Rinaudo Christian

« Mestizaje

Mexico » Veracruz

In E Cunin

Éds O Hoffmann

Mestizaje and ethnicity in the city of Veracruz, Mexico

The research presented here is the result of fieldwork conducted between 2007 and 2010 in the city of Veracruz, Mexico. 1 It bears on social, political and cultural usages of ethnic categories linked to the local history of slavery and the historical presence in the region of Black populations of African origin. Focusing on the evolution of local cultural policy, the ways of describing the city, activities striving to enhance the status of what is claimed to be its historical ‗heritage', the popular music scene as well as forms of social domination expressed in daily life, this study aims to examine the conditions and modalities in which differences relating to the ‗African roots' of mestizaje (ethnic mixing) -such as the perception of skin colour and phenotypic traits that are seen as more or less ‗negroid' or cultural characteristics defined as ‗Black' or ‗African' -are marked, displayed, debated, eluded or argued.

, they do not account for other social usages among categories referring to their African heritage in contemporary and urban Mexico. 2 These usages appear in historical processes of transformation of representations of mestizaje and regional identities, but also emerge in concrete situations as observed in urban areas where the ethnic dimension is not always present and may acquire different meanings according to context, ranging from prestige to stigma.

Despite its long history as a port for bringing in and trading in African slaves and the process of political recognition of the importance of ‗African roots' in local culture, there are today in the city of Veracruz no collective activities aiming to develop awareness of belonging to a single social group self-identified as ‗Black' or ‗Afrodescendant', nor are there any organisations that seek to speak or take action on behalf of ‗people of African origin'. Here, the idea of the existence of a specific group, like ‗Afro-Veracruzans' or a ‗Black community of Veracruz', has little meaning in an urban space where the inhabitants identify

themselves largely on the basis of class distinctions or regional differences. The term Jarocho, referring, often indiscriminately, to people from the South of the State, the Sotavento coast and the city of Veracruz, relates more to the issue of mestizaje, or even ‗specific mestizaje' and its different ‗roots' than to ‗Black identity', even if it did evoke, at one time in local history, the ‗Black' ‗rural' population in the southern State of Veracruz [START_REF] Delgado Calderón | Historia, cultura e identidad en el Sotavento[END_REF]. Moreover, somatic variations among inhabitants (skin colour, morphology, facial features...) are unimportant, which may cause confusion among ‗Black' militants when they come to Veracruz in search of a ‗Black population' sharing their condition. True, another characteristic of urban life in Mexico is surely the constant use of qualifiers referring to mestizaje and somatic and cultural differences (Indígena, Mestizo, Creollo, Moreno, Pardo, Negro, Mulato, Güero, etc), even if there are few visible differences in terms of distinctive features.

AN EMPIRICAL APPROACH

This study adopted the inductive non-substantialist approach now shared by many researchers working in this field3 and refraining from presupposing the presence of existing entities. It considers the matter of Afro-descent as one factor among others for understanding: the broader processes of racism and discrimination; maintenance of ethnic boundaries and ethnicisation processes; empowerment phenomena and the development of militant organisations; the stakes of patrimonialisation and construction of collective memory; the emergence of multicultural policies and redefinitions of citizenship; ‗ethnic' or ‗non-ethnic' framing of events and specific courses of action; and moments when specific identities crystallise.

In this perspective, the survey work involved studying the moments, modalities and reasons for which ethnic boundaries are marked, maintained, signified, proclaimed, disclaimed or valued. The ethnographic approach adopted bears on: contexts or moments of more or less strong expression of ‗groupness'4 definable in cultural terms; on the social uses of categories referring to colour or presumed African origin; on civic organisations involved in processes of patrimonialisation of an urban memory of slavery and mestizaje; on enhancing the ‗Third Root' in defining cultural policy; and on social actors who, especially in the cultural field, emphasise the city's ‗African roots' and its place in the Caribbean, as well as individual and collective staging of to relationships with Africa, the Caribbean and the ‗Black Atlantic'. Special attention has also been granted to dealing with physical appearance, gait, dance and dress inspired by globalised, relocalised models that could be identified as ‗Black culture'whether or not this has led to any cultural movement (promotion of ‗Black music', ‗Afro dances'...) or identity claims (demand for recognition by entrepreneurs of ‗Black identity') that are specifically, continually or unanimously labelled as ‗Black' or ‗Afrodescendant'.

Thus, the hypothesis guiding this approach is that such phenomena, sometimes insubstantial or inconsequent in certain situations, actually have a major role in maintaining ethnic boundaries. The political changes in the past twenty years contributed to orienting research towards collective forms of action aiming to produce or crystallise ethnic solidarity by relying on cultural (sharing common attributes) or political bases (sharing a common condition of being minimised or victims of racial discrimination). However, the ethnicisation processes underway in certain regions do not account for the space, time and fields in which ethnic categories -often created in the colonial period -express themselves and are transformed, and in which some distinctions are reinforced while others fade or appear more flexible and sometimes less necessary. Besides cases where awareness of identities develops against the ideology of mestizaje and its principles of indifference to differences that deny social inequality and ethnic discrimination, other relationships between mestizaje and ethnicity can be observed that also have their social importance. For example, relations that develop in the city involve the power balance and forms of domination on the basis of distinctions constantly noted and reproduced between on the one hand light-skinned, welldressed ‗beautiful people' (gente bonita) who live in residential areas, dine in expensive restaurants and go to fashionable discotheques, and on the other hand poor people, ‗Indians', ‗Black', peasants or workers. Guillermo Bonfil made mention of this schism between ‗imaginary Mexico' (modern, urban, cosmopolitan, heir of the Spanish conquest) and ‗México profundo' [START_REF] Batalla | México profundo. Una civilización negada[END_REF]. Once again, the relationship between mestizaje and ethnicity constantly means thinking, justifying, reasserting the idea of mestizaje and describing and structuring its different ‗roots' and associated ‗populations'.

To account for the heuristic value of this approach, three orientations will be presented for this research referring to three distinct levels of analysis: one bearing on social and socially situated activities that entail ‗describing the city'; another aiming to account for definitions and transformations of localised cultural policy; and a third highlighting modes of expression, displaying and experimenting with ‗elective Africanity' in urban public space.

THE TOURIST DISCOURSE AROUND THE CITY

Veracruz, one of the main ports for reaching the continent or departing towards other destinations, was not spared the social phenomenon represented throughout the 19th century by the passage of foreign travellers and chroniclers visiting Latin America. There are many written traces of descriptions of the region and its inhabitants, most of which were published in a collection of eleven works, coordinated by Martha Poblett Miranda under the title Cien viajeros en Veracruz. Crónicas y relatos [START_REF] Miranda | Cien viajeros en Veracruz. Crónicas y relatos[END_REF]. These travel diaries have already been amply commented and analysed,5 recreating a picture of Veracruz seen at the time above all as a dangerous inhospitable unwholesome town, a dismal place that was hard to reach, whose local population was often described as indolent, ‗sleeping all day in frightful sunshine,' and belonging ‗to all Mexican races, their colour ranging from ochre to ebony' (García Díaz, 2002: 215-238).

The development of tourism as such did not begin until the 1920s-1930s, when yellow fever ceased to be perceived as a threat and with the arrival from Europe and the United States of a sport-loving hygienist movement that hailed the benefits of bathing in the sea and sunshine (García Díaz, 1998: 48). It was also at this time, with the arrival in Veracruz of several railways, the modernisation of urban transport, the increasingly widespread use of electricity and the will to make Veracruz a tourist destination, that there was the revitalisation of Carnival, thought to date from the 17th century. Taken up by social clubs, and shopkeepers and hotel associations who were funded by the municipality and the Veracruz State government, the aim was to attract more visitors to the city and transform this Carnival into a popular media event of national and international renown [START_REF] Olivera | Los carnavales del puerto de Veracruz‖[END_REF].

At the end of the 1930s and in the early 1940s, Miguel Alemán Valdés, then governor of the State of Veracruz before becoming President of Mexico in 1946, stimulated the development of tourism in the region, inciting investors from the capital to buy land and build tourist infrastructures, like Hotel Mocambo described today as an ideal place ‗for the rich and famous who enjoyed staying in places full of romance and tranquillity'. 6 This was precisely the type of clientele targeted by one of the State of Veracruz's first tourist guides, published in 1940 by the Mexican Tourism Association and the Department of Tourism of the Gobernación Secretariat. This guidebook proposed an itinerary called ‗Travel in the land of flowers,' from Mexico City to Veracruz. México, 1940Source: Anonymous, 1940 On reaching the coast, a chapter titled ‗Rejoicing in Veracruz' clearly emphasises the entertainment typical of seaside tourism and describes the town through the special atmosphere prevailing there, in a setting characterised by both its colonial charm and its modern infrastructures (Anonymous, 1940). The many photographs illustrating the text include, on the one hand, stereotypic representations of a happy local population -Jarochos singing, playing music or dancing, who, as Ricardo Pérez Montfort clearly showed at the time of post-revolutionary nationalism, gradually moved away from the connotation of poverty and peasantry, as well as relations with an African heritage, to highlight a whitened image evoking aristocratic ascent in terms of skin colour, posture and dress (Pérez Montfort, 2007: 200). On the other hand, the photographs in this guidebook also show tourists in swimsuits and clothing fashionable among urban upper and middle classes visiting the coast to enjoy the merry festive atmosphere, tropical climate and bathing in the sea in an idyllic setting of fine sand, palm trees and lively terraces by the sea. Thus, the guide develops a vision of tropical coasts seen from Mexico. Stripped of any mention of African presence or indigenous roots, not yet stamped as part of the Caribbean cultural area, it was then entirely dedicated to satisfying the whims of wealthy refined ‗modern' summer visitors from the capital.

Cover of Veracruz

It is in this context that the city of Veracruz is often described as a tourist destination, a place to ‗have a good time,' ‗have fun,' ‗make merry'. And this ‗specificity' of the port as a place one visits -for its ‗joyous nature' attributed to its inhabitants' ancestral rootscontributed to the stereotypes attributed to the local population, something widely ‗overplayed' by the inhabitants themselves to foreigners, and by professionals who earned their living from this urban folklore, especially around the city's main square [START_REF] Flores Martos | Una etnografía del puerto de Veracruz[END_REF].

