Yvon Le Caro

Andrew W Gilg

L E Caro

Yvon Gilg

Farmland as a Leisure Space Exclusive and Public Access in Ille-et-Villaine (France) and Devon (England)

Introduction

a) A conceptual model of farm based recreation and some differences between France and England Countryside recreation is a very diverse activity and even walking in the countryside has been classified by [START_REF] Kay | Paths for whom? Countryside access for recreational walking[END_REF] into 20 different types. This paper thus focuses on one type of countryside recreation, namely, outdoor recreation on farms.

In more detail, Figure One, which is based on a conceptual model developed by [START_REF] Evans | A conceptual framework for investigating farm-based accomodation and tourism in Britain[END_REF], conceptualises three types of countryside recreation depending on the environment they are practised in. The first type is focussed on one farm as shown in Box A in Figure One. The second type is focussed on several farms in the farmed countryside in general as shown in Box B. The third type is practised in the less-farmed landscape of woodland, water, parks, moors and mountains as shown in Box C where the land may be farmed but only in an extensive way.

Based on a conceptual framework proposed by [START_REF] Evans | A conceptual framework for investigating farm-based accomodation and tourism in Britain[END_REF] Figure 1

: A conceptual model of farm based recreation

Six main recreational activities can be seen in Box A of Figure One. Four of these activities are not considered by this paper, namely: A1-Accommodation; A2-Events and Visits; A3-Indoor attractions; and A4a-Special areas like golf courses because they either depend on purpose built facilities or don't depend on multiple land use. Instead this paper concentrates on the two boxes A4b, that is outdoor recreation provided by a farmer on his farmed land (for instance the hunting lease), and A5, that is outdoor recreation using a wider countryside area including farmland (as walking along a trail).

The paper thus focuses on the problems caused by recreational use of the farmed landscape in terms of both multiple land use management conflicts and the social relationship between farmers and the general public. In particular it concentrates on how farmer attitudes to unpaid recreational use may have become more negative as recreation has become to be seen as a business opportunity to be exploited rather than as an activity to be tolerated.

In this regard there is a big difference between France and England. In France farm based recreational facilities are poorly developed compared to England with a general acceptance that only about 2% of French farmers provide farm-based accommodation compared to the 10% of English farmers who have been estimated to have developed a farm based recreation facility according to [START_REF] Chaplin | Establishing the significance of farm-based recreation in post-productivist agriculture in England and Wales, in Environnement et nature dans les campagnes[END_REF]. When both business recreation and non-business recreation are combined the English percentage of farms involved rises to 40% [START_REF] Chaplin | Establishing the significance of farm-based recreation in post-productivist agriculture in England and Wales, in Environnement et nature dans les campagnes[END_REF] but the survey by Le Caro on which much of this paper is based produced a figure of nearly 100% for non-business recreational use of the French countryside.

Thus, in England recreational access to the farmed environment is restricted to only about a third of such farms and much of this is based on payment of a fee to enter a purpose built facility. In contrast, in France the public has access to virtually the entire farmed countryside but is unlikely to pay a fee to enter a relatively rare purpose built facility. There is one main reason for this. In France access to the countryside is socially considered to be a right not just for walking but for other activities too, for example, shooting, fishing and picking wild fruits and flowers. In contrast, in England there is no right of access to the farmed countryside or even to open land. Furthermore, shooting and fishing are controlled by legal rights of access which are a very expensive and highly saleable commodity which exclude all but the wealthy in most cases.

b) The concept of exclusion applied to farmland access Farmers control most of the countryside in both France and England and since most members of the public live in towns they are axiomatically excluded from the countryside by the friction of distance except during weekends and holidays. Even during these times many people are excluded by behavioural and attitudinal factors rather than locational factors. For example, some people exclude themselves from countryside recreation even to those areas where they are lawfully allowed to have access because they lack confidence in themselves having been 'excluded' (Paugam, 1996) elsewhere in society. Other factors according to [START_REF] Burgess | Focussing on fear: the use of focus groups in a project for the Community Forest Unit[END_REF] include fear of open spaces, fear of woodland and its potential to hide danger, fear of wild creatures and fear of being told off by farmers and their agents.

As farming has become more technical and has eliminated more and more semi-natural features, notably hedgerows, the general public can only feel more excluded from an environment that often feels more like a factory floor than a natural environment. Ironically, the intensification of agriculture has been a prime factor in the production of surpluses and a decline in farm profitability which in turn has encouraged many farmers to diversify into farm based recreational facilities, notably in urban fringe areas. Any other use of the countryside by non paying visitors has thus become less amenable to farmers who have diversified while those who continue to invest in 'hitech' farming are also less and less likely to welcome recreational access.

Nonetheless, the provision of some low price recreational facilities in urban fringe areas may have opened up the countryside to some social groups who cannot afford to travel to more distant sites. Furthermore, the surveys by Le Caro (forthcoming) often found that farmers were happy to provide access when a member of the public or a group asked politely. In this case recreational use may prove a route to breaking down barriers between farmers and the public and reducing exclusion through informal non-business arrangements.

c) Methodology

The paper is largely based on research carried out in France (Le [START_REF] Caro | Les loisirs en espace agricole: modalités d'une approche géographique[END_REF] where 176 farmers completed a face to face questionnaire between 1993 and 1999. This sample represented 75% of farmers in the study area in Ille-et-Vilaine. The sampled farms were in three types of countryside: the urban fringe; open countryside; and coastal countryside with tourism attractions. Sixteen farmers were also interviewed in depth during a two hour walk through their farm about their feelings towards the conflicts generated between farm productivity, landscape management and recreational access. This walking interviews allow a more comprehensive understanding of how farmers manage relationships with recreationists, with respect to the place configuration. Some factors as land use, distance from the farmhouse, ownership of the particular place are better considered than in desk interviews. And ideology and stereotypes thus loose a part of their power on farmers individual attitude.

In England, data were collected from 6 interviews with farmers in December 1998 and from a variety of qualitative and quantitative sources, for example, [START_REF] Denman | Recreation and tourism on farm crofts and estates, Scottish Tourist Board[END_REF], [START_REF] Evans | A conceptual framework for investigating farm-based accomodation and tourism in Britain[END_REF] and notably [START_REF] Chaplin | Establishing the significance of farm-based recreation in post-productivist agriculture in England and Wales, in Environnement et nature dans les campagnes[END_REF] major survey of recreation on 2237 farms in the mid 1990s.

A comparison of usage between France and England a) Introduction

This section is based on evidence provided by the French surveys carried out by Le Caro and on a review of the literature on English recreational usage carried out by Gilg as a comparator. The key findings of the French survey are shown in Table One. Three activities from this table have been picked out for a more detailed discussion in this paper: shooting; angling; and walking/rambling since these allow the main comparisons to be made. They are now considered in turn. In France, shooting rights were at the centre of a debate during the French Revolution with a catalogue of greivances concerning both the abuses of aristocrats shooting over farmed land and the excessive severity of the courts against poor poachers (Duby and Wallon, 1976). The result was that citizens were given widespread rights to have a gun and for farmers to be able to shoot on their farmland. Nevertheless, the new definition of private property still gave the remaining gentry the right to shoot over let land on their estates. Elsewhere, the division of landownership first arising from the Revolution and then from divided inheritance under the 'Code Napoleon' meant that shooting rights linked to land ownership became so widespread throughout the community that in practice a 'right' to shoot over farmland came to be exercised collectively by citizens of the commune and their friends.

