Claudine Chamoreau

Equative and similative constructions in Chibchan languages

INTRODUCTION

Equative and similative constructions have been understudied in linguistic literature, contrary to comparative constructions of inequality, which have been studied from areal and typological perspectives, in particular, the comparison of superiority [START_REF] Andersen | Word order typology and comparative constructions[END_REF][START_REF] Stassen | Comparison and universal grammar[END_REF][START_REF] Heine | Cognitive foundations of grammar[END_REF][START_REF] Cuzzolin | Comparison and gradation[END_REF][START_REF] Dixon | Comparative constructions: A cross-linguistic typology[END_REF]. [START_REF] Haspelmath | Equative and similative constructions in the languages of Europe[END_REF], [START_REF] Henkelmann | Constructions of equative comparison[END_REF] and [START_REF] Haspelmath | the Leipzig Equative Constructions Team[END_REF] have proposed the first typologies of equative and similative constructions. Haspelmath & Buchholz (1998: 278) distinguish both types of constructions and indicate that "equatives express equal extent, and similatives express equal manner". The former constructions express quantity, that is, "Equative constructions express situations in which two referents have a gradable property to the same degree" (Haspelmath et al. 2017:9), as in Max is as tall as John, while the latter structures express quality, as in Max is tall like John. This distinction, semantically and formally present in European languages, is not attested in all languages (see Pakendorf, this volume). This is also the case for various languages of the Chibchan family, which consists of sixteen understudied languages spoken in Central America and in the north of South America, from Honduras to Colombia.

The aims of this paper are to offer a first description of equative and similative constructions in twelve of the sixteen extant Chibchan languages and to illustrate two typological particularities within this family. The languages were chosen on the basis of the existence of reliable data (texts and/or grammars). They all share a construction with a similative standard marker and the majority have only one construction for expressing similarity and equality.

So, on the one hand, the construction shared in the Chibchan languages contains a comparee (C), a parameter (P) that is a property, a standard entity (SE) and a standard marker 'like' (SM), but no equative degree marker. The SE and SM form a standard phrase (SP). This construction type can be analysed as:

(1)

Max is tall like John

C P [SM SE]SP
The construction in (1), which is Type 1 in Haspelmath et al.'s (2017) typology, is the basic type in these twelve Chibchan languages. In many languages, this is the sole construction, and it is used for both equative and similative comparison. However, although many Chibchan languages exhibit several features associated with SOV order (Constenla [START_REF] Umaña | The Chibchan languages[END_REF][START_REF] Quesada | The Chibchan languages, Costa Rica[END_REF], the constituent order in the standard phrase shows a crucial distinction. In the SP, the standard marker always follows the standard entity, but some languages (Pesh, the Votic languages in the north and the Magdalenian languages in the south) present the expected order for SOV languages, that is, the SP precedes the parameter. An SP has the same position in relation to a parameter as an object does in relation to a verb. But other languages (notably the central Isthmian languages) show the reverse order, that is, the SP follows the parameter and therefore has a different position in relation to a parameter than an object does in relation to a verb. The variation is thus between the standard phrase and the parameter. The Isthmian languages are in contradiction with two generalisations presented in Haspelmath et al. (2017:26-27): Generalisation 2, "If the parameter follows the standard, then the language generally has dominant object-verb order"; and Generalisation 3, "If the standard precedes the parameter, then the standard marker generally follows the standard, and if the standard follows the parameter, then the standard marker generally precedes the standard".

On the other hand, the majority of Chibchan languages under investigation have only one construction for expressing similarity and equality, but four languages have the possibility of distinguishing equality from similarity by using different constructions. In addition to the construction with a marker 'like', Malecu, Cabecar, Ika and Pesh have dedicated constructions for the expression of equality.

This paper is organised as follows: Section 2 introduces some of the main morphosyntactic features of Chibchan languages. Section 3 describes the basic construction type with the SP containing a similative standard marker attested in all the languages for expressing similarity and equality. This encompasses two subtypes with a split in constituent order between northern and southern Chibchan on the one hand, and central Chibchan on the other. Section 4 discusses various dedicated constructions in four Chibchan languages for expressing either equative or similative meanings. Section 5 shows the conceptual relations between similative markers and equative, deictic and simultaneity markers and advances hypotheses on the sources of these markers. Section 6 concludes the paper.

MAIN MORPHOSYNTACTIC FEATURES

The Chibchan family encompasses sixteen living languages (Constenla [START_REF] Umaña | Comparative Chibchan phonology[END_REF][START_REF] Umaña | Las lenguas del área intermedia: introducción a su estudio areal, San José[END_REF][START_REF] Quesada | Gramática del Buglere[END_REF][START_REF] Holt | The development of the Paya sound system[END_REF][START_REF] Quesada | The Chibchan languages, Costa Rica[END_REF][START_REF] Uhle | Verwandtschaften und Wanderungen der Tschibtscha[END_REF] spoken from Honduras to Colombia (see figure 1). The languages studied in this paper are underlined (in this introduction and in figure 1). Pesh (also known as Paya) is the northernmost of the Chibchan languages, the only one spoken in Honduras, and the only one to be classified as an isolate within the Chibchan family: Some languages, such as Buglere and Costa Rican Guaymi, have nominative-accusative alignment (known as marked nominative), Teribe (with an inverse morpheme), and perhaps Rama and Cuna (although the available descriptions are unclear). Others, such as Malecu, have ergative alignment. Of the ergative languages some, such as Cabecar, Panamanian Guaymi, Bribri and Damana have split ergativity in TAM, voice or verb type; others, such as Ika, Kogi and Pesh, have split ergativity in NP type or in the free-bound distinction [START_REF] Quesada | The Chibchan languages, Costa Rica[END_REF]:193, Chamoreau 2017a).

All Chibchan languages have SOV constituent order (Constenla [START_REF] Umaña | Las lenguas del área intermedia: introducción a su estudio areal, San José[END_REF]:38, 2012:407, Quesada 2007:29-31) and postpositions. The order within the noun phrase is noun-numeral and genitive-noun. The transitive clauses in (2) show that the NPs that function as subject and object generally precede the predicate. Some languages have rigid OV order while other languages allow OV/VO variation. For example, in Bribri and Cabecar, OV order is constrained but S can either precede or follow the verb, that is, SOV or OVS [START_REF] Quesada | Ergativity in Chibchan[END_REF]. In other languages, the OV constraint is not attested and O usually occurs before the verb but may also be attested after it, this movement usually being motivated by information structure as in Pesh for focalisation. Arguments may be case marked as ergative in Cabecar in (2c) or differentially as in Pesh (2a) and Ika (2e) (see Chamoreau 2017a and[START_REF] Frank | Ika syntax[END_REF]). In Chibchan languages only oblique and adjunct phrases (X), that is postpositional phrases, are overtly marked for their syntactic function. Depending on the language, grammatical markers can have varying morphological status. We find independent postpositions in Teribe (3) and Costa Rican Guyami (4), enclitics in Pesh (5) and Damana (6), and suffixes in Ika (7) and Malecu (8).

( Examples (3-8) reveal a fact that is relevant for our study: even though the constituent order is usually OV in all these languages (VO is always motivated by information structure), the postpositional phrases occur in different positions. Two tendencies exist: In Pesh (5), Malecu (8), Damana (6) and Ika ( 7), the PP usually precedes the verb, that is XV, while in the Isthmian languages, Teribe (3) and Guaymi (4), the PP is usually postposed to the verb, that is VX. These tendencies may be modified by information structure.

ONE CONSTRUCTION FOR SIMILARITY AND EQUALITY

The twelve Chibchan languages studied in this paper display one construction with a standard phrase for both equative and similative comparison. This is the sole possibility in eight languages (Rama, Bribri, Teribe, Costa Rican Guaymi, Panamanian Guaymi, Buglere, Cuna, Damana). This standard phrase consists of a standard entity and a standard marker, this latter marker uniformly following the standard entity (section 3.1). However, the order between the standard phrase and the parameter shows a variation: the standard phrase may precede the parameter (9) or follow it (10). This trait is relevant for the later discussion of constituent order generalisations in Chibchan languages (section 3.2.).

Basic construction with a standard phrase

The basic construction type with only a similative standard marker is attested in all languages investigated. This construction is used both for similatives, (e.g. Max is tall like John), and for equatives (e.g. Max is as tall as John). In the majority of Chibchan languages, this construction is the only way to express both meanings (except in languages that will be described in section 4).

