all t ∈ (0, T ), if γ(t) ∈ ∂C α , then for all x ∈ C α , γ(t), xγ(t) ≥ 0, (iii) if t ′ > t and γ(t) ∈ C α , then γ(t ′ ) ∈ C α .

(6.6)

Condition (ii) in (6.6) is a nonsmooth generalization of orthogonality: if ∂C α is smooth at γ(t) and γ is differentiable at t then γ(t) is orthogonal to the tangent space of ∂C α at γ(t). Further, condition (iii) guarantees that the curve γ(t) enters into each convex set of the foliation. In this context, one has the following result. Proposition 6.6 (Orbits orthogonal to a convex foliation). The curve γ is a bounded selfcontracted curve, thus, of bounded length.

Proof. The curve γ is clearly bounded. 0 ≤ t 1 < T . Then, for almost all t ∈ [0, t 1 ], we have

f (x) =    x, if -2 ≤ x ≤ 0, 0, if 0 ≤ x ≤ 1, x -1, if 1 ≤ x ≤ 2.
Then the sublevel sets of f do not define a foliation on [-2, 2] ⊂ R since property (iii) of (6.5) fails at the level set [f = 0]. This drawback appears whenever the level sets of such functions have "flat" parts outside the set of their global minimizers. Actually, it follows from [9, Theorem 3.1] that the sublevel sets of a continuous quasiconvex coercive function f define a convex foliation if and only if the function is semi-strictly quasiconvex. (We refer to [START_REF] Daniilidis | Some remarks on the class of continuous (semi-)strictly quasiconvex functions[END_REF] for the exact definition and basic properties of semi-strictly quasiconvex functions.) (ii) Let f : R 2 → R be a coercive C 1,1 quasiconvex function and γ : [0, +∞) → R 2 be the solution of (6.1). Let x ∞ be the limit of γ(t) as t → +∞ and assume that f has no critical point in {f (x ∞ ) < f ≤ f (x 0 )}. Then, it is not difficult to see that {f ≤ α} α∈[f (x∞),f (x 0 )] is a family of C 1 convex compact subsets which satisfies (6.5) (in fact, f is semi-strictly quasiconvex in [f (x ∞ ), f (x 0 )]) and γ satisfies (6.6).

(iii) Despite the first remark, Proposition 6.6 can be used to obtain the result of Corollary 6.3 without the extra assumption made in the second remark. The reason is that the trajectory of the gradient flow will not pass through the flat parts of f anyway (if it does, then it stops there). We leave the technical details to the reader. One can check that f is smooth, positive away from O, with no critical point except at the origin. The gradient trajectory of f issued from the point (r, θ) = (( 3π 2 ) -1 , 0) remains close to the spiral given by r = 3π 2 + t -1 θ = -t where t runs over [0, ∞). Therefore, it converges to the origin and its length is infinite. 7.2. Thom conjecture fails for convex functions. Let f : R 2 → R be a convex continuous function which admits a minimum. Then, Corollary 6.5 guarantees that the orbits of the gradient flow of f have finite length (thus, a fortiori, are converging to the global minimum of f ). However, it may happen that each orbit turns around its limit infinitely many times. In particular the corresponding statement of the Thom conjecture fails in the convex case.

We construct below a counter-example using a technique due to D. Torralba [START_REF] Torralba | Convergence épigraphique et changements d'échelle en analyse variationnelle et optimisation[END_REF] which allows us to build a convex function with prescribed level-sets given by a sequence of nested convex sets. Let us recall his result.

For any convex set C ⊂ R n , the support function of C is defined as δ C (x * ) = sup x∈C x, x * for all x * ∈ R n . Let {C k } k∈N be a decreasing sequence of convex compact subsets of R 2 such that C k+1 ⊂ int C k . Set

K k = max ||x * ||=1 δ C k-1 (x * ) -δ C k (x * ) δ C k (x * ) -δ C k+1 (x * )
.

Then Torralba's theorem [START_REF] Torralba | Convergence épigraphique et changements d'échelle en analyse variationnelle et optimisation[END_REF] asserts that for every real sequence {λ k } k∈N satisfying 0 < K k (λ kλ k+1 ) ≤ λ k-1λ k for every k ≥ 1, (

there exists a continuous convex function f such that for every k ∈ N, {f ≤ λ k } = C k . Moreover, λ k converges to min f and, for any k ≥ 0 and λ ∈ [λ k+1 , λ k ], we have

{f ≤ λ} = λ -λ k+1 λ k -λ k+1 C k + λ k -λ λ k -λ k+1 C k+1 (7.2)
(i.e., the level-sets of f are convex interpolations of the two nearest prescribed level-sets).

Step (see Fig. 7 for the first steps of the construction). The sequence of convex sets {C k } satisfies the assumptions of Torralba's theorem and we can define a sequence {λ k } as in (7.3) which satisfies (7.1) (note that {K k } is 4-periodic since, for all convex set C ⊂ R 2 and x * ∈ R 2 , δ T (C) (x * ) = 1 2 δ C (r -1 (x * ))). We obtain a convex continuous function f : C 0 → R + with argmin f = {O}. The trajectory starting from the top of C 0 spirals around the origin and converges to O (see Fig. 7 where the beginning of the trajectory is drawn with a deflection of 3θ).

