Emmanuel Gheerbrant

Christian De Muizon

Didier Néraudeau

The oldest modern therian mammal from Europe and its bearing on stem marsupial paleobiogeography

Keywords:

We report here the discovery of mammalian tribosphenic teeth from the basal Cenomanian of SW France that we refer to a new primitive marsupial-like form identified as a basal taxon of Marsupialiformes, a new clade recognized here to include the crown group Marsupialia and primitive stem lineages more closely related to Marsupialia than to Deltatheroida. Arcantiodelphys marchandi gen et sp nov. shares several significant marsupial-like features (s.l.) with marsupialiform taxa known from the North American mid-Cretaceous. Among marsupialiforms, it shows closer resemblances to Dakotadens. These resemblances, which are plesiomorphic within "tribotherians", make Arcantiodelphys one of the most archaic known Marsupialiformes. Arcantiodelphys is moreover characterized by an original and precocious crushing specialization. Both the plesiomorphic and autapomorphic character states of Arcantiodelphys among Marsupialiformes might be explained by an Eastern origin from Asian stem metatherians, with some in situ European evolution. In addition, the presence of a mammal with North American affinities in Western Europe during the early Late Cretaceous provides further evidence of a large Euramerican biogeographical province at this age or slightly before. Concerning the paleobiogeographical history of the first stem marsupialiforms during the Albian-Cenomanian interval, two possible dispersal routes can now be proposed from an Asian metatherian ancestry: (H1) Asia to Europe via North America, or (H2) Asia to North America via Europe. The main significance of the Archingeay-Les Nouillers mammal discovery is that the beginning of the stem marsupialiforms history involved not only North America, but also Europe, and that this early history in Europe remains virtually unknown.

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The early evolution of modern therians (i.e., metatherian and eutherian boreosphenidan mammals) is mainly documented in North America and Asia, where numerous rich mid-Cretaceous microvertebrate localities are known. In Europe, a major gap occurs in the mammal fossil record for the Late Cretaceous, and especially for the totally blank Aptian-Santonian interval [START_REF] Kielan-Jaworowska | Mammals from the age of dinosaurs: origins, evolution, and structure[END_REF]. New Cenomanian localities in western France (Charentes) have yielded some rich coastal assemblages, including continental vertebrates [START_REF] Vullo | Vertebrates from the Late Cretaceous of Charentes (SW France): biodiversity, taphonomy, palaeoecology and palaeobiogeography[END_REF][START_REF] Vullo | Cenomanian vertebrate assemblages from southwestern France: a new insight into the European mid-Cretaceous continental fauna[END_REF]. Some rare, indeterminate mammal teeth were briefly reported from Charentes [START_REF] Vullo | Cenomanian vertebrate assemblages from southwestern France: a new insight into the European mid-Cretaceous continental fauna[END_REF][START_REF] Néraudeau | Stratigraphy and palaeontology (plants, vertebrates) of the paralic deposits from the Uppermost Albian-Lower Cenomanian of Tonnay-Charente (Charente-Maritime, France)[END_REF], attesting to the presence of this group in the terrestrial ecosystems of the mid-Cretaceous European archipelago. The present study describes a new mammal from basal Cenomanian deposits of the Font-de-Benon quarry, at Archingeay-Les Nouillers, Charentes (SI Fig. 1). The new mammal material reported here includes four marsupial-like tribosphenic fragmentary teeth, which provide information on the early European boreosphenids. The earliest therian mammals [boreosphenidans sensu Luo et al. [START_REF] Luo | Dual origin of tribosphenic mammals[END_REF]] from Europe are Aegialodon and Tribactonodon from the Early Cretaceous of England [START_REF] Kermack | Aegialodon dawsoni, a new trituberculosectorial tooth from the lower Wealden[END_REF][START_REF] Sigogneau-Russell | The oldest tribosphenic mammal from Laurasia (Purbeck Limestone Group, Berriasian, Cretaceous, UK) and its bearing on the "dual origin" of Tribosphenida[END_REF]. The mammal from Archingeay-Les Nouillers is the oldest modern therian known so far in Europe, i.e. the oldest European representative of the clade including metatherians and eutherians. Systematic Paleontology. Systematic paleontology: Boreosphenida Luo, Cifelli and Kielan Jaworowska, 2001; Metatheria Huxley, 1880; Marsupialiformes supercohort nov.; Family indet.; Arcantiodelphys marchandi gen et sp nov. Comments: Marsupialiformes [= Marsupialia sensu Kielan-Jaworoswka et al. [START_REF] Kielan-Jaworowska | Mammals from the age of dinosaurs: origins, evolution, and structure[END_REF]] is erected to account for the crown group Marsupialia (extant marsupials and related extinct fossil taxa) plus all stem marsupialiform taxa that are more closely related to them, as their sister taxa, than to Deltatheroida and basal Metatheria. Basal marsupialiforms such as the North American taxa from the Early/Late Cretaceous (e.g., ref. [START_REF] Cifelli | Marsupial mammals from the Albian-Cenomanian (Early-Late Cretaceous) boundary, Utah[END_REF]) includes the stem groups of the crown marsupials. Basal marsupialiforms include primitive Cretaceous taxa previously gathered in the paraphyletic taxon "Ameridelphia" (e.g., refs. 1 and 8). Metatheria includes Marsupialiformes and Deltatheroida (presumed sister groups), plus basal metatherians such as Sinodelphys.

Holotype. The holotype MNHN CCH4 is as follows: right upper (anterior?) molar, possibly an anterior one, with mesial part broken; Charentes Cenomanian collections of the Muséum national d'Histoire naturelle (MNHN CCH), Paris, France (Fig. 1). Hypodigm: holotype MNHN CCH4; MNHN CCH1, trigonid of left lower molar, M 1 ; MNHN CCH2, talonid of right lower molar; MNHN CCH3, trigonid of right lower molar, M 2 or more posterior molar..

Etymology.

The generic name is based on Arcantius (Latin), name of the gallo-roman domain from which the village of Archingeay developed, and delphys (Greek), term used for marsupials. The specific name is dedicated to Joseph Marchand, who sorted part of the material and discovered the holotype, and to the Marchand family, owner of the Font-de-Benon quarry.

Locality and Horizon. The hypodigm was collected "Font-de-Benon" quarry, Archingeay-Les Nouillers, Charente-Maritime, SW France, Cenomanian. The horizon corresponds to the sand at the base of the unit B (subunit B1; level B1cs), which is early Cenomanian in age [START_REF] Néraudeau | Variation in echinoid biodiversity during the Cenomanian-early Turonian transgressive episode in Charentes (France)[END_REF].

Diagnosis. Tribosphenic mammal with marsupialiform characters: paraconid enlarged (but smaller than metaconid) and lingual (posterior molars), hypoconulid lingual, approximated to entoconid (but not twinned), well developed entoconid, paracone and metacone of similar size and well separated, wide stylar shelf, stylar cusps D and C present (but small), protocone well developed (uncompressed), no lingual cingulum. Molar morphology similar to earliest North American basal marsupialiforms such as Dakotadens, Iugomortiferum, Adelodelphys, Synbadelphys, and Pariadens [START_REF] Cifelli | Marsupial mammals from the Albian-Cenomanian (Early-Late Cretaceous) boundary, Utah[END_REF][START_REF] Cifelli | Marsupial from the earliest Late Cretaceous of western U[END_REF][START_REF] Cifelli | Cretaceous mammals of Southern Utah. II. Marsupials and Marsupiallike mammals from the Wahweap Formation (Early Campanian)[END_REF][START_REF] Eaton | Therian mammals from the Cenomanian (Upper Cretaceous) Dakota Formation, southwestern Utah[END_REF]. It shares especially with Dakotadens and Iugomortiferum the robust construction (cusps low and bulbous, low trigonid, wide talonid, large protocone), and the wear pattern, indicating an early specialized crushing function. Differs from Cretaceous stem marsupialiforms, except Dakotadens, in primitive features such as the absence of postcingulid, the postprotocrista which does not extend labially at the posterior base of the metacone (no metacingulum and no double rank postvallum-prevallid shearing), absence of conules (traces in Dakotadens) and their internal crest, and absence of carnassial notch on the postmetacrista. Iugomortiferum differs in these traits (although conules are reduced), and in the protocone less lingual, less inflated, and more anterior, the cristid obliqua more lingual on the trigonid, and the hypoconulid and entoconid twinned. Differs from Dakotadens in the following traits: more inflated cusps, absence of conules, hypoconulid and entoconid approximated but not twinned (separated by a small disto-lingual notch), protocone more lingual and more inflated, protoconid more inflated, metaconid less distal, hypoconid more compressed, hypoconulid smaller with respect to the entoconid. This is among the smallest known marsupialiforms with Adelodelphys and Sinbadelphys. Dimensions: upper molar CCH4: Minimal Estimated Length > 1.14 mm; width = 1.86 mm (length of the protocone = 0.7 mm); lower molars: see Table 1.

Description and Comparisons. CCH4 (Fig. 1, SI Fig. 3 A and C), is a three-rooted upper molariform tooth. The paracone and parastylar area are broken and missing. The prominent metastylar lobe and asymmetric ectoflexus suggest an anterior molar. The tooth is characterized by a robust and bunodont morphology, with low and inflated cusps and crests. The occlusal outline seems short and wide, but this might be unreliable because of the broken mesial part of the crown (including the paracone). The metastylar lobe is well developed, with long and predominantly transversal postmetacrista, and a wide stylar shelf labial to the metacone. The ectoflexus is deep and asymmetric with wider metastylar area according our comparative reconstruction. The ectocingulum and stylar cusps are worn, but the cusps D and C are distinctly inflated. A slight but distinct oblique crest links the cusp D to the base of the metacone. The postmetacrista is robust, and bears no carnassial notch. The paracone is broken, but is apparently similar in size to the metacone, or even slightly larger judging from its greater lingual extension. Paracone and metacone are more compressed mesiodistally than labiolingually. These are well separated, but not deeply. The conules are absent, as are their internal crests. The postprotocrista links the lingual flank of the metacone, so that there is no metacingulum. The protofossa (trigon basin) is large, and nearly as long as wide. The protocone is remarkably well developed, both longitudinally and transversely. Its labial flank is inflated at the apex. It is lingual to the paracone-metacone notch, and slightly procumbent. There are no lingual cingula. CCH1 (Fig. 2 A, SI Fig. 2A-D), is a trigonid of a tribosphenic left lower molar. It preserves only the most anterior part of the talonid. CCH1 is identified as an M 1 , based on its mesiodistally expanded trigonid and its paraconid shifted mesially. The trigonid is low and displays a robust and bunodont morphology: the cusps are inflated, low and bulbous, and the wear is dominated by abrasion. Only the most distal part of the precingulid is preserved. The occlusal outline is triangular, with a very transverse protocristid and a paraconid well-shifted mesially. The trigonid crests show wide notches, but not carnassial notches. The paraconid is well-developed, nearly as large and high as the metaconid. It is located lingually; its mesiolingual flank is very convex and salient mesially above the root (paraconid keel). The paraconid and metaconid are linked lingually by a small crest (premetacristid), which closes the trigonid basin lingually. The protoconid apex is truncated by the abrasion. It is slightly elongated mesiodistally. The trigonid basin is large and mesiodistally extended (longer than wide). The mesial flank of the protoconid is slightly inflated labially toward the paraconid, close to the paracristid notch. Judged from the preserved part of the cristid obliqua, the talonid is wider than the trigonid. The cristid obliqua abuts noticeably labially against the trigonid, between the protoconid apex and the protocristid notch. The hypoflexid is consequently shallow. A postmetacristid is present at the base of the metaconid. The ectocingulid is absent.