This construction type is illustrated in three Isthmian languages below. Here, the suffix -re is used for manner and similarity. The constructions are analysed by the different authors as similative constructions, in Costa Rican Guaymi (11), Panamanian Guaymi (12), and Buglere (13). Furthermore, in the two varieties of Guaymi, the -re marker is used to form postpositions that express meanings such as 'same', 'like '. In Costa Rican Guaymi, Murillo Miranda (2016:84) indicates that the combination of the suffix -re 'like' with ere 'much' forms the postposition erere, which means 'same/like' ( 14). The same phenomenon is observed in Panamanian Guaymi, wherere may be used alone (12) or in combination with kwere 'much', with the resultant postposition kwerere, meaning 'way of, like, as' ( 15). No equative constructions are described in the two works on Guaymi languages. However, in Buglere, the equative construction in ( 16) contains the postposition kare. This marker is clearly related to the suffix -re 'like' in (13) from Buglere as well as the suffix -re 'like' found in Guaymi (11,12). According to Quesada (2012:132), the additive postposition kare 'too, also' is used in equative constructions in which the two referents, Maria and Pedro in ( 16), have a gradable property (speed) to the same degree. [START_REF] Quesada | Gramática del Buglere[END_REF] only mentions the use of kare in Buglere equative constructions but not in constructions with a similative meaning. By contrast, [START_REF] Quesada | Information structure and the expression of givenness in Buglere (Chibchan)[END_REF] shows that kare can express similarity (17). One could use 'likewise' to translate this term to mirror its morphological structure. The use of the same marker in similative and equative constructions is also attested in other Chibchan languages, such as Damana. In ( 18), the suffix -ia 'like' and the adverbialiser -ai are used as derivational suffixes to form the adverb 'yesterday', literally 'like the place inside'. In ( 19), these two suffixes are used to form an equative construction, according to [START_REF] Williams | A grammar sketch of Damana[END_REF]. Nevertheless,in (19), the similative meaning also seems possible, cf. the free translation 'His hair is bright like cotton'. In Pesh, the same marker =kan is used for similarity and equality but the position of the standard phrase that contains the standard entity and the standard marker differs depending on the meaning. In (20), the similative marker =kan is used as an enclitic after nouns and pronouns to build phrases that express similarity of manner between two entities. The SP pokupoku=kan occurs before the parameter nast 'jump'. In (21), this enclitic is also used for equatives. In this construction, the SP ũtʃã=kan 'fish' appears after the parameter tapeh 'swim'. Generally, the distinction between similative meaning (20) and equative meaning ( 21) is made by the position of the standard phrase marked by =kan: this occurs before the predicate in similative constructions (20) and after it in equative constructions (21). The parameter is expressed by the predicate (that may be a verbal or a nominal predicate). The latter position is unexpected. The postpredicate position of the equative phrase seems to reflect the need to distinguish between similative and equative phrases because both use the marker =kan. If we compare ( 20) and ( 21), only the position of the standard phrase makes it possible to distinguish the two construction types. The similative phrase preserves the typical and expected position of the postpositional phrase, while the equative phrase looks like an innovation which does not reflect the constituent order expected in a SOV language.

This pattern is attested in the majority of the examples found in my corpus (Chamoreau 2017b:328-329). Quantitatively (based on 47 tokens), in 52.5% of prepredicate postpositional phrases marked by =kan, the phrase can only have a similative meaning; in 17.5%, it can only have an equative meaning; in 30%, both interpretations are possible. As for postpredicate postpositional phrases marked by =kan, in 70.2% of instances the meaning is always equative, and in 29.8% the distinction between similative and equative is not clear, either because there are no semantic differences depending on the standard phrase position so both interpretations are acceptable, or because speakers do not establish a difference between them. However, the postpredicate postpositional phrase marked by =kan has never been interpreted as having a purely similative meaning. In Pesh, the prepredicate position containing the standard entity and the standard marker that is the standard phrase ( 20) is most common even though the postpredicate position is possible.

In some Chibchan languages, such as Pesh (20), Rama (22), Malecu (21), Damana (19) and Ika (24) the standard phrase precedes the parameter. In contrast, in the Isthmian languages, such as Costa Rican Guaymi ( 14), Panamanian Guaymi (12), Buglere (13), ( 16) and ( 17), Bribri (25), Cabecar (26), Teribe (27) and Cuna (28), the standard phrase usually follows the parameter. Variations in the position of the postpositional and standard phrases may exist and are usually determined by information structure (Quesada 2007:195)

Constituent order in similative and equative constructions

As shown in section 3.1, the basic construction type for expressing similarity and equality in Chibchan languages is a standard phrase which contains a standard and a postposed standard marker. Across Chibchan languages, the standard phrase that contains the standard entity and the standard marker may precede the parameter or follow it. Given that Chibchan languages have verb-final constituent order, the expected order is standard phraseparameter. This is Haspelmath et al.'s (2017:26-27) Generalisation 2: "If the parameter follows the standard, then the language generally has dominant object-verb order". Some Chibchan languages, such as Pesh (20), the Votic languages Rama (22) and Malecu (23), and the Magdalenian languages Damana (19) and Ika (24) abide by this generalisation. These languages also support Haspelmath et al.'s Generalisation 3: "If the standard precedes the parameter, then the standard marker generally follows the standard, and if the standard follows the parameter, then the standard marker generally precedes the standard". In these languages, the standard phrase does indeed precede the parameter.

In contrast, the Isthmian languages, such as Costa Rican Guaymi ( 14), Panamanian Guaymi (15), Buglere (13,16,17), Bribri (25), Cabecar (26), Teribe (27) and Cuna (28), contradict the two generalisations. These languages usually have parameterstandard phrase order, which is in disagreement with Generalisation 2. Furthermore, they do not abide by Generalisation 3 because the standard phrase follows the parameter and the standard marker follows the standard entity.

A crucial issue for Chibchan languages is that the two possible positions of the standard phrase are consistent with the position of the postpositional phrase that follows two different tendencies (as shown in section 2). In Pesh (5) and in the Magdalenian languages ( 6), ( 7), the PP usually precedes the predicate, while in Isthmian languages (3, 4), the PP usually follows the predicate. The case of Malecu and Rama, both Votic languages, is noteworthy as the PP generally precedes the predicate, as shown by Malecu in (8). However, Rama also permits the reverse order; in ( 22) the PP nguu-ki 'in the house' follows the predicateeven though this position is less frequent [START_REF] Craig | A grammar of Rama[END_REF].

A consistent order is clearly mapped in Chibchan languages: PP-verb and SP-parameter orders are used in some languages while verb-PP and parameter-SP orders occur in others. I therefore postulate that: "If a language has different orders for [verb, object], and [verb, PP] then the order of [parameter, SP] follows the pattern for [verb, PP] rather than the pattern for [verb, object]".

This section has shown that the first type of Haspelmath et al.'s typology, i.e. constructions with only a standard marker, is attested in the twelve Chibchan languages discussed in this paper and is used for expressing both equative and similative meanings. In certain languages (Rama, Bribri, Teribe, Costa Rican Guaymi, Panamanian Guaymi, Buglere, Cuna, Damana), this construction is the only one used for both meanings; in other languages (Ika, Pesh, Cabecar, Malecu), alternative constructions exist. The data also show the relevance of a property that is part of a larger typological phenomenon, namely that the SP-parameter order corresponds to PP-verb order and not to object-verb order.

DISTINGUISHING EQUATIVES FROM SIMILATIVES

Four languages (Ika, Pesh, Cabecar, Malecu) usually use the construction described in section 3 for the expression of similarity and equality but in addition have a dedicated construction for the expression of equality. In this construction, a degree marker occurs. However, a distinction exists between Ika and the other three languages because in Ika the construction with the degree marker is mono-clausal (Section 4.1) while Pesh, Cabecar and Malecu have bi-clausal constructions (Section 4.2).

Ika: unified degree marker construction for equatives

As shown in example (23) repeated as ( 29), the Ika construction for similative meaning only contains the marker -i 'while' suffixed to the copula nar. The equative construction ( 30) is more complex than the construction used to express similative meanings because the comparee entity buru mouga nʌʔʌn zei 'my two burros' and the standard entity múra 'the mule' are expressed in a single conjoined NP unified by the marker -sin 'with'. The predicative parameter džuma aʔnikuʌža 'are strong' is preceded by the degree maker dikin 'same'. This marker is very likely cognate with the standard marker dik 'like' in Teribe ( 27).

(29) Similative construction in Ika (Magdelanian ;Frank 1990:111 This Ika equative construction corresponds to Type 3 of Haspelmath et al.'s typology (2017:14), in which there is "an ordinary predicative parameter with an equative degree-marker" but "the comparee and standard referents are unified, i.e. they are expressed as a single conjoined or plural noun phrase".

Pesh, Cabecar and Malecu: bi-clausal constructions for equatives

As we have shown above, Pesh uses the similative marker =kan in similative (20) and equative (21) constructions, reproduced in ( 31) and ( 32), respectively. In these constructions, the position of the standard phrase with the standard entity and the marker =kan is usually sufficient to distinguish between similatives and equatives. Nevertheless, when the standard phrase precedes the parameter (31), the equative meaning is also possible for some speakers. This is probably why another dedicated construction for expressing equality exists. The construction in question has two juxtaposed clauses (33). The first includes the comparee jose and the standard ugo, linked by the coordinator -rih; both function as the subject of the predicate, as shown by the presence of a third person plural subject suffix on the predicate. The marker of equality =kan is encliticised to the numeral as 'one' which functions as the nominal predicate of this clause, accompanied by the copular verb. The equality relationship between the comparee and the standard is expressed by the predicate ('one'). The second clause is juxtaposed with no overt coordinator and contains the gradable property, the parameter, expressed by the adjective wawista 'thin', also followed by the copular verb. The subject of this clause is the third person plural and refers to the comparee jose and the standard ugo. This is the syntactic way to indicate the coordination of two clauses in Pesh, the coordinator -rih only being used for coordinating noun phrases.

(33) Equative construction in Pesh (Isolate ;Chamoreau 2017b:328) jose-rih ugo-rih as=kan=r-er-wa wawista=r-er-wa José-COORD Hugo-COORD one=sim=be-S3PL-PRS thin=be-S3PL-PRS 'José is as thin as Hugo.' (Lit. 'José and Hugo are equal, they are thin.') As in Ika, Pesh expresses the comparee and the standard entities as a single conjoined or plural noun phrase, but, contrary to Ika, the Pesh construction is bi-clausal. This is type II in Henkelmann's description (2006:377), that is, a type with an equative degree as predicate. The Pesh construction constitutes a sub-type of type II in which the two predicates (the parameter and the degree) are related by parataxis (2006:391).