Thus, in contemporary travel guides, agency brochures, online presentations and other advertising material vaunting the merits of the place, the first thing that appears from now on is the association, explicit or implicit, between the festive, merry, convivial, rowdy definition of Veracruz and its inhabitants, its identification with the Caribbean and the contribution of cultural, physical and psychological traits inherited from the African presence highlighted to describe the specificity of this old colonial town. For example, the Guide Bleu, a French reference for upscale cultural tourism, insists on the commercial importance of the colonial port and the character of its inhabitants: -Its inhabitants, the Jarochos, of mixed Spanish, Indian and African descent, display a gaiety they demonstrate more specifically during Carnival, one of the most brilliant after those of Rio de Janeiro and Trinidad.‖ (Guide Bleu, 2005) Similarly, in the Guide du Routard, published by the same press group, Hachette, and targeting French tourists through a more laid-back vision of cultural tourism (carrying rucksacks, staying with local people...), the description focuses above all on the charm, exciting nightlife and tropical hedonism of the place and the local population: -There isn't much to see in Veracruz, but the city has a certain charm, with its squares lined with palm trees and its Promenade on the malecón (boardwalk) along the port. A hot ambiance reigns around the Zócalo after sunset and way into the night. This is the city of music and dance (Afro-Cuban influence).‖ (Guide du Routard, 2006) Paradoxically, while tourism is above all a national phenomenon in Mexicoespecially Veracruzas in many Latin American countries [START_REF] Nathalie | Las interrogantes que platea América Latina al estudio del fenómeno turístico‖[END_REF], there are still few national guidebooks. In an online guide titled TravelByMexico, All About Mexico, the introductory chapter on Veracruz describes the town as the cradle of the Mexican nation, emphasising the ‗mixing of two worlds,' Spanish and Indian, the ‗cultural diversity'which, in Mexico, refers essentially to the diversity of Indian culturesand the joyous nature of the inhabitants, the Jarochos. 7If national descriptions emphasise that Veracruz is a historic centre for the meeting of two worlds, a local thematic guide dedicated to the culture of Son (sound) presents a more complex landscape in which the different influences have to do with the different roots of mestizaje and cultural features from the Caribbean:

-The culture of Son covers practically the entire coast of Veracruz, although notable differences appear according to region: to the North, Son Huasteco has a slow beat and a melancholy mood with the violin while, in Los Tuxtlas region, in the south of the State, Son Campesino (peasant) or Abajeño (coastal) is much more cheerful because of the influence of African and Caribbean culture introduced to the region.‖ (Jiménez Illescas, 2002: 6) Concerning the city of Veracruz in particular, a historical paragraph retraces the context of the arrival of populations of African origin and their subsequent importance in the formation of local culture resulting from the mestizaje of ‗three worlds': Spanish, Indian and African. The author of La Ruta del Son also insists on relations between the rural coastal area and the city and on the former's influence in what is described at the Porteño nature:

-Another substantial migration is made up of sequestered persons from Africa, which forms the third ethnic root of the coast of Veracruz. Because of the dearth of native people, manpower held in slavery was brought in. These slaves were distributed from the port to perform hard labour in mines and sugarcane plantations. A century after the start of this migration, in 1681, half of the port's thousand inhabitants were Black. Erotic exchanges between Blacks and Indians produced the Jarochos, a term describing the rural population of the coastal plains of the Sotavento. And it was in Jarocha music, a genre to which this mestizaje gave rise, [...] that the three worlds present in countryside of Veracruz since colonial times melded. Its culture, clearly Vaquero and Campesino, was central to being Porteño.‖ (Jiménez Illescas, 2002: 58) Different analytical elements may be drawn from these few examples. First, the change in the description of the 1940s and what is said today is obvious. Now, all insist on the joyous welcoming atmosphere of the coast and city, in contrast with most 19th-century travel accounts of Veracruz, like the view given by the English merchant William Bullock in 1824, as ‗the most disagreeable place on earth' and ‗the most unhealthy spot in the world', making ‗the stranger shudder every hour he remains within its walls' (Bullock, 1992: 42). Since all tourist discourse characteristically highlights the destination's assets, the assets in this case have been, quite consistently since it arose in the 1930s-1940s, the fun, the friendly festive atmosphere, the people's joy of living, described as elements central to the destination's appeal. But it is only recently that this local characteristic of the city and coastal region is described by highlighting the mestizaje of the population, the diversity of its origins, the African and Afro-Cuban influences linked to the history of slavery and cultural movements in the Caribbean, as well as the resulting wealth of forms of expression.

AFRICAN ROOTS AND CARIBBEAN INSERTION IN CULTURAL POLICY

Another research orientation entails studying social rationales having contributed in Veracruz to valuing ‗Afro-Caribbean' culture and understanding the institutional, academic and partisan stakes involved in its inclusion in local cultural policy.

In his work on Mexican popular culture and nationalist stereotypes, Ricardo Pérez Montfort, who studied the Jarocho Fandango and its cultural position of the ‗Afro-Andalusian Caribbean' in the 17th to 19th centuries, clearly shows how the construction of national cultural symbols entailed downplaying regional diversity in the 1920s-1930s in Mexico (Pérez Montfort, 2003). The 1970s-1980s marked the end of this post-revolutionary period and centralist homogenising cultural nationalism [START_REF] Lucina | Políticas culturales en transición. Retos y escenarios de la gestión cultural en México[END_REF]. This was the start of a cultural decentralisation process initiated by the federal administration and gradually instituted in all Mexican States, Secretariats, Institutes and Councils for Culture.

Thus, the years following the creation, in 1987, of the Instituto Veracruzano de la Cultura (IVEC) gave rise to a definition of the main orientations and implementation of what was to be a decentralised cultural policy in the State of Veracruz. Three elements are of interest here because they profoundly marked what can be analysed as a ‗Caribbean' and ‗Afro-Caribbean' flavour to the cultural policy of Veracruz.

The first entailed, from the creation of IVEC, institutionalising work begun at the end of the 1970s for the retrieval and diffusion of Son Jarocho as the ‗authentic musical tradition of Veracruz' and Fandango as ‗the region's traditional community celebration.' This movement that initially aimed to be a return to the rural Afro-Andalusian Caribbean tradition described by Antonio García de León (García de León, 1992) contributed to producing a new cultural and artistic style that, though very different from the past reality it sought, created new ties with this cultural area, introducing or reintroducing Caribbean instruments and blending the rhythmic, instrumental and harmonic bases of Son Jarocho and Afro-Caribbean music (García de León, 2006: 58). In so doing, musical experimentation, exchange and the circulation of knowledge made possible by the intensification of meetings and festivals revealed made the different influences of this music more visible and audible, in particular those from Africa.

Another element marking local cultural policy impelled by IVEC in the first years of its existence was the work of relaunching in the city of Veracruz Danzón and Son Montuno 8 , two styles of dance and music of Cuban origin introduced in the late 19th and early 20th centuries thanks to permanent contacts between the ports of Havana and Veracruz, but also to the presence of a small Cuban community, which probably facilitated their very rapid adoption in lower-class neighbourhoods and in public places in Veracruz to the point of becoming of the city's main attractions, mentioned in all tourist guides. At the time of IVEC's creation, the Danzón tradition in Veracruz was waning, the golden age of Son Montuno was a thing of the past with many bands and musicians having moved to Mexico City to pursue their careers and eke out a living from their music. This is how one of the first measures taken was to ask those in charge of cultural promotion to locate the musicians and groups of that time to offer them new possibilities for a career in the city. It was from this project begun in the 1980s that several major events, including the Festival Internacional Afrocaribeño, then Festival Internacional Agustín Lara and Festival del Son Montuno, were created a few years later.

The third element in this cultural policy was precisely that of stimulating academic reflection on the Caribbean and its African heritage, which led to creating the Festival Internacional Afrocaribeño. It started with the organisation of two academic forums in 1989 and 1990: Veracruz también es Caribe. 9 The purpose of these meetings was to present research results to promote a representation of Veracruz as a region culturally turned towards the Caribbean. At the same time, there were similar dynamics in Cancún, Quintana Roo, with what was called the International Caribbean Cultural Festival in 1988. But what would actually become a characteristic of the cultural policy of Veracruz was the association of this regional identification with the Caribbean and a national programme also launched in 1989 under the name Nuestra Tercera Raíz (Our Third Root). Its goal was to study and add value to the African presence in Mexico by recognising it as the Third Root in the formation of the country's ‗mestizo culture'. It led to the realisation of specific research, exhibitions, symposia and workshops, as well as the organisation of National Afro-Americanist meetings. This combination of a definition of local cultural policy focusing both on the positioning of Veracruz in the ‗Caribbean cultural basin' and on awareness on a local level of the national ‗Our Third Root' programme largely resulted from the presence and commitment of Luz María Martínez Montiel at the head of the Cultural Heritage at IVEC at the time of its foundation. She was in charge of renovation of the City Museum with the first display dedicated to slavery in a Mexican museum, while also coordinating the ‗Our Third Root' programme for the Dirección General de Culturas Populares. On the one hand, the Caribbean dimension of Veracruz developed around other rationales in its recent history, in particular the growth of festive and sexual tourism not directly linked to the Third Root issue and focusing on the cultural heritage from the Caribbean area. From this standpoint, this dimension is not strictly local in that it is part of tourist and heritage policy aiming to identify the east coast of Mexico with this regional area, positioning places of entertainment and cultural interest in a highly competitive international market. On the other hand, the programme focusing on the Third Root of mestizaje is more strictly national and does not correspond to rationales of tourist appeal. In this context, it is interesting to see how, at any given time, valuing the Caribbean dimension and the theme of the Third Root converge in Veracruz, in the context of implementation of local cultural policy. And it is precisely following this consideration begun by the IVEC Caribbean Studies Centre and for the fourth National Afro-Americanist meeting organised in June 1994 in the city of Veracruz that this convergence was reinforced thanks to the first International Afro-Caribbean Festival.

Consequently, this Festival results from the implementation of cultural policy that itself results from academic discussions conducted in this key period in the history of Mexico and Latin America in which issues relating to diversity, multiculturalism, the cultural heritage, decentralisation and cultural globalisation emerged in the debates and public agendas. This policy entailed bringing together various orientations that began to emerge, in both academic research and public action. The former aimed to study the African cultural heritage in Latin America. Directly linked to Luz María Martínez Montiel's influence, it arose from political and academic interest specifically focusing on the history of slavery. It was a matter of promoting the Third Root of mestizaje, the historic contribution of slaves and descendants of slaves to Mexican national culture.

A second orientation present in the issue of defining and implementing local cultural policy is borne by researchers close to militant Afro-Mexican movements (Cruz [START_REF] Carretero | Yanga and the Black Origins of Mexico‖[END_REF] in connection with Black Studies departments in North American universities. In this case, it is a matter of accepting the cultural and somatic specificity linked to the African presence in Mexico, not only as a common heritage for everyone, but also as a characteristic that can be reasserted today in the context of viewing Mexico as pluriethnic with ‗Afro-Mexicans' as one of the ethnic groups forming the national society.

A third orientation prevailing in Veracruz emphasises the diversity of cultural heritage in the ‗Afro-Andalusian Caribbean'. It promotes mestizaje as the essence of folk culture, mestizaje recognising the importance of the African heritage without making it exclusive. Thus, in his texts, García de León evokes the ‗folk civilisation' that generated mestizaje in the colonial period against the Spanish conquistadors and the city's white elite, a ‗folk culture with various influences, African, Caribbean, European' that developed on the fringe of the dominant class's elitist culture. He presents Veracruz as the ‗tropical Babel of the Indies where all possible races and mixtures coexist,' which differs from militant ‗Afro-Mexican' rationales that seek to relate these traits solely to ‗Black identity', as opposed to the ideology of mestizaje (García de León, 1996).