Location of farms

As a result, most game disappeared until an attempt to control overshooting was provided by the 'Loi Verdeille, 1964'. This created Authorized Communal Shooting Associations (Associations Communales de Chases Agrées: ACCA) which were enabled to exercise the accepted 'rights' to shoot over small agricultural properties as long as they are able to manage game and have respect for farming and local inhabitants. Under an arbitrary threshold of 20 hectares (raised to 27 hectares when a house is located on the property) landowners are unable to refuse access.

Even if only 28 of the 99 French Départements adopted the ACCA system (including Ille-et-Vilaine) the philosophy of an extensive 'right' to shoot over farmland is accepted as an unofficial norm across most of rural France. In contrast, some landowners, often on the largest estates, chose to keep shooting rights for themselves and their friends on the English model, outlined below. As the survey by Le Caro has found this behaviour is not very well thought of and one farmer expressed the following paraphrased sentiment: Some large farmers, that is more than 100 hectares keep shooting for themselves, but they can't ever find the time to shoot all over it, preferring to keep it for themselves. As for me I allow people to shoot on the 3 hectares that I have retained now that I've retired.

Overall, the survey found that farmers in Ille-et-Vilaine do accept communal shooting even if few of them shoot themselves. On average about 6% of Frenchmen shoot, compared to 26% of all French farmers, although our french survey only found about 13% of the surveyed farmers shot.

In England shooting rights are a saleable commodity and some areas can command prices of over £1,000 a day at certain times of the year. What can be shot is also closely controlled by season and by conservation legislation. Shooting in England is thus a very exclusive activity confined by and large to a wealthy elite, mainly large landowners with private capital which means they can manage their land for shooting purposes rather than solely for production. In these cases, usually landed estates, gamekeepers rear game birds, mainly pheasants, which are then shot on highly ritualistic occasions when the pheasants are driven by beaters into a line of 'guns'for a mass slaughter on a few days in the autumn. Another elite is represented by wealthy individuals who can afford to purchase shooting rights from those farms that have preserved habitats suitable for shooting rather than seeking productivity gains by 'hi-tech' farming [START_REF] Winter | Game Management in England: Report of a Socio-Economic Survey[END_REF].

These two type of farms are however, very much the exception and shooting is very much an adjunct to farming on most farms. For example, [START_REF] Chaplin | Establishing the significance of farm-based recreation in post-productivist agriculture in England and Wales, in Environnement et nature dans les campagnes[END_REF] found that 9% of farms had 'game shooting', and 8% 'rough shooting' as a formal recreation activity. On the majority of farms however 'rough shooting' might take place on a few days in the year involving the farmer and a few of his friends, often in a loose syndicate.

The Devon survey confirmed this view, with most farmers pointing out that the lack of cover for game due to 'hi-tech' farming meant that shooting rights were hard to sell, because there was little to shoot and few environments in which to do it. Three comments from farmers which have been paraphrased exemplified this.

We own the shooting rights, but we let our neighbours shoot. This is free, because we haven't got very much cover, so there aren't many pheasants but some friends go rabbiting.

We are about to stop renting out the shooting rights because fewer of the people coming were our friends. We are going back to putting down a few birds ourselves and inviting our friends to a shooting party. I normally shoot with my neighbour, but then only maybe on Boxing day. Some people have asked to shoot on the farm but I usually refuse.

In summary, there is an enormous difference between England and France in both accessibility to shooting and attitudes to shooting. In France it is still very much a part of rural culture, but in England it is seen as an elitist and perhaps a rather unsavoury practice. These differences are also reflected in the type of birds shot. In England only a few 'game' birds are shot, but in France, a much wider variety of birds are shot. To some extent this is a reflection of the longer established urban society in England allied with a much higher proportion of large farms and thus a more excluded population compared to France where far more people have a relative who still owns farmland. It is also a reflection of a much less densely populated France where there are far more genuinely wild places to provide habitats suitable for hunting away from settlements.

c) Angling

In France, the public have fishing rights to all navigable waters and the public have access to 1.5 metres of land parallel to the watercourse. Along the 250,000 kms of private water (nonnavigable water) the public can fish if they are given a right of access by the landowner (Guilbaud, 1992). However, on most rivers and streams (but not ponds) fishing rights are either provided legally by Authorized Associations or informally by the tacit approval of landowners. If farmers want to stop fishing they have to go through an explicit public process which many are unwilling to do since it would make them unpopular in the local community.

In England, angling on most water bodies is controlled by fishing rights [START_REF] Bonyhady | The law of the countryside: The rights of the public[END_REF] which like shooting rights may be very expensive for certain waters, notably the main Salmon rivers like the Wye and Severn and some famous trout streams notably in Hampshire. This type of angling, known as 'fly' fishing or 'game' fishing is thus very exclusive and restricted mainly to non-farmers.

However, angling is a very popular sport in England with over a million participants [START_REF] Glyptis | Countryside Recreation[END_REF] who have traditionally been confined to canals and stretches of river where local councils or clubs have been able to purchase the less desirable stretches of water. The general mass of anglers are also confined to 'coarse fishing' using a hook suspended by a float aimed at catching so called 'coarse' fish like perch and carp. [START_REF] Chaplin | Establishing the significance of farm-based recreation in post-productivist agriculture in England and Wales, in Environnement et nature dans les campagnes[END_REF] found that only 7% of farmers offered formal angling facilities for 'coarse fishing'. In recent years some farmers have invested in trout ponds which have widened access to 'game' fishing albeit in artificial environments where the catch may be limited to one or two fish and fish may even have to be returned. However, [START_REF] Chaplin | Establishing the significance of farm-based recreation in post-productivist agriculture in England and Wales, in Environnement et nature dans les campagnes[END_REF] found that only 3% of farmers offered a 'game fishing' facility.

Two paraphrased comments from the Devon survey examplify these observations. I'm the tenant farmer but I think my landlord sold the fishing rights for £300,000. That's enormous and its only for the fishing rights, no ground, and for only half a mile too.

We've actually made more ponds for carp. If people ask we let them fish especially if we know them like, three or four local young children we let fish. But if we did it commercially we'd have to have somebody there on overtime, taking money, which wouldn't be worth it.

In summary, angling is in common with shooting more restricted in England than in France but the division is less strong. In England, angling is a part of urban working class culture and is thus focussed in urban areas alongside the main rivers and canals. In the more remote countryside English angling is the preserve of a wealthy elite. As with shooting the difference between the two countries is also largely a reflection of resource availability, in terms of both waters mileage and conservation.

d) Walking/Rambling

In France, 'rural ways' (chemins ruraux) provide a compact network through the countryside, nearing a million of kilometres. They are generally linear features, 3 to 10 metres wide and are owned by the commune. They were created for general rural life and not for recreation. A lot of them were sold to farmers after a 1959 ordinance from the Government which aimed to reduce the maintenance costs of an underused network. Some 'rural ways' have also disappeared under individual restructuring of rural properties, notably 'remembrement' which has consoldated fields and cut down many hedgerows notably in the bocage areas of north west France. Those that do remain are often poorly maintained and a number of farmers may wilfully restrict access by blocking access, ploughing up paths and fencing land.