Cabecar displays another strategy in which the construction contains two clauses. As we have shown in (23) repeated in (34) Cabecar uses the standard marker kä͂í in similative constructions, see also (35). The equative construction in ( 36) is bi-clausal, the parameter shká that expresses the intensity/distance walked is repeated in each clause. The comparee entity bá '2sg' is expressed in the first clause while the standard entity yı ͂ś '1sg' occurs in the second clause. The two clauses are juxtaposed.

( Malecu, a Votic language, displays a construction with the similative marker inhá 'like' introduced in (23) and repeated in (37). In this construction, the standard phrase na-inhá is made up of the standard na first person exclusive singular and the standard marker inhá. This standard phrase precedes the parameter. Malecu also displays a bi-clausal strategy (38). The comparee entity ton '1sg' is expressed in the first clause with the parameter ampenhé 'is tall' while the standard entity Oyanhqui 'Oyanhqui' occurs in the second clause which is introduced by ninháfa 'also'. The adverb ninháfa is a combination of ninhá 'thus' and the marker =fa. Called an 'attributive theme' (Constenla Umaña 1998:148, 192), it occurs at the beginning of a juxtaposed clause (38). In the second clause of the type of construction that begins with ninháfa, the predicate may be repeated (39) or the elements that are identical in the two clauses may be deleted in the second clause, for example i-pcorro-ye unhé 'is working' (40) and ampenhé 'is tall' (38). The Cabecar and Malecu bi-clausal constructions used for equative constructions are very briefly reported in Haspelmath et al.'s typology (2017:23) as infrequently attested types. But they are described as conjoined comparative in Stassen's typology (1985:183-188), polarity schema in Heine's description (1997:116-118) and paratactic construction (type V) in Henkelmann's typology (2006:377-378). For equative comparison, they could be described as the juxtaposition of two clauses in which the parameter is repeated in Cabecar and may be deleted in Malecu, as an economic stripping process since it is identical in the two clauses: The comparee appears in the first clause and the standard entity occurs in the second. The equal meaning is not totally implicit as Henkelmann (2006:395) claims because even if the two clauses seems to be parallel to each other, the standard marker kä͂í 'like' in Cabecar and the expression ninháfa 'also' are used in the second clause.

The bi-clausal constructions seem to be a common structure for expressing comparison in various Chibchan languages as in older Pesh comparison of equality (41), or in Malecu comparison of superiority (42). Distinguishing similarity from equality seems not to be relevant in the majority of the Chibchan languages investigated in this study. The distinction has only been found in some of the languages that have been presented in section 4. For the other languages, the descriptions do not report any formal distinction and the free translations include both similative and equative meanings.

SAME FORM IN DIFFERENT CONSTRUCTIONS WITH DIFFERENT MEANINGS

This section highlights possible conceptual relations between different constructions with different meanings which use the same form in the Chibchan languages surveyed.

In most languages, the similative morpheme (same manner) is also used to encode equality (same extent). The relation between standard and equative markers comes from the fact that a conceptual relation exists between similative and equative. The most probable diachronic hypothesis is that the similative morpheme has been extended in use from similative to equative constructions. This statement follows from a general assumption that the equative meaning is a restricted case that should be traced back to the similative meaning. Pesh is a clear example of this diachronic process (Chamoreau 2017b In Malecu and Buglere, the relation between similative and equative also exists but the similative marker occurs with another morpheme. In Malecu, the degree marker ninháfa 'also' is formed by the adverb ninhá 'thus' plus the emphatic marker =fa. In Buglere, -re is the similative morpheme (13) and kare is an additive morpheme which possibly contains the similative morpheme -re (16). No etymological explanation is known for the morpheme ka. In these two languages, the probable source of one of the morphemes that composes the standard or equative marker is a similative morpheme (44).

(44) Markers for which the probable source of one of their components is a similative marker Malecu ninháfa 'also' Buglere kare 'too'

Another well-known relation is between standard markers and manner deictics [START_REF] König | The deictic identification of similarity[END_REF]. In Bribri, the standard marker eʔs in (25) reproduced in (45) comes from the distal demonstrative eʔ, which is used to refer to distal entities in space and time that are not accessible to the speakers (46).

( Example (22) reproduced in (47) shows that eʔs is used in an equative construction (Constenla Umaña et al. 1998:7). The same morpheme is also used in a similative construction (48). The marker s added to the demonstrative eʔ could come from the adjectiviser is (Constenla Umaña et al. 1998:4) The last relation between similative marker and other meanings in the Chibchan languages addressed in this paper is found in Ika. This language uses the -i marker 'while' plus a copula for similative constructions (21, 49): This -i 'while' marker is also used in adverbial clauses with a verb indicating a simultaneously occurring action (50). It seems that in Ika the similative marker 'COP-while' shows the relation between similarity and simultaneity.

(50) Ika (Magdalenian, Frank 1990:110) perɨ-seʔ-ri win-was-i žun-na dog-ERG-TOP 3S-chase-while go.down-DIST 'The dog went down chasing it'

CONCLUSION

This paper has shown that the domains of equality and similarity in various Chibchan languages show formal similarities among the constructions used to express these two meanings. First, in most languages (Rama, Bribri, Teribe, Costa Rican Guaymi, Panamanian Guaymi, Buglere, Cuna, Damana) no formal distinction is made between equality and similarity. Only four Chibchan languages (Ika, Pesh, Malecu, Cabecar) make this distinction. In three of these languages (Pesh, Malecu, Cabecar), equality shows a complex bi-clausal construction. Second, this paper has shown that the most frequent conceptual relation is between similative markers and equative, deictic and simultaneity markers.

Third, all the Chibchan languages investigated in this paper have a construction with a similative standard marker to express similative and equative meanings. In this type of construction, it is interesting to distinguish between the languages that have the expected constituent order for SOV languages, that is, the ones that are consistent with Generalisation 2 and Generalisation 3 in Haspelmath et al. (2017:26-27), from those that do not abide by these generalisations. This distinction reveals a split in the Chibchan family between the northern Pesh, Votic and Southern Magdalenian languages on the one hand and the Central Isthmian languages on the other. This split is relevant for two reasons. First, cross-linguistically and typologically, the order of verb and object is generally the same as the order of verb and PP (Dryer 2007:89). This is true for Pesh, and the Votic and Magdalenian languages, where we find OV and XV. In contrast, Isthmian languages do not follow this tendency, as object and PP have different positions, that is OV and VX respectively. Nevertheless, in a coherent way, in the languages that have XV, the SP precedes the parameter and in the languages that have VX, the parameter precedes the SP. Thus, beyond Haspelmath et al.'s (2017) two generalisations, the Chibchan languages favour internal constituent order coherence. This leads me to postulate that: "If a language has different orders for [verb, object], and [verb, PP] then the order of [parameter, SP] follows the pattern for [verb, PP] rather than the pattern for [verb, object]".

Second, at the areal level, the split reveals a geographical particularity. The central Isthmian languages constitute a first subtype. They have VX and parameter-SP constituent order, and they are spoken in Costa Rica and Panama (see figure 1). This VX subtype separates the two areas where the second subtype of languages with XV and SP-parameter order are found. This second subtype regroups the northern Pesh and Votic languages with the geographically distant southern Magdalenian languages. The northern languages are located in Honduras, Nicaragua and north of Costa Rica while the southern languages are located in Colombia. Despite their geographical distance, Pesh, the northernmost of the Chibchan languages, and Ika and Kogi, which are the southernmost, also show some other notable similarities, such as split ergativity governed by a free-bound distinction (Chamoreau 2017a[START_REF] Quesada | The Chibchan languages, Costa Rica[END_REF]).

  it does not belong to Core Chibchan. The other languages are distributed across several branches. Rama in Nicaragua and Malecu in Costa Rica constitute the Votic branch (also known as the Pota branch). The Isthmian branch is divided into three sub-branches: Western Isthmian in Costa Rica and Panama (Cabecar, Bribri, Teribe, Boruca), Doracic with two extinct languages (Changuena, Dorasque), and Eastern Isthmian located in Costa Rica and Panama (Costa Rican Guaymi, Panamanian Guaymi, Buglere, Cuna). Finally, the Magdalenian branch is split into two sub-branches located in Colombia (and also in Venezuela for Bari): Southern (Tunebo, Bari and the extinct Muisca, Duit) and Northern (Kogui, Damana, Ika, Chimila and the extinct Atanques).

Figure 1 .

 1 Figure 1. Chibchan language family (adapted from Constenla Umaña 2012)

( 11 )

 11 Similative construction in Costa Rican Guaymi (Isthmian; Murillo Miranda 2016:42) , as though they were one person.'

( 18 )

 18 Similative construction in Damana (Magdalenian; Williams 1993:85) ša-ia-ai place.in-like-ADZ 'like the place inside, yesterday' (19) Equative construction in Damana (Magdalenian; Williams 1993:148) na-že ša zidua buruna ia-ai bo 3S-POSSA hair careful cotton like-ADZ light k-ə-ža-n-ka kəma na-a-áADV-BEN-be-VBZ-3S.PRTC INFW be-PFV-ST 'His hair is as bright as cotton.' / 'His hair is bright like cotton.'

( 20 )

 20 Similative construction in Pesh (Isolate; Chamoreau 2017b:326) jumps like a frog.'

( 21 )

 21 Equative construction in Pesh (Isolate; Chamoreau 2017b:328) un-tas=ma tapeh-ber-wa ũtʃã=kan EXCL-PRO1=TOP swim-S1PL.EXCL-PRS fish=SIM 'We swim as much as fish.'