The difficulty of maintaining these different orientations is apparent in the design of posters for the Festival. Most of them emphasise phenotypic traits and physical postures representing Africanity, which, though stylised, may seem quite remote from the very idea of the Third Root's contribution to contemporary Mexico's national mestizo culture:

Poster (1996) Logo ( 1997) Poster ( 2001)

Poster ( 2004) Poster ( 2008) Poster (2009) On the other hand, after various changes at the head of IVEC, this illustration of Africanity has sometimes been replaced by another, more in stride with the focus on the many origins of mestizaje and cultural diversity in the Caribbean. This was the case in particular in 1999 when the Festival changed its name to Festival of the Caribbean, or in 2005 when, after restoring its original name, a subtitle was added, specifying that this was a ‗Festival for everyone.' In the former case, the poster evoked the Caribbean in a very stylised way around the letter ‗C' and the sun. In the second, it sought to express diversity through the representation of a multiplicity of faces, each different from all the others: Poster ( 1999) Poster (2005) It is possible to conclude that this policy of institutional promotion of the Third Root has not succeeded locally in creating a community feeling around Afrodescent [START_REF] Christina | Race and national ideology in Mexico: an ethnographic study of racism, color, mestizaje and blackness in Veracruz, Ph D of Sociology[END_REF][START_REF] Christina | Race and national ideology in Mexico: an ethnographic study of racism, color, mestizaje and blackness in Veracruz, Ph D of Sociology[END_REF]. Not particularly influential in the city of Veracruz, the position supported by Afro-Mexican militant movements has been able to develop, in a very limited way, in some of the region's villages. It is also possible to hypothesise that all those years of promoting the Third Root of Mexican mestizaje, punctuated by academic meetings, exhibitions and many concerts with groups invited from the entire Caribbean region performing alongside local music groups, have produced some effects. From the start, the different players involved in implementing cultural policy agreed, for example, on unequivocal criticism of the dominant position of a conservative local elite that rejected all forms of recognition of a common heritage linked to African origins and the cultural proximity of Veracruz with the Caribbean. From this standpoint, thirty years of public policy highlighting ‗traditional' Son Jarocho, ‗Afro-Caribbean' music, the Caribbean as a specific cultural area and the Third Root of mestizaje did bring about profound changes in the criteria for cultural legitimation. They contributed further to moving from a historical period characterised by accepted negation of the African heritage and ‗whitening' operations for stereotyped notions of Jarocha identity (Pérez Montfort, 2007) at a time when African roots and mestizaje had become part of the representation of local society. This does not mean that this representation is accepted as such, but that it has now become a legitimate norm for defining local identity. Consequentially, the stakes for social actors in concrete situations involve positioning themselves with respect to this norm which they can accept or reject, assume or endure.

ELECTIVE AFRICANITY AND CALLEJERA CULTURE

To account for all these possible stakes, the third research orientation presented entails examining identity-related expression as it appears in public places as the stage for urban life. From this point of view, it is in the interest of many establishmentsfrom the city of Veracruz and its historic centre's pedestrian squares and streets, to nightlife establishments and the malecón (boardwalk) where passers-by meet -that all of these places are occupied daily by musicians and dancers, regular visitors and passing tourists who, by their very copresence, scenically produce modes of social identification combining different levels of affiliation and social distinction, thereby making them interpretable. We will broach this issue here through an ethnographic study aiming to describe the forms of expression of ‗elective Africanity'.

The study bears on a cultural programme called Noches de Callejón (night in the streets) organised every weekend at El Portal de Miranda, a small pedestrian street in the historic centre of Veracruz. This programme was instigated by the artistic director, singer and guitarist of the Juventud Sonera group whose initial project was to bring back the tradition of Son Montuno that existed in Veracruz since the end of the 1920s and include modern elements to attract younger people (drums, electric guitar...). Thus, since the release of the recording ¡Guarachero!10 very close to the Afro-Cuban tradition, the group's leader has been paying more attention to his pop-star look while publically declaring his interest in Son. After letting his hair grow and playing electric guitars sporting prestigious American brands, he looks more and more like Jimi Hendrix, Lenny Kravitz or Ben Harper as his fans keep reminding him, thereby intensifying the references to black American culture.

Record sleeve for ¡Guarachero! Gobierno del Estado de Veracruz, IVEC, 2004 Thanks to its location, at the crossroads of tourist routes and local inhabitants' evening strolls, this urban stage in Veracruz is a place where ordinary passers-by interested in the local scene can meet a group of mutual acquaintances made up of regular patrons who greet musicians as they arrive, form small social groups, start dancing and strike up conversations... This does not mean they form a homogeneous social group or a single age class. Many are between 16 and 30 years old, but there are also older individuals, alone or with partners. Some are from the city's working-class neighbourhoods, while others belong to a local cultural elite who enjoy this type of music and urban atmosphere.

These regular patrons, despite differences -between young people and those who are not so young, men and women, passing dancers and musicians, give rise to a form of expression that entails sharing certain cultural elements whose codes may be subject to interpretation and signification. It is not so much a matter of defining oneself as ‗Black', but of behaving ‗like a Black,' dancing according to certain rhythmic sequences, overplaying choreographic gestures and movements of different bodily expressions referring to the Afro-Caribbean world. Most of the time, these demonstrations are identified by the public and musicians as a ‗show' that differs from more traditional dancing in couples. A circle forms gradually around the person, shouting encouragements, regular visitors exchange remarks and the musicians stray from their routine with more emphatic improvisation and solos. Some people regularly put on such ‗shows', like a 20-year-old girl living in a lower-class neighbourhood of Veracruz, whose skin colour and features are not particularly ‗Black' or ‗mulatto', who often starts dancing when musicians play Carlos Oliva's Pelotero a la Bola. At the end, one of the musicians in the group always thanks her by name with a short comment like ‗Look at that Black girl's wild dance!' (June 2009). According to the people interviewed on her performance, her style resembles both that of the Rumberas who perform in cabarets and the erotic dances practised with table dancing stereotypically associating unbridled sexuality and Africanity.

Other people may also put on such shows, like an old lady of modest origin from Veracruz who often dances for tourists near restaurants at the Zócalo. In this case, regular patrons and musicians describe her way of dancing as ‗Creole' (Criollo). This category is used locally to describe old rural Caribbean traditions, at the origin of Son, a mixture of Spanish guitar and the syncopated beats brought by African slaves. Thus, in addition to inciting admiration on the part of these knowledgeable audiences, this lady's performance is viewed as an expression of the old Afro-Caribbean heritage of Veracruz.

Young boys and girls from the city or region accustomed to more alternative cultural venues, but regularly present at Noches de Callejón, may also join in such self-display, combining emblematic signs of Caribbean Blacks (dreadlocks, beard, colour and style clothing...) and ways of moving and dancing, which, there again and in a different style, are described as forms of expression of elective Africanity. This is what is explained by a 23year-old man from a lower-case neighbourhood of Veracruz who plays drums and sings with friends in public transportation when he needs money:

-I really enjoy coming here to El Portal in the late evening when there is a special atmosphere and everyone joins a little in the Something else that comes out of these observations is that the expression of such elective Africanity is sometimes combined with a social boundary: one that entails positioning oneself with respect to distinctive practices that leave a strong mark on the organisation of nightlife in Veracruz. For example, during a collective discussion on the subject in one of the cafés at the Zócalo, a young artist who is a regular customer of the different landmark nightlife establishments in the historic centre of Veracruz describes this expression in his own way:

-I really prefer going there (Portal de Miranda) rather than to trendy (Fresas) discotheques on the Boulevard (coastal road). First, here the music is live, and it's free; you're in the street... You can come and go, have a beer, have a look at what's happening elsewhere, come back... Let's say it corresponds much better to my state of mind, the Callejero (street) spirit... That's the true spirit of Veracruz... isn't it? And most of all, what I can't stand in trendy places (antros) is the overly codified way you have to dress, walk, talk... look at others, while others make remarks about your clothes, the people you hang around with... Okay, they're not all like that; you have nicer places, more open you might say, but it's generally rather like that...

-And what's it like here? -Well, in this kind of place, you do what you like; you don't have to ask yourself if you're okay or not; you can even enjoy doing the opposite, speaking bad, dressing wrong, doing all the things ‗nice people' think is ‗bad'.

-Meaning? -Well, going to lower-class places, not being with a girl who spends all her time getting ready to go out... When Julia [his girlfriend] dresses nice, 11 -you see, with a dress, high heels, nail polish -I warn her: You're becoming Fresa, 12 and we laugh about it...

-And is it more a matter of listening to this kind of music with Afro-Caribbean influences than anything else? -Well, this music recalls above all the past of the port (Porteño), slaves and pirates, contraband, all the nightlife around the port, the wharves, the dockers, the popular cantinas around the market.. In other words, the issue here is sharing certain cultural practices with others to transform them into signs, signs that refer not only to ‗popular' culture, but a way of life that strays somewhat from the norms set out by local ‗polite society' (gente bien). In other words, signs of non-alignment with ‗nice' people who go to certain trendy bars and ‗speak with a lisp' 13 to emphasize their social distinction, or who have chosen to move away from the centre of Veracruz to the cosier environment of the nearby municipality of Boca del Río, where the region's Fresas now gather in their clearly expressed will to emphasise social distance through greater spatial separation. And also signs that, by adopting the sexually explicit body movements of African dance and Rumbera, denote a common deviation from those who are also called ‗decent people' (gente decente), one of whose characteristics involves precisely showing their moral distance from ‗ordinary' folk 14 and rubbing out all that can appear as signs of Africanity in their way of displaying themselves to others.

Here, the display of physical traits, postures, gestures and aesthetics referring to the different cultural styles evoking the African heritage is a way of expressing one's empathy with Callejera (street) culture that develops its own reinterpretation of ethnic boundaries and maintains an elective relationship with Africa and the black Americas. And it is also a way of positioning oneself in a class through signs attributed locally to the different ‗origins' of mestizaje. Like ‗speaking with a lisp' or dancing ‗like a Black' are social markers based on a shared belief in the common origins forming local society. This belief can be more or less well accepted or rejected, and the cultural and physical traits associated with it more or less displayed or concealed, blackened or whitened. It is indeed in this way that expressions of ‗Black identity' observable in such contexts are not despite, but rather because of a representation of mestizaje that entails constant reference to one or the other of these different ‗roots' that leads individuals to position themselves socially, physically and culturally with respect to supposed ‗African roots'.

11 English word used locally to refer to others or oneself as la gente nice (nice people) featured in celebrity magazines or, more modestly, in the VIP pages of regional daily newspapers. 12 ‗Well-born young people' or, more derogatorily, ‗rich kid' [‗mama's boy'] in the local taxonomy. 13 The Spanish expression 'hablar con la zeta' is used locally to refer to Spaniards who pronounce the letters ‗c' and ‗z' with a lisp like the English ‗th', unlike most Mexicans, except to evoke one's Spanish origins with some snobbery. 14 Such expressions as gente fea, gente corriente, gente vulgar, gente coloniera are all used in the local taxonomy to refer to people who are poor, unrefined and vulgar from the working-class districts of Veracruz.