Around 18% of french 'rural ways' are signposted and maintained for recreation, often under the 1985 PDIPR regulation (Plan Départemental des Itinéraires de Promenade et de Randonnée) according to national figures. In Ille-et-Vilaine, 5137 kms have been registered under the PDIDR. Some more 1236 kms are signposted for off-road cycling (see Table Two). Much of the network is intended also for tourist use. Another innovation is for various groups to provide circular routes of between 2 and 15 kms often intended to boost tourism. The rest of the network remains neglected and used only by a few local residents.

Devon (UK)

Ille-etVilaine (France) In England, there is no right of access to any land, even open unenclosed farmland, unless a right of way has been negotiated [START_REF] Winter | Game Management in England: Report of a Socio-Economic Survey[END_REF]. Furthermore, the laws of trespass allow landowners to remove people from their land and to prosecute trespassers if they cause damage. The underlying ethos according to [START_REF] Curry | Enjoyment of the countryside[END_REF] has been one of restricting access and of diverting public pressure into highly managed sites like country parks and picnic sites as part of an unholy alliance between farmers and nature conservationists to keep the public out of the countryside [START_REF] Harrison | Countryside Recreation in a Changing Society[END_REF].

In England, public rights of way are the accidental product of links between villages that were important enough to be made roads. They have been legally incorporated as public rights of way since the 1949 National Parks and Access to the Countryside Act gave local councils the power to make definitive maps showing each public right of way. However, most rural local councils have been dominated by farmers or landowning interests and progress in defining rights of way and updating the network, let alone looking after it and signposting it has been lethargic. The Countryside Commission (Countryside Agency since 1999) has attempted to speed up the process and the advent of more and more non-farming people in rural areas has led to the election of less farmer dominated councils. In turn this has led to more interest in the rights of way network. New rights of access are being created as part of agri-environment schemes with for example, the Countryside Stewardship Scheme providing 250,000 hectares in its first year [START_REF] Ravenscroft | New access initiatives: the extension of recreation opportunities or the diminution of citizen rights?[END_REF] this has been criticised by [START_REF] Shoard | Robbers versus revolutionaries: What the battle for access is really all about[END_REF] who argues that exclusion has been turned into a tradeable assett. However, [START_REF] Curry | Access:Policy directions for the late 1990s[END_REF] argues that in the absence of non-excludability that such opportunities are more efficient and equitable.

Because 85% of the network is paths [START_REF] Ravenscroft | Horses in the countryside: conflict or co-operation[END_REF] which are threaded through farmland, there is a good deal of conflict between users and farmers. Users accuse farmers of deliberately obstructing paths, ploughing up paths and leaving dangerous animals in fields crossed by paths. Farmers accuse users of leaving gates open, letting dogs interfere with livestock and leaving litter including glass and plastic which are dangerous to livestock. Not surpisingly, [START_REF] Chaplin | Establishing the significance of farm-based recreation in post-productivist agriculture in England and Wales, in Environnement et nature dans les campagnes[END_REF] found that only about 2% of farmers had created formal access facilities to farmland via access agreements or nature trails. Typical attitudes were revealed by survey of Devon farmers who expressed the following paraphrased comments:

Its a problem because these footpaths used to be functional, they weren't created for recreation.

There are too many people that don't understand country life and think that milk comes from a bottle and not from a cow. Another problem is that people who own land are afraid that you might get an influx of people from the towns and more interference from the government.

In summary, the system in England does not encourage people to ramble but to follow paths closely and less than one half of one per cent is covered by non-linear access agreements [START_REF] Shoard | Robbers versus revolutionaries: What the battle for access is really all about[END_REF]. This causes conflicts with not only farmers but also other users, notably betwen dog owners, cyclists, horse riders and motorised visitors [START_REF] Kay | Paths for whom? Countryside access for recreational walking[END_REF]. [START_REF] Jenkins | Rural landholder attitudes: the case of public recreational access to 'private' rural lands[END_REF] in their review of access have argued that English landowners have a long developed tradition of exclusive control of the countryside, but also note that recent policy developments have resulted in a change from the perceived need to control the public to their wholesale encouragement as agriculture is no longer seen by policy makers as the primary user of rural space. In France there is less conflict largely because of the lower population density and also because the wider nature of the 'ways' allows a more diverse range of uses, notably: rambling; horse riding, cycling and motorised access (Table Two able a comparison between Ille-et-Vilaine and Devon).

A comparison of farmer attitudes to recreation between France and England

a) The attitudes of French farmers to farmland access The attitudes of French farmers about access to their farms were surveyed as shown in Table Three. This demonstrates that about half of French farmers agree to allow access to the land for some resource based activities as shooting, angling or picking mushrooms, with a better tolerance with shooting. Table Three demonstrates that farmers are less than happy with some aspects of the status quo as recorded in Table One. For example, in Table One farmers report that shooting access is virtually universal yet only half are happy with this type of access. Perhaps more significant in terms of social inclusion is the attitude about linear access : some 80 per cent of the surveyed farmers agree with public use of rural ways, even when they are not included in referenced routes for leisure. The more detailed comments from farmers revealed that they were becoming less happy about access, as their farms became more productive and more dependent on 'hi-tech' systems. They did however continue to allow access as part of a tradition of access, as part of a wider community to which they felt a responsibility, and because of their legal obligations.

Location of farms

It was also found that some farmers thought that access was a price worth paying as an antidote to opposition to intensive farming and that public grants and business opportunities could develop as a result of informal access.

b) Comparison between French and English attitudes

As shown above French farmers are less able to exclude people from their farms and thus don't see recreation as an obvious business venture. In contrast, English farmers, except for the restricted network of public rights of way can not only exclude people from their land, but can charge for access. Consequently, most farm diversification in England is based on recreation and tourism usually focused around some sort of activity or accommodation. In both countries however we might begin to see a polarisation of attitudes between two types of farms. First, in rural areas 'hitech' farms will want to exclude recreation because it interferes with productivity and because it is not financially necessary. Second, in urban fringe, tourism or marginal areas, recreation might offer greater financial rewards and may already be interfering with commercial farming enough already for recreation to be reluctantly accepted as a 'cuckoo in the nest'.

Trends in exclusion a) Local people and tourists

The French data have shown that local people have a great advantage over urban visitors to the countryside, because they can negotiate informal access with local farmers on the basis of personal relationships developed as part of the local community. There is some evidence that similar patterns exist in England but probably less so far there are fewer farmers and more incomers in the English environment. In contrast urban based visitors tend to use public or privately provided access facilities.

In France, an April 1999 decision by the European Court of Human Rights decreed that French landowners could ban hunting on their property even if they own less than 20 hectares under the Loi Verdeille. This was perceived as an urban attack against field-sports. In England, the Government after some prevarication published a consultation paper in the summer of 1999 concerning proposals to give a 'right to roam' over open land, that is land not farmed, except very extensively, for say sheep grazing [START_REF] Kay | The right to roam-a restless ghost[END_REF]. Several surveyed farmers spoke about that in terms of urban invasion.

Nonetheless, many French urban dwellers do shoot on French farms, similarly, in England many people from the towns support and follow fox and stag hunting. And a lot of rural people would enjoy the benefits of a priori urban rambling claims. That's why debates, both in France and England, about hunting and rambling cannot be seen as town against country debates, but more as conflicts between two cultural trends in rural consumption.

b) Rich and poor

Poverty imposes three types of exclusion: travel problems; access problems; and lack of selfconfidence. First, travel may be a problem because the lack of a car makes access very difficult except to urban fringe sites. Second, access to private recreational facilities is also difficult if a fee has to be paid. Even access to the network of informal recreational sites in France may also be difficult. For example, many of the local networks have been negotiated by wealthier people able to buy second homes in villages but such properties are out of the reach of the urban poor. Third, and notably in England, the laws of trespass and the antagonistic attitude of many farmers may deter many people who lack self-confidence from using rural spaces.