( 22 )

 22 Similative construction in Rama (Votic; Craig 1992:195) kiina isii i-kalmk-i nguu-ki man like 3-stand.up-TNS house-in 'He stood up in the house like a man.' (23) Similative construction in Malecu (Votic; Constenla Umaña 1998:109) mad like a dog (does).'

( 37 )

 37 Similative construction in Malecu (Votic; ConstenlaUmaña 1998:109)

  as tall as you.' (Lit. 'I am tall, you are tall'.) (42) Comparison of superiority in Malecu (Votic; Constenla Umaña 1998:193) taller than you.' (Lit. 'I am tall, you (are) not.')

  OBL: oblique, PFV: perfective, PL: plural, POSS: possessive, POSSA: alienable possession, PRO: pronoun, PRS: present, PRTC: participle, PST: past, REA: realis, REFL: reflexive, REM: remote, S: subject, SG: singular, SIM: similative, ST: sentence terminator, TERM: terminative, TNS: tense , TOP: topic, TPST: testimonial past, VBZ: verbaliser.

	3) (4) (5) (6) 2 Abbreviations: 3D: three dimensional noun class , ADV: advancement, ADZ: adverbialiser, AP: antipassive, Teribe (Isthmian; Quesada 2000:10) ta be-no sök u shko 1SG stay-PFV sit house in 'I stayed in the house.' Costa Rican Guaymi (Isthmian; Murillo Miranda 2016:102) Jamaraga ruga-ba ngobo ben friend arrive-EV son with 'My friend arrived with his son.' Pesh (Isolate; Chamoreau 2017b:324) a ͂ =yã=ma ta-suwa=yo a-katiʃ-k-a-ri DEM.DIST=LOC=TOP POSS1-grandmother=COM/INS refl-bring.up-k-s1sg-pst 'There, I was raised by my grandmother.' Damana (Magdalenian; Williams 1993:34) inguna zukuaiga=ba nəngui trail good=OBL stand 'He is standing on the good road.' BEN: benefactive, CAUS: causative, CERT: certainty, COM/INS: comitative/instrumental, COMP: comparative marker, COORD: coordinator, COP: copula, DEIC: deictic, DEM: demonstrative, DES: desiderative, DIST: distal, E: exclusive-absolutive, ERG: ergative, ESS: essive, EV: evidential, EXCL: exclusive, FOC: focus, FT: formative, FUT: future, GEN: genitive, INCL: inclusive, INF: infinitive, INFW: information with proof of witness, IPFV: imperfective, K: unknown function, LOC: locative, MED: medial, MID: middle, NEG: negation, NMLZ: Ika (Votic; Frank 1990:3) perɨ-sin keiwɨ zoža-na-rua dog-with right.away go-DIST-1S 'I went with my dog.' (8) Malecu (Votic; Constenla Umaña 1998:112) i-yu-to na-lhai=nh 3-with-FUT 1E-speak=REA nominaliser, O: object, (7) 'I will speak with him.'

  also bathing on the sand (like you).' / 'I am bathing on the sand as well.'

	(16) Equative construction in Buglere (Isthmian; Quesada 2012:132)
	Maria blire	bliga	Pedro		kare
	Maria eat	rapidly Pedro		too
	'Maria eats as rapidly as Pedro.' / 'Maria eats rapidly, Pedro likewise'
	(17) Similative construction in Buglere (Isthmian; Quesada 2011:295-297)
	cha	kle	basale ugoba gidi	jobe	kare
	1SG	be	tour	sand	on	bathe	too
	'I am

  .

	(25) Equative construction in Bribri (Isthmian; Quesada 2007:97)
	beʔ	datsiʔ	dör	mâtk	ieʔ	datsi	eʔs
	2POSS dress	be	red	3POSS	dress	like
	'Your dress is as red as hers.'
	(26) Similative construction in Cabecar (Isthmian; Guillermo González Campos p.c.)
	jír	bá	shká	yís	kä́i
	today you	walk.PFV 1SG	ESS
	'Today you walk like me.'
	(27) Equative construction in Teribe (Isthmian; Quesada 2000:139)
	Maria		e	plú	Juan	dik
	Maria		DEM	good	Juan	like
	'Maria, she is as good as Juan.'
	(28) Similative construction in Cuna (Isthmian; Llerena Villalobos 2010:61)
	we	nekakuepule	muulusuli	naipe-yoppi
	DEIC	village		long		snake-same
	'This village is long like a snake.'

  My two burros are as strong as the mule.' (Lit. 'My two burros with (and) the mule are equally (same) strong.')

	(30) Equative construction in Ika (Magdelanian; Frank 1990:73)
	buru mouga nʌʔʌn zei múra-sin dikin džuma aʔnikuʌ-ža
	burro two	I	GEN mule-with same strong 3D.be-MED
	'
						)
	perɨ	nar-i		o	a'zɨna	ni
	dog	COP-while	mad	think	CERT
	'It gets mad like a dog (does).'

  ). Example (43) presents a list of equative standard markers that apparently come from a similative marker:

	(43) Similative standard marker -equative standard marker
	Pesh	kan	'like, as'
	Rama	isii	'like, as'
	Teribe	dik	'like, as'
	Panamanian Guaymi	kwerere	'like, as'
	Damana	ia-ai	'like, as'
	Costa Rican Guaymi	erere	'like, same'
	Cuna	yoppi	'same, like'
	Ika	dikkin	'same, like'

  .

	(47) Equative construction in Bribri (Isthmian; Quesada 2007:97)
	beʔ	datsiʔ	dör	mâtk	ieʔ	datsi	eʔs
	2POSS dress	be	red	3POSS	dress	like
	'Your dress is as red as hers.'
	(48) Similative construction in Bribri (Isthmian; Constenla Umaña et al. 1998:4)
	wëm eʔs	dör	buaʔ
	man like	be	good
	'He is good like the man.'


Patrick Iglesias-Zemmour

REFRACTION AND REFLEXION ACCORDING TO THE PRINCIPLE OF GENERAL COVARIANCE

Keywords: Mathematics Subject Classification. 83C10, 78A05, 37J10. Key words and phrases. General Covariance, Geometric Optics, Symplectic Geometry. 1

We show how the principle of general covariance introduced by Souriau in smoothly uniform contexts, can be extended to singular situations, considering the group of diffeomorphisms preserving the singular locus. As a proof of concept, we shall see how we get again this way, the laws of reflection and refraction in geometric optics, applying an extended general covariance principle to Riemannian metrics, discontinuous along a hypersurface.

Introduction

In his paper "Modèle de particule à spin dans le champ électromagnétique et gravitationnel" published in 1974 [START_REF] Souriau | Modèle de particule à spin dans le champ électromagnétique et gravitationnel[END_REF], Jean-Marie Souriau suggests a precise mathematical interpretation of the principle of General Relativity. He names it the Principle of General Covariance. Considering only gravitation field 1 , he claimed that any material presence in the universe is characterized by a covector defined on the quotient of the set of the Pseudo-Riemannian metrics on space-time, by the group of diffeomorphisms. This principle being, according to Souriau, the correct statement of the Einsteins's principle of invariance with respect to any change of coordinates. Actually, Souriau's general covariance principle can be regarded as the active version of Einstein invariance statement, where change of coordinates are interpreted from the active point of view as the action of the group of diffeomorphisms. Now, for reasons relative to the behavior at infinity and results requirement, the group of diffeomorphisms of space-time is reduced to the subgroup of compact supported diffeomorphisms.

In consequence, after some identifications, Souriau interprets a material presence for a Pseudo-Riemannian metric g on the space-time M, responding to this principle of covariance, as a tensor distribution whose test functions are compacted supported covariant 2-tensors δ g defined on M, which vanishes along the infinitesimal action of the compact supported diffeomorphisms group Diff cpct (M). That is, in Souriau's notations:

(δ L g ) = 0,
where δ L g is the Lie derivative of the metric g by any compact supported vector field δx. In a more standard notation, denoting by ξ the compact supported vector field on M, δ L g = £ ξ (g ). A tensor distribution satisfying that last equation has been called by Souriau, an Eulerian distribution.

Among others, there are two main examples of application of this principle.

The first one is a continuous distribution of matter described by a C 1 -smooth symmetric contravariant 2-tensor T µν ,

(δ g ) = 1 2 M T µν δ g µν vol, (1)
where vol is the Riemannian volume associated with g , and we use the Einstein's convention on (up-down) repeated indices. We recall that the Lie derivative of a metric g by a vector field ξ is given, in terms of coordinates, by

δ L g µ,ν = ∂µ ξ ν + ∂ν ξ µ ,
where ∂ denotes the covariant derivative with respect with the Levi-Civita connection relative to g (sometimes denoted also by ∇). Then, the Eulerian condition above writes now:

1 2 M T µν ( ∂µ ξ ν + ∂ν ξ µ ) vol = 0.
By reason of symmetries and integrating by parts, recalling that ξ is compact supported, one gets:

M ∂µ (T µν )ξ ν vol = 0,
for all compact supported vector field ξ. Therefore the Eulerian condition writes:

∂µ (T µν ) = 0, or div(T) = 0,
where div(T) is the Riemannian divergence of the tensor T. Hence, the Eulerian character of the distribution translates simply in the Euler conservation equations of the energy-momentum tensor. Which is the first principle in General Relativity. And that justifies a posteriori the chosen vocabulary.