CCH2 (Fig. 2C, SI Fig. 3 B andC) is the talonid of an unidentified left lower tribosphenic molar, but not of the most distal as illustrated by the hypoconulid morphology. Although the base of the crown is broken, it was probably low. In occlusal view, this talonid is short and wide. The postcingulid is absent. The hypoconulid is shifted lingually, but it is not twinned with the entoconid. The two cusps are separated by a deep notch, which is distinct in distolingual view of the postcristid. As a result, the hypoconulid is salient distolingually. Because of the strong abrasion of the hypoconulid with respect to the entoconid, the two cusps seem less approximated than they probably were. The cristid obliqua is slightly more lingual mesially than on CCH1, and the hypoflexid seems deeper.

CCH3 (Fig. 2B, SI Figs. 2 E and F, 3 B and C) is the trigonid of a right lower molar similar in size to CCH1. It belongs to a more posterior molar (M 2? , M 3? , or M 4? ) than CCH1, based on its trigonid being more compressed mesiodistally and its paraconid more lingual. The trigonid is low. The paraconid is only slightly smaller than the metaconid, and slightly more labial. Its mesiolingual flank is strongly salient mesially, forming a prominent vertical mesiolingual "keel", somewhat inflated or crested. The protoconid apex is even more truncated by abrasion than in CCH1. The trigonid basin is opened lingually, with no trace of lingual crest as seen in CCH1. The precingulid, although damaged was well-developed mesiodistally. The protocristid is very transverse. It bears a deeper notch than the paracristid. The postmetacristid is well developed. The cristid obliqua abuts labially against the trigonid, as in CCH1.

Wear is illustrated by attrition facets resulting from the labial phase [nomenclature of Crompton (13)] and by apical abrasion. CCH1 shows a small shearing wear facet 2 (postvallum-prevallid shearing facet) on the mesial flank of the paraconid, and apical abrasion of the paraconid and protoconid. The shearing wear facet 1 is not distinct. CCH2 shows strong apical abrasion of the hypoconulid. On the internal flank of the entoconid is a well-developed wear facet 6, resulting from the postprotocrista of opposite upper molar. On CCH3 the shearing wear facet 2 is extended on the mesial flank of the protoconid. The wear facet 1 (prevallum-postvallid shearing facet) is distinct on the labial margin of the distal flank of the protoconid. There is also a well-developed wear facet 5 on the postmetacristid. Abrasion is well developed at protoconid apex, and slight on the protocristid. On CCH4, abrasion strongly affects apices of protocone, metacone, and postmetacrista. A triangular wear facet 9 is distinct in the lingual flank of the protocone, denoting developed crushing-grinding function. Other wear facets are indistinct.

The four mammal specimens from Archingeay-Les Nouillers present close morphological affinities suggesting that they belong to the same mammal species. Especially, they share an overall robust construction with low and inflated cusps, a similar wear pattern with developed abrasion and are comparable in size. Affinity of the tribosphenic upper and lower molars from Archingeay-Les Nouillers is illustrated by resemblance in robust general morphology and related crushing (horizontal) wear pattern, in the absence of carnassial notches, and by their overall consistent occlusal relative pattern as illustrated by their occlusal relationship (SI Fig. 3C). The occlusal relationship shows slight size difference (CCH4 is slightly smaller), but it remains in the range of the intraspecific variation. However, it is noteworthy that this specific association is based on very fragmentary material and needs to be confirmed by additional material, including more complete teeth and tooth rows. In the meantime, our interpretation, based on anatomical data, is regarded as most consistent.

The Archingeay-Les Nouillers material documents the typical tribosphenic molar pattern of therian mammals. These molars have a remarkably robust construction, with low and bulbous cusps, low trigonid and wide talonid, broad protofossa and large and inflated protocone. Together with the developed abrasion, and development of the grinding wear facet 9 on the holotype, this suggests an early therian species with developed crushing diet. The Archingeay-Les Nouillers material belongs to a new therian mammal, Arcantiodelphys marchandi gen et sp nov., which is characterized by an original association of features that are summarized and compared in SI Tables 12. These features are also included in the character list used for the phylogenetic analysis (SI Text).

Among all known Cretaceous tribosphenic mammals, Arcantiodelphys marchandi most closely recalls the most primitive known marsupialiforms and some peculiar "tribotherians" (i.e., stem boreosphenidans). It differs from primitive eutherians in the paraconid enlarged and lingually placed, in the paraconid keel, in the paracone and metacone similar in size and well separated, and in the occurrence of stylar cusps D and C. It differs from the deltatheroidans (Metatheria) in the large talonid and protocone, the narrower stylar shelf, the shorter postmetacrista, and the smaller paraconid and paracristid.

Among all Cretaceous therians, Arcantiodelphys marchandi most closely resembles Dakotadens, which was initially described as an indeterminate "tribotherian" with marsupiallike features [START_REF] Eaton | Therian mammals from the Cenomanian (Upper Cretaceous) Dakota Formation, southwestern Utah[END_REF], and which was recently referred to Marsupialia (= Marsupialiformes here) by Kielan-Jaworowska et al. [START_REF] Kielan-Jaworowska | Mammals from the age of dinosaurs: origins, evolution, and structure[END_REF]. It is noteworthy that Dakotadens is of contemporaneous Cenomanian age with Arcantiodelphys marchandi, as is Pariadens. Arcantiodelphys marchandi also closely resembles several other early North American stem marsupialiforms, previously classified as "Ameridelphia" (e.g., ref. 8). Its marsupial-like features (sensu lato, i.e., marsupialiform features) are the paracone and metacone of similar size and well separated [character K10(1-2)], the wide stylar shelf [K3(0)] and occurrence of stylar cusps D and C [K6(1)-K7( 1)] (although small on known material), the well developed (uncompressed) protocone [K4(1)], the hypoconulid lingual approximated (but not connate) to entoconid [K(1)], the large and lingual paraconid [K21(1), 22(1)], the low trigonid [K17(1)], the wide talonid [K30(2)], the large entoconid [K32(3)], and the cristid obliqua, which joins the trigonid labially, below the protoconid [K27(2)] and defines a shallow hypoflexid [K28 [START_REF] Vullo | Vertebrates from the Late Cretaceous of Charentes (SW France): biodiversity, taphonomy, palaeoecology and palaeobiogeography[END_REF]]. Other features mostly known in basal stem marsupialiforms are the low and bulbous cusps [K1(2)], the low protoconid [K23(1)], the distinct postmetacristid [K29(0)], the mesiolingual flank of the paraconid inflated and prominent above the root [K19(0)], the weak ectocingulum and stylar cusps D and C, the wide stylar shelf bearing a long and transverse postmetacrista [K12(1)] (but which is less developed than in deltatheroidans). The small size also reminds one of the smallest known Cretaceous marsupialiforms such as Sinbadelphys and Adelodelphys. The paraconid keel [K19(0)] might be primitive, corresponding to a probable trace of the eupantotherian and "tribotherian" cusp e. Features 6(1), 10(1-2), 12(0), 21(0), 22(0), 29(0) (see SI Text) are also found in "tribotherians" and might be primitive. However, most of these traitsand especially the paraconid large (but smaller than metaconid) and lingualare also reminiscent of marsupialiforms. Features 4(1), 27(2), 28(2), 30(2), 31(1), 32 (3) (see SI Text) are the most significant, derived, marsupialiform features of Arcantiodelphys marchandi.

It is noteworthy that several of the derived features shared with stem marsupialiforms are remarkably poorly developed in Arcantiodelphys. This is especially true for the paraconid development, and the hypoconulid-entoconid relation. Several other remarkable features shared with Dakotadens are unknown in other stem marsupialiforms, including in Kokopellia which is the oldest known one [START_REF] Cifelli | Early Cretaceous mammal from North America and the evolution of marsupial dental characters[END_REF][START_REF] Cifelli | Dentition and jaw of Kokopellia juddi, a primitive marsupial or near-marsupial from the medial Cretaceous of Utah[END_REF]: the absence of the postcingulid [K34(0)] and of the conules [K8(0)] and their internal crests, the postprotocrista which does not extend labially at the base of the metacone [K9(0)], and the protocone poorly shifted mesially [K14(0)]. These character states of Arcantiodelphys are known in "tribotherians" and most probably correspond to symplesiomorphies retained in some of the most basal Marsupialiformes. Dakotadens was not formally included in "Marsupialia" (i.e., Marsupialiformes) by its author mainly because of some of these features, such as especially the absence of the postcingulid [START_REF] Eaton | Therian mammals from the Cenomanian (Upper Cretaceous) Dakota Formation, southwestern Utah[END_REF]. However, these primitive traits are associated in Arcantiodelphys marchandi, with a suite of other features that are more common in early stem marsupialiforms than in any other early tribosphenic mammals. We should also emphasize that deltatheroidan metatherians lack the postcingulid and the labial extension of the postprotocrista, and also retain a less mesially shifted protocone than in marsupialiforms. As a whole, the morphology of Arcantiodelphys marchandi, although still poorly known, clearly suggests an archaic stem Marsupialiformes. The species might belong to one of the most primitive marsupialiform metatherians known (Fig. 3A). It is even more primitive than the earlier Kokopellia in some remarkable traits [K8(0), 9(0), 14(0), 34(0)], although it also shows some specialized features in relation to it such as the more lingual hypoconulid, and the robust, crushing molars which bear especially a large and inflated protocone, that additionally displays a wear facet 9.

Among primitive marsupialiforms, Arcantiodelphys marchandi is characterized by an overall striking robust molar construction, with low and bulbous cusps, large and inflated protocone, low trigonid and protoconid, large hypoconid and wide talonid bearing large basin. It indicates a predominantly crushing function, which is also illustrated by the wear pattern with developed apical abrasion of the cusps (especially protoconid and protocone). This construction is also known in Dakotadens and Iugomortiferum, but in a less advanced way, and most probably as a parallelism. Crushing molar with developed "horizontal wear" (abrasion) is considered by Fox and Naylor (16) as a distinctive feature of early "marsupials" (i.e., stem marsupialiforms) with respect to other therian mammals. In Arcantiodelphys marchandi, the crushing function associated to a large and inflated protocone bearing a wear facet 9 is a noteworthy early specialization, possibly corresponding to an autapomorphy of the European lineage illustrated by Arcantiodelphys.