Step 3. Smoothing f . Actually, the function f built above is C ∞ except at the origin and at the boundaries ∂C k . It is possible to smooth out f in order to obtain a function which is C ∞ everywhere except at the origin and C m at the origin (for any fixed m ≥ 1). The smoothing procedure is quite involved from a technical point of view and is omitted. We refer the interested reader to [START_REF] Bolte | Characterizations of Lojasiewicz inequalities and applications, 57p[END_REF]Section 4.3] where such a smoothing is realized (in a different context). This procedure does not modify significantly neither the function nor its gradient trajectories (i.e. they remain a spiral). This concludes the construction.

Remark 2 . 4 (

 24 Basic properties).(i) Inequality(1.3) shows that the image of a segment (a, b) by a self-contracted curve γ lies in a ball of radius ρ := dist(γ(a), γ(b)).

Proof.

  The line passing through p and the origin O together with the circle of radius |Op| centered at the origin define a decomposition of the circle of radius |pm| centered at p into four arcs. One of these arcs, denoted by C, contains the point m. Let m ′ be the endpoint of C lying in the circle of radius |Op| centered at the origin, cf. Fig. 3 below. By construction, length(σ) = 2 |pm| = 2 |pm ′ |. (4.3) Since m ′ is at the same distance from the origin as p, there exists a rotation ρ centered at the origin which takes p to m ′ . This rotation sends the ray [O, p] to [O, m ′ ] and preserves distances and angles. Therefore, it also sends ∆ p to ∆ m ′ . In particular, the rotation ρ maps π(p) to π(m ′ ). From Thales' formula, we derive |π(p)π(m ′ )| |pm ′ | = |Oπ(p)| |Op| . Hence, |π(p)π(m ′ )| ≥ λ |pm ′ |. (4.4) Since the endpoints of the segment [π(p), π(m ′ )] lie in the arc π([pm ′ ]), we have |π(p)π(m ′ )| ≤ length(π([p, m ′ ])). (4.5)

Fig. 3 .Figure 3 . 3 . 4 . 4 (

 33344 Figure 3.

Figure 4 .

 4 Figure 4.

Lemma 4 . 5 (

 45 Total horizontal length upper bound).If {σ i } k+1 i=1 is a polygonal approximation of γ in the annulus A, cf. Definition 3.3, then i∈H length(σ i ) ≤ 8π 1λ ∆Rwhere H is the set of indices corresponding to the horizontal segments.

Proposition 5 . 1 (

 51 Length estimate in the annulus A). Every continuous self-contracted planar curve γ converging to the origin O satisfies length(γ ∩ A) ≤ (8π + 2) ∆R.

7 . 1 .

 71 Two counter-examples 7.Absence of Convexity. The second conclusion of Corollary 6.3 fails if f is not quasiconvex, even when the function is C ∞ and has a unique critical point at its global minimum. Let us give an explicit example below: Define a function f : R 2 → R in polar coordinates as f (r, θ) = e -1/r (1 + r + sin( 1 r + θ)) with f (O) = 0. The graph of f in the plane θ = 0 looks like the graph of Fig. 5.

Figure 5 .

 5 Figure 5.

1 . 4 Figure 6 .

 146 Figure 6.We then obtain a positive function f 0 : C 0 → R with argmin f 0 := {f 0 = min f 0 } = C 4 . Consider the subgradient trajectory γ 0 starting from the point A 0 of C 0 (see Fig.6). It reaches A 4 ∈ ∂C 4 . From (7.2) this trajectory is radial (pointing towards the origin) between ∂C 0 = {f 0 = λ 0 } and ∂C 1 = {f 0 = λ 1 } and between ∂C 3 = {f 0 = λ 3 } and ∂C 4 = {f 0 = λ 4 }. Due to the presence of the ellipse C 2 , the trajectory deflects with an angle θ := A 0 OA 4 > 0 in the clockwise direction. Step 2. Construction of the function from the previous step. Consider the transformation T = r • h, where h is the homothety of center O and coefficient 1/2 and r is the rotation of center O and angle θ. We define, for all k ∈ N and k ∈ {0, 1, 2, 3} C k = T [k/4] (C k ) where [k/4] is the integer part of k/4 and k = k (modulo 4)

Figure 7 .

 7 Figure 7.

  ∂C α for some α. By (6.6) (iii) and since t 1 ≥ t, we also have γ(t 1 ) ∈ C α . Therefore, (6.6) (ii) implies that the right-hand side of (6.7) is nonpositive. It follows that t ∈ [0, t 1 ] → ||γ(t)γ(t 1 )|| 2 is nonincreasing and γ is self-contracted. Applying Theorem 1.3 finishes the proof.

	1 2	d dt	||γ(t) -γ(t 1 )|| 2 = γ(t), γ(t) -γ(t 1 ) .	(6.7)
	By (6.6) (i), we have γ(t) ∈

Remark 6.7. (i) The sublevel sets of a continuous quasiconvex function need not define a convex foliation. Indeed, consider the quasiconvex function f : [-2, 2] → R given by

Acknowledgment This work has been realized during a research stay of the first author in the University of Tours (Spring 2008). The authors acknowledge useful discussions with G. Barles (Tours), J. Bolte (Paris 6), H. Giacomini (Tours), N. Hadjisavvas (Aegean) and M. Hassaine (Talca).

Supported by the MEC Grant No. MTM2005-08572-C03-03 (Spain).