The second example is a preamble to the subject of our paper. It concerns the construction of geodesics as Eulerian distributions supported by a curve. So, let t → T µν be a C 2 -smooth curve in the fiber bundle of symmetric contravariant 2-tensor over M, over a curve γ : t → x. Let then,

(δ g ) = 1 2 +∞ -∞
T µν δ g µν dt .

After some astute manipulation [START_REF] Souriau | Modèle de particule à spin dans le champ électromagnétique et gravitationnel[END_REF], Souriau shows that this distribution is Eulerian if and only if the supporting curve γ is a geodesic, that is,

d dt dx dt = 0.
And then:

(δ g ) = k 2 +∞ -∞ dx µ dt dx ν dt δ g µν dt ,
where k can be any real constant. In general k is interpreted as the mass m of the particle moving along the geodesic γ with velocity v = dx/dt . But for classical geometric optics k will be interpreted as the color coefficient hν/c 2 , where h is the Planck constant, ν is the frequency of the ray and c is the speed of light in vacuum.

Thus, the condition to be geodesic is interpreted through this approach as an immediate consequence of the principle of General Relativity, which is quite satisfactory. This Souriau's construction has been also commented in a great way, and applied unexpectedly by Shlomo Sternbeg, to derive the Schrödinger's equation from the same covariance principle [START_REF] Sternberg | General Covariance and the Passive Equations of Physics[END_REF].

In this paper, we shall see how, this general covariance principle can also be extended to the case of an interface. That is, a singular situation where the metric is not anymore smooth, but has a discontinuity along a hypersurface. This is interpreted as a refraction/reflexion problem, as we shall see. And we shall get then, as an application of this extended general covariance principle, the Snell-Descartes law of reflection and refraction of light.

Thanks. -I am thankful to the referee for the careful reading of the manuscript, and for his remarks which contributed to enhance the content of this paper.

The General Covariance Principle with interface

In the following we shall consider the following situation, described by Fig. 1.

A manifold M is split in two parts M 1 and M 2 by an embedded hypersurface Σ. We can consider that M 1 and M 2 are two manifolds with a shared boundary Σ, but their union is the smooth manifold without boundary M:

M 1 ∩ M 2 = Σ and M 1 ∪ M 2 = M.
Next, on each part M 1 and M 2 we define two smooth Riemannian (or Pseudo-Riemannian) metrics g 1 and g 2 . That means precisely that g 1 is the restriction to M 1 of a metric defined on a small open neighborhood of M 1 , idem for g 2 with M 2 . The two metrics have then a limit on Σ which may not coincide. We shall denote by g = (g 1 , g 2 ) this metric on M, having a discontinuity on Σ.

Next, we introduce the subgroup of compact supported diffeomorphisms of

M preserving the hypersurface Σ, Diff cpct (M, Σ) = {φ ∈ Diff cpct (M) | φ(Σ) = Σ}.
Actually we want φ to preserve separately the two parts

M 1 and M 2 . That is, φ ∈ Diff cpct (M), φ M 1 ∈ Diff cpct (M 1 ), φ M 2 ∈ Diff cpct (M 2 )
, and of course φ Σ ∈ Diff cpct (Σ). To say that φ preserves Σ and is connected to the identity would be sufficient. Now, we introduce the test tensors for the distribution tensors relative to this configuration. Let g = ( g 1 , g 2 ) be a metric on M = (M 1 , M 2 ) as described above and let s → g s = (g 1,s , g 2,s ) be two smooth paths in the set of metrics, pointed at g , that is, g 0 = g . We shall denote δ g = (δ g 1 , δ g 2 ) the variation of g s at s = 0,

δ g 1 = ∂ g 1,s ∂ s s=0 and δ g 2 = ∂ g 2,s ∂ s s=0 .
Therefore, an infinitesimal diffeomorphism which preserves the figure gives a variation of g which is a Lie derivative:

δ L g = (δ L g 1 , δ L g 2 ) with δ L g i = ∂ (e sξ ) * (g i ) ∂ s s=0 ,
where ξ is a smooth compact supported vector field on M such that ξ(x) ∈ T x Σ for all x ∈ Σ. Its exponential e s ξ belongs to Diff cpct (Σ). We have then

(δ L g i ) µν = ∂i,µ ξ ν + ∂i,ν ξ µ ,
where ∂i is the covariant derivative with respect to g i . Let us come back to the tensor distribution: according to what was said until now, by linearity splits in the sum of two tensors, each relative to a part of M. That is,

(δ g ) = 1 (δ g 1 ) + 2 (δg 2 ).
(2)

The tensor distribution 1 is equal to (δ g ) for δ g 2 = 0, as well for 2 with δ g 1 = 0. Therefore, for to be Eulerian means:

(δ L g ) = 0 ⇔ 1 (δ L g 1 ) + 2 (δ L g 2 ) = 0.
(3)

Crossing the Border

As a first case, let us consider a continuous medium described by a tensor T µν , extended on M 1 and M 2 . That is,

(δ g ) = 1 2 M T µν 1 δ g 1,µν vol + 1 2 M T µν 2 δ g 2,µν vol,
with T a = T M a and δ g a = δ g M a . Considering a point x ∈ M but not on Σ we can find a vector field ξ with support contained in M 1 or M 2 , but avoiding Σ. Thus, the condition established above applies and we have divT a = 0 on M a -Σ, and by continuity, on M a including Σ. Therefore, the Euler equations are still satisfied for each side of the continuous medium.

∂µ T µν 1 = 0 and ∂µ T µν 2 = 0.

Next, let us consider a vector field ξ with compact support, extending over the two parts M 1 and M 2 . The Eulerian condition writes:

0 = M 1 T µν 1 ∂µ ξ ν vol 1 + M 2 T µν 2 ∂µ ξ ν vol 2 .
Integrating by parts, and after applying Euler equations, we get:

M 1 ∂µ T µν 1 ξ ν vol 1 + M 2 ∂µ T µν 2 ξ ν vol 2 = 0.
Introducing the vector T a (ξ) defined by its components T µ a,ν ξ ν = T µν a ξ ν , with

T µ a,ν = g a,νλ T λµ a , we get M 1 div T 1 (ξ) vol 1 + M 2 div T 2 (ξ) vol 2 = 0.
That is, using the identity 2 div(θ) vol = d [vol(θ)] and Stoke's theorem,

0 = M 1 d vol 1 T 1 (ξ) + M 2 d vol 2 T 2 (ξ) = ∂ M 1 vol 1 T 1 (ξ) + ∂ M 2 vol 2 T 2 (ξ)
Then, we can orient M and Σ such that: ∂ M 1 = Σ, and then ∂ M 2 = -Σ.

Hence,

Σ vol 1 T 1 (ξ) = Σ vol 2 T 2 (ξ) ,
which implies and is equivalent to the condition:

vol 1 T 1 (δx) Σ = vol 2 T 2 (δx) Σ.
for all x ∈ Σ and δx ∈ T x Σ. Hence, the tensor T = (T 1 , T 2 ) is Eulerian in presence of the interface Σ if the Euler equations above are satisfied, together with this boundary condition.

Let J be a vector transverse to Σ at the point x, complete with a basis = (e 1 , . . . , e m ) of T x Σ to get a basis (J, ) of T x M. In that basis vol 1 and vol 2 write

| det(G 1 )| det and | det(G 2 )| det .
where G 1 ,G 2 are the Gramm matrices of the metrics g 1 , g 2 . Let (J * , * ) be the dual basis of (J, ), with * = (e * 1 , . . . , e * m ). That is, J * J = 1, J * e i = e * i J = 0, e * i e j = δ i j . We introduce the projectors on RJ and T x Σ, associated with the basis (J, ) :

JJ * and * = m i=1 e i e * i ,
where we identified the vectors with their matrix representations. Then, the condition above writes, where the letters T a denote the matrices associates with the tensors denoted by the same letter:

J * | det(G 1 )| T 1 -| det(G 2 )| T 2 * = 0. (♦)
Now, in the framework of relativity (special or general), this condition must be specified for each kind of medium: elastic, fluid or dust. We leave the discussion of these cases for future work. But note already that the case of dust, for which particles follow geodesics, leads to the special treatment that we shall address in the following part.

2

The notation vol(θ) denotes the contraction of the form vol by the vector θ. The Broken Geodesics consider now a curve γ in M that cuts Σ at some point x for t = 0, and let γ 1 and γ 2 be the two pieces of γ,

γ 1 : ]-∞, 0] → M 1 and γ 2 : [0, +∞[ → M 2 , with γ 1 (0) = γ (0) = x.
Let be a tensor distribution supported by γ,

(δg ) = 1 2 0 -∞ T µν 1 δ g 1,µν dt + 1 2 +∞ 0 T µν 2 δ g 2,µν dt .
Solving the Eulerian condition (δ L g ) = 0, by considering a vector field ξ whose supports lie first anywhere in M 1 , then in M 2 we get, thanks to Souriau's computation, that γ 1 and γ 2 are geodesic for the respective metric and also that:

(δ g ) = k 1 2 0 -∞ dx µ dt dx ν dt δ g µν dt + k 2 2 +∞ 0 dx µ dt dx ν dt δ g µν dt .
Where k 1 and k 2 are a priori two arbitrary real constants. Applied to δ L g for a compact supported vector field ξ on M, tangent to Σ on Σ, one gets, by regroupings terms, using the fact that the curves are geodesic, and integrating by parts:

0 = k 1 0 -∞ dx µ dt dx ν dt ∂1,ν ξ µ , dt + k 2 +∞ 0 dx µ dt dx ν dt ∂2,ν ξ µ , dt = k 1 0 -∞ ∂1 ξ µ ∂ x ν dx ν dt dx µ dt dt + k 2 +∞ 0 ∂2 ξ µ ∂ x ν dx ν dt dx µ dt dt = k 1 0 -∞ d1 ξ µ dt dx µ dt dt + k 2 +∞ 0 d2 ξ µ dt dx µ dt dt
where y = (t , x, v) ∈ Y and δy ∈ T y Y. Let Z 1 and Z 2 be the space of geodesics in (M 1 , g 1 ) and (M 2 , g 2 ). In the two cases a parametrized geodesic that cut (transversally) the interface Σ is completely determined by (t , x, v), where x ∈ Σ is the intersection of the geodesic with the interface, t ∈ R is the time of the impact and v ∈ T x M is the velocity of geodesic at the point x. This defined two open subsets Z 1 ⊂ Z 1 and Z 2 ⊂ Z 2 . These two subspaces are equipped naturally with the restriction of their Cartan form ϖ 1 and ϖ 2 , as stated above.