CONCLUSION

The approach to this research, of which some empirical elements have been presented here, seeks to portray a broader perspective than those focusing solely on the political embryos that strive towards collective mobilisation of ‗Black identity' awareness which is associated with the experience of domination and discrimination. The analysis focuses on the processes whereby ethnic definitions referring to the ‗African roots' of mestizaje, phenotypic traits more or less perceived as ‗Black' and ‗Black' or ‗African' cultural characteristics are carried, interpreted or circumvented by different types of social actors in more or less stable, durable or variable ways according to context and situation. Since Fredrik Barth's seminal text (Barth, 1969), the processes of creation-transformation of boundaries and the activities of agents and agencies striving for their recognition and social legitimacy or, conversely, their disappearance, have been the subject of many studies and theoretical analyses. Recently, Andreas Wimmer listed the different contributions to a comparative approach focusing on a ‗boundary-making approach' in ethnicity studies and proposed a new taxonomy that attempts to organise conceptually the ways in which social actors participate in the transformations of ethnic boundaries (Wimmer, 2008a, b). However, this overinvestment in historical and politital processes aiming to stabilise or obscure the social meaning of such boundaries does not account for situations where boundaries take on meaning for the social actors and contribute to social life. From this standpoint, the focus on scenes from urban life, definitions of cultural policy or rationales of social distinction suggests research orientations to help better understand: the ambiguity of such boundaries, various interpretations of social scenes, ethnic categorisations that function only in certain areas of social life all while contributing to its organisation, vague stereotypes that are sometimes mobilised in courses of action, and plays on appearances.

On the basis of Bastide's dialectical approach, it is possible to account for the processes observed by analysing them in terms of ‗stable instability' and ‗unstable stability'. 15 The former can be taken to mean the relative permanence of the fluidity of ethnic boundaries. In public policy concerning ‗African roots', as in touristic descriptions of Veracruz or in the ordinary situations of social life, nothing can support the idea that this fluidity is threatened by rationales for radicalising differences or toughening identities. As we have seen, here ethnic boundaries find meaning in their relationships with representations of mestizaje and its ‗roots'. They are not merely ‗surviving' the reality of mestizaje until they disappear; quite the contrary, they are constantly fuelled by a representation of mestizaje recalling the ‗heritages', ‗peoples' and ‗races' composing it.

At the same time, the ‗stable instability' of ethnic boundaries is accompanied by ‗unstable stability'. It is by moving from negation of the African heritage and the undertaking to whiten the stereotyped frames of Jarocha identity to recognition of the African roots in mestizaje that most of the phenomena observed in this study take on their meaning. And if they have found a certain historical permanence, nothing suggests that this will persist. Transnational trends towards the formation of collective awareness of belonging to the same group, supported by international development agencies, may get the better of this specific relationship observed in the city of Veracruz between the representations of mestizaje and enacted ethnicity.

BIBLIOGRAPHIC REFERENCES

  . and even I also like other musical genres and atmospheres, what I like most is the idea of being attracted to what ‗decent folk' reject... (June 2008).

  fun of showing off what's Black in you (laughs) -Do you define yourself as Black? -No, not at all, but we all have some Black ancestors, don't we? So, when you hear this music and you're plunged in this ambiance, you can't help but start moving in a certain way... -Just how? Could you be more explicit? -Not really. I don't see myself... But if you take my buddies, like Ricardo: he feels it more like a Jamaican, you see, reggaeman, easy [...]. While Fallo'll really be influenced by the rap movement, hip hop and all that [...]. Sara, my girlfriend, lets loose with African dance. You see, that's really another style... (April 2008)

This research was undertaken in the context of an IRD-INAH-CIESAS research contract, the ANR Suds AFRODESC programme ‗Afrodescendants and slavery: domination, identification and heritage in the Americas,15th-21st centuries' (http://www.ird.fr/afrodesc/) and the 7th PCRD EURESCL European programme on ‗Slave Trade, Slavery, Abolitions and their Legacies in European Histories and Identities' (http://www.eurescl.eu/).

Some authors have recently begun showing interest in this[START_REF] Hettie | Creative Standardization: Danzon and the Port of Veracruz[END_REF][START_REF] Figueroa | The Complexities of the Visible: Mexican Women's Experiences of Racism, Mestizaje and National Identity[END_REF][START_REF] Christina | Race and national ideology in Mexico: an ethnographic study of racism, color, mestizaje and blackness in Veracruz, Ph D of Sociology[END_REF].

In particular, it is part of research conducted from 2002 to 2006 in the context of the IDYMOV programme, Identidades y movilidades, las sociedades regionales frente a los nuevos contextos políticos y migratorios. Una comparación entre México y Colombia (IRD-CIESAS-ICANH).

In Brubaker's terms, unlike ‗groupism', the tendency to consider ethnic groups, nations and races as substantial entities, the notion of ‗groupness' refers to an event, something that may occur -or not -in the social world, may be crystallized -or not -despite identity entrepreneurs' efforts to impose their existence[START_REF] Brubaker | Ethnicity without Groups‖[END_REF].

See, for example, García Díaz, 2002;[START_REF] Pasquel | Viajeros en el Estado de Veracruz, México, Citlaltépetl, Colección Suma Veracruzana[END_REF][START_REF] Montfort | El jarocho y sus fandangos vistos por viajeros y cronistas extranjeros de los siglos XIX y XX. Apuntas para la historia de la formación de un estereotipo regional‖, Veracruz y sus viajeros[END_REF]

See Historia del Hotel Mocambo, information leaflet distributed by Hotel Mocambo Veracruz.

http://www.travelbymexico.com/veracruz (consulted on 26 November 2009).

The name given to this Son of Cuban origin to distinguish it from Son Jarocho. The different episodes since its arrival in the 1920s and its adoption in Veracruz with the formation of bands are recounted by many analysts of local cultural life(Figueroa Hernández, 2002; García Díaz, 2002a; González, 2007; Mac Masters, 1995).

Literally, ‗Veracruz is the Caribbean, too'.

In Cuba, the word guarachero refers to a musician who composes and sings la guaracha, a satirical musical genre, viewed as a playful person who falls in love easily.

Roger Bastide borrowed the terms continuous discontinuity and discontinuous continuity from Georges Gurvitch to attempt to exceed the opposition structuring the field of Black studies in the Americas around analyses in terms of continuity, dear to Melville Herskovits, or discontinuity, according to Franklin Frazier[START_REF] Roger | Continuité et discontinuité des sociétés et des cultures afro-américaines‖, Bastidiana, n°13-14[END_REF].


Claire Chavaudret
email: chavaudret@unice.fr

Laurent Stolovitch

Analytic reducibility of resonant cocycles to a normal form

We consider systems of quasi periodic linear differential equations associated to a 'resonant' frequency vector ω, that is a vector whose coordinates are not linearly independent over Z. We give sufficient conditions that ensure that a small analytic perturbation of a constant system is analytically conjugate to a 'resonant cocycle'. We also apply our results to the non resonant case : we obtain sufficient conditions for reducibility.

Introduction

Let θ = (θ 1 , . . . , θ d ) be the coordinates on the torus T d . Let x = (x 1 , . . . , x n ) be coordinates on R n . Let ω ∈ R d and A 0 = (a i,j ) 1≤i,j≤n be a n × n-matrix. Let us consider the following vector field on T d × R n ,

D := ω ∂ ∂θ + A 0 x ∂ ∂x
Here, the notation ω ∂ ∂θ stands for d i=1 ω i ∂ ∂θ i and A 0 x ∂ ∂x stands for the vector field 1≤i,j≤n a i,j x j ∂ ∂x i . We consider a linear perturbation of D of the form

X := D + R = ω ∂ ∂θ + A(θ)x ∂ ∂x (1)
where A = A 0 + a(θ), a being an analytic matrix-valued function on T d with zero mean value: â(0) = 0. This corresponds to the differential equation

θ = ω ẋ = A(θ)x,
or, in other words, it corresponds to the linear differential equations with quasi periodic coefficients

dx dt (t) = A(θ 0 + tω)x(t) (2)
for some initial θ 0 ∈ T d . On this vector field, there is a natural action by linear diffeomorphisms over the torus: if Z : T d → gl n (C) is a matrix-valued function with zero mean value, we can consider the (linear) change of variables Z = (Id n + X(θ))Y over the torus. It conjugates (2) to dx dt (t) = A ′ (θ 0 + tω)x(t).

(3)

where A ′ is the matrix valued function satisfying to DX(θ).ω = A(θ)(Id n + X(θ)) -(Id n + X(θ))A ′ (θ).

This can readily be seen by differentiating with respect to t, the quantity Z(θ 0 + tω) = (Id n + X(θ 0 + tω))Y (θ 0 + tω)

We will denote DX(θ).ω by ∂ ω X(θ).

We are interested in the problem of analytic classification with respect to such a group action. A case of particular interest is the one for which the cocycle A can be transformed to a constant matrix. In that situation, we say that A is reducible. The regularity class of the transformations used is important. This reducibility phenomenon is well known for cocycles over the circle (Floquet theory, see [START_REF] Arnold | Chapitres supplémentaires de la théorie des équations différentielles ordinaires[END_REF][chap. 26]). When the base is a higher dimensional torus, the situation is much more complicated. The frequencies ω = (ω 1 , . . . , ω d ) are usually assumed to be linearly independent over Z. In that case, they are also assumed to satisfy a 'Diophantine' condition.

Most known results consider the case where the 'fiber' has dimension n = 2. In that case, the rotation number [START_REF] Johnson | The rotation number for almost periodic potentials[END_REF] of the cocycle can be defined. The rotation number is a very useful quantity since its arithmetical properties determine whether the cocycle can be reduced or not: in the analytic category, [START_REF] Eliasson | Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation[END_REF] proved reducibility for Schrödinger cocycles with a large energy or a small potential, under a Siegel-type Diophantine condition on the frequency vector and on the rotation number; a similar result was obtained in [START_REF] Chavaudret | Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition[END_REF] under weaker arithmetical conditions, namely, Brjuno-Rüssmann conditions, on the frequencies and on the rotation number. Non-perturbative versions of Eliasson's results were also obtained: the article [START_REF] Avila | Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles[END_REF] gives a description of Schrödinger cocycles with a recurrent Diophantine frequency vector for almost every energy.

In a higher dimensional fiber, reducibility results and reducibility criteria were obtained in [START_REF] Krikorian | Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts[END_REF][START_REF] Krikorian | Réductibilité presque partout des flots fibrés quasi-périodiques à valeurs dans des groupes compacts[END_REF] when the fiber is in a compact Lie group; in particular, reducibility in full measure for a oneparameter family of cocycles under non-degeneracy conditions and the equivalence between reducibility and compactness of the iterates. The authors of [START_REF] He | Full measure reducibility for generic one-parameter family of quasi-periodic linear systems[END_REF] proved a similar result in the non compact case. Almost reducibility of cocycles in any dimension, that is, the possibility of conjugating them to a cocycle arbitrarily close to a constant, was also proved in a perturbative framework in [START_REF] Eliasson | Almost reducibility of linear quasi-periodic systems[END_REF] and improved in [START_REF] Chavaudret | Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles[END_REF]. However, positive reducibility results still seem to depend on the existence of a rotation number, or on strong spectral assumptions as in [START_REF] Ju | On constructing solutions of linear differential equations with quasiperiodic coefficients by the method of improved convergence[END_REF].