In contrast, more affluent people will own a car and increasingly as in the United States will also own a recreational vehicle. Such people will also often be a member of a rural recreational club, notably golf in England, which allows exclusive use to a stretch of countryside. Axiomatically, wealthier people will also tend to be self-confident and not be deterred by unwelcoming signs or attitudes.

Conclusion Figure Two presents a typology of exclusion and inclusion in England and

France. The typology is created from four concepts. First, business recreation where a full commercial charge is made. Second, informal recreation where a charge may be made but more often than not access is provided on an exchange basis. For example, gifts may be exchanged as a token of appreciation or a proportion of the catch/kill will be handed over at the end of the day. Third, exclusive recreation whereby access is by wealth, social contacts or by the self-confidence to ask for access. Fourth, inclusive recreation where the availability of access may range from implicit to explicit and from passive encouragement to active encouragement to participate. The failure of english farmers to increase access via voluntary agreements which might often include some sort of financial payment marks a big difference with their French counterparts where the survey has found that farmers have a greater sense of public conscience and social obligation to their wider community.

In summary, in England there is a generally antagonistic attitude to countryside recreation for everybody, everywhere and at all times, unless some money can be made, but in France the attitude is more generally welcoming to recreational activity and less likely to be financially motivated.

In order to improve the comparative data the two surveys carried out in France could be repeated in England. In particular attention could be drawn to the attitudes of the many millions of people who have moved to rural settlements and market towns in the last two decades, notably in southern England and around the major French conurbations. In both countries more work could also be done on the more conceptual issues of why people perceive rural recreation to be beneficial to them and how recreation and the countryside are socially constructed. Finally, the economic aspects of farm-based recreation could be studied by comparing the situation in the urban fringe, the farmed countryside and the marginal fringe.

Figure 2 :

 2 Figure 2 : A typology of exclusion and inclusion in England and France

Table 2 : Paths and ways networks (Ille-et-Vilaine and Devon comparison)

 2 Surveys by Le Caro / Devon County Council / Conseil Général d'Ille-et-Vilaine

	Surface

Table 3 : The attitudes of French farmers about access to their farmland

 3


Mahdi Jasim

Hasan Al-Kaabi

Kurusch Ebrahimi-Fard
email: kurusch.ebrahimi-fard@ntnu.no

Dominique Manchon
email: dominique.manchon@uca.fr

Mahdi J Hasan

Post-Lie-Magnus expansion and BCH-recursion

Keywords: post-Lie algebra, pre-Lie algebra, Rota-Baxter algebra, Magnus expansion, BCH-formula, rooted trees 2020 Mathematics Subject Classification: 16T05, 16T10, 16T30, 17A30

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

Introduction

The present paper consists in a quick survey of post-Lie algebras, Baker-Campbell-Haudorff recursion, Rota-Baxter algebras and post-Lie Magnus expansion (Sections 2, 3 and 4), followed by a new result in Section 5, which establishes an equality between two seemingly different formal series: the Baker-Campell-Hausdorff recursion in a weight-one Rota-Baxter algebra, and the post-Lie Magnus expansion relative to the associated post-Lie algebra structure described in [START_REF] Bai | Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras[END_REF]. Our motivation comes from a result obtained in 2006 by the second and the third author together with Li Guo [START_REF] Ebrahimi-Fard | Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion[END_REF], identifying the BCH-recursion with the pre-Lie Magnus expansion in the weight-zero case.

Interest in Magnus-type expansions results from their appearance in the context of numerical integration methods for Lie group valued problems [START_REF] Iserles | On the solution of linear differential equations in Lie groups[END_REF]. In [START_REF] Curry | The Magnus expansion and post-Lie algebras[END_REF], we studied the relation between the classical and the post-Lie Magnus expansions by looking at a non-autonomous matrix-valued initial value problem from the viewpoint of the theory of numerical Lie group integrators. Post-Lie algebras and the post-Lie Magnus expansion play a central part in the latter. In this context post-Lie algebras characterise the relation between two Lie algebras (one coming from the Jacobi Lie bracket and the other from the torsion Lie bracket, in the context of a flat connection with constant torsion). This relation can be lifted to the level of the (completed) enveloping algebra (of the Lie algebra implied by the torsion Lie bracket). In a nutshell, the post-Lie Magnus expansion naturally appears in the context of backward error analysis for the Lie-Euler method. This is consistent with the fact that the pre-Lie Magnus expansion plays an analogous role with respect to backward error analysis for the Euler method.

The well-known Baker-Campbell-Hausdorff (BCH) formula BCH(x, y) is a formal power series, which lives in the completion of the free Lie algebra L(x, y) generated (over a base field K of characteristic zero) by the two non-commutating variables x and y. It is defined by exp(x) exp(y) = exp BCH(x, y) = exp x + y + BCH(x, y) or BCH(x, y) = log exp(x) exp(y) = x + y + BCH(x, y).

It plays a prominent role in modern mathematics [START_REF] Arnal | A note on the Baker-Campbell-Hausdorff series in terms of right-nested commutators[END_REF][START_REF] Bonfiglioli | Topics in noncommutative algebra[END_REF]. 1A fruitful connection between the BCH-series and the notion of Rota-Baxter algebra has been explored in [START_REF] Ebrahimi-Fard | Spitzer's identity and the algebraic Birkhoff decomposition in pQFT[END_REF][START_REF] Ebrahimi-Fard | Integrable renormalization. II. The general case[END_REF][START_REF] Ebrahimi-Fard | Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion[END_REF]. The latter originated in the seminal 1960 article [START_REF] Baxter | An analytic problem whose solution follows from a simple algebraic identity[END_REF] by the American mathematician G. Baxter, which in turn was motivated by F. Spitzer's 1956 article [START_REF] Spitzer | A combinatorial lemma and its application to probability theory[END_REF]. Baxter's algebra was further developed foremost in the commutative realm in the 1960s and '70s by P. Cartier, G.-C. Rota and F.V. Atkinson, among others, from algebraic, combinatorial and analytic viewpoints. We refer the reader to the review article [START_REF] Ebrahimi-Fard | From iterated integrals and chronological calculus to Hopf and Rota-Baxter algebras[END_REF] as well as the monograph [START_REF] Guo | An introduction to Rota-Baxter algebra[END_REF] for details.