A cladistic analysis of the few known features of Arcantiodelphys marchandi, with the program TNT [START_REF] Goloboff | TNT, a free program for phylogenetic analysis[END_REF] was developed in order to test and formalize the position Arcantiodelphys marchandi with and among marsupialiforms (see SI Text). The analysis only includes characters observed in Arcantiodelphys, i.e. 36 molar features, with four additional key characters (K37-40) related to the dental formula, which are diagnostic of the major therian clades (Eutheria, Metatheria, Deltatheroida, Marsupialiformes). The exact analysis confirms the relationships of Arcantiodelphys to marsupialiforms. However, it yields unusual results with respect to previous studies, such as a basal stem therian position of the Deltatheroida versus Metatheria (i.e., sister group of Sinodelphys + Marsupialiformes), and the derived position of primitive bunodont taxa such as Dakotadens and Arcantiodelphys, which are closer to Didelphis, i.e. to crown Marsupialia, than to the base of the Marsupialiformes. The stem therian position of the Deltatheroida (based for instance on the central position of the hypoconulid) was already suggested in the parsimony analysis of Cifelli [START_REF] Cifelli | Mesozoic Differentiation, Multituberculates, Monotremes, Early Therians, and Marsupials[END_REF], also based on molar features, which led the author to question their relationships with Marsupialia (Marsupialiformes here). The convergence of our results with Cifelli's analysis ( 18) is unexpected given our addition of the recently discovered major taxa such as Sinodelphys, Asiatherium and Kokopellia. The derived position of Dakotadens and Arcantiodelphys is mainly the result of noticeably and unlikely reversions favoured by the parsimony analysis, such as the postprotocrista lingual, the absence of conules, the protocone central, and the absence of postcingulid. Other shared features known in Dakotadens and Arcantiodelphys, and also Iugomortiferum (e.g., bunodonty, low protoconid, extended abrasion) are probably convergent.

Calculation of the Bremer support value (as well as bootstrap value) shows in fact that only three nodes are well supported and reliable in this tree (Fig. 3B), among which the Marsupialiformes including Arcantiodelphys. Otherwise, it shows that several important points are poorly resolved in the parsimony analysis: 1) relationships of Arcantiodelphys among Marsupialiformes; and 2) relationships of basal taxa to Marsupialiformes, with the polytomy including especially Marsupialiformes, Eutheria, and Holoclemensia. These limitations of our analysis are probably linked to our poor knowledge of Arcantiodelphys, on which it is focused (e.g., mostly molar characters).

Conclusions: Implications for marsupial and marsupialiform origin and paleobiogeography

The new mammal from Archingeay-Les Nouillers is the first fossil discovery that fills within the huge gap in the fossil record of mid-Cretaceous of Europe (e.g., ref. [START_REF] Kielan-Jaworowska | Mammals from the age of dinosaurs: origins, evolution, and structure[END_REF]. Arcantiodelphys marchandi is the oldest known European representative of the metatherian-eutherian clade. It is also the oldest known European stem marsupialiform, and one of the oldest known Marsupialiformes. The only older stem marsupialiform known is Kokopellia from the Early Cretaceous (latest Albian). Sinodelphys from the Early Cretaceous (Barremian) of China is a much more primitive and basal metatherian that predates the deltatheroidan divergence [START_REF] Luo | An Early Cretaceous tribosphenic mammal and metatherian evolution[END_REF]. Moreover, Arcantiodelphys marchandi is one of the most primitive stem marsupialiform known so far (Fig. 3A; although its primitive features are interpreted as reversals in the parsimony analysis). It is most closely reminiscent of its sub-contemporaneous Dakotadens, with which it shares plesiomorphic character states such as the absence of the postcingulid, the protocone central and the postprotocrista restricted lingually to the metacone, but also the derived and probably convergent robust crushing molars.

The Archingeay-Les Nouillers discovery sheds new light on an unknown early center of the marsupialiform evolution: the Cretaceous of Europe. The only other known stem marsupialiform described in the Late Cretaceous of Europe is Maastrichtidelphys from the Maastrichtian of the Netherlands, an advanced form related to herpetotheriids [START_REF] Martin | A new European marsupial indicates a Late Cretaceous high-latitude transatlantic dispersal route[END_REF]. This family has been regarded as the sister group of Marsupialia and would therefore include the most derived stem marsupialiforms [START_REF] Sánchez-Villagra | Exceptionally preserved North American Paleogene metatherians: adaptations and discovery of a major gap in the opossum fossil record[END_REF]. Arcantiodelphys marchandi emphasizes a major gap in our knowledge of the early evolution of marsupialiforms in Europe. The discovery in SW France (Charentes) of a new mammal most closely related to early North American stem marsupialiforms provides the first mammalian evidence for the wide Euramerican paleobiogeographical province (s.l.) in the mid-Cretaceous. The early Euramerican continental province is well supported by affinities of other vertebrate taxa. Among dinosaurs, the early Cenomanian deposits of Charentes have yielded teeth of Probactrosaurus-grade iguanodontians (similar to Protohadros and Eolambia), nodosaurids, brachiosaurids, and troodontids [START_REF] Vullo | Dinosaur teeth from the Cenomanian of Charentes, western France: evidence for a mixed Laurasian-Gondwanan assemblage[END_REF]. All these forms are well represented in the North American Cenomanian assemblages, in particular in the Mussentuchit local fauna (upper Cedar Mountain Formation, Utah) [START_REF] Cifelli | High-precision 40 Ar/ 39 Ar geochronology and the advent of North America's Late Cretaceous terrestrial fauna[END_REF][START_REF] Cifelli | Vertebrate Paleontology in Utah, ed Gillette DD[END_REF]. The same observation can be made concerning the crocodyliform taxa, with atoposaurids, cf. bernissartiids, goniopholidids, and pholidosaurids occurring in both Utah and Charentes localities. Some of these dinosaurian and crocodyliform taxa are also present together with an indeterminate stem marsupialiform in the Woodbine Formation of Texas [START_REF] Lee | The Archosauria from the Woodbine Formation (Cenomanian) in Texas[END_REF][START_REF] Jacobs | Advances in Vertebrate Paleontology and Geochronology[END_REF], which also shares numerous coastal selachians with Charentes [START_REF] Cappetta | Additions to the Cretaceous selachian faunas of Texas (upper Albian-Campanian)[END_REF].

The discovery of Arcantiodelphys marchandi shows that Europe was an important area in the basal radiation of marsupialiforms, and fits well the current models of metatherian and marsupial paleobiogeographical history which assume an early dispersal from a Laurasian and most probably an Asian center of origin (metatherian ancestry) to the modern Australian center of diversification, via the North and South American early centers of radiation (e.g., refs. 19 and 28). Affinities of the Archingeay-Les Nouillers species suggest that the marsupialiform initial center of radiation involved a wide Euramerican continental province.

Although metatherians seem to have originated in Asia (e.g., Sinodelphys, deltatheroidans), the paleobiogeographical history and origin of the Cretaceous European and Euramerican marsupialiformsand of the therians in generalremains mostly unresolved. Direct dispersals from Asia to North America are illustrated by several vertebrate taxa (e.g., ref. [START_REF] Averianov | Mammals from the mid-Cretaceous Khodzhakul Formation, Kyzylkum Desert, Uzbekistan[END_REF], some of which colonized Europe from North America, especially in the Late Cretaceous (e.g., ref. [START_REF] Martin | A new European marsupial indicates a Late Cretaceous high-latitude transatlantic dispersal route[END_REF]. But this does not exclude direct early dispersals from Asia to Europe, especially because the Ouralian seaway (Turgai strait) developed later, in the Turonian. A Cretaceous dispersal from Asia explains indeed the origin of the European zhelestids, Gobiconodon, kogaionids, amphilestines, and Probactrosaurus-grade iguanodontians [START_REF] Averianov | Mammals from the mid-Cretaceous Khodzhakul Formation, Kyzylkum Desert, Uzbekistan[END_REF][START_REF] Gheerbrant | The Upper Cretaceous mammals from Laño (Spanish Basque Country)[END_REF][START_REF] Averianov | Mammals from the Upper Cretaceous Aitym Formation, Kyzylkum Desert, Uzbekistan[END_REF][START_REF] Cuenca-Bescós | A new gobiconodontid mammal from the Early Cretaceous of Spain and its palaeogeographic implications[END_REF]. Comparison with Cretaceous North American marsupialiforms suggests that Arcantiodelphys marchandi is a very primitive stem marsupialiform, despite the (unlikely) reversals implied by our parsimony analysis. Our view of the primitiveness of Arcantiodelphys would favour an Asian origin and a trans-Ouralian dispersal (versus a North American origin and a North Atlantic dispersal) (Fig. 4).

It is noteworthy that basal marsupialiforms have a distribution on both sides of the Western Interior Seaway since the Cenomanian [START_REF] Kielan-Jaworowska | Mammals from the age of dinosaurs: origins, evolution, and structure[END_REF][START_REF] Cifelli | Marsupial from the earliest Late Cretaceous of western U[END_REF][START_REF] Jacobs | Advances in Vertebrate Paleontology and Geochronology[END_REF], suggesting that the latter was not a major barrier to the dispersal of some of the North American and Euramerican Cenomanian taxa (see Fig. 4 and SI Table 3). This is also true for the other European taxa with Euramerican affinities reported above. However, most of the described basal marsupialiforms, including the oldest known form (i.e., Kokopellia), occur in the western part of North America (see SI Table 3). If this North American fossil record is representative, it might be the best evidence for an eastern dispersal of the Marsupialiformes from their Asian metatherian stem group, and accordingly for a North American origin of Arcantiodelphys. However, the origin of the primitive Euramerican Marsupialiformes remains, in our view, a widely open question, especially because of the major gaps in the European Cretaceous. New fossil discoveries might further enlighten the paleobiogeographical significance of both the primitive and autapomorphic state of Arcantiodelphys among marsupialiforms.

The Archingeay-Les Nouillers discovery shows that the beginning of the Marsupialiformes' history also involved Europe, and that the European part of this history remains widely enigmatic. Current data cannot exclude any paleobiogeographical scenario (especially with respect to the involved dispersal route) in the American marsupialiform colonization. Given the scarcity of the Cretaceous mammal record of Europe, any new discovery in the mid-Cretaceous of western France is likely to provide key data to our knowledge of early marsupialiform phylogeny and paleobiogeography.