The previously mentioned articles all require Diophantine or Brjuno conditions on the frequency vector. In [START_REF] Zhou | Reducibility results for quasiperiodic cocycles with Liouvillean frequency[END_REF] and in [START_REF] Avila | A KAM scheme for SL(2, R) cocycles with Liouvillean frequencies[END_REF], the case of a liouvillean frequency vector was considered. However, the results are only stated for a one or two-dimensional frequency vector.

In this article, we shall consider analytic cocycles over a torus T d with a fiber of any dimension n. We consider them as 'small' perturbations of constant cocycles. We consider the general case where the frequencies might be linearly dependent over Z, that is there are resonances. We cannot expect such a general cocycle to be reducible over the torus. We shall give sufficient conditions that ensure that the cocycle is analytically conjugate to an analytic resonant cocycle, that is a cocycle in which Fourier modes e i k,θ , depend only on the resonances relations of the frequencies, that is k, ω = 0. In particular, a resonant cocycle is constant along resonant trajectories of the torus.

Our first result concerns perturbations of a diagonal matrix with separated spectrum, and can be viewed as a generalization of [START_REF] Ju | On constructing solutions of linear differential equations with quasiperiodic coefficients by the method of improved convergence[END_REF] and [START_REF] Ja | The reducibility of systems of n linear differential equations with quasiperiodic coefficients[END_REF].

Theorem 1.1. Let A 0 be a diagonal matrix with eigenvalues λ 1 , . . . , λ n having distinct real parts and let C = min j =k Reλ j -Reλ k >0. Assume that ω is very weak with exponent R > 0 (see definition 2.8). Let r > R and let a be a C ω r matrix valued function with zero mean value. There exists ε 0 (r, n, d, ω, C) such that if |a| r ≤ ε 0 , then the system whose coefficients are A 0 + a can be analytically conjugated to a system whose coefficient matrix is resonant and diagonal.

Corollary 1.2. Under assumptions of the previous theorem, equation (1) is analytically reducible to a constant cocycle on each level set

∩ i {θ | m i , θ = constant i }.
Corollary 1.3. Under assumptions of the previous theorem and if there is no resonances relations (i.e. R = {0}), then the cocycle A is analytically reducible. This result is due to L. Adrianova [START_REF] Ja | The reducibility of systems of n linear differential equations with quasiperiodic coefficients[END_REF] and to Mitropolskii-Samoilenko ( [START_REF] Ju | On constructing solutions of linear differential equations with quasiperiodic coefficients by the method of improved convergence[END_REF]) when the frequencies satisfy a Siegel's type small divisors condition. In our result, we only require a much weaker condition (even much weaker than Brjuno-Rüssmann condition).

The second main result considers triangular perturbations of a constant system.

Theorem 1.4. Let S be a diagonal matrix which satisfies a second Melnikov condition away from the resonances (see definition 2.9). Let F be a C ω r matrix valued function, with upper triangular and nilpotent values. There exists ε 0 (n, d, ω, S, r) such that if |F -F (0)| r ≤ ε 0 , then the system whose coefficients are given by S + F can be analytically conjugated to a system whose coefficients matrix is resonant, upper triangular and commuting with S.

As a consequence, we obtain a decomposition into analytic invariant subbundles of the initial cocycles defined by the unperturbed constant cocycle.

Next we will consider strongly commuting perturbations, i.e perturbations which commute with the constant part and also such that their various Fourier modes all commute with each other (see definition 5.1).

Theorem 1.5. Assume that the vector ω is very weak with exponent R > 0. Let A 0 ∈ gl(n, C). Let r > R and let F ∈ C ω r be matrix valued and strongly commuting, and such that all Fourier modes of F commute with A 0 . There exists C(ω, A 0 , r, n, d) such that if |F | r ≤ C, then the system A 0 + F is reducible to a resonant system A ∞ (θ) which commutes with A 0 .

Notice that, under this strong algebraic condition on the perturbation, no second Melnikov condition is needed since most small divisors will be avoided. In the last section, we shall consider similar statements under assumptions that the cocycles belongs to a Lie sub algebra of matrices.

We shall also apply our results to a non-resonant situation to obtain sufficient conditions that ensure analytic reducibility. Our proofs are based on a Newton-KAM scheme whose convergence is due to the control of the small divisors of the initial unperturbed constant cocycle.

Acknowledgement We would like to thank Nikos Karaliolios who suggested the scheme of this final proof of theorem 1.1. It happens that, finally, this allowed us to strongly weaken the diophantine condition used. We would also like to thanks Hakan Eliasson who suggested the very short and elegant proof of theorem 1.5. We also would like to thank the annonymous referees for helping us providing a better text.

Notations and assumptions

Definition 2.1. The vector ω is said to be resonant if there exists m 1 , . . . m s ∈ Z d linearly independent over Q such that m i , ω = 0, i = 1, . . . , s, and if k, ω = ω i then k = E i + s j=1 l j m j where the l j 's are integers and E i is the ith-unit vector of Z d . We shall denote by R the Z-module generated by the mi's. We define an equivalence relation ∼ on Z d by

k 1 ∼ k 2 ⇔ ∃l ∈ Z s , k 1 = k 2 + l • m ⇔ k 1 -k 2 ∈ R
The equivalence class of an integer vector v will be denoted by v .

Remark 2.2. A sum is defined on equivalence classes as follows:

u + v = u + v ; this is well defined since u ∼ u ′ , v ∼ v ′ ⇒ u + v ∼ u ′ + v ′ .
Definition 2.3. A function f defined on the torus is said to be resonant if its Fourier series has only modes which are proportional to the m j 's, i.e. f = l∈Z s f l e i l•m,θ , where l • m stands for l 1 m 1 + • • • + l s m s . In that case, we shall write f res to notify this fact. Definition 2.4. Let H v be the subspace of functions f defined on the torus whose elements only have Fourier modes in v .

Remark 2.5. For all function f continuous on the torus, there exists a unique decomposition f = f res + f nr where f res is resonant and f nr does not have any harmonics in the lattice R generated by the m j 's. More generally, there is a natural decomposition of the space of continuous functions on the torus into subspaces

H v : f = v ∈Z d /∼ f v . With this notation, f res coincides with f 0 . Note that H v • H v ′ ⊂ H v+v ′ .
Definition 2.6. For a function f defined on T d (be it matrix-valued or scalar-valued), if the Fourier modes of f belong to a finite number of equivalence classes, then we shall denote by deg F the degree of f , i.e the smallest integer such that all equivalence classes of Fourier modes of f have a representative

v = (v 1 , . . . , v d ) with length |v 1 | + • • • +|v d | ≤ deg F .
When necessary, for a matrix-valued function F , we will denote by F diag its diagonal part.

We will consider the Banach space C ω r of (matrix valued) functions which are analytic on a neighborhood {|Im θ| ≤ r} of the torus such that

|f | r := v ∈Z d /∼ |f v | r < +∞, |f v | r := sup |Im θ|≤r ||f v (θ)||
where ||.|| denotes the operator norm on R n or C n . We notice that this norm is sub-multiplicative,i.e. |f g| r ≤ |f | r |g| r .

Assumption 2.7. The function A in equation (1) is analytic in a neighborhood of the torus (in fact, in a complex strip around the real d-dimensional plane) so there exists r > 0 such that A ∈ C ω r .

Definition 2.8. We shall say that the frequency vector ω is very weak if there exist R > 0, K > 0 such that for all k ∈ Z d with k, ω = 0, then

| k, ω | ≥ Ke -R|k| . ( 5
)
If this holds, the number R is simply called the exponent of ω.

This kind of very weak condition also appeared in different context in [START_REF] Stolovitch | Forme normale de champs de vecteurs commutants[END_REF].

Definition 2.9. We shall say that A 0 is a Melnikov matrix or satisfies the second Melnikov condition away from the resonances if there exist a constant κ ′ and a strictly increasing differentiable function

g ′ : [1, +∞[→ R * + satisfying ∞ 1 ln(g ′ (t)) t 2 dt < +∞
such that for all pair (α, β) of eigenvalues of A 0 , and all

m ′ ∈ Z d such that i m ′ , ω -α + β = 0, then |α -β -i m ′ , ω | ≥ κ ′ g ′ (|m ′ |) . ( 6
)
We shall also say that A 0 is Melnikov up to order N ∈ N if Equation (6) holds for all m ′ with |m ′ | ≤ N .

Remark 2.10. If A 0 is a Melnikov matrix, then in particular ω satisfies a Brjuno-Rüssmann arithmetical condition away from the resonances, which is weaker than Siegel's diophantine condition, and was used for instance in [START_REF] Chavaudret | Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition[END_REF].

3 Analytic conjugation to a normal form: the separated diagonal case

Let us make an extra assumption:

Assumption 3.1. A 0 = diag(λ 1 , . . . , λ m ) is diagonal with eigenvalues having distinct real parts.
The main result of this section is the following :

Theorem 3.2. Assume that ω is very weak with exponent R > 0. Let r > R and let A = A 0 + a be a C ω r -analytic cocycle (1) where a is a zero-mean valued matrix function. There exists ε 0 depending only on A 0 , n such that if |a| r ≤ ε 0 , then there exists a convergent transformation conjugating (1) to a normal form in a neighborhood of the torus:

N F = ω ∂ ∂θ + D res (θ)x ∂ ∂x .
where D res is a resonant diagonal matrix-valued function.