A weight-λ Rota-Baxter operator on an associative K-algebra A is a K-linear map R: A -→ A, satisfying the Rota-Baxter identity of weight λ ∈ K:

R(x)R(y) = R R(x)y + xR(y) + λxy , x, y ∈ A. (1.1)
The pair (A, R) is a weight λ Rota-Baxter algebra. 2 For example, the indefinite Riemann integral satisfies (1.1) when the weight λ = 0 (integration by parts). The linear map R := -λid A -R is also Rota-Baxter of weight λ, and satisfies together with R the mixed identity

R(x) R(y) = R R(x)y + R x R(y) , x, y ∈ A.
Starting from a Rota-Baxter operator R of weight λ, the BCH-recursion [START_REF] Ebrahimi-Fard | Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion[END_REF] is defined by

χ λ (a) := a + 1 λ BCH R χ λ (a) , R χ λ (a) , a ∈ A. (1.2)
It lies at the heart of the solution of an exponential factorisation problem [START_REF] Ebrahimi-Fard | Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion[END_REF] and thereby permits the generalisation of a classical result for commutative Rota-Baxter algebras, known as Spitzer's identity [START_REF] Spitzer | A combinatorial lemma and its application to probability theory[END_REF], to non-commutative Rota-Baxter algebras. The resulting non-commutative Spitzer identity says that for a ∈ A the exponential

X := exp R χ λ log(1 + tλa) λ
solves the fixed point equation

X = 1 + tR(aX) (1.3) in the algebra A[[t]
] of formal series with coefficients in A. Here the formal parameter t commutes with all elements in A. More precisely, iterating the fixed point equation (1.3) yields the rather non-trivial equality

1 + tR(a) + t 2 R aR(a) + t 3 R aR aR(a) + • • • = exp R χ λ log(1 + tλa) λ .
Thanks to the commuting parameter t, the last equality can be seen as between formal power series and therefore encompasses at each order a specific relation between coefficients. For instance, at order two, that is, comparing the coefficients of t 2 , we have the identity

2R aR(a) = R(a)R(a) -R [R(a), a] + λa 2 ,
which is easily verifiable in a Rota-Baxter algebra of weight λ by using the Rota-Baxter identity (1.1) on the right-hand side. We note that the fixed point equation (1.3) is reminiscent of the integral fixed point equation naturally associated to a linear matrix-valued initial value problem; the indefinite Riemann integral is a weight-zero Rota-Baxter map. Indeed, the series (1.2) turns out to be closely related to a well-known Lie algebra expansion due to W. Magnus [START_REF]On the exponential solution of differential equations for a linear operator[END_REF]. This connection to the so-called Magnus expansion was studied in reference [START_REF] Ebrahimi-Fard | Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion[END_REF] in the case of the weight being zero (λ = 0). The adequate algebraic setting is provided through the notion of pre-Lie algebra, which is naturally defined on any non-commutative Rota-Baxter algebra. In [START_REF] Ebrahimi-Fard | A Magnus-and Fer-type formula in dendriform algebras[END_REF] it was shown that the pre-Lie Magnus expansion can be expressed in terms of the BCH-recursion as follows

Ω (a) := a + n>0 B n n! L (n) Ω (a) (a) = χ λ log(1 + λa) λ . (1.4)
Here B n is the n-th Bernoulli number and L [x](y) = L (1) [x](y) := x y is the left-multiplication operator defined in terms of the aforementioned (left) pre-Lie product, denoted , on a noncommutative Rota-Baxter algebra. Note that the weight λ is absorbed in the definition of the pre-Lie product. In the weight-zero case, (1.4) boils down to

Ω (a) = χ 0 (a). (1.5)
In particular, for the indefinite Riemann integral, the pre-Lie product is defined for -matrixvalued -functions A, B as (A B)(t) := t 0 A(s)ds, B(t) . When inserted in (1.4), one recovers Magnus' original expansion [START_REF]On the exponential solution of differential equations for a linear operator[END_REF].

Recall that any Rota-Baxter algebra with nonzero weight gives rise to a post-Lie algebra structure [START_REF] Bai | Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras[END_REF]. In this work, we describe a close relationship between the BCH-recursion (1.2) in the nonzero weight case and the Magnus expansion in its post-Lie version [START_REF] Ebrahimi-Fard | On the Lie enveloping algebra of a post-Lie algebra[END_REF][START_REF] Ebrahimi-Fard | Post-Lie algebras, factorization theorems and isospectral flows[END_REF][START_REF] Ebrahimi-Fard | Post-Lie algebras and factorization theorems[END_REF][START_REF] Mencattini | Post-symmetric braces and integration of post-Lie algebras[END_REF]. Our main result (Theorem 5.3) shows that the post-Lie Magnus expansion and the BCH-recursion in (1.2) coincide in the context of a Rota-Baxter algebra of weight 1 endowed with its naturally associated post-Lie structure. This is an extension to nonzero weight of one of the main results of [START_REF] Ebrahimi-Fard | Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion[END_REF], resumed by (1.5), identifying the weight zero BCH-recursion with the pre-Lie Magnus expansion. The special role of weight one here simply comes from the definition of the post-Lie structure (2.9), (2.10), and any Rota-Baxter algebra with nonzero weight can be set to weight one by an appropriate rescaling of the Rota-Baxter operator.

We close this introduction by noting that the Magnus expansion, in its various forms (classical [START_REF]On the exponential solution of differential equations for a linear operator[END_REF][START_REF] Mielnik | Combinatorial approach to Baker-Campbell-Hausdorff exponents[END_REF], pre-Lie [START_REF] Agrachev | Chronological algebras and nonstationary vector fields[END_REF][START_REF] Chapoton | Enveloping algebras of preLie algebras, Solomon idempotents and the Magnus formula[END_REF][START_REF] Ebrahimi-Fard | A Magnus-and Fer-type formula in dendriform algebras[END_REF] and post-Lie [START_REF] Ebrahimi-Fard | On the Lie enveloping algebra of a post-Lie algebra[END_REF][START_REF] Ebrahimi-Fard | Post-Lie algebras, factorization theorems and isospectral flows[END_REF][START_REF] Ebrahimi-Fard | Post-Lie algebras and factorization theorems[END_REF][START_REF] Mencattini | Post-symmetric braces and integration of post-Lie algebras[END_REF]), has been studied in applied mathematics, control theory, physics and chemistry. See reference [START_REF] Blanes | The Magnus expansion and some of its applications[END_REF] for details on the classical Magnus expansion in applied mathematics. The reader can also find a brief summary in the recent work [START_REF] Curry | The Magnus expansion and post-Lie algebras[END_REF].

This paper consists of four sections accompanied by two appendices. In Section 2, we review some basic topics related to post-Lie algebras and their universal enveloping algebras. The post-Lie structure defined on any Rota-Baxter algebra is recalled from [START_REF] Bai | Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras[END_REF]. Section 3 contains the description of the Baker-Campbell-Hausdorff recursion and its inverse, as well as their properties. Several important details on the post-Lie Magnus expansion and its inverse are included in Section 4. Section 5 is the main part of this work, in which the identification of the post-Lie Magnus expansion with the BCH-recursion is proven. Finally, the two Appendices A and B contain low-order computations of the post-Lie Magnus expansion and its inverse.

Post-Lie algebras

A post-Lie algebra is a Lie algebra (L, [• , •]) together with a bilinear mapping : L × L -→ L, which is compatible with the Lie bracket in the following sense

x [y, z] = [x y, z] + [y, x z],
(2.1)

[x, y] z = a (x, y, z) -a (y, x, z), (2.2)
for any x, y, z ∈ L. Here, a (x, y, z) is the associator defined by

a (x, y, z) = x (y z) -(x y) z.
Any Lie algebra can be seen as a post-Lie algebra by setting the second product to zero. Another possibility is to take for the second product the opposite of the Lie bracket.

A (left) pre-Lie algebra is an abelian post-Lie algebra, i.e., a post-Lie algebra with Lie bracket set to zero. The defining relation is the left pre-Lie identity 0 = a (x, y, z) -a (y, x, z).