Materiel and Methods

The early Cenomanian deposits of Charentes, in the northern margin of the Aquitaine Basin (SW France), display estuarine to shallow coastal facies corresponding the lithostratigraphic units A2, B1-3 defined by Néraudeau et al. [START_REF] Néraudeau | Variation in echinoid biodiversity during the Cenomanian-early Turonian transgressive episode in Charentes (France)[END_REF]. The vertebrate-bearing layer of the Font-de-Benon quarry corresponds to the shelly sand at the base of the unit B1. It is earliest Cenomanian in age according to the presence of the benthic foraminifera Orbitolina concava (33, 34) (SI Fig. 1). Intensive sediment screen-washing of about 5 tons of matrix, from 2003 to 2005, allowed to collect a rich vertebrate assemblage, including sharks and rays, bony fishes, frogs, turtles, crocodyliforms, theropod dinosaurs, pterosaurs, snakes and mammals [START_REF] Vullo | Vertebrates from the Late Cretaceous of Charentes (SW France): biodiversity, taphonomy, palaeoecology and palaeobiogeography[END_REF][START_REF] Vullo | Dinosaur teeth from the Cenomanian of Charentes, western France: evidence for a mixed Laurasian-Gondwanan assemblage[END_REF][START_REF] Néraudeau | A new fossil locality with insects in amber and plants (likely uppermost Albian): Archingeay (Charente-Maritime, France)[END_REF]35). The sedimentology, paleontology, and paleoenvironment are described in detail in Vullo et al. (34). Supporting Information includes the character description and the cladistic analysis with TNT program [START_REF] Goloboff | TNT, a free program for phylogenetic analysis[END_REF]. [START_REF] Fox | A Late Cretaceous taeniodont (Eutheria, Mammalia) from Alberta, Canada[END_REF] based on 36 molar characters plus four major diagnostic characters among tribosphenidans, related to the dental formula (see SI Text). Only the nodes with Bremer support value ≥ 3 are reported and retained here from the consensus tree. Other nodes are fragile and do not fit previous analysis (e.g., Deltatheridium and Sinodelphys relative position; see text); they need to be further tested with additional data on Arcantiodelphys. Fig. 4. Possible scenario on the early paleobiogeographical history of Marsupialiformes. 0: Marsupialiformes originated from an Asian Early Cretaceous metatherian stem group; 1: they reached North America via Beringia before the Albian-Cenomanian where they underwent the Late Cretaceous diversification; 2: this was followed by a dispersal from North America to Europe that occurred around Albian-Cenomanian boundary, explaining the origin of Arcantiodelphys marchandi. Hypothesis 1-2 is supported by the distribution of early Marsupialiformes in the western part of North America. Alternative hypothesis 1'-2': the metatherian stem group of Marsupialiformes dispersed from Asia to North America via Europe, which is supported by the primitive features of the Charentes taxon Arcantiodelphys marchandi. The Cenomanian marsupialiforms occurrences of the Euramerican biogeographical province (dashed area) discussed in the present study are represented by stars (black star: Charentes, France; grey star: Woodbine Formation, Texas; white star: upper Cedar Mountain and Dakota formations, Utah).

Table Legends

Tab. 1. Dimensions of the lower teeth of Arcantiodelphys marchandi gen et sp nov. (in mm.)

  34. Vullo R, Néraudeau D, Videt B (2003) A "falun" facies from the earliest Cenomanian of Charente-Maritime (France). Ann Paléont 89:171-189 (in French). 35. Vullo R, Cappetta H, Néraudeau D (2007) New sharks and rays from the Cenomanian and Turonian of Charentes, France. Acta Pal Pol 52:99-116.

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig. 1. Arcantiodelphys marchandi gen et sp nov., holotype, MNHN CCH4, anterior(?) right upper molar in occlusal (A: stereophotography), distal (B) and labial (C) views; SEM photographs. Arrows: stylar cusps C and D. (Scale bar: 1 mm.)

ACKNOWLEDGEMENTS

We thank C. Chancogne (MNHN) for SEM photographs, M. Forissier (Saga), J. Marchand (Saga) for the sorting of the sediment, and the Marchand family for letting us work in their quarry. We thank R. Cifelli for providing several important casts of North American Cretaceous mammals, which were critical for our systematic comparisons.


B Fernandez
email: bastien.fernandez@cpt.univ-mrs.fr

A Quas
email: aquas@uvic.ca

STATISTICAL PROPERTIES OF INVARIANT GRAPHS IN DISCONTINUOUS FORCED SYSTEMS

When a contractive map is forced by a chaotic discontinuous system, the asymptotic response function that defines the attracting invariant set can be highly irregular. In this context, it is natural to ask whether the invariant distributions of the base and factor systems share the same characteristics and in particular, whether the factor distribution of an absolutely continuous SRB measure in the base can be absolutely continuous. Here, we address this question in a basic example of linear real contractions forced by (generalized) baker's maps and we prove absolute continuity for almost every value of the factor contraction rate.

Introduction

Generalized synchronization is an essential feature of forced dissipative systems [START_REF] Rulkov | Generalized synchronization of chaos in directionally coupled chaotic systems[END_REF]. This term refers to the fact that the asymptotic dynamics of (discrete time) skew-product systems in Banach spaces with dissipative factor, i.e.

(w n+1 , z n+1 ) = (F (w n ), G(w n , z n )) where G(w, z) -G(w, z ) ≤ λ z -z for some λ < 1 independent of w, is globally attracted by an invariant graph in phase space composed of points (w, h(w)), viz. for arbitrary z 0 , the iterated variable z n approaches h(w n ) uniformly in w 0 as n → ∞. The corresponding synchronization function h, which conjugates the skew-product on the invariant graph to its base system F , naturally depends both on this forcing and on the factor G.

A standard problem in this context is to analyze the properties of the synchronization function. One aims to identify whose features of the forcing dynamics that carry over to the response system; see e.g. the introduction in [START_REF] Stark | Regularity of invariant graphs for forced systems[END_REF] for a fairly complete overview. In the case where F is an homeomorphism, h turns out to be continuous. Furthermore, it can be Hölder or Lipschitz continuous depending upon additional assumptions on the Lyapunov exponents [START_REF] Hirsch | Invariant Manifolds[END_REF]. Moreover, h was proved to be smooth when F is a diffeomorphism [START_REF] Stark | Regularity of invariant graphs for forced systems[END_REF]. However, when the forcing is not continuous and the response is sensitive to discontinuities, h may fail to be continuous [START_REF] Afraimovich | Synchronization in unidirectionally coupled systems: some rigorous results[END_REF] and it could become singular in a measure-theoretic sense.

To the best of our knowledge, no study has been reported in the literature where properties of response statistical distributions have been described in the case of discontinuous synchronization functions. This paper investigates this issue in a basic example of discontinuous system. Assuming that the forcing is given by the iterations (x n+1 , y n+1 ) = F a (x n , y n ) (n ∈ Z and (x n , y n ) plays the role of w n here) of the generalized baker's transformation on the unit square [0, 1] 2 :

F a (x, y) = (ax, y/a) if 0 ≤ y < a (a + (1 -a)x, (y -a)/(1 -a)) if a ≤ y < 1
where 0 < a < 1, we study the invariant distribution of the response variable in the case where G(x, y, z) = x + λz depends on the forcing variable only via the first coordinate x.

Since F a is invertible and the previous values of x of preimages of (x, y) are determined by x alone, the synchronization function h only depends on x and its explicit expression is given by [START_REF] Afraimovich | Synchronization in unidirectionally coupled systems: some rigorous results[END_REF] (1)

h(x) = ∞ t=0 λ t T t+1 a (x), x ∈ [0, 1]
where

T a (x) = x a if 0 ≤ x < a x-a 1-a if a ≤ x ≤ 1
This function h appears to be extremely choppy. Indeed, there is no interval where it is monotonic and the function is discontinuous with negative jumps h(x) < h(x -0) for every x in the set of pre-images of a under iterations of T a . Yet, the image h([0, 1]) coincides with the interval [h(0), h(1)].

In this context, we would like to know whether the distribution of the (z n ) inherits the absolute continuity of the system to which it is synchronized. It is well known that two-dimensional Lebesgue measure is an absolutely continuous invariant measure for T a , and hence is the unique Sinai-Ruelle-Bowen (SRB) measure. Accordingly we ask whether the measure Leb • h -1 is absolutely continuous where Leb denotes the Lebesgue measure on [0, 1].

Notice that while h can be regarded as the uniform limit of non-singular functions -take the approximation h n obtained by truncating the series in [START_REF] Afraimovich | Synchronization in unidirectionally coupled systems: some rigorous results[END_REF] to the first n + 1 terms -this property does not suffice to guarantee that h is non singular. For instance a sequence of linear functions on a bounded interval with slopes tending to zero provides a counter-example.

To address absolute continuity of Leb•h -1 , we need to separate the cases λ < 1/2 and λ ≥ 1/2. In the first case, we use that h is of bounded variation to show that it has finitely many pre-images almost everywhere and to obtain lower bounds on the measure of specific neighborhoods. In the case λ ≥ 1/2, we rely on the symbolic dynamics of the map T a to rewrite h as a random series and to regard the distribution Leb • h -1 as a Bernoulli convolution. Then we adapt the proof in [START_REF] Peres | Absolute continuity of Bernoulli convolutions, a simple proof[END_REF] to show absolute continuity for a.e. λ.

The Case λ < 1/2

In this section, we deal with functions of bounded variation and take advantage of their multiple properties. Recall in particular that any discontinuity is necessarily of jump type, i.e. the left and right limits exist at every point.

In addition, we only consider functions that are right continuous, so that they are cadlag. The jump of a cadlag function f that occurs at x is denoted by

J f (x) = f (x)-f (x-0). A cadlag function f on an interval [a, b] is said to be a step function if f (x) = f (a) + a<t≤x J f (t), ∀x ∈ [a, b]
viz. if all changes in the value of f as the argument changes may be accounted for by the jumps.

Any cadlag function f of bounded variation has a unique decomposition into three parts: f = f d + f c + f s , a step function f d , an absolutely continuous part f c and a singular continuous part f s . In our case, when λ < 1/2 the synchronization function h defined by ( 1) is of bounded variation and has no singular continuous part in its decomposition:

Lemma 1. When λ < 1/2, we have h(x) = h c (x) -h d (x) for all x ∈ [0, 1]
where h c is strictly increasing and absolutely continuous and h d is a strictly increasing step function with a dense set of discontinuities.

Proof. We already know from [START_REF] Fernandez | Discontinuous generalized synchronization of chaos[END_REF] that h d is a strictly increasing step function with a dense set of discontinuities and h c is strictly increasing and continuous. There remains only to prove that h c is absolutely continuous.

Consider the approximation h n obtained by truncating the series in (1) to the first n + 1 terms. The function h n has a similar decomposition h n,c -h n,d where the continuous part h n,c consists of finitely many affine pieces; hence it is absolutely continuous. By the fundamental theorem of calculus, we have

(2) h n,c (x) -h n,c (0) = x 0 h n,c (y)dy, ∀x ∈ [0, 1]
where the derivative is given by

h n,c (x) = n t=0 λ t (T t+1 a (x)) , for a.e. x
By monotony, each sequence {h n,c (x)} converges as n → ∞ to

h c (x) = ∞ t=0 λ t (T t+1 a (x)) Moreover, each h n,c (x) converges to h c (x) (and h n,c (0) = h c (0) = 0). Thus relation (2) implies that each sequence { x 0 h n,c (
y)dy} is bounded. By Lebesgue's monotone convergence theorem, we conclude that h c is integrable over [0, 1] and we have

h c (x) = x 0 h c (y)dy, ∀x ∈ [0, 1]
which implies that h c is absolutely continuous.

The main result in the case where λ < 1/2 is given by the following statement.

Theorem 2. For every a ∈ (0, 1) and every λ < 1/2, the push-forward distribution Leb • h -1 is absolutely continuous with density given a.e. by x → y∈h -1 (x) 1 h (y) .

2.1.

Proof of Theorem 2. The crucial point is to obtain a lower bound for the Lebesgue measure of the pre-image h -1 U of small neighborhoods U of points in an appropriate subset of h([0, 1]) with full measure. In order to select this subset, we need a series of preliminary results.