We shall prove Theorem 3.2 in two steps: first, the system can be diagonalized without any arithmetical assumption on ω; second, such a diagonal system can be conjugated to a resonant one under the very mild arithmetical assumption (5) on ω (it is much weaker than the Brjuno-Rüssmann condition contained in the definition 2.9). The precise statements are as follows:

Proposition 3.1. Suppose that A 0 has eigenvalues with distinct real parts and that a is a matrix-valued function, analytic on the torus, with zero mean value. Then there exists ε 0 depending only on A 0 , n such that if |a| r ≤ ε 0 , then there exists an analytic transformation conjugating the system whose coefficient matrix is A 0 + a to a diagonal system. Proposition 3.1 will be proved in the following sections. The second step is much easier: Proposition 3.2. Let D be a C ω r -analytic function on the torus whose values are diagonal matrices; if there exist

r ′ > 0, C ′ > 0 such that for all k ∈ Z d with k, ω = 0, | k, ω | ≥ C ′ e -|k|(r-r ′ )
then the system with coefficient matrix D can be C ω r ′ -conjugated to a resonant diagonal system. Proof. Consider the system with analytic coefficients given by D, that is,

     ẋ1 (θ, t) = D 1,1 (θ + tω)x 1 (θ, t) . . . ẋn (θ, t) = D n,n (θ + tω)x n (θ, t) (7)
Each line can be solved separately and immediately by

x j (θ, t) = exp t 0 D j,j (θ + sω)ds x j (θ, 0)
By setting D j,j (θ) = D j,j,res (θ) + D j,j,nr (θ), one has

x j (θ, t) = exp(tD j,j,res (θ)) exp t 0 D j,j,nr (θ + sω)ds x j (θ, 0). Indeed, we have D j,j,res (θ) = l∈Z s D j,j,l e i l•m,θ , where l • m stands for l 1 m 1 + • • • + l s m s . Therefore, D j,j,res (θ + sω) = D j,j,res (θ). One can explicit t 0 D j,j,nr (θ + sω) = k∈Z d D j,j,nr (k) e i k,θ+tω -e i k,θ i k, ω
Letting now D res (θ) = diag(D j,j,res (θ)) and letting Z be the diagonal-valued function given by

Z j,j (θ) = exp   k∈Z d , k,ω =0 D j,j (k) e i k,θ i k, ω  
gives the desired conjugation

X(θ, t) = Z(θ + tω)e tDres(θ) Z(θ) -1
The question therefore reduces to finding an arithmetical condition on ω under which Z is analytic. Since D is assumed to be analytic, then for every k,

| D j,j (k)| ≤ |D j,j | r e -|k|r
thus Z is analytic whenever there exist r ′ > 0 and C > 0 such that for all k ∈ Z d , k, ω = 0,

e -|k|r | k, ω | -1 ≤ Ce -|k|r ′ that is to say, | k, ω | ≥ C ′ e -|k|(r-r ′ ) .

Cohomological equation and iteration process

The proof of Proposition 3.1 is based on the following:

|A -A 0 | r < δ 8C n
then there exists a solution X :

T d → gl(n, C) of the equation ∂ ω X(θ) = [A(θ), X(θ)] + F (θ) -F diag (θ) (8)
where ∂ ω stands for the derivative in the direction ω.

Moreover, X is analytic and belongs to C ω r , with the estimate

|X| r ≤ 4C n |F | r 3δ -8C n |A -A 0 | r ≤ 2C n δ |F | r .
Proof. Equation ( 8) is equivalent to

(D + R)(X)(θ) = F (θ) -F diag (θ) (9)
where

D : X → ∂ ω X -[A res , X] and R : X → -[A nr , X]. One has the bound |RX| r ≤ 2|A -A 0 | r |X| r . Concerning D, an equation of the form DX = G -G diag is equivalent to its decomposition along the H v 's: for all v ∈ Z d / ∼ such that v, ω = 0, i v, ω X v (θ) = [A res (θ), X v (θ)] + G v (θ) -G v ,diag (θ). (10)
By decomposing by matrix coefficients, we obtain, if j = k, that (G -G diag ) v ,j,j = 0 and we can set X v (θ) j,j := 0. For j = k, and all v, θ,

X v (θ) j,k = G v (θ) j,k i v, ω -A(θ) j,j,res + A(θ) k,k,res , If v ∈ Z d / ∼ is such that v, ω = 0, then for j = k, we select X v (θ) j,j := 0. For j = k, we have X v (θ) j,k = G v (θ) j,k -A(θ) j,j,res + A(θ) k,k,res
.

Since for all j = k and for all v ∈ Z d , we have

inf |Im θ|≤r |A(θ) j,j,res -A(θ) k,k,res -i v, ω | ≥ inf |Im θ|≤r |ReA(θ) j,j,res -ReA(θ) k,k,res |,
|A(θ) j,j,res -A(θ) k,k,res -i v, ω | ≥ δ - C n δ 4C n = δ 3 4
as well as

|X v j,k | r = sup |Im θ|≤r |X v j,k (θ)| ≤ 4 3 sup |Im θ|≤r |G v ,j,k (θ)| δ ≤ 4 3 C n |G v | r δ .
Therefore, the operator D is invertible and one has the bound Let C n be the constant defined in the assumptions of Proposition 3.3. Assume that A satisfies to

|D -1 | C ω r →C ω r ≤ 4Cn 3δ . Therefore, if |A -A 0 | r < δ 8Cn , then |(D + R) -1 | r ≤ |D -1 | r |(Id + D -1 R) -1 | r ≤ 4C n 3δ j≥0 4C n 3δ 2|A -A 0 | r k ≤ 4C n 3δ -8C n |A -A 0 | r ≤ 2C n δ , thus
inf |Im θ|≤r |ReA(θ) j,j -ReA(θ) k,k | ≥ δ ( 12
)
for all v ∈ Z d , and to

|A -A 0 | r ≤ ε 2 ≤ δ 8C n . (13)
There exists A ′ : T d → gl(n, C) a diagonal C ω r -analytic perturbation of A 0 , there exists F ′ : T d → gl(n, C) which belongs to C ω r , and there exists Z :

T d → GL(n, C) which belongs to C ω r such that for all θ ∈ T d , ∂ ω Z(θ) = (A(θ) + F (θ))Z(θ) -Z(θ)(A ′ (θ) + F ′ (θ))
and

|F ′ | r ≤ 8C n δ |F | 2 r Moreover |Z -I| r ≤ 2C n δ |F | r (14)
and

A ′ satisfies inf |Im θ|≤r |ReA ′ (θ) j,j -ReA ′ (θ) k,k | ≥ δ -2C n ε (15)
for all v ∈ Z d , as well as

|A ′ -A 0 | r ≤ ε 2 + ε (16)
Proof. Let X be the solution of ∂ ω X = [A, X] + F -F diag given by Proposition 3.3 and let us set Z := I + X. Then, according to (4), we have

∂ ω Z(θ) = (A(θ) + F (θ))Z(θ) -Z(θ)(A ′ (θ) + F ′ (θ)),
where

A ′ := A + F diag F ′ := (I + X) -1 F X + l≥1 (-X) l F diag .
By definition, A ′ is diagonal and the estimate (16) comes directly from (13). According to (12), applying Proposition 3.3, we obtain

|X| r ≤ 2C n δ |F | r ≤ 1 2
where C n only depends on n (which also gives the estimate ( 14)).

Thus we have

|F ′ | r ≤ |(I + X) -1 (F X -XF diag )| r ≤   l≥0 |X| l r   4C n δ |F | 2 r (17) ≤ 8C n δ |F | 2 r . (18)
Moreover, for any scalar f ∈ C ω r , we have

sup |Im θ|≤r |f (θ)| ≤ |f | r .
As a consequence, we have, for |Im θ| ≤ r and for all j = k,

|ReA ′ (θ) j,j -ReA ′ (θ) k,k | ≥ (|ReA(θ) j,j -ReA(θ) k,k |) -(|Re(F (θ)) j,j | + |Re(F (θ)) k,k |) ≥ (|ReA(θ) j,j -ReA(θ) k,k |) -(|(F ) j,j | r + |(F ) k,k | r )
Hence, we have

inf |Im θ|≤r |ReA ′ (θ) j,j -ReA ′ (θ) k,k | ≥ δ -2C n ε

Proof of Proposition 3.1

We prove this by iterating Proposition 3.4 and constructing a sequence of changes of variables, all C ω ranalytic, conjugating the system A 0 + a to something which is arbitrarily close to the system given in the statement. Let C n be the constant defined in Proposition 3.3. Let δ = δ 0 and ε = ε 0 ≤ 1/2 be such that

ε 0 ≤ ε 1/2 0 ≤ δ 0 8C n and min j =k |ReA 0,j,j -ReA 0,k,k | ≥ δ 0 . Let us set ε k := ε (3/2) k 0
as well as

δ k := δ k-1 -2C n ε k-1 = δ 0 -2C n k-1 j=0 ε j , k ≥ 1.
Notice that, if ε is small enough, we have δ k > 16C n ε 0 , for all k ≥ 0. Indeed, for k ≥ 1 we have ε

(3/2) k 0 ≤ ε k 0 so that j≥0 ε j ≤ ε 0 + ε 0 1 -ε 0 ≤ 3ε 0 .
Therefore, we have, if ε 0 is small enough

δ k ≥ 2C n (4ε 1/2 0 -3ε 0 ) ≥ 16C n ε 0 .
Assumptions 2.7 and 3.1 make it possible to apply Proposition 3.4 with A = A 0 and F = a : if |a| r ≤ ε 0 , we obtain a conjugation Z 1 to a new vector field

D 1 + R 1 with D 1 = ω ∂ ∂θ + A 1 (θ)x ∂ ∂x , where A 1 (θ) is diagonal with δ 1 = δ 0 -2C n ε 0 -separated spectrum, and R 1 is C ω r -analytic on the torus with |R 1 | r ≤ 8 C n δ 0 |a| 2 r ≤ ε 3 2 0 = ε 1 and |A 1 -A 0 | r ≤ 3 2 ε 0 ≤ δ 1 2C n
The change of variable Z 1 is itself C ω r -analytic on the torus, and

ε 1 2 0
4 -close to the identity since

2C n δ 0 |a| r ≤ ε 1 2 0 4 .
Now suppose that we have a change of variables Z k which conjugates the vector field D + R given by (1) to

D k + R k where D k (θ) = ω ∂ ∂θ + A k (θ)x ∂ ∂x and A k is diagonal with δ k -separated spectrum, with the estimates |R k | r ≤ ε k , |A k -A 0 | r ≤ k-1 j=0 ε j < 2ε 0 ≤ δ k 8C n and | Zk -Id| r ≤ 2C k-1 j=0 ε 1/2 j
We remark, if ε 0 is small enough, then, for all k ≥ 1, we have ε

1/2 k < δ k 8Cn
. Indeed, we have

δ k 8C n ≥ 2C n ε 0 ≥ ε (1/2)(3/2) k 0 .
Thus, one can apply again Proposition 3.4 to conjugate D k + R k by some change of variable Z k such that

|Z k -Id| r ≤ 2C n δ k ε k ≤ ε 1 2
k to a vector field D k+1 + R k+1 with analogous properties:

D k+1 (θ) = ω ∂ ∂θ + A k+1 (θ)x ∂ ∂x where A k+1 is diagonal, with δ k+1 -separated spectrum, |R k+1 | r ≤ 8C n δ k ε 2 k ≤ ε k+1 , |A k+1 -A 0 | r ≤ |A k -A 0 | r + ε k ≤ k j=0 ε j ≤ δ k+1 8C n so that D + R is conjugated to D k+1 + R k+1 by Zk+1 = Z k Zk .
In C ω r , R k tends to 0 and {D k } k is a Cauchy sequence thus it converges to a limit D ∞ which is diagonal and analytic. Since Zk is C ω r -analytic for all k and defines a Cauchy sequence,then it has an analytic limit

Z ∞ conjugating D + R to D ∞ .

Analytic reduction of an upper triangular perturbation

In this section, we assume that A 0 = S + M , where S = diag(λ j ) is diagonal and M is upper triangular and commutes with S. This assumption is not restrictive since A 0 can be conjugated from the start to its Jordan normal form. In Section 6, we will see how to preserve real structures while still using the Jordan normal form.