Because of the indetermination we have no clear map 1 → 2 , but a relation that we can describe by its graph in Z 1 × Z 2 . Let y 1 = (t 1 , x 1 , v 1 ) and y 2 = (t 2 , x 2 , v 2 ) represent two geodesics in each of these spaces:

y 1 y 2 ⇔    t 1 = t 2 = t , x 1 = x 2 = x and k 1 g 1 (v 1 , δx) = k 2 g 2 (v 2 , δx) for all δx ∈ T x Σ.
The first condition translates the fact that the two geodesics cut the interface at the same point at the same time, and the second condition describes the Eulerian nature of the scattering, expressed in (♠).

Actually, we shall ask to preserve the Cartan form. It is a stronger request but usual in this kind of problems. Let us precise that, for the relation , to preserve the Cartan form means that the form ϖ 1 ϖ 2 , defined on Z 1 × Z 2 , vanishes on . That is, for all y ∈ , if δy ∈ T y then ϖ 1 ϖ 2 (δy) = 0.

But ϖ 1 ϖ 2 (δy) = k 1 g 1 (v 1 , δx) - k 1 2 g 1 (v 1 , v 1 ) δt -k 2 g 2 (v 2 , δx) + k 2 2 g 2 (v 2 , v 2 ) δt .
Therefore, preserves the Cartan form between the two spaces of geodesics if and only if

k 1 g 1 (v 1 , v 1 ) = k 2 g 2 (v 2 , v 2 ). (♣)
We notice here that k 1 g 1 (v 1 , v 1 )/2 is the moment of the symmetry group of time translation (t , x, v) → (t + e, x, v), for all e ∈ R. And preserving the Cartan form is equivalent to the preservation of the energy-moment when crossing the interface (which is not surprising). It describes a conservative i.e. non-dissipative -process, which is the rule in symplectic mechanics.

δg γ in γ out γ y 1 = (t 1 ,x 1 ,v 1 ) y 2 = (t 2 ,x 2 ,v 2 ) Figure 2. A Metric Blob
Thus, the symplectic framework decreases the level indetermination in the scattering process, but it remains the arbitrary of the ratio k 2 /k 1 which maybe lifted by a state proper to the system considered.

We shall see that, for geometric optics, the physical context lifts indeed the indetermination.

Note 1. -This construction also answers the question of reflexion, when γ 2 lies in M 1 too. In this case k 1 = k 2 and the symplectic condition states that v 1 and v 2 have the same norm relatively to g 1 , which solves the issue without other considerations.

Note 2. -The requirement of symplectic scattering is justified by the following situation. Consider a manifold M, equipped with some metric g . Its space of parametrized geodesics is a symplectic manifold for the pushforward ω of the differential d ϖ of the Cartan form. Now consider a perturbation g of g , such that the two metrics differ only on a compact K ⊂ M. For the sake of simplicity, we assume that every geodesic for g leaves the compact K in the past and the future. The set of parametrized geodesics of (M, g ) is symplectic for ω . Consider a geodesic γ ∈ , before entering in the compact K it defines a geodesic γ in ∈ , and after exiting the compact it defines a geodesic γ out ∈ , as it is described in Fig. 2. That defines a scattering φ : γ in → γ out on a subset of , the geodesics crossing the compact K. Let us check that φ is a local symplectomorphism. Let γ ∈ , and δγ and δγ be two tangents vectors at γ ∈ . Let y = (t , x, v) be an initial condition for γ, let δy = (δt , δx, δv) and δ y = (δ t , δ x, δ v) be two tangent vectors projecting on δγ and δ γ. Then, ω (δγ, δγ ) = d ϖ (δy, δ y). But we can evaluate d ϖ (δy, δ y) at a point y 1 = (t 1 , x 1 , v 1 ) before γ entering the compact K. In this case ω (δγ, δγ ) = d ϖ (δy 1 , δ y 1 ) = d ϖ(δy 1 , δ y 1 ) = ω(δγ in , δ γ in ). And we can also evaluate d ϖ (δy, δ y) at a point y 2 = (t 2 , x 2 , v 2 ) after γ leaving the compact K. in this case ω (δγ, δγ ) = d ϖ (δy 2 , δ y 2 ) = d ϖ(δy 2 , δ y 2 ) = ω(δγ out , δ γ out ). Therefore, ω(δγ out , δ γ out ) = ω(δγ in , δ γ in ), that is φ * (ω) = ω. The scattering φ : γ in → γ out , which is bijective (by t → -t ), is then a symplectomorphism. Hence, assuming that the scattering of geodesics by an interface -dividing two different media -is symplectic, is completely legitimate.

The Case of Light

Let us consider now the special case of light rays propagating in two homogeneous media with different indices n 1 and n 2 , separated by the hypersurface Σ. We shall see that the particular context of geometric optics will achieve to lift the indetermination in the rule (♠), with the condition of symplectic scattering (♣). Let v 1 and v 2 be the speed of light in two different media.

Then,

g 1 = n 2 1 g 0 and g 2 = n 2 2 g 0 with n 1 = c v 1 and n 2 = c v 2 ,
where g 0 is an ambient smooth metric on M, the vacuum metric. Assuming M ⊂ R 3 , we chose g 0 to be the standard scalar product, and we note as usual

g 0 (u, v) = u • v and g 0 (v, v) = v 2 .
Let us recall that we defined the color coefficients as

k 1 = hν 1 c 2 and k 2 = hν 2 c 2 ,
and let,

v 1 = v 1 u 1 and v 2 = v 1 u 2 with u 1 2 = u 2 2 = 1.
Next, the symplectic condition (♣) writes

k 1 n 2 1 v 1 2 = k 2 n 2 2 v 2 2 , that is, k 1 c 2 v 2 1 v 2 1 = k 2 c 2 v 2 2 v 2 2 .
Hence,

k 1 = k 2 i.e. ν 1 = ν 2 .
The frequency of the light ray, its color, does not change when crossing the interface. Now, the covariant condition (♠) writes, for all δx ∈ T x Σ,

n 2 1 v 1 •δx = n 2 2 v 2 •δx ⇔ c 2 v 2 1 v 1 u 1 •δx = c 2 v 2 2 v 2 u 2 •δx ⇒ c v 1 u 1 •δx = c v 2 u 2 •δx.
That is, (n 1 u 1 -n 2 u 2 ) • δx = 0 for all δx ∈ T x Σ. And then,

n 2 u 2 = n 1 u 1 + αn,
where n is the unit normal vector to Σ at x, and α some number. By decomposing the unit vectors u 1 and u 2 in an orthonormal basis completing n -in the plane spanned by u 1 and n -and denoting by θ i the angles between the u i and the normal n, we deduce the classical Snell-Descartes law of refraction of light, n 1 sin(θ 1 ) = n 2 sin(θ 2 ).

(♥)

In Conclusion, the law of refraction of light becomes a consequence of the general covariance principle, conjugated with the symplectic structure of the space of rays. The case of reflexion is also covered and gives as usual sin(θ 1 ) = sin(θ 2 ).

It will be noted again that in these cases, the distribution tensor defines a scattering map 1 → 2 , from the inward geodesic γ 1 to the outward geodesic

γ 2 .

Behind the Scenes

It is very intriguing that the evolution of material media, that are generally described by a principle of least action, can be also described by a principle of invariance (actually covariance) by diffeomorphisms. The medium can be elastic, or a particle, or even a light ray refracted by an interface.

There is always some kind of mystery behind the principle of least action, sort of God willing principle. Or the Nature which -by a hidden willdecides at each instant what is the best for the system in its immediate future. Every student who has been there once, knows this feeling of some magic that governs the world of matter.

And suddenly, we have the same behavior described just by an invariance principle, where all the magic has disappeared: that material presence and its evolution do not depend on the way they are described, they have an objective nature. And it is this objectivity which is captured by this principle of general covariance: the laws of nature are described by a covector on the space of geometries modulo diffeomorphisms. How is that possible?

Actually, these two principles are not as independent as they seem. The principle of least action and the principle of general covariance are indeed the two faces of the same coin. And to bring that to light, we shall consider the following toy model.

Let be a space, we shall call the space of fields. Let be a space, we shall call the space of geometries. Here and will be manifolds. Let G be a Lie group acting on the fields and the geometries. Let c ∈ , g ∈ and φ ∈ G. Let φ * (c) and φ * (g ) be the action of φ on c and g . Assume that the action of G on is transitive. Let : × → R be a smooth map we call the action function. Assume that the action is invariant under the diagonal action of G on × . That is (φ * (c), φ * (g )) = (c, g ) for all φ ∈ G.