1st step: Selecting a suitable subset in h([0, 1]). Image measure of sets with zero measure. Since h is of bounded variation, it has a finite derivative a.e. Since h d is a step function, its derivative vanishes almost everywhere and thus h = h c a.e. Let E be the exceptional set of points where either the derivative of h fails to exist or we have h = h c . The following statement guarantees that h(E) has zero Lebesgue measure. Lemma 3. Let f and g be real functions defined on [0, 1] where f is absolutely continuous and increasing and g is an increasing step function. Then defining i(x) = f (x) -g(x), we have Leb(i(A)) = 0 whenever Leb(A) = 0.

Proof. Let > 0. By definition of absolute continuity, there exists δ > 0 such that if (a 1 , b 1 ), (a 2 , b 2 ), . . . are disjoint intervals whose lengths sum to at most δ, then

i |f (b i ) -f (a i )| < /2.
On the other hand, since g is increasing and bounded, it has at most countably many jump discontinuities. Let {x : g(x) -g(x -0) > 0} be enumerated as

{x 1 , x 2 , . . .}. Let n be chosen such that i>n g(x i ) -g(x i -0) < /2.
Since Leb(A) = 0, the set A is contained in a countable union of open intervals whose lengths sum to at most δ. Removing {x 1 , . . . , x n }, A is contained in the union of a finite set and a countable collection of open intervals whose lengths sum to at most δ.

For any x < y we have |i(y

) -i(x)| ≤ f (y) -f (x) + x<z≤y (g(z) -g(z -0)
). Applying this inequality to the intervals covering A we obtain that i(A) is contained in the union of a collection of intervals whose lengths sum to at most with a finite set of points. It follows that i(A) has outer measure 0.

Determining pre-images. It is convenient to use notions from the symbolic dynamics associated with T a . Given a finite word θ 0,N := θ 0 θ 1 • • • θ N ∈ {0, 1} N +1 , let I θ 0,N be the corresponding cylinder, that is

I θ 0,N = N k=0 T -k a (I θ k )
where I 0 = [0, a) and I 1 = [a, 1). Let also I 0,N := [x - θ 0,N , x + θ 0,N ) implicitly introduce notations for the cylinder boundaries. Consider the quantity

h -1 θ 0,N (y) = inf{t ∈ I θ 0,N : h(t) ≥ y}.

Right continuity of h implies

h • h -1 θ 0,N (y) ≥ y. Let {θ i 0,N } k(N )
i=1 be an enumeration of words for which we indeed have equality, i.e. h • h -1 θ 0,N (y) = y. The next statement claims that this collection remains bounded as N → ∞. Lemma 4. There are finitely many pre-images h -1 (y) for a.e. y ∈ h([0, 1]).

Proof. Using symbol concatenation we have

I θ 0,N = I θ 0,N 0 ∪ I θ 0,N 1 and x + θ 0,N 0 = x - θ 0,N 1
It easily follows from the definition (1) of h that the discontinuities are negative jumps that lie at the x + θ 0,N . Write D θ 0,N = [h(x + θ 0,N ), h(x + θ 0,N -0)) for the corresponding gaps.

Lemma 5. There can be pre-images h -1 (y) on both sides of x + θ 0,N 0 in I θ 0,N only if y ∈ D θ 0,N 0 .

Proof. Let x ∈ I θ 0,N 1 be such that x - θ 0,N 1 < x. Both points belong to the same cylinder I θ 0,N 1 . Hence we have

T k a (x - θ 0,N 1 ) < T k a (x), for k = 0, • • • , N + 1
Moreover, the point x - θ 0,N 1 is a pre-image of 0 by definition, i.e.

0 = T k a (x - θ 0,N 1 ) ≤ T k a (x), ∀k ≥ N + 2 hence h(x - θ 0,N 1 ) < h(x) for all x ∈ I θ 0,N 1 . Therefore, if y < h(x - θ 0,N 1 ) there cannot be any pre-image h -1 (y) in [x - θ 0,N 1 , x + θ 0,N
). Similarly, one proves that h(x) < h(x + θ 0,N 0 -0) for all x ∈ I θ 0,N 0 and then there cannot be any pre-image

h -1 (y) in [x - θ 0,N , x + θ 0,N 0 ) if y ≥ h(x + θ 0,N 0 -0).
The lemma easily follows.

Since N θ 0,N Leb(D θ 0,N ) < ∞ the first Borel-Cantelli lemma implies that Lebesgue a.e. y belongs to finitely many gaps D θ 0,N , i.e. for almost every y, there exists N y ∈ Z + such that no gap D θ 0,N with N ≥ N y contains y.

It follows from Lemma 5 that any pre-image h -1 (y) ∈ I θ 0,N with N ≥ N y , must belong to either I θ 0,N 0 or to I θ 0,N 1 . By induction, for each word θ 0,Ny there exists a unique sequence {I θ 0,N k } k≥Ny of nested cylinders whose diameters asymptotically vanish such that any pre-image in I θ 0,Ny belongs to every I θ 0,N k . By the Nested Balls theorem, it follows that there is at most one pre-image h -1 (y) in each interval clos(I θ 0,Ny ) and thus there are finitely many pre-images for a.e. y ∈ h([0, 1]).

We will make use of the following approximate version of h in which small-scale discontinuities are removed. Given x ∈ [0, 1) and N ∈ Z + , let h N (x) be the function defined by h N (x) = sup{h(t) : t ≤ x, t ∈ I θ 0,N (x)}, where I θ 0,N (x) denotes the cylinder of length N + 1 containing x. This function is increasing on each I θ 0,N . It follows that there can be at most countably many y such that h -1 N (y) contains an interval for some N . Let

F 1 = {y ∈ h([0, 1
]) : for each N, h -1 N (y) contains no interval} and let F 0 be the subset of h([0, 1]) composed of points with finitely many preimages under h. Each y ∈ F 0 ∩ F 1 has only finitely many pre-images under h and each h N .

Let

G = (h([0, 1]) \ h(E)) ∩ F 0 ∩ F 1 . Since we have h c (x) ≥ ∞ t=0 λ t (a ∧ (1 -a)) t+1 ≥ a ∧ (1 -a)
for all x ∈ E, the derivative h exists and is bounded below at each pre-image of every y ∈ G under h and h N . Accordingly the following functions are well-defined in this set

ψ N (y) = {x : h N (x)=y} 1 h (x)
and ψ(y)

= {x : h(x)=y} 1 h (x)
Notice that if h N (x) = h(x) then h N +1 (x) = h(x) also so that for y ∈ G, {x : h N (x) = y} is a subset of {x : h N +1 (x) = y}. It follows that the ψ N are pointwise increasing. Further since the preimages of y all lie in distinct cylinders for N ≥ N y it follows that for N ≥ N y , ψ N (y) = ψ(y).

2nd step: Lower bound on the measure of neighborhood pre-images and absolute continuity.

Let x ∈ h -1 (y) for y ∈ G and > 0. Since by assumption h is differentiable at x there exists µ > 0 such that

|h(x + η) -y -h (x)η| < h (x)|η| for all 0 < |η| < µ.
Since y has finitely many pre-images, there exists a μ for which the above equation holds simultaneously for all x ∈ h -1 (y). As a consequence, for every 0 < δ < μ(1 + ) min x ∈E h (x) and each pre-image

x ∈ h -1 (y), the inequality |t -x| < δ/(h (x)(1 + )) implies |h(t) -y| < δ. It follows that Leb(h -1 (y -δ, y + δ)) ≥ 2δψ(y)/(1 + ) for all δ < μ(1 + ) min x ∈E h (x).
We will use the following weaker version valid for all N ≥ 0:

Leb(h -1 (y -δ, y + δ)) ≥ 2δψ N (y)/(1 + ) for all δ < μ(1 + ) min x ∈E h (x).
Being finite sums of uniformly bounded functions, the ψ N are integrable over h([0, 1]). Then the Lebesgue differentiation theorem implies that for a.e. y ∈ h([0, 1]), we have 2δψ N (y) ≥ y+δ y-δ (ψ N (t) -) dt for sufficiently small δ.

By Lemma 4, the set G has full Lebesgue measure. Let the subset of H ⊂ G with full measure for which the previous inequality holds. Combining the previous two inequalities we conclude that for y ∈ H, there exists a δ > 0 such that

(3) Leb(h -1 (y -δ, y + δ)) ≥ 1 1 + y+δ y-δ (ψ N (t) -) dt when 0 < δ < δ.
For any subinterval J of h([0, 1]), the collection of intervals U δ,y ⊂ J with y ∈ J ∩ H and δ < ∆ is a Vitali cover. By the Vitali Covering theorem, there exists a countable collection of such intervals that disjointly cover all of J up to a set of measure 0. By countable additivity of the two sides of (3) we deduce that

|h -1 J| ≥ 1 1 + J (ψ N (t) -) dt
Since this inequality holds for every > 0 we get

(4) |h -1 J| ≥ J ψ N for each subinterval J of h([0, 1]).
Since this holds for all N , we deduce that ψ (the increasing pointwise limit) is integrable and ( 5)

|h -1 J| ≥ J ψ for each subinterval J of h([0, 1]).
To continue the proof, we now make the following claim ( 6)

h([0,1]) ψ = 1
and we proceed as follows. For any interval

J = [a, b] ⊂ h([0, 1]), let J -= [0, a) and J + = (b, h (1)
]. Then by ( 4) and ( 6), we have

1 = |h -1 (h([0, 1]))| = |h -1 J -| + |h -1 J| + |h -1 J + | ≥ J- ψ + J ψ + J+ ψ = h([0,1]) ψ = 1.
Since the left and right sides are equal, all of the inequalities in the middle are equalities; so we see that

|h -1 J| = J ψ
for any interval J. Since the left and right sides define two measures that agree on all intervals, they must be the same measure and hence we see that

|h -1 B| = B ψ
for any measurable set B, as desired. It remains to prove [START_REF] Peres | Self-similar measures and intersections of Cantor sets[END_REF]. Since the functions ψ N pointwise increase to ψ it is enough to show that lim N →∞ ψ N = 1. Let S N = {x ∈ [0, 1) : h N (x) > h(x)}. We show that ψ N = 1-Leb(S N ) and that Leb(S N ) → 0. Firstly, since h (x) = h N (x) for every x ∈ E, the push-forward under h N of the Lebesgue measure restricted to [0, 1] \ S N is absolutely continuous with density exactly ψ N . It follows that ψ N = 1 -Leb(S N ) as claimed. Secondly, notice that S N consists of a union of countably many intervals. Indeed if h N (x) > h(x), then by right continuity there exists δ > 0 such that

h N (t) = h N (x) > h(t) ∀t ∈ [x, x + δ). Let J be a maximal interval in S N and let J = [a, b]. Since we must have h(b -0) = h(a) it follows that h c (b) -h c (a) = h d (b -0) -h d (a). Since the left side exceeds m(b -a) where m = ess inf h > 0, we deduce that b -a ≤ 1 m x∈J h(x -0) -h(x)
Moreover, the function h is discontinuous at x ∈ I θ 0,N iff T n a (x) = a for some n > N and we have h(x -0) -h(x) = λ n 1-λ [START_REF] Fernandez | Discontinuous generalized synchronization of chaos[END_REF]. Summing the previous inequality over all intervals included in the cylinders of length N + 1, we deduce that Leb(S N ) ≤

1 m(1-λ) ∞ n=N +1 2 n λ n .
Since this converges to 0 as N → ∞ the proof is complete.