The main result of this section is :

Theorem 4.1. Let A 0 = S + M ∈ gl(n, C) where S = diag(λ j ) is diagonal, M is nilpotent and upper triangular and [M, S] = 0. Let F ∈ C ω r be upper triangular valued. Suppose S is Melnikov (see definition 2.9). There exists

ε 0 (n, d, κ ′ , g ′ , A 0 , r) such that if |F | r ≤ ε 0 , then there exists r ∞ > 0 and Z ∞ ∈ C ω r∞ and an upper triangular, resonant R ∞ ∈ C ω r∞ such that ∂ ω Z ∞ = (A 0 + F )Z ∞ -Z ∞ R ∞ .
Remark 4.2. If A 0 is not in Jordan normal form, the assumption on F is that it is conjugate to an upper triangular valued function by the conjugation which takes A 0 to its Jordan normal form.

Cohomological equation and iteration process

Definition 4.3. Let l ≤ n and let F l be the set of n × n-matrices M = (m i,j ) 1≤i,j≤n such that m i,j = 0 if j < i + l. We have F n = {0}.

Let us start with an elementary lemma :

Lemma 4.4. We have [F l , F p ] ⊂ F p+l .
Proof. Let M ∈ F l and N ∈ F p . Then, we have

[N, M ] i,j := n k=1 n i,k m k,j - n k=1 m i,k n k,j = n k=i+l n i,k m k,j - n k=i+p m i,k n k,j .
In the first sum, m k,j = 0 if j < k + p so the sum is zero if j < (i + l) + p ≤ k + p. By the same argument, the second sum is zero for j < (i + p) + l ≤ k + l.

We will now give a lemma (appearing in [START_REF] Eliasson | Almost reducibility of linear quasi-periodic systems[END_REF] for the non resonant case) which shows that, up to a simple analytic change of variables, one can assume that S = diag(λ j ) satisfies i v, ωλ k + λ l = 0 ⇒ i v, ω = 0 Lemma 4.5. Let S = diag(λ j ) be a diagonal matrix and N commuting with S. There exists an analytic change of variables Φ and a diagonal matrix S commuting with N such that

∂ ω Φ(θ) = (S + N )Φ(θ) -Φ(θ)( S + N ) and every pair ( λj , λk ) of eigenvalues of S satisfies i v, ω -λk + λl = 0 ⇒ i v, ω = 0 (19)
Proof. Suppose there exist indices i 1 , . . . , i r and j 1 , . . . , j s such that

λ i 1 = • • • = λ ir , λ j 1 = • • • = λ js and
for some v such that v, ω = 0, one has λ i 1λ j 1i v, ω = 0. Let Φ 1 (θ) be the diagonal matrix with diagonal coefficients (Φ 1 1 (θ), . . . ,

Φ 1 n (θ)) such that Φ 1 i 1 (θ) = • • • = Φ 1 ir (θ) = e i v,
θ and all other diagonal coefficients are equal to one. Then

∂ ω Φ 1 (θ) = D • Φ 1 (θ)
where D is the diagonal matrix with coefficients D i 1 ,i 1 = • • • = D ir,ir = i v, ω and all other coefficients equal to 0. By construction, Φ 1 commutes with S + N and so does D, therefore

∂ ω Φ 1 (θ) = (S + N )Φ 1 (θ) -Φ 1 (θ)(S -D + N )
and the coefficients (λ 1 j ) of S -D are the same as those of S, except that λ i 1 , . . . , λ ir are shifted to

λ 1 i 1 = λ i 1 -i v, ω , . . . , λ 1 ir = λ ir -i v, ω , so that λ 1 i 1 = • • • = λ 1 ir = λ 1 j 1 = • • • = λ 1 js .
Therefore, the change of variables has merged two groups of identical coefficients into one.

In the new matrix S -D, if another resonance appears, one can perform a similar change of variables in order two merge the two groups of coefficients which are in resonance with each other. Thus, in a finite number of steps (at most n -1), a new diagonal matrix S is obtained, which commutes with N , and in which every pair of coefficients are either identical or non-resonant with each other.

From now on, we will assume that the eigenvalues of S satisfy (19). In the following statement, which gives the solution of the cohomological equation, recall that T N F stands for a truncation that preserves the equivalence classes of Fourier modes.

Proposition 4.6. Let S = diag(λ j ) be a constant Melnikov diagonal matrix. Let R(θ) = S + M (θ) where M (θ) is resonant, upper triangular. Assume that for all 1 ≤ k, l ≤ n and for all v ∈ Z d ,

i v, ω -λ k + λ l = 0 ⇒ i v, ω = 0 and |M | r ≤ 2||A 0 || + 2ε 0 where ε 0 = κ ′ 2n||A 0 || n . Then for every N ≥ 1 and if F ∈ C ω r is upper triangular, there is an upper triangular X ∈ C ω r such that ∂ ω X = [R, X] + T N F nr (20)
and

|X| r ≤ C ′′ (n)(||A 0 || + 1) κ ′n+1 N d g ′ (N ) n+1 |F | r
where C ′′ (n) only depends on n. Moreover, if T N F nr is upper triangular, then so is X.

Proof. Developing (20) into the subspaces H v yields

(L v + N (θ))X v (θ) = T N F v (θ)
where

L v : gl(n, C) → gl(n, C), X → i v, ω X -[S, X] and N (θ) : X → -[M (θ), X]. Since M is upper triangular, then M ∈ F 1 .
According to the previous lemma, N (F k ) ⊂ F k+1 . Hence, the operator N is nilpotent.

If i v, ω -(λ kλ l ) = 0 for all 1 ≤ k, l ≤ n, then L v is invertible and this equation amounts to

(Id + L -1 v N (θ))X v = L -1 v T N F v (θ) (21)
One has the estimate

||L -1 v || ≤ C(n) min j,k |i v, ω + λ j -λ k | ≤ C(n)g ′ (|v|) κ ′ Each F k is left invariant by L v . Therefore, L -1 v N (F k ) ⊂ F k+1 . Hence, the restriction to F k of L -1 v N is nilpotent. Hence, if T N F v (θ) belongs to F k , so does X v and there is a p ≤ n such that (I + L -1 v N (θ)) -1 = p l=0 (L -1 v N (θ)) l . Therefore, |X v | r ≤ n l=0 ||L -1 v || l+1 |N | l r |T N F v | r where |N | r = sup X∈C ω r |N X|r |X|r (so that |N | r ≤ 2n|M | r : recall that |M | r = sup θ |M (θ)| since M is resonant). Let us set X res := 0 and X := X res + v / v,ω =0 X v ; then X is solution of (20) and |X| r ≤ C(n) v n l=0 ( κ ′ g ′ (|v|) ) -(l+1) |M | l r |T N F v | r ≤ C ′ (n)|M | n r N d g ′ (N ) n+1 κ ′n+1 |F | r ≤ C ′′ (n)(||A 0 || + 1) κ ′n+1 N d g ′ (N ) n+1 |F | r (22)
Proposition 4.7. Let S be a constant Melnikov diagonal matrix. Let R(θ) = S + M (θ) where M (θ) is resonant, upper triangular. Let ε > 0 and F ∈ C ω r be upper triangular with |F | r ≤ ε. Let r ′ ∈ (0, r) be defined by

|ln ε| r -r ′ = g ′′-1 κ ′n+1 C ′′ (n)(||A 0 || + 1) √ ε (23)
where g ′′ (x) = x d g ′ (x) n+1 and C ′′ (n) is given by Proposition 4.6. Suppose also that

ε ≤ 1 4C 2 N (r -r ′ ) 2d+2 (24)
where

C N = N d + dN d-1 + • • • + d!. Then there exists Z ∈ C ω r such that ∂ ω Z = (R + F )Z -Z(R ′ + F ′ )
where F ′ is upper triangular, R ′ (θ) = S + M ′ (θ) with M ′ resonant and upper triangular. We have

|F ′ | r ′ ≤ ε 3 2 , and |Z -I| r ≤ √ ε. Proof. Let N be such that 1 2 ε -1 2 = C ′′ (n)(||A 0 ||+1) κ ′n+1 N d g ′ (N ) n+1 .
By assumption (23) on r ′ , e -N (r-r ′ ) = ε. By Proposition 4.6, there exists

X ∈ C ω r such that ∂ ω X = [R, X]+ T N F nr and |X| r < 1. Let Z = I + X. Then ∂ ω Z = [R, Z] + T N F nr = (R + F )Z -Z(R + Z -1 F Z -Z -1 T N F nr ) (note that Z is invertible since |X| r < 1.) Let R ′ = R + T N F res . Then R + Z -1 F Z -Z -1 T N F nr = R ′ + Z -1 F Z -Z -1 T N F nr -T N F res Let F ′ = Z -1 F Z -Z -1 T N F nr -T N F res ; thus F ′ = l≥0 (-X) l F (I + X) - l≥0 (-X) l T N F nr -T N F res = F -T N F + l≥1 (-X) l F (I + X) - l≥1 (-X) l T N F nr + F X (25) Now |F -T N F | r ′ ≤ |k|>N || F (k)||e |k|r ′ ≤ |k|>N |F | r e -|k|(r-r ′ ) ≤ C N |F | r (r -r ′ ) d+1 e -N (r-r ′ ) (where C N = N d + dN d-1 + • • • + d!), thus |F ′ | r ′ ≤ C N |F | r (r -r ′ ) d+1 e -N (r-r ′ ) + 6|F | r |X| r ≤ C N |F | r (r -r ′ ) d+1 e -N (r-r ′ ) + C ′′ (n)(||A 0 || + 1) κ ′n+1 N d g ′ (N ) n+1 |F | 2 r ≤ C N |F | r ε (r -r ′ ) d+1 + 1 2 |F | 3 2 r ( 26
)
(the last inequality holds because of the choice of the parameter N ). By the assumption (24), one has

|F ′ | r ′ ≤ ε 3 2 .
Moreover, if F is upper triangular, then so is X, thus F ′ as well since it is product of upper triangular matrices. As required, R ′ (θ) = S + M ′ (θ) where M ′ (θ) = M (θ) + F N res (θ) is resonant and upper triangular.

We need the following (trivial) lemma:

Lemma 4.8. Let g ′′ (x) = x d g ′ (x) n+1 . If g ′′ satisfies: |ln ε| g ′′-1 κ ′n+1 C ′′ (n)(||A 0 ||+1) √ ε ≥ (4C 2 ε) 1 2d+2 (27)
and if r ′ ∈ (0, r) is defined by

|ln ε| r -r ′ = g ′′-1 κ ′n+1 C ′′ (n)(||A 0 || + 1) √ ε (28) then ε ≤ 1 4C 2 (r -r ′ ) 2d+2 (29)
The Brjuno-Rüssmann assumption on ω gives a control on the loss of analyticity:

Lemma 4.9. There exists ε 0 > 0 which depends only on n, d, r, κ ′ , g ′ such that if, for all k ∈ N,

ε k = ε (3/2) k 0 and r k+1 = r k - |ln ε k | g ′′-1 κ ′n+1 C ′′ (n)(||A 0 ||+1) √ ε k
with r 0 = r, then (r k ) has a positive limit r ∞ .