Next, consider a 1-parameter subgroup s → φ s in G and let us put Thus, obviously, if (δ L g ) = 0, then (δ L c), and conversely. Hence:

(1) If c is a critical point for g : c → (c, g ), then (δ L g ) = 0 for all δ L g . That is, is Eulerian. (2) If is Eulerian, then (δ L c) = 0 for all δ L c. But since G is transitive on , we get (δc) = 0 for any variation δc. Which means that

δ g = ∂ g (c s ) ∂ s s =0
= 0 with g : c → (c, g ), for all smooth path s → c s . Therefore, c is a critical point of the action g , and a solution of the least action principle for this action.

Of course, for the cases treated in general the space of fields and the space of geometries are not manifolds but infinite dimensional spaces. There is certainly a right way to treat these situations rigorously, from the point of view of diffeology for example. But that is not the point here. The idea here is just to see how the principle of general covariance withdraws the magic from the principle of least action, by giving the main role to the laws of symmetries and the principle of objectivity/relativity which definitely deserve it. At the same time the principle of general covariance is a well founded a priori generalization of the least action principle.

Figure 1 .

 1 Figure 1. Refraction Geodesics

δ

  L c = ∂ φ s * (c) ∂ s s=0 and δ L g = ∂ φ * s (g ) ∂ s s=0 ,where φ * = φ -1 * . Therefore, taking the derivative of the invariance identity of the action above, we get that for all infinitesimal action δ L of the group G we have(δ L c) -(δ L g ) = 0 with =

Then, denoting by v 1 and v 2 the limits of the velocities

, we get the Eulerian condition for the tensor distribution at the point x ∈ Σ which writes

We get then a general condition which must be satisfied by the two geodesics γ 1 and γ 2 at the interface, to define an Eulerian distribution. But the system is underdetermined due to the arbitrary choice of constants k 1 and k 2 . We can diminish the arbitrary by considering the symplectic structure of the spaces of geodesics.

Remark. -The case above admits a solution when there is no part 2, and just the curve γ 1 . In that case the equation (♠) becomes g 1 (v 1 , δx) = for all δx ∈ T x (Σ), that is,

) Hence, the Eulerian distributions are the geodesics which end orthogonally at the intersection with the interface. It is the case in the Poincaré half-plan, for example. That has been observed in real world, with light rays in a saline solution for which the concentration tends to infinity near the border.

The Symplectic Scattering

We shall use now the symplectic structure of the space of geodesics and we shall request the distribution to be a symplectic scattering 1 → 2 , in order to lower the degree of indetermination in the equation (♠).

Let us recall now that the space Z of geodesics curves in a Riemannian (or Pseudo-Riemannian) manifold (M, g ) has a canonical symplectic structure. It is the projection of the exterior derivative d ϖ of the Cartan 1-form ϖ defined on Y = R × (TM -M) by ϖ(δy) = k g (v, δx) -k 2 g (v, v)δt ,


Serap Gürer

Patrick Iglesias-Zemmour

DIFFERENTIAL FORMS ON STRATIFIED SPACES II

Keywords: . 1991 Mathematics Subject Classification. 58A35, 58A10 Stratification, Diffeology, Differential Forms

We prove that, for a conelike stratified diffeological spaces, a zeroperverse form is the restriction of a global differential form if and only if its index is equal to 1 for every stratum.

Introduction

In the previous paper "Differential Forms On Stratified Spaces" [START_REF] Gürer | Differential Forms On Stratified Spaces[END_REF], for each zero-perverse form defined on the regular part of a diffeological stratified space, we introduced an index for every stratum, counting the number of different forms generated by the zero-perverse form around the stratum. We showed that under some natural condition on the stratification -if two points could be connected by a path cutting the singular subset in a finite number of points -the zero-perverse form is the restriction of a differential form, defined globally on the space, if and only if the index is equal to 1 for every stratum.

In this paper, we prove this statement for locally conelike diffeological stratified spaces, without accessory conditions.

Thanks. -We are thankful to the referee who, by his relevant remarks, allowed us to improve the readability and the content of the paper.

Locally Conelike Diffeological Stratified Spaces

We suggest Benoît Kloeckner's survey [START_REF] Kloeckner | Quelques Notions d'Espaces Stratifiés[END_REF] for the classical topological approach of stratified spaces. About diffeology, we refer to the textbook [START_REF] Diffeology | Mathematical Surveys and Monographs[END_REF], and about conelike stratified spaces, Larry Siebenmann's paper [START_REF] Siebenmann | Deformation of Homeomorphisms on Stratified Sets[END_REF].

Let us begin by specifying what we understand by Stratified Diffeological Spaces. Note that the topology of a diffeological space we referring to in the following is its natural D-Topology, defined in [PIZ13, §2.8].

1. Stratified Diffeological Spaces. A stratification of a diffeological space X by manifolds is a finite family of subspaces = {S i } i∈ , called strata, such that:

(1) The strata form a partition of X.

(2) Each stratum is a manifold for the subset diffeology 1 .

(3) The strata satisfy the frontier condition:

S i ∩ S j = ∅ ⇒ S i ⊂ S j . (4) Let m = max{dim(S i )} i∈ . Then, X reg = i∈ |dim S i =m S i
is an open dense subset, called the regular part of the stratification.

(5) Let be the subset of indices i ∈ such that dim(S i ) < m. If is not empty, then let

= {S i } i ∈ and X = ∪ i∈ S i , equipped with the subset diffeology. Then is a stratification of X .
The space X is assumed to be Hausdorff, metrisable and connected for the D-topology. The subset X wich is equal to X-X reg is called the singular part of the stratification, and can be sometimes denoted by X sing . The elements of are called the singular strata. The first four axioms can be called the structure axioms and the fifth, the recursion axiom. For topological stratified spaces, the definition can be found in Pflaum survey [START_REF] Pflaum | Analytic and geometric study of stratified spaces[END_REF]. By this recursion we get a filtration of diffeological subspaces:

X 0 ⊂ X 1 ⊂ • • • ⊂ X ⊂ • • • ⊂ X k-1 ⊂ X k .
where X k = X, X k-1 = X etc. Each term X of the filtration is a stratified space with the according subset of strata, and the main strata has some dimension n .

Each term being the singular part of the term above. The next step consists in specializing the structure near the singular strata. We said that the stratification is locally fibered [GIZ18] -or the space is a locally fibered stratified spaces -if there exists a tube system {π S : TS → S)} S∈ such that:

(A) TS is an open neighborhood of S, called a tube over/around S. (B) The map π S : TS → S is a smooth retraction which is a diffeological fibration. (C) For all x ∈ TS ∩ TS ∩ π -1 S (TS), one has π S (π S (x)) = π S (x). The fiber F x = π -1 S (x), with x ∈ S, is a diffeological space itself which is stratified inherited by the ambient stratification. Its intersection with the stratum is reduced to one point: F x ∩ S = {x}. We call it the apex of the fiber.

1 See [PIZ13, §4.1] for manifolds as diffeologies. Note. -To be locally fibered is not a priori recursive. Do we need a natural recursion on the local fibration? This is a legitimate question, but we do not need this property for the case treated in this paper.

2. Diffeological Cones. Let Σ be a space. We call a cone over

Σ the quotient space Cone(Σ) = Σ × [0, ∞[ / Σ × {0}
: Cone(Σ) → F such that φ [Cone(Σ) -] is a diffeomorphism onto its image, where = class(Σ × {0}
) is the apex of the cone. Thus, without their apex, these spaces are equivalent to Σ × ]0, ∞[. Then, F will be called a diffeological cone, or simply a cone, Σ is called the base of the cone. The cone F differs from the cone over Σ only by the germ of the diffeology at the apex.

Σ × [0, ∞[ Cone(Σ) F class Φ φ
This consideration is necessary because diffeology discriminates between cones, as

shows for example the positive cone in R 3 defined by x 2 + y 2 = z 2 & z ≥ 0, equipped with the subset diffeology, compared with the cone over

S 1 , Cone(S 1 ) = S 1 × [0, ∞[/S 1 × {0} see [GIZ18, §3].
Note. -An equivocal situation arises with this definition, related to the different types of stratified spaces [START_REF] Gürer | Differential Forms On Stratified Spaces[END_REF]Introduction]. We have called geometric stratified spaces, the stratified spaces for which the strata are the (connected components of the) orbits of the pseudogroup of local diffeomorphisms. The formal stratified spaces are the other ones. The same situation happens with diffeological cones when we compare the two examples sketched in Fig. 1 above: the hemisphere (left cone) defined by x 2 + y 2 +(z -1) 2 = 1 with 0 ≤ z < 1, and the embedded cone (right cone) defined by x 2 + y 2 -z 2 = 0, both equipped with the subset diffeology of R 3 . What differentiates these two cones is clearly the behavior of the apex . On the left cone it is interchangeable with any other point. Precisely, the pseudogroup of local diffeomorphisms does not distinguish the apex from the other points. On the other hand, the apex of the right cone is alone in its orbit. That observation leads us to distinguish at least two classes of cones:

Definition We shall say that a cone is a geometric cone if its apex is alone in its orbit by the pseudogroup of local diffeomorphisms. Otherwise, we shall say that it is a formal cone.

This remark has a real content, related to the idea about what is a cone. Intuitively, the apex of a cone should be distinguished. In diffeology it is the role of local diffeomorphisms to discriminate between points.