(2.3)

We refer the reader to [START_REF] Manchon | A short survey on pre-Lie algebras[END_REF] for a short survey on pre-Lie algebras. The post-Lie operation permits to produce two other operations:

[[x, y]] := x y -y x + [x, y], x y := x y + [x, y],
for all x, y ∈ L. From (2.1) and ( 2. For more details on post-Lie algebras, we refer to [START_REF] Curry | What is a post-Lie algebra and why is it useful in geometric integration[END_REF][START_REF] Ebrahimi-Fard | On the Lie enveloping algebra of a post-Lie algebra[END_REF][START_REF] Ebrahimi-Fard | Post-Lie algebras, factorization theorems and isospectral flows[END_REF][START_REF] Munthe-Kaas | On post-Lie algebras, Lie-Butcher series and moving frames[END_REF] and references therein.

The universal enveloping algebra of a post-Lie algebra

Inspired by the work of J.-M. Oudom and D. Guin in the pre-Lie context [START_REF] Oudom | On the Lie enveloping algebra of a pre-Lie algebra[END_REF], the authors in [START_REF] Ebrahimi-Fard | On the Lie enveloping algebra of a post-Lie algebra[END_REF] consider the enveloping algebra

U(L), • of the Lie algebra L, [• , •] underlying a post- Lie algebra L, [• , •], . The post-Lie product is then extended to L ⊗ U(L) → U(L) by requiring x 1 := 0 and x (x 1 • • • x n ) := n i=1 x 1 • • • x i-1 (x x i )x i+1 • • • x n ,
for all x, x 1 , . . . , x n ∈ L. Here, 1 denotes the unit in U(L). Recall that the enveloping algebra U(L) together with the product • and the unshuffle coproduct has the structure of a noncommutative, co-commutative Hopf algebra. The unshuffle coproduct ∆ is defined for all letters x ∈ L → U(L), by ∆(x) := x ⊗ 1 + 1 ⊗ x and extended multiplicatively. We employ Sweedler's notation, ∆(X) := X (1) ⊗ X (2) , for the coproduct of any X ∈ U(L). The final definition of the extended post-Lie product on U(L), together with its properties, is given by the next two propositions.

Proposition 2.1 ([16, Proposition 3.1]). There is a unique extension of the post-Lie product from L to U(L) satisfying:

1 X = X, xX y = x (X y) -(x X) y, X Y Z = X (1) Y X (2) Z ,
for all x, y ∈ L, and X, Y, Z ∈ U(L). Proposition 2.2 ([16, Proposition 3.2]). The extended post-Lie product on U(L) possesses the following properties:

X 1 = (X), (X Y ) = (X) (Y ), ∆(X Y ) = (X (1) Y (1) ) ⊗ (X (2) Y (2) ), xX Y = x (X Y ) -(x X) Y, X (Y Z) = (X (1) X (2) Y ) Z,
for all x ∈ L and X, Y, Z ∈ U(L), where : U(L) → K is the counit map.

From the last equality in Proposition 2.2, an associative product, known as Grossman-Larson product, can be defined on U(L) as follows

X * Y := X (1) X (2) Y , (2.4)
for all X, Y ∈ U(L). As a main example, for any x ∈ L and Y ∈ U(L), we find Remark 2.3. Conversely, the product of the enveloping algebra can be expressed in terms of the Grossman-Larson product and the unshuffle coproduct as follows

x * Y = x Y + xY, (2.5
XY = X (1) * S * X (2) Y . (2.6)
This is seen by plugging (2.4) into the right-hand side of (2.6).

Free post-Lie algebras

F. Chapoton and M. Livernet presented in [START_REF] Chapoton | Pre-Lie algebras and the rooted trees operad[END_REF] the free pre-Lie algebra in terms of (non-planar) decorated rooted trees. Similarly, H. Munthe-Kaas and A. Lundervold gave in [START_REF] Munthe-Kaas | On post-Lie algebras, Lie-Butcher series and moving frames[END_REF] an explicit description of the free post-Lie algebra in terms of formal Lie brackets of planar decorated rooted trees. Let us briefly review this construction: a magma is a set M together with a binary operation, without any further properties. For any (non-empty) set E, the set of all parenthesized words on the alphabet E is the free magma over E, denoted M (E). A practical presentation of it can be given in terms of planar rooted trees. Indeed, consider the set T pl E of all planar rooted trees with vertices decorated by E, and let • denote the left Butcher product defined on T pl E as

σ • τ = B e + (στ 1 τ 2 • • • τ k ), for σ, τ 1 , τ 2 , . . . , τ k ∈ T pl E and τ := B e + (τ 1 τ 2 • • • τ k ).
Here, B e + is the operation defined by grafting a monomial τ 1 τ 2 • • • τ k of E-decorated rooted trees on a common root decorated by some element e in E, to obtain a new tree. For example (in the undecorated context)

T pl E as: σ • ցτ = B e + (στ 1 τ 2 • • • τ k ), for σ, τ 1 , τ 2 , . . . , τ k ∈ T pl E and τ := B e + (τ 1 τ 2 • • • τ k ).
Here, B e + is the operation defined by grafting a monomial τ 1 τ 2 • • • τ k of E-decorated rooted trees on a common root decorated by some element e in E, to obtain a new tree. For example (in the undecorated context):

• ց = , • ց = , • ց = .
Denote by T pl E the linear span of the set T pl E . Besides the left Butcher product, • ց, this space has another magmatic product defined through left grafting, denoted ց and defined by

σ ց τ = v vertex of τ σ ց v τ, (2.9)
where σ ց v τ is the tree obtained by grafting the root of the tree σ onto the vertex v of the tree τ, such that σ becomes the leftmost branch starting from vertex v. See for example references [START_REF] Al-Kaabi | Monomial bases for free pre-Lie algebras[END_REF][START_REF] Butcher | An algebraic theory of integration methods[END_REF]. Computing some examples (in the undecorated context) we find:

ց = + , ց = + + .
By freeness universal property, there is a unique morphism of magmatic algebras

Ψ : (T pl E , • ց) -→ (T pl E , ց) τ -→ e τ := Ψ(τ)
such that Ψ(• a ) = • a for any a ∈ E, which is a linear isomorphism. A detailed account of the map Ψ can be found in [START_REF] Al-Kaabi | Monomial bases for free pre-Lie algebras[END_REF].