Proof. One has

r - k≥1 (r k -r k+1 ) = r - k |ln ε k | g ′′-1 κ ′n+1 C ′′ (n)(||A 0 ||+1) √ ε k
and by the change of variables g ′′ (Y ) =

κ ′n+1 C ′′ (n)(||A 0 ||+1) √ ε k , r - k≥1 (r k -r k+1 ) ≥ r - ∞ g ′′-1 κ ′n+1 C ′′ (n)(||A 0 ||+1) √ ε 0 C ln g ′′ (Y ) Y dY where C = 2 √ ε 0 |ln ε 0 | . Therefore r ∞ > 0 if ε 0 is small enough as a function of r, g ′′ , κ ′ , n, A 0 .
4.2 Proof of theorem 4.1

Let us define the following sequences: for all k ∈ N,

ε k = ε ( 3 2 ) k 0 r k+1 = r k - |ln ε k | g ′′-1 κ ′n+1 C ′′ (n)(||A 0 ||+1) √ ε k
(by Lemma 4.9 , r k has a positive limit). Let N k be defined by the relation

1 2 ε -1 2 k = C ′′ (n)(||A 0 || + 1) κ ′n+1 N d k g ′ (N k ) n+1
The assumption (27) of Lemma 4.8 is satisfied, thus one can apply Proposition 4.7. Applying Proposition 4.7 with R is the constant map equal to S + N , one obtains

Z ∈ C ω r 1 conjugating R + F to R 1 + F 1 where • |F 1 | r 1 ≤ ε 1 , • R 1 is resonant, R 1 (θ) = S + M 1 (θ) where M 1 (θ) is upper triangular, • F 1 is upper triangular, • |Z -I| r 1 ≤ √ ε 1 . Now suppose, by induction, that A 0 + F is conjugated by Z k ∈ C ω r k to R k + F k with the properties • |F k | r k ≤ ε k , • R k (θ) = S + M k (θ)
, where M k is resonant with upper triangular values,

• F k has upper triangular values,

• |Z k -I| r k ≤ 2 k l=1 √ ε l .
Then applying again Proposition 4.7 (by means of Lemma 4.8 ), one has

Z k+1 ∈ C ω r k+1 conjugating A 0 + F to R k+1 + F k+1 with • |F k+1 | r k+1 ≤ ε k+1 , • R k+1 (θ) = S + M k+1 (θ)
, where M k+1 is resonant with upper triangular values,

• F k+1 has upper triangular values,

• |Z k+1 -I| r k+1 ≤ 2 k+1 l=1 √ ε l .
Thus, for all k ∈ N, the system

A 0 + F is conjugated, in C ω r 2 , to R k + F k , where where R k is resonant, |F k | r
2 ≤ ε k , and |Z k -I| r k ≤ 2. By Lemma 4.9, r k has a strictly positive limit r ∞ . Let R ∞ be a limit point, in C ω r∞ , of (R k ) k∈N (thus R ∞ is a resonant map); let (k l ) be a sequence such that R k l tends to R ∞ , and let Z ∞ be a limit point, in C ω r∞ , of the subsequence

Z k l . Then ∂ ω Z ∞ = (A 0 + F )Z ∞ -Z ∞ R ∞ .
5 Analytic reduction of a strongly commuting perturbation

Definition 5.1. Let F ∈ C ω r ; F is strongly commuting if for all equivalence classes v, v ′ ∈ Z d / ∼, one has [F v , F v ′ ] = 0.
The aim of this section is to prove the following: Theorem 5.2. Let F ∈ C ω r be strongly commuting. Assume that there exists C > 0, 0 < R < r such that for all k ∈ Z d with k, ω = 0,

| k, ω | ≥ Ce -|k|R (30)
Then the system with coefficient matrix F is analytically reducible to a resonant system.

Proof. The solution can be written as

X(t, θ) = exp t 0 F (θ + sω)ds X(0, θ)
The strong commutation assumption implies that the solution of the initial system can be directly computed and written in a reduced form:

X(t, θ) = e tFres(θ) v ∈Z d /∼, v,ω =0 exp   e it v,ω -1 i v, ω k∈Z d ,k∈ v Fk e i k,θ   = e tFres(θ) v ∈Z d /∼, v,ω =0 exp e it v,ω -1 i v, ω F v (θ) = Z(θ + tω)e tFres(θ) Z(θ) -1 (31)
where

Z(θ) = v,ω =0 exp F v (θ) i v,ω
. Thus we seek an arithmetical condition on ω under which Z is analytic. This will hold if the function

Y (θ) := v,ω =0 F v (θ) i v, ω is itself analytic. Now Y (θ) = k∈Z d , k,ω =0 F (k)e i k,θ i k, ω thus Ŷ (k) = F (k) i k,ω if k, ω = 0. Therefore Y is C ω r ′ -analytic if there is C > 0 such that for all k with k, ω = 0, || Ŷ (k)|| ≤ Ce -|k|r ′ which holds if |F | r e -|k|(r-r ′ ) ≤ C| k, ω |.
Now this holds if ω satisfies the condition (30) with R ≤ rr ′ .

Preservation of Lie structures

Assuming that the initial system takes its values in a Lie algebra g among gl(n, R), sl(n, C) or sp(n, R) (for n even), a slight modification of the proofs will make the reduced system have its values in the same Lie algebra and the reducing transformation have its values in the corresponding Lie group G.

Firstly, notice that the homological equations ( 8) and (20) have a solution in the Lie algebra where the coefficients A, R and F have their values. This was used in [Cha13] (Proposition 2.8) in the non resonant case and works identically even if the frequency vector ω is resonant, since by construction the solution is unique (since resonances are removed from the right-hand side of the homological equation). This comes from the fact that the Lie algebras considered here are defined by an equation of the form L(F ) = 0, where L is a linear operator on matrices, and such that for all X, whenever A is in the Lie algebra, either L([A, X]) = 0 (for instance if L is the trace operator) or L([A, X]) = [A, L(X)] (for instance if L(X) = J(X * J + JX) where J = 0 -I I 0 ).

Then one has to define a change of variables which takes its values in the Lie group. The change of variables defined above was I + X, one would have to take exp(X) instead. Since, at the first order, I + X and exp(X) coincide, the difference in the estimates will be quadratic, thus the new estimates will change only by a universal constant and will not prevent the convergence of the KAM scheme. Also, the structure will be preserved by integration in the proofs of Proposition 3.2 and Theorem 5.2: integration preserves the zero trace property, commutes with the operator X → X * J +JX and preserves the space of integrable real functions. Therefore, Theorem 3.2 can be restated as follows: Theorem 6.1. Assume that ω is very weak with exponent R > 0 and that the eigenvalues of A 0 have distinct real parts. Let r > R. Let A = A 0 + a be a C ω r -analytic cocycle (1) with values in g where a is a zero-mean valued function satisfying |a| r ≤ ε 0 (n, A 0 ). Then there exists a convergent transformation with values in G conjugating (1) to a normal form in a neighborhood of the torus:

N F = ω ∂ ∂θ + D res (θ)x ∂ ∂x .
where D res is a resonant diagonal matrix valued function with values in g.

and Theorem 5.2 becomes the following: Theorem 6.2. Assume that ω is very weak with exponent R. Let r > R. Let A 0 ∈ g. Let F ∈ C ω r be g-valued, strongly commuting and such that for all v, θ, [A 0 , F v (θ)] = 0. There exists C(A 0 , n) such that if |F | r ≤ C, then the system A 0 +F is reducible, by a G-valued transformation, to a resonant system A ∞ (θ) which commutes with A 0 and has its values in g.

In Section 4, it was assumed that the constant part A 0 was in a Jordan normal form and it was said that this assumption is not restrictive. In order to remain in gl(n, R), however, two more arguments are needed.

First of all, the Lie structure can slightly affect the way Lemma 4.5 is applied. Indeed, if S + N is the Jordan normal form of a real matrix, one will have to preserve the pairs of eigenvalues which are complex conjugate, and therefore it will be necessary to double the period. While doubling the period, new resonances might appear, namely k, ω with k ∈ 1 2 Z d . However, after doubling the period a finite number of times (at most n -1 times), all resonances will be deleted. So we will find the transformation not defined on the original torus but rather on a 2 n-1 covering.

While preserving the real structure, this way of eliminating the resonances also produces a transformation with determinant 1, thus the structure of SL(n, R) is also preserved.

On the other hand, let P be such that A 0 = P (S + N )P -1 where S is diagonal and N is nilpotent. Assume that P RP -1 and P F P -1 are real valued. If X is solution of (20), then P XP -1 is solution of ∂ ω P XP -1 = [P RP -1 , P XP -1 ] + P T N F nr P -1 , thus it is real. The change of variables P e X P -1 is thus also real, and its iteration is real (recall that taking e X instead of I + X does not essentially change the estimates). The estimates are not changed except by the constant ||P -1 || • ||P || since the change of basis is unchanged through the iteration. Finally, the system with constant part A 0 can be reduced by a real-valued change of variables, and the reduced system is then automatically real.

Moreover, the trace is invariant by matrix conjugation. Thus, Theorem 4.1 can be restated as follows: Theorem 6.3. Let g = gl(n, R) or g = sl(n, R). Let A 0 = P (S + M )P -1 ∈ g where S = diag(λ j ) is diagonal, M is nilpotent and upper triangular, and [M, S] = 0. Let F ∈ C ω r (T d , g) be such that P -1 F P is upper triangular valued. Assume that S is Melnikov and that |F | r ≤ ε 0 (n, d, κ ′ , g ′ , A 0 , r).Then there exists an r ∞ > 0 and a Z ∞ ∈ C ω r∞ (2 n-1 T d , G) and a resonant R ∞ ∈ C ω r∞ (2 n-1 T d , G), such that P -1 RP has upper triangular values and such that

∂ ω Z ∞ = (A 0 + F )Z ∞ -Z ∞ R ∞ .

  Proposition 3.3. Let A : T d → gl(n, C) be a diagonal perturbation of A 0 . Assume inf |Im θ|≤r |ReA(θ) j,j -ReA(θ) k,k | ≥ δ > 0 for all v ∈ Z d and all j = k. Let F : T d → gl(n, C) be a C ω r -analytic matrix-valued function. There exists a constant C n only depending on n such that if

  since we have A(θ) j,j,res = A(θ) j,j + (A(θ) j,j,res -A(θ) j,j ) and sup |Im θ|≤r |A(θ) j,j,res -A(θ) j,j | ≤ |A(θ) j,j,res -A(θ) j,j | r ≤ C n |A 0 -A(θ)| r , then we have inf |Im θ|≤r

  the equation (8) has a solution X satisfying |X| r ≤ 2Cn δ |F | r . Proposition 3.4. [Induction argument]Let A : T d → gl(n, C) be a diagonal analytic perturbation of A 0 . Let δ and ε < δ 4Cn be positive numbers. Let F : T d → gl(n, C) belong to C ω r with |F | r ≤ ε (11)

This research was supported by ANR grant "ANR-10-BLAN 0102" for the project DynPDE