The Case λ ≥ 1/2

When λ ≥ 1/2, the variation of h becomes infinite [START_REF] Fernandez | Discontinuous generalized synchronization of chaos[END_REF] and a decomposition into the difference of bounded functions h = h c -h d as in Lemma 1 no longer exists. In order to circumvent this difficulty, we use the symbolic dynamics associated with the map T a to regard h as a random series.

The map T a is semi-conjugate to the left shift σ on {0, 1} Z + , i.e. we have

T a •π = π • σ where π : {0, 1} Z + → [0, 1] is given explicitly by π(θ) = a ∞ k=0 θ k k-1 j=0 α(θ j ), ∀θ = {θ k } ∞ k=0 ∈ {0, 1} Z +
where α(0) = a and α(1) = 1 -a (and by convention -1 j=0 α(θ j ) = 1). The coding map π -1 (x) is unique for a.e. x and the push forward of Leb (which is T a -invariant) under π -1 is the (σ-invariant) Bernoulli measure µ a where the symbol 0 appears with probability a and 1 appears with probability 1 -a, viz. µ a = Leb • π. Therefore, the measure Leb • h can be regarded as µ a • g -1 where the symbolic function g = h • π is obtained explicitly by inserting the expression of π into (1)

g(θ) = a ∞ n=1 θ n λ n n k=1 λ -k n-1 j=n-k+1 α(θ j ), ∀θ ∈ {0, 1} Z +
Using instead of θ, the sequence ω ∈ {-1, +1} Z + of independent identically distributed random variables taking the values +1 and -1 with probability 1 -a and a respectively, the question of absolute continuity of the measure µ a • g -1 reduces to the following: what can be said about the set of λ for which the random variable

Z = ∞ n=1 ω n λ n n k=1 λ -k n-1 j=n-k+1
α((ω j + 1)/2) has an absolutely continuous distribution?

In [START_REF] Solomyak | On the random series Σ ± λ i (an Erdős problem)[END_REF] Solomyak showed that for a = 1/2 and Lebesgue a.e. λ ∈ [1/2, 1), the random variable ∞ n=1 ω n λ n has an absolutely continuous distribution. His proof made use of Fourier transform arguments and a transversality condition. Shortly afterwards Peres and Solomyak [START_REF] Peres | Absolute continuity of Bernoulli convolutions, a simple proof[END_REF] gave a new proof based on correlation dimension. This approach allowed them to extend later their results to arbitrary a ∈

[1/3, 2/3] [6].
Here we are able to establish the transversality condition of the function g in the case where a = 1/2. By adapting the argument of [START_REF] Peres | Absolute continuity of Bernoulli convolutions, a simple proof[END_REF], we obtain the following conclusion. Theorem 6. Assume that a = 1/2. Then Leb • h is absolutely continuous for a.e. λ ∈ (1/2, 1).

Proof. When a = 1/2, the random variable Z takes the following simpler form

2 λ -2 -1 ∞ n=1 ω n (λ n -2 -n )
The first thing to note is that it is sufficient to demonstrate the absolute continuity of a random variable of the form

Zλ = ∞ n=M ω n (λ n -2 -n )
-that is we can omit any subset of terms -as the convolution of an absolutely continuous random variable with another random variable is always absolutely continuous.

For ω, τ in {±1} Z + following Peres and Solomyak we define

φω,τ (λ) = ∞ n=M (τ n -ω n )(λ n -2 -n ).
For comparison with [START_REF] Peres | Absolute continuity of Bernoulli convolutions, a simple proof[END_REF] we also define

φ ω,τ (λ) = ∞ n=M (τ n -ω n )λ n . Notice that φω,τ (λ) = φ ω,τ (λ) -C ω,τ where C ω,τ = ∞ n=M (τ n -ω n )2 -n .
The absolute continuity of the random series ∞ n=1 ω n λ n was shown in [START_REF] Peres | Absolute continuity of Bernoulli convolutions, a simple proof[END_REF] to follow from an estimate of the form

S := lim inf r→0 Ω Ω Leb {λ : φ ω,τ (λ) ≤ r} dµ(τ ) dµ(ω) < ∞.
We want to show a similar estimate for the quantity S obtained in a similar way by using φω,τ instead of φ ω,τ .

For now fix ω and τ and let k(ω, τ ) = min{n ≥ M : ω n = τ n } (which we will denote by k when ω and τ are fixed). We assume without loss of generality that τ k = 1 and ω k = -1 and we define g(λ) := φω,τ (λ)/(2λ k ).

In particular, we have g(λ) = g(λ) -C τ,ω /(2λ k ) where the function g is as in [START_REF] Peres | Absolute continuity of Bernoulli convolutions, a simple proof[END_REF] g(λ) = 1 + The proof works by showing |g (λ)| exceeds δ when g(λ) is δ-close to 0. Fix λ 0 ∈ (1/2, 2 -2/3 ) and choose an M such that 2(2λ 0 ) -M < δ/2. For λ ∈ [λ 0 , 2 -2/3 ], g differs from g by a constant that is at most 2(2λ) -M < δ/2 so that |g (λ)| exceeds δ when |g(λ)| < δ/2. This is sufficient to allow us to conclude in the same way as [START_REF] Peres | Absolute continuity of Bernoulli convolutions, a simple proof[END_REF] that S < ∞ and hence Zλ is absolutely continuous for almost every λ in [λ 0 , 2 -2/3 ]. Since this holds for any λ 0 > 1/2 this establishes absolute continuity of Zλ for almost every λ in [2 -1 , 2 -2/3 ].

For the range [2 -2/3 , 2 -1/2 ] the same modifications as in [START_REF] Peres | Absolute continuity of Bernoulli convolutions, a simple proof[END_REF] give the result on [2 -1 , 2 -1/2 ].

For λ in the range J k = [2 -1/2 k , 2 -1/2 k+1 ] we need to make a small modification to the Peres-Solomyak argument (in their case it was argued that this case followed immediately from the previously understood cases). For us if λ ∈ J k we take only the terms of the sequence whose indices are multiples of k. Showing absolute continuity of the partial series arising in this way is sufficient to prove absolute continuity of the entire series Zλ . Setting ρ = λ k , the partial series that we are now summing is ∞ n=M ω kn (ρ n -2 -kn ).

The previous proof works verbatim to show that this series is absolutely continuous. In fact, as pointed out in [START_REF] Peres | Absolute continuity of Bernoulli convolutions, a simple proof[END_REF], applying this technique to each of the k parts separately shows that the original random variable is the sum of k parts that are each absolutely continuous. This implies that the resulting random variable is absolutely continuous with a continuous density function.

∞b

  n=1 n λ n with b n ∈ {-1, 0, 1} Notice that 0 ≤ C ω,τ /(2λ k ) ≤ 2(2λ) -M since 0 ≤ C ω,τ ≤ 2 2-k .Peres and Solomyak then showed for an explicit δ > 0 thatLeb{λ ∈ [2 -1 , 2 -2/3 ] : |g(λ)| ≤ ρ} ≤ 2δ -1 ρ ∀ρ > 0.


A V Kazeykina

R G Novikov

Absence of exponentially localized solitons for the Novikov-Veselov equation at negative energy

HAL is

Introduction

In the present paper we are concerned with the following (2 + 1)-dimensional analog of the Korteweg-de Vries equation:

∂ t v = 4Re(4∂ 3 z v + ∂ z (vw) -E∂ z w), ∂ z w = -3∂ z w, v = v, E ∈ R, v = v(x, t), w = w(x, t), x = (x 1 , x 2 ) ∈ R 2 , t ∈ R, (1.1)
where

∂ t = ∂ ∂t , ∂ z = 1 2 ∂ ∂x 1 -i ∂ ∂x 2 , ∂ z = 1 2 ∂ ∂x 1 + i ∂ ∂x 2 .
(1.2)

Equation (1.1) is contained implicitly in the paper of S.V. Manakov [M] as an equation possessing the following representation

∂(L -E) ∂t = [L -E, A] + B(L -E) (1.3) (Manakov L-A-B triple)
, where L = -∆ + v(x, t), ∆ = 4∂ z ∂ z , A and B are suitable differential operators of the third and zero order respectively, [•, •] denotes the commutator. Equation (1.1) was written in an explicit form by S.P. Novikov and A.P. Veselov in [START_REF] Novikov | Finite-zone, two-dimensional, potential Schrödinger operators. Explicit formula and evolutions equations[END_REF], [START_REF] Novikov | Finite-zone, two-dimensional Schrödinger operators. Potential operators[END_REF], where higher analogs of (1.1) were also constructed. Note that both Kadomtsev-Petviashvili equations can be obtained from (1.1) by considering an appropriate limit E → ±∞ (see [ZS], [G]).

In the case when v(x 1 , x 2 , t), w(x 1 , x 2 , t) are independent of x 2 , (1.1) can be reduced to the classic KdV equation:

∂ t u -6u∂ x u + ∂ 3 x u = 0, x ∈ R, t ∈ R. (1.4)
It is well-known that (1.4) has the soliton solutions u(x, t) = u κ,ϕ (x-4κ 2 t) = -2κ 2 ch 2 (κ(x -4κ 2 tϕ))

, x ∈ R, t ∈ R, κ ∈ (0, +∞), ϕ ∈ R.

(1.5) Evidently,

u κ,ϕ ∈ C ∞ (R), ∂ j
x u κ,ϕ (x) = O(e -2κ|x| ) as x → ∞, j = 0, 1, 2, . . .

(1.6)

Properties (1.6) imply that the solitons (1.5) are exponentially localized in x.

For the 2-dimensional case we will say that a solution (v, w) of (1.1) is an exponentially localized soliton if the following properties hold:

v(x, t) = V (x -ct), x ∈ R 2 , c = (c 1 , c 2 ) ∈ R 2 , V ∈ C 3 (R 2 ), ∂ j x V (x) = O(e -α|x| ) for |x| → ∞, |j| 3 and some α > 0 (where j = (j 1 , j 2 ) ∈ (0 ∪ N) 2 , |j| = |j 1 | + |j 2 |, ∂ j x = ∂ j 1 +j 2 /∂x j 1 1 ∂x j 2 2 ), w(•, t) ∈ C(R 2 ), w(x, t) → 0 as |x| → ∞, t ∈ R.
(1.7) In [N1] it was shown that, in contrast with the (1 + 1)-dimensional case, the (2 + 1)-dimensional KdV equation (1.1), at least for E = E f ixed > 0, does not have exponentially localized solitons. More precisely, in [N1] it was shown that the following theorem is valid for E = E f ixed > 0:

Theorem 1.1. Let (v, w) be an exponentially localized soliton solution of (1.1) in the sense (1.7). Then v ≡ 0, w ≡ 0.

The main result of this paper consists in the proof of Theorem 1.1 for the case E = E f ixed < 0. This proof is given in Section 3 and is based on Propositions 3.1 and 3.2. In addition: Proposition 3.1 is an analog of the result of [N1] about the transparency of sufficiently localized solitons for equation (1.1) for E > 0; Proposition 3.2 is an analog of the result of [N2], [START_REF] Grinevich | Transparent potentials at fixed energy in dimension two. Fixed energy dispersion relations for the fast decaying potentials[END_REF] that there are no nonzero bounded real-valued exponentially localized transparent potentials (that is potentials with zero scattering amplitude) for the Schrödinger equation (2.1) for E = E f ixed > 0.