3. Differential Forms On Conelike Stratifed Diffeological spaces. We have introduced the general notion of locally fibered stratified space in [START_REF] Gürer | Differential Forms On Stratified Spaces[END_REF]. They are diffeological spaces, together with a stratification, such that the space looks like a diffeological fiber bundle [PIZ13, §8.8, 8.9] on the neighborhood of each stratum. This is a priori a two step generalization of the standard situation in the classical theory of stratified spaces, where the fiber is assumed to be conelike over a stratified base, as it is stated in [START_REF] Siebenmann | Deformation of Homeomorphisms on Stratified Sets[END_REF]. Hence, we have a diffeological intermediary to consider, the analogue of the usual conelike stratified spaces:

Definition -We call a Locally Conelike Diffeological Stratified Spaces any locally fibered stratified diffeological space with fibers some diffeological cones.

Note first that, in a locally conelike diffeological stratified space, the bases of the cones are stratified spaces as well. Secondly, the apex of the fibers is their intersection with the stratum.

Then, let α ∈ Ω k 0 [X] be a zero-perverse form defined on the regular part X reg of a diffeological stratified space X, and let ν the index function on zero-perverse forms, defined in [GIZ18, §5].

Proposition -If X is a locally conelike diffeological stratified space, then there exists a (unique) differential form α ∈ Ω k (X) such that α = α X reg , if and only if ν S (α) = 1 for all strata S. Note 1. Actually, the proposition applies to every stratum which is in the closure of the regular part. But since, by definition, X reg is dense in X then the property applies for all strata.

Note 2. The proof of the proposition could be easily extended to the case of locally fibered diffeological stratified spaces with contractible fibers, maybe with some condition on the retraction with respect to the connected components of the regular part. But that would be a minor generalisation.

Proof. If ν S (α) = 1 for all S ∈ , then there exists a (unique) differential form α ∈ Ω k (X) such that α = α X reg . That has been proven in a general context in [START_REF] Gürer | Differential Forms On Stratified Spaces[END_REF].

Let then α ∈ Ω k (X). For all strata S, let π S : TS → S be the retraction of the tube around a singular stratum S, which is by hypothesis a local fibration with fiber a cone F with base Σ. Recall that S ⊂ TS and for all x ∈ S, π S (x) = x.

Let α = α X reg , and assume that α is zero-perverse with respect to these local structures. To follow the construction of the index ν we consider the pullback of the tube TS by the projection of the universal covering pr: S → S, even if it is not completely necessary, since our considerations will be local. Let us denote by T Ŝ the pullback pr * (TS), that is,

T Ŝ = {(x, y) ∈ S × TS | pr(x) = π S (y)}.
This pullback is actually a tube around the embbeding Ŝ of the universal covering S by the smooth section σ : x → (x, pr(x)).

Ŝ = {(x, pr(x)) | x ∈ S} ⊂ T Ŝ,
which justifies the notation. Indeed, since pr(x) ∈ S, pr(x) = π S (pr(x)) which is the condition for (x, pr(x)) to belong to T Ŝ, and obviously pr 1

• σ = 1 S. Now, π Ŝ : (x, y) → (x, π S (y)),
is a retraction from T Ŝ to Ŝ. The tube T Ŝ is also a fibration with the same fiber F, it is the deployment of the tube TS along the embedded universal covering Ŝ.

Hence, T Ŝ is everywhere locally diffeomorphic to a product × F, where ⊂ Ŝ is some open subset, and the projection π Ŝ : (x, y) → (x, π S (y)) locally equivalent to the first projection pr 1 : ×F → . That means that there exists a local diffeomorphism ψ, defined on × F, into T Ŝ such that π Ŝ • ψ = pr 1 , We choose always to be connected and simply connected. Now, the regular part of π -1 Ŝ ( ) is the image by ψ of the regular part of the product × F, that is, × F reg . But F reg is naturally identified to ]0, ∞[ × Σ reg , by construction. Therefore we have the following chain of square commutative diagrams.

× Σ reg × ]0, ∞[ × F reg × F T Ŝ TS × { } Ŝ S S S pr 1 pr 1 π ψ π Ŝ pr 2 π S pr 1 ψ| ×{ } pr 1 pr 2 1 1 pr
We need to make a remark here about the arrow ψ| ×{ } : ×{ } → Ŝ that associates (x, ) with (x, x = pr(x)). For geometric cones, this arrow exists by construction since the apex is alone in its orbit by the pseudogroup of local diffeomorphisms. For formal cones only, the local triviality should be understood in the category of stratified spaces. Let us now introduce the following notations:

α = pr * 2 (α) ∈ Ω k (T Ŝ), and α = α T Ŝreg = pr * 2 (α) ∈ Ω k (T Ŝreg ).
Next, we decompose F reg in terms of connected components and have:

× F reg = × a F a reg = × a Σ a reg × ]0, ∞[ with a ∈ π 0 (Σ reg ).
And we get a two-squares diagram summarizing the situation we will deal with: where λ is the smashing function described by Fig. 2. Now:

× a Σ a reg × ]0, ∞[ × F T Ŝ S
(1) For all s ∈ ]-∞, ε], γ(s ) = is the apex of F.

(2) For all s ∈ ]ε, ∞[, γ(s ) ∈ F a reg .

(3) In particular γ(0) = and γ(1) = y.

Next, let Q × γ be the plot (r, s ) → (Q(r ), γ(s )), into × F, when (r, s) runs over V × R. Let us fix (u i ) k i =1 with u i ∈ R n , and (τ i ) k i =1 with τ i ∈ R. We consider then the real smooth function defined on V × R,

f (r, s) = α(ψ • (Q × γ)) (r,s) (u i , τ i ) k i =1 Indeed: ψ • (Q × γ) is a plot in × Y, ψ is a local diffeomorphism into T Ŝ, α is a k-form on T Ŝ, and the (u i , τ i ) k i=1 have been fixed. (1) Consider firstly s ∈ ]-∞, ε[, thus γ(s ) = . Let us denote xr = Q(r ), then ψ(Q(r ), γ(s )) = ψ(x r , ) = (x r , x r
), with x r = pr(x r ). Hence, we have: Therefore,

f (r, s) = α((r, s ) → ψ(x r , )) (r,s) (u i , τ i ) k i=1 f (r, s) = α((r, s ) → (x r , x r )) (r,s) (u i , τ i ) k i=1 = α((r, s ) → r → (x r , x r )) (r,s) (u i , τ i ) k i=1 = pr * 1 α(r → (x r , x r )) (r,s ) (u i , τ i ) k i=1 = α(r → (x r , x r )) r (u i ) k i=1 = α(r → x r ) r (u i ) k i=1 , because α = pr * 2 (α) = α(pr •Q) r (u i ) k i=1 . Therefore f V × ]-∞, ε[ = pr * (α)(Q) r (u i ) k i=1 . (
f V × ]ε, ∞[ = ᾱa (Q) r (u i ) k i=1 .
(3) In conclusion -For all r ∈ f r : s → f (r, s) is a smooth function defined on R. On the interval ]-∞, ε[, f r is constant and equal to pr * (α)(Q) r (u i ) k i=1 , and on ]ε, -∞[, f r is constant and equal to ᾱa (Q) r (u i ) k i=1 . Since f r is smooth, then continuous, these two constants must be equal. Thus, since that is true locally for any r and any vectors u i in R n , we get ᾱa = pr * (α S), for all indices a. That is, ν(S) = 1. Now, that applies to every stratum which is in the closure of X reg . Since by definition X reg is dense in X, ν(S) = 1 for all strata.

Figure 1 .

 1 Figure 1. -A formal cone (left) and a geometric cone (right).

Figure 2 .

 2 Figure 2. -The smashing function λ.

1

  Now, consider a n-plot P in S defined on some open set U, and r 0 ∈ U. There exists an open neighborhood V ⊂ U of r 0 such that Q = P V takes its values in some trivialization open set , which we assume to be connected and simply connected.Now let y ∈ F reg = a Σ a reg × ]0, ∞[, then y = Φ(z, t ) for some z ∈ Σ a reg and t ∈ ]0, ∞[,where Φ has been introduced in (art. 2). Consider then the (stationary) smooth path γ : s → Φ(z, λ(s )t ),

  2) Consider secondly s ∈ ]ε, +∞[. Then, γ(s ) ∈ F a reg , Q×γ is a plot in ×F a reg andψ • (Q × γ) is a plot of {T Ŝ} a , the connected component of T Ŝ = pr * (TS) associated with the component F a reg = Σ a reg × ]0, ∞[, see [GIZ18, §5, Step 1]. Hence, according to [GIZ18, (♠)], α {T Ŝ} a = pr * 1 (ᾱ a ), where ᾱa ∈ Ω k ( S). Thus, f (r, s) = α((r, s) → ψ(x r , γ(s ))) (r,s) (u i , τ i ) k i=1 = pr * 1 (ᾱ a )((r, s) → ψ(x r , γ(s ))) (r,s ) (u i , τ i ) k i=1 = ᾱa (pr 1 •[(r, s) → ψ(x r , γ(s ))]) (r,s) (u i , τ i ) k i=1 = ᾱa ((r, s ) → pr 1 •ψ(x r , γ(s ))) (r,s) (u i , τ i ) k i=1 = ᾱa ((r, s ) → xr ) (r,s) (u i , τ i ) k i=1 , since pr 1 • ψ = pr 1 = ᾱa (r → xr ) r (u i ) k i=1 .

This research is partially supported by Tübitak, Career Grant No. 115F410, and the French-Turkish Research Fellowships Program, Embassy of France in Turkey 2017. The authors thank the «Espace Forbin» of the Faculty of Economy from Aix-Marseille University, and the people there, for their constant hospitality.