Let L(T pl E ) be the free Lie algebra generated by T pl E . It can be endowed with a structure of post-Lie algebra by extending the aforementioned left grafting, ց, as follows:

σ ց [ τ, τ ′ ] = [ σ ց τ, τ ′ ] + [ τ, σ ց τ ′ ] ,
Denote by T pl E the linear span of the set T pl E . Besides the left Butcher product, • , this space has another magmatic product defined through left grafting, denoted and defined by

σ τ = v vertex of τ σ v τ, (2.7)
where σ v τ is the tree obtained by grafting the root of the tree σ onto the vertex v of the tree τ , such that σ becomes the leftmost branch starting from vertex v. See for example references [START_REF] Al-Kaabi | Monomial bases for free pre-Lie algebras[END_REF][START_REF] Butcher | An algebraic theory of integration methods[END_REF]. Computing some examples (in the undecorated context) we find T pl E as:

σ • ցτ = B e + (στ 1 τ 2 • • • τ k ), for σ, τ 1 , τ 2 , . . . , τ k ∈ T pl E and τ := B e + (τ 1 τ 2 • • • τ k ).
Here, B e + is the operation defined by grafting a monomial τ 1 τ 2 • • • τ k of E-decorated rooted trees on a common root decorated by some element e in E, to obtain a new tree. For example (in the undecorated context):

• ց = , • ց = , • ց = .
Denote by T pl E the linear span of the set T pl E . Besides the left Butcher product, • ց, this space has another magmatic product defined through left grafting, denoted ց and defined by

σ ց τ = v vertex of τ σ ց v τ, (2.9)
where σ ց v τ is the tree obtained by grafting the root of the tree σ onto the vertex v of the tree τ, such that σ becomes the leftmost branch starting from vertex v. See for example references [START_REF] Al-Kaabi | Monomial bases for free pre-Lie algebras[END_REF][START_REF] Butcher | An algebraic theory of integration methods[END_REF]. Computing some examples (in the undecorated context) we find:

ց = + , ց = + + .
By freeness universal property, there is a unique morphism of magmatic algebras

Ψ : (T pl E , • ց) -→ (T pl E , ց) τ -→ e τ := Ψ(τ)
such that Ψ(• a ) = • a for any a ∈ E, which is a linear isomorphism. A detailed account of the map Ψ can be found in [START_REF] Al-Kaabi | Monomial bases for free pre-Lie algebras[END_REF].

Let L(T pl E ) be the free Lie algebra generated by T pl E . It can be endowed with a structure of post-Lie algebra by extending the aforementioned left grafting, ց, as follows:

σ ց [ τ, τ ′ ] = [ σ ց τ, τ ′ ] + [ τ, σ ց τ ′ ] ,
By freeness universal property, there is a unique morphism of magmatic algebras

Ψ : T pl E , • -→ T pl E , , τ -→ e τ := Ψ(τ ),
such that Ψ(• a ) = • a for any a ∈ E, which is a linear isomorphism. A detailed account of the map Ψ can be found in [START_REF] Al-Kaabi | Monomial bases for free pre-Lie algebras[END_REF]. Let L T pl E be the free Lie algebra generated by T pl E . It can be endowed with a structure of post-Lie algebra by extending the aforementioned left grafting, , as follows

σ [τ, τ ] = [σ τ, τ ] + [τ, σ τ ], [σ, τ ] τ = a (σ, τ, τ ) -a (τ, σ, τ ), for all σ, τ, τ ∈ T pl E . The triple L T pl E , [• , •],
is the free post-Lie algebra generated by E [START_REF] Vallette | Homology of generalized partition posets[END_REF] (see also [START_REF] Munthe-Kaas | On post-Lie algebras, Lie-Butcher series and moving frames[END_REF][START_REF] Munthe-Kaas | On the Hopf algebraic structure of Lie group integrators[END_REF]).

Recall that an E-decorated

planar forest f = τ 1 • • • τ n is a (non-commutative) product of E-decorated planar rooted trees τ i ∈ T pl E , i = 1, . . . , n.
Denote by F pl E the set of all E-decorated planar forests, and by F pl E its linear span. The space F pl E forms together with the concatenation product the free associative algebra generated by T pl E . The left grafting, , defined by (2.7) on T pl E can be generalized to a grafting of forests as follows:

• Left grafting a tree on a forest is also defined by (2.7). We thus have

σ f f = (σ f )f + f (σ f ),
for any tree σ ∈ T pl E and any two forests f, f ∈ F pl E . • The left grafting of a forest f = τ 1 • • • τ k onto a forest f is the sum of forests obtained by summing over all ways of successively left grafting the trees τ k , . . . , τ 1 to any node of f .

The well-known (planar) Grossman-Larson product on F pl E is defined by [START_REF] Grossman | Hopf-algebraic structure of families of trees[END_REF] f

f := B -f B e + (f ) , (2.8)
where B -is the left inverse operation of B e + , which removes the root of a tree and thus produces a forest. This product endows the space F pl E with a structure of an non-commutative associative unital algebra, called the Grossman-Larson algebra. This algebra acts naturally on T pl E by extended left grafting

(f f ) τ := f (f τ ),
for all f, f ∈ F pl E and τ ∈ T pl E . The universal enveloping algebra U L T pl E of the free post-Lie algebra L T pl E is the free associative algebra on T pl E , and can therefore be identified with F pl E . The terminology is justified by the following Proposition 2.4 ([16, Proposition 3.5]). With the identification recalled above, the Grossman-Larson product * on U L T pl E is identical to the Grossman-Larson product on F pl E , .

Post-Lie structure on a Rota-Baxter algebra

Recall that a unital algebra is said to be complete filtered if it is equipped with a separating complete filtration

A = A 0 ⊇ A 1 ⊇ A 2 ⊇ • • • ⊇ A n ⊇ • • •
by ideals [START_REF] Ebrahimi-Fard | Spitzer's identity and the algebraic Birkhoff decomposition in pQFT[END_REF]. Separation means that the intersection of the A n 's is equal to {0}, and completeness refers to the topology associated with the filtration, so that any series a = n≥1 a n with a n ∈ A n converges in A. The filtration is moreover supposed to be compatible with the product, i.e., A n 1 A n 2 ⊆ A n 1 +n 2 for any n 1 , n 2 ≥ 0. A Rota-Baxter algebra is said to be complete filtered if it is equipped with a separating complete filtration by Rota-Baxter ideals, i.e., by ideals A n stable by the Rota-Baxter operator. The Rota-Baxter Algebra A, R of weight λ has a structure of a post-Lie algebra defined by the following operations:

[x, y] λ := λ[x, y],
(2.9)

x y := [R(x), y],

(2.10) for all x, y ∈ A. We leave it to the reader to show that the operations in (2.9), (2.10) satisfy the post-Lie identities (2.1) and (2.2) (see [START_REF] Bai | Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras[END_REF]Section 5.2]). As expected, the post-Lie algebra A, [• , •] λ , reduces to a (left) pre-Lie algebra in the case of a weight zero Rota-Baxter algebra. Indeed, if λ = 0, the product defined by (2.10) verifies the left pre-Lie identity (2.3).

Baker-Campbell-Hausdorff (BCH)-recursion

We give here a brief account of the Baker-Campbell-Hausdorff recursion, which was defined and explored in [START_REF] Ebrahimi-Fard | Spitzer's identity and the algebraic Birkhoff decomposition in pQFT[END_REF][START_REF] Ebrahimi-Fard | Integrable renormalization. II. The general case[END_REF][START_REF] Ebrahimi-Fard | Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion[END_REF]. Let A = K x, y be the free complete associative K-algebra of formal power series generated by non-commuting variables x and y. The Baker-Campbell-Hausdorff expansion BCH(x, y) is the element in A satisfying the following equation exp(x) exp(y) = exp BCH(x, y) .

The first terms are given by BCH(x, y) = x + y + BCH(x, y)

= x + y + 1 2 [x, y] + 1 12 [x, [x, y]] - 1 12 [y, [x, y]] - 1 24 [x, [y, [x, y]]] + • • • ,
where [x, y] := xy -yx is the usual commutator of x and y in A. See, e.g., [START_REF] Bonfiglioli | Topics in noncommutative algebra[END_REF] for details.