Note that nonzero bounded algebraically localized solitons for equation (1.1) for E < 0 are also unknown (see [G]), but their absence is not proved.

As regards integrable systems in 2+1 dimensions admitting exponentially decaying solitons in all directions on the plane, see [BLMP], [FS].

As regards integrable systems in 2 + 1 dimensions admitting nonzero bounded algebraically decaying solitons in all directions on the plane, see [FA], [BLMP], [G], [KN] and references therein.

2 Inverse scattering for the 2-dimensional Schrödinger equation at a fixed negative energy

Consider the scattering problem for the two-dimensional Schrödinger equation at a fixed negative energy:

-∆ψ + v(z)ψ = Eψ, E = E f ixed < 0, ∆ = 4∂ z ∂ z , z = x 1 + ix 2 , x ∈ R 2 , (2.1)
where ∂ z , ∂ z are the same as in (1.2). We will assume that the potential v(z) satisfies the following conditions

v(z) = v(z), v(z) ∈ L ∞ (C), |v(z)| < q(1 + |z|) -2-ε for some q > 0, ε > 0. (2.2)
In this paper we will be concerned with the exponentially decreasing potentials, i.e. with the potentials v(z) satisfying (2.2) and the following additional condition v(z) = O(e -α|z| ) as |z| → ∞ for some α > 0.

(2.3) Direct and inverse scattering for the two-dimensional Schrödinger equation (2.1) at fixed negative energy under assumptions (2.2) was considered for the first time in [START_REF] Grinevich | Two-dimensional inverse scattering problem for negative energies and generalized-analytic functions. I. Energies below the ground state[END_REF]. For some of the results discussed in this section see also [N2], [G].

First of all, we note that by scaling transform we can reduce the scattering problem with an arbitrary fixed negative energy to the case when E = -1. Therefore, in our further reasoning we will assume that E = -1.

It is known that for λ ∈ C\(0 ∪ E), where E is the set of zeros of the modified Fredholm determinant ∆ for the integral equation (2.10), (2.4) there exists a unique continuous solution ψ(z, λ) of (2.1) with the following asymptotics

ψ(z, λ) = e -1 2 (λz+z/λ) µ(z, λ), µ(z, λ) = 1 + o(1), |z| → ∞.
(2.5)

In addition, the function µ(z, λ) satisfies the following integral equation

µ(z, λ) = 1 + ζ∈C g(z -ζ, λ)v(ζ)µ(ζ, λ)dζ R dζ I , (2.6) g(z, λ) = - 1 2π 2 ζ∈C exp(i/2(ζ z + ζz)) ζ ζ + i(λ ζ + ζ/λ) dζ R dζ I , (2.7) where z ∈ C, λ ∈ C\0, ζ R = Reζ, ζ I = Imζ.
In terms of ψ of (2.5) equation (2.6) takes the form

ψ(z, λ) = e -1/2(λz+z/λ) + ζ∈C G(z -ζ, λ)v(ζ)ψ(ζ, λ)dζ R dζ I , (2.8) G(z, λ) = e -1/2(λz+z/λ) g(z, λ), (2.9) where z ∈ C, λ ∈ C\0. In terms of m(z, λ) = (1+|z|) -(2+ε)/2 µ(z, λ) equation (2.6) takes the form m(z, λ) = (1+|z|) -(2+ε)/2 + ζ∈C (1+|z|) -(2+ε)/2 g(z-ζ, λ) v(ζ) (1 + |ζ|) -(2+ε)/2 m(ζ, λ)dζ R dζ I , (2.10) where z ∈ C, λ ∈ C\0. In addition, A(•, •, λ) ∈ L 2 (C × C), |TrA 2 (λ)| < ∞,
where A(z, ζ, λ) is the Schwartz kernel of the integral operator A(λ) of the integral equation (2.10). Thus, the modified Fredholm determinant for (2.10) can be defined by means of the formula:

ln ∆(λ) = Tr(ln(I -A(λ)) + A(λ))
(2.11)

(see [GK] for more precise sense of such definition).

Taking the subsequent members in the asymptotic expansion (2.5) for ψ(z, λ), we obtain (see [N2]):

ψ(z, λ) = exp - 1 2 λz + z λ 1 -2πsgn(1 -λ λ)× × iλa(λ) z -λ 2 z + exp - 1 2 1 λ -λ z + 1 λ -λ z λb(λ) i( λ2 z -z) +o 1 |z| , (2.12) |z| → ∞, λ ∈ C\(E ∪ 0).
The functions a(λ), b(λ) from (2.12) are called the "scattering" data for the problem (2.1), (2.2) with E = -1. It is known that for a(λ), b(λ) the following formulas hold (see [N2]):

a(λ) = 1 2π 2 z∈C µ(z, λ)v(z)dz R dz I , (2.13) b(λ) = 1 2π 2 z∈C exp - 1 2 λ - 1 λ z -λ - 1 λ z µ(z, λ)v(z)dz R dz I , (2.14)
where λ ∈ C\(0 ∪ E), z R = Rez, z I = Imz. In addition, formally, formulas (2.13), (2.14) can be written as

a(λ) = h(λ, λ), b(λ) = h λ, 1 λ , (2.15)
where

h(λ, λ ′ ) = 1 2π 2 z∈C exp 1 2 (λ ′ z + z/λ ′ ) ψ(z, λ)v(z)dz R z I , (2.16) and λ ∈ C\(0 ∪ E), λ ′ ∈ C\0. (Note that, under assumptions (2.2), the integral in (2.16) is well-defined if λ ′ = λ of if λ ′ = 1/ λ but is not well- defined in general.) Let T = {λ ∈ C : |λ| = 1}.
(2.17)

From (2.15), in particular, the following statement follows:

Statement 2.1. Let (2.2) hold and ∆ = 0 on T . Then

a(λ) = b(λ), λ ∈ T.
(2.18)

The following properties of functions ∆(λ), a(λ), b(λ) will play a substantial role in the proof of Theorem 1.1.

Statement 2.2. Let (2.2) hold. Then:

1. ∆(λ) ∈ C(C); 2. ∆(λ) → 1 as λ → 0 or λ → ∞; 3. ∆(λ) ≡ const for λ ∈ T ; 4. ∆ is real-valued: ∆ = ∆. 5. ∆(λ) = ∆(1/ λ), λ ∈ C\0. Statement 2.3. Let conditions (2.2)-(2.
3) be fulfilled. Then:

• ∆(λ) is a real-analytic function on D + , D -, where D + = {λ ∈ C : 0 < |λ| 1}, D -= {λ ∈ C : |λ| 1}. (2.19) • a(λ) = A(λ) ∆(λ) , b(λ) = B(λ) ∆(λ)
, where A(λ), B(λ) are real-analytic func-

tions on D + , D -.
Items 1-4 of Statement 2.2 are either known or follow from results mentioned in [HN], [N2] (see page 129 of [HN] and pages 420, 423, 429 of [N2]). In particular, item 1 of Statement 2.2 is a consequence of continuous dependency of g(z, λ) on λ ∈ C\0; item 3 of Statement 2.2 is a consequence of (2.11) and of the formula (see pages 420, 423 of [N2]) G(z, λ) = (-i/4)H 1 0 (i|z|), z ∈ C, λ ∈ T , where G is defined by (2.9), H 1 0 is the Hankel function of the first type. In addition, item 5 of Statement 2.2 follows from item 4 of this statement and from symmetry

G(z, λ) = G(z, 1/ λ), z ∈ C, λ ∈ C\0.
Statement 2.3 is similar to Proposition 4.2 of [N2] and follow from: (i) formulas (2.13), (2.14), (ii) Cramer type formulas for solving the integral equation (2.10), (iii) the analog of Proposition 3.2 of [N2] for g of (2.7).

Under assumptions (2.2), the function µ(z, λ), defined by (2.6), satisfies the following properties:

µ(z, λ) is a continuous function of λ on C\(0 ∪ E);
(2.20)

∂µ(z, λ) ∂ λ = r(z, λ)µ(z, λ), (2.21a) r(z, λ) = r(λ) exp 1 2 λ - 1 λ z -λ - 1 λ z , (2.21b)
r(λ) = πsgn(1 -λ λ) λ b(λ) (2.21c) for λ ∈ C\(0 ∪ E); µ → 1, as λ → ∞, λ → 0. (2.22)
The function b possesses the following properties (see [START_REF] Grinevich | Two-dimensional inverse scattering problem for negative energies and generalized-analytic functions. I. Energies below the ground state[END_REF], [N2]):

b ∈ C(C\E), (2.23) b - 1 λ = b(λ), b 1 λ = b(λ), λ ∈ C\0, (2.24) λ -1 b(λ) ∈ L p (D + ) (as a function of λ) if E = ∅, 2 < p < 4.
(2.25)

In addition, the following theorem is valid: ]). Let v satisfy (2.2) and E = ∅ for this potential. Then v is uniquely determined by its scattering data b (by means of (2.20), (2.21) and equation (1.1) for E = -1 and ψ of (2.5) ).

Theorem 2.1 ([GN2], [ N2
Finally, if (v(z, t), w(z, t)) is a solution of equation (1.1) with E = -1, where (v(z, t), w(z, t)) satisfy the following conditions: v, w ∈ C(R 2 × R) and for each t ∈ R the following properties are fulfilled:

v(•, t) ∈ C 3 (R 2 ), ∂ j x v(x, t) = O |x| -2-ε for |x| → ∞, |j| ≤ 3 and some ε > 0, (2.26) w(x, t) → 0 for |x| → ∞,
then the dynamics of the scattering data is described by the following equations

a(λ, t) = a(λ, 0), (2.27) b(λ, t) = exp λ 3 1 λ 3 -λ3 - 1 λ3 t b(λ, 0), (2.28)
where λ ∈ C\0, t ∈ R.

3 Proof of Theorem 1.1

The proof of Theorem 1.1 is based on Proposition 3.1 and Proposition 3.2 given below.

a ζ (λ) = a(λ), (3.1) b ζ (λ) = exp - 1 2 λ - 1 λ ζ -λ - 1 λ ζ b(λ), (3.2) where z, ζ ∈ C, λ ∈ C\0.
Proof. We first note that ψ(z -ζ, λ) satisfies equation (2.1) with the operator

L = -∆ + v ζ (z). Then the function ψ ζ (z, λ) corresponding to v ζ (z) and possessing the asymptotics (2.5) is ψ ζ (z, λ) = e -1 2 (λ ζ+ζ/λ) ψ(z -ζ, λ). In terms of function µ this relation is written µ ζ (z, λ) = µ(z -ζ, λ) Thus we have a ζ (λ) = 1 2π 2 z∈C v(z -ζ)µ(z -ζ, λ)dz R dz I = a(λ),
and, similarly,

b ζ (λ) = 1 2π 2 z∈C exp - 1 2 λ - 1 λ z -λ - 1 λ z × × v(z -ζ)µ(z -ζ, λ)dz R dz I = = exp - 1 2 λ - 1 λ ζ -λ - 1 λ ζ b(λ).
Proposition 3.1. Let (v(z, t), w(z, t)) be an localized soliton of (1.1) in the sense (1.7). Let b(λ, t) be the scattering data for v(z, t) for some E = E f ixed < 0. Then b(λ, t) ≡ 0 in the domain where it is well-defined, i.e. in C\E, where E is defined by (2.4).

Proof. In virtue of (2.28) and Statement 2.3 it is sufficient to prove that b(λ, 0) ≡ 0 in some neighborhoods of 0 and ∞. Let U 0 , U ∞ be the neighborhoods of 0 and ∞, respectively, such that ∆ = 0 in U 0 , U ∞ (such neighborhoods exist in virtue of item 2 of Statement 2.2). For λ ∈ U 0 ∪ U ∞ the function b(λ, 0) is well-defined and continuous. As (v(z, t), w(z, t)) is a soliton, the dynamics of the function b(λ, t) can be written as

b(λ, t) = exp - 1 2 λ - 1 λ c -λ - 1 λ c t b(λ, 0) (3.3) (see Lemma 3.1).
Combining this with formula (2.28), we obtain exp -

1 2 λ - 1 λ c -λ - 1 λ c t b(λ, 0) = = exp λ 3 + 1 λ 3 -λ3 - 1 λ3 t b(λ, 0). As functions λ, λ, 1 λ , 1 λ , λ 3 , λ3 , 1 λ 3 , 1 λ 3 , 1 are linearly independent in any neighborhood of 0 and ∞, we obtain that b(λ, 0) ≡ 0 in U 0 ∪ U ∞ . Proposition 3.2. Let v(z) satisfy (2.2)-(2.
3) and b(λ) be its scattering data for some E = E f ixed < 0. If b(λ) ≡ 0 in the domain where it is well-defined, i.e. in C\E, where E is defined by (2.4), then v ≡ 0.

Note that Proposition 3.2 can be considered as an analog of Corollary 3 of [START_REF] Grinevich | Transparent potentials at fixed energy in dimension two. Fixed energy dispersion relations for the fast decaying potentials[END_REF].

Proof of Proposition 3.2.

1. First we will prove that from the assumptions of this proposition it follows that a(λ) ≡ 0 in U 0 ∪ U ∞ , where U 0 and U ∞ are such neighborhoods of 0 and ∞, respectively, that ∆(λ) = 0 for λ ∈ U 0 ∪ U ∞ . We note that from (2.13), (2.14), (2.21a), (2.22) it follows that

∂a(λ) ∂ λ = πsgn(1 -λ λ) λ b(λ)b(λ), λ ∈ C\(E ∪ 0), (3.4) a(λ) → v(0) as λ → ∞ or λ → 0, where (3.5) v(p) = 1 2π 2 z∈C e i 2 (pz+pz) v(z)dz R dz I , p ∈ C. (3.6) It means that a(λ) is holomorphic in C\(E). (3.7)
According to item 3 of Statement 2.2, ∆(λ) ≡ const for λ ∈ T . We will consider separately two cases: ∆ ≡ C = 0 on T and ∆ ≡ 0 on T .

(a) ∆(λ) ≡ C = 0 on T : From item 1 of Statement 2.2 it follows that there exists U T , a neighborhood of T , such that ∆(λ) = 0 in U T . Thus a(λ) is holomorphic in U T . From Statement 2.1 we obtain that a(λ) = b(λ) = 0 on T . It follows then that a(λ) ≡ 0 in U T . Using statement 2.3, we obtain that a(λ) ≡ 0 in U 0 ∪ U ∞ .

(b) ∆(λ) ≡ 0 on T :

In [HN] the ∂-equation for ∆ was derived. In variables λ, λ it is written as

∂ ln ∆(λ) ∂ λ = - πsgn(λ λ -1) λ a 1 λ -v(0) . (3.8) Equation (3.8) and properties (3.5), (3.7) imply that ∂ ln ∆ ∂ λ is an antiholomorphic function in U 0 ∪ U ∞ ,
where ∆ is close to 1 and, thus, ln ∆ is a well-defined one-valued function. As ∆ is a realvalued real analytic function, it follows that

ln ∆ = f (λ) + f (λ) (3.9)
for some holomorphic function f (λ), or

∆ = F (λ)F (λ) (3.10)
for some holomorphic function F (λ) on U 0 ∪ U ∞ . Now we will use the following lemma (the proof of this lemma is given in Section 4):

Lemma 3.2. Let ∆(λ) be real-analytic in D + = {λ ∈ C : 0 < |λ| ≤ 1}.
Suppose that ∆(λ) can be represented as

∆(λ) = F (λ)F (λ), λ ∈ U 0 , (3.11)
for some function F (λ) holomorphic on U 0 , a neighborhood of zero. Then the representation (3.11) holds on D + , i.e. F (λ) can be extended analytically to D + .

Thus, the representation (3.10) is valid separately on D + and on D -, where we used also item 5 of Statement 2.2. As ∆(λ) ≡ 0 on T , it follows that F (λ) ≡ 0 on T and, further, F (λ) ≡ 0 on C. This contradicts with item 2 of Statement 2.2. Thus we have shown that under the assumptions of Proposition 3.2 the case ∆(λ) ≡ 0 on T cannot hold.

2. Our next step is to prove that ∆(λ) ≡ 1 for λ ∈ C.

Formula (3.5) states that a(0) = a(∞) = v(0). Thus from equation (3.8) it follows that ∂ ln ∆ ∂λ = 0, and ln ∆ is holomorphic in some neighborhood of 0 and ∞. As ∆(λ) is a real-valued function and item 2 of Statement 2.2 holds, we conclude that ∆ ≡ 1 in some neighborhood of 0 and ∞. Now using Statement 2.3, we obtain that ∆ ≡ 1 in C and, as a corollary, E = ∅.

3. From the previous item it follows that equation (2.21a) holds for ∀λ: λ ∈ C\0. Due to the assumptions of Proposition 3.2 and the property that E = ∅, we have that b ≡ 0 on C which means that µ(z, λ) is holomorphic on D + , D -. As it is also continuous on C and property (2.22) holds, we conclude that µ(z, λ) ≡ 1 and v(z) ≡ 0.

Proof of Theorem 1.1 for E = E f ixed < 0. The result follows immediately from Propositions 3.1, 3.2.

4 Proof of Lemma 3.2

As F (λ) is analytic in U 0 , it can be represented in this domain by a Taylor series. Let us consider the radius of convergence R of this Taylor series. Suppose that the statement of Lemma 3.2 is not true and R < 1.

Let us take a point λ 0 , such that |λ 0 | = R. In this point ∆(λ) can be represented by the following series

∆(λ) = ∞ k,j=0 b k,j (λ -λ 0 ) k ( λ -λ0 ) j (4.1)
uniformly convergent in U λ 0 , some neighborhood of λ 0 . We will prove that the coefficients b j,k satisfy the following properties:

b k,k ∈ R, b k,k ≥ 0; (a) b k,j = b j,k ; (b) b k,j b m,l = b k,l b m,j . (c) Indeed, (a) : b k,k = 1 (k!) 2 ∂ k λ ∂ k λ ∆(λ) λ=λ 0 = lim λ→λ 0 1 (k!) 2 ∂ k λ ∂ k λ ∆(λ) = lim λ→λ 0 1 (k!) 2 |∂ k λ F (λ)| 2 ∈ R, ≥ 0. (b) : b k,j = lim λ→λ 0 1 k!j! ∂ k λ F (λ)∂ j λ F (λ) = lim λ→λ 0 1 k!j! ∂ k λ F (λ)∂ j λ F (λ) = b j,k . (c) : b k,j b m,l = lim λ→λ 0 1 k!j!m!l! ∂ k λ F (λ)∂ j λ F (λ)∂ m λ F (λ)∂ l λ F (λ) = b k,l b m,j .
From properties (a)-(c) it follows that there exist such

a k ∈ C, k = 0, 1, . . ., that b k,j = a k āj . (4.2)
We will prove this statement by considering two different cases:

1. b k,k = 0 ∀k ∈ N ∪ 0
In this case from properties (b), (c) it follows that b k,j = 0 ∀k, j ∈ N∪0, and we can take a k = 0 ∀k ∈ N ∪ 0.

2. b k,k = 0 for some k ∈ N ∪ 0.

In this case we take l to be the minimal number such that b l,l = 0. Then we set a 0 = a 1 = . . . = a l-1 = 0 and we take an arbitrary From convergence of series (4.1) it follows that the following series The same reasoning can be applied to any point λ 0 on the boundary of the ball B R = {λ ∈ C : |λ| ≤ R}, i.e. F (λ) can be continued analytically to some larger domain. Hence we obtain a contradiction to the assumption that R < 1 is the radius of convergence of the Taylor series for F (λ). Thus R = 1 and F (λ) can be extended analytically to D + .

F 1 (λ) =
[ZS] Zakharov V.E., Shulman E.I.: Integrability of nonlinear systems and perturbation theory // What is integrability? Berlin: Springer-Verlag. 185-250 (1991)

Lemma 3. 1 .

 1 Let v(z) satisfy (2.2) and a(λ), b(λ) be the scattering data corresponding to v(z). Then the scattering data a ζ (λ), b ζ (λ) for the potential v ζ (z) = v(zζ) are related to a(λ), b(λ) by the formulas

  complex number a l satisfying |a l | 2 = b l,l . For the rest of the coefficients we set a n = b n,l āl , (4.3) where n = l + 1, l + 2, . . .. Now let us prove property (4.2). Let us suppose that k < l. Then a k = 0, b k,k = 0 and from properties (b), (c) it follows that b k,j = 0 ∀j ∈ N ∪ 0. Thus property (4.2) holds when k < l. A similar reasoning can be carried out when j < l. Now let us suppose that k ≥ l, j ≥ l. Then a k āj = b k,l bj,l āl a l = b k,l b l,j b l,l = b k,j . (4.4) Representation (4.2) is proved.

  U λ 0 (indeed, the case when b k,k = 0 ∀k ∈ N ∪ 0 is trivial, and in the case when ∃l : b l,l = 0 we take the sum of the members of series (4.1) with coefficients b k,l , k = 0, 1, . . ., and obtain series (4.5) multiplied by āl ( λ -λ0 ) l ). Thus there exists the function F 1 (λ) analytic in U 0 such that ∆(λ) = F 1 (λ)F 1 (λ). Consequently, we have two functions F (λ) and F 1 (λ) analytic in a common domain lying in {λ ∈ C : |λ| ≤ R} ∩ U λ 0 and such that |F (λ)| = |F 1 (λ)|. It means that F (λ) and F 1 (λ) are equal up to a constant factor: F (λ) = µF 1 (λ), |µ| = 1. It follows then that µF 1 (λ) is an analytic continuation of F (λ) to U λ 0 .

Centre des Mathématiques Appliquées, Ecole Polytechnique, Palaiseau, 91128, France; email: kazeykina@cmap.polytechnique.fr

CNRS (UMR 7641), Centre des Mathématiques Appliquées, Ecole Polytechnique, Palaiseau, 91128, France; email: novikov@cmap.polytechnique.fr