Proposition 3.1 ([15, Proposition 1]). Let A be a complete filtered K-algebra, and let R be a K-linear map preserving the filtration of A. There exists a unique (usually non-linear) map χ :

A 1 -→ A 1 , such that (χ -id A )(A n ) ⊂ A 2n , for all n ≥ 1, and BCH R(χ(x)), R(χ(x)) = x, (3.1)
for all x ∈ A 1 , where R := id A -R. This map is bijective, and its inverse is

χ -1 (x) = BCH R(x), R(x) = x + BCH R(x), R(x) . (3.2)
As a consequence of (3.1), we have the exponential

factorization exp R(χ(x)) exp R(χ(x)) = exp(x), (3.3)
for any x ∈ A 1 . Note also that (3.2) yields the non-linear BCH-recursion

χ(x) := x -BCH R(χ(x)), R(χ(x)) , (3.4)
for all x ∈ A 1 .

Lemma 3.2 ([15]

). Let A be a complete filtered algebra, and let R : A -→ A be a linear map preserving the filtration. The following holds:

1. The map χ given by (3.4), can be simplified:

χ(x) = x + BCH -R(χ(x)), x ∀x ∈ A 1 . (3.5)
2. If R is an idempotent algebra homomorphism, then the map χ in (3.5) is further simplified, namely χ(x) = x + BCH(-R(x), x).

Proof . See [START_REF] Ebrahimi-Fard | Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion[END_REF]Lemmas 6 and 7].

The BCH-recursion in the Rota-Baxter algebra framework is given as follows in the case where the weight λ is different from zero: Proposition 11]). Let A, R be a complete filtered Rota-Baxter algebra of weight λ = 0, and set R := -λid A -R. The λ-weighted BCH-recursion is written

Proposition 3.3 ([15,
χ λ (x) = x + 1 λ BCH R(χ λ (x)), R(χ λ (x)) , (3.6)
for all x ∈ A 1 . It can be simplified to

χ λ (x) = x - 1 λ BCH -R(χ λ (x)), λx .
Its inverse is given by

χ -1 λ (x) = x - 1 λ BCH R(x), R(x) .
Moreover, the factorization obtained in

(3.3) becomes exp R(χ λ (x)) exp R(χ λ (x)) = exp(-λx). (3.7)
The expansion χ λ can be written as the infinite sum

χ λ (x) = n≥1 χ (n) λ (x), where χ (n)
λ ∈ A n is the n-th homogenous component of the BCH-recursion. Here, we write the components χ (n) λ up to order n = 4 using the post-Lie algebra notation (2.9) and (2.10)

χ (1) λ (x) = x, χ (2)
λ (x) = 1 2λ R χ (1) λ (x) , R χ (1)
λ (x) = 1 2λ R χ (1)
λ (x) , -λid A -R χ (1)
λ (x) = - 1 2 R(x), x = - 1 2 x x, χ (3)
λ (x) = 1 2λ R χ (1)
λ (x) , R χ (2)
λ (x) + R χ (2) λ (x) , R χ (1) λ (x)
+ 1 12λ R χ (1)
λ (x) , R χ (1)
λ (x) , R χ (1) λ (x)
-R χ

(1)

λ (x) , R χ (1)
λ (x) , R χ (1)
λ (x) = 1 4 (x x) x + 1 12 x (x x) + 1 12 [x x, x] λ , χ (4) λ (x)
= λ -1 24 x (x x) x - λ + 1 24 (x x) (x x) + λ -3 24 (x x) x x - λ + 1 24 x (x x) x + 1 24 x, x (x x) + (x x) x λ .
These coefficients are recursively computed using (3.4).

Magnus expansion

W. Magnus [START_REF]On the exponential solution of differential equations for a linear operator[END_REF] considered the problem of expressing the solution of the matrix-valued linear initial value problem Ẏ (t) = M (t)Y (t), Y (0) = Y 0 as an exponential [START_REF] Blanes | The Magnus expansion and some of its applications[END_REF][START_REF] Mielnik | Combinatorial approach to Baker-Campbell-Hausdorff exponents[END_REF]

Y (t) = exp Ω(M )(t) Y 0 .
The Magnus expansion, Ω(M )(t) = log(Y (t)), is determined by the particular differential equation

Ω(M ) := M + n>0 B n n! ad (n) Ω(M ) (M ) (4.1) = dexp -1 Ω(M ) (M ) := ad Ω(M ) e ad Ω(M ) -1 (M ), (4.2)
with Ω(M )(0) = 0. Here, B n are the Bernoulli numbers and ad

(n) M 1 (M 2 ) := ad (n-1) M 1 ([M 1 , M 2 ]), ad (0) M 1 (M 2 ) = M 2 .
Defining the pre-Lie product, (M 1 M 2 )(t) := t 0 M 1 (s)ds, M 2 (t) , we can rewrite (4.2) using the left-multiplication operators L (x) := x -defined in terms of the pre-Lie product:

Ω(M ) = L [ Ω(M )] e L [ Ω(M )] -1 (M ).

Post-Lie Magnus expansion

We consider now the universal enveloping algebra F pl E := U L T pl E of the free post-Lie algebra

L(T pl E ), [• , •],
, graded by the number of vertices of the forests. Denote by U L T pl E its completion with respect to the grading. Any element of the completion can be written as a so-called Lie-Butcher series [START_REF] Ebrahimi-Fard | On the Lie enveloping algebra of a post-Lie algebra[END_REF][START_REF] Munthe-Kaas | On post-Lie algebras, Lie-Butcher series and moving frames[END_REF][START_REF] Munthe-Kaas | On the Hopf algebraic structure of Lie group integrators[END_REF]

α = f ∈F pl E α, f f , where • , • : U L T pl E ⊗ U L T pl E → K is the natural pairing defined on any pair (f, f ) of forests by f, f = 0, f = f , 1, f = f .
The unshuffle coproduct, ∆, is naturally extended to the completion. The set Prim F pl E consists in primitive elements (infinitesimal characters), whereas G F pl E denotes the set of group-like elements (characters)

Prim(F pl E ) := α ∈ U L T pl E | ∆(α) = 1 ⊗ α + α ⊗ 1 = L T pl E , G(F pl E ) := α ∈ U L T pl E | ∆(α) = α ⊗ α .
(f ) = ∞ n=0 f * n n! = 1 + f + 1 2 f * f + 1 6 f * f * f + • • • , exp(f ) = ∞ n=0 f n n! = 1 + f + 1 2 f f + 1 6 f f f + • • • .
Both these exponential functions map Prim F pl E bijectively onto G F pl E . See [START_REF] Ebrahimi-Fard | On the Lie enveloping algebra of a post-Lie algebra[END_REF] for details. The post-Lie Magnus expansion χ is the bijective map from L T pl E onto itself defined by

exp * χ(f ) = exp(f ), namely, χ(f ) = log * exp(f ) . (4.3)
Introducing a formal commuting indeterminate t, it can also be described as

χ(f t) = n≥1 χ (n) (f )t n ,
where χ (n) (f ) is the n-th order component of the post-Lie Magnus expansion χ. The latter is defined recursively by χ (1) (f ) = f , and [START_REF] Ebrahimi-Fard | On the Lie enveloping algebra of a post-Lie algebra[END_REF][START_REF] Ebrahimi-Fard | Post-Lie algebras, factorization theorems and isospectral flows[END_REF][START_REF] Ebrahimi-Fard | Post-Lie algebras and factorization theorems[END_REF]]

χ (n) (f ) := f n n! - n k=2 1 k! p 1 +•••+p k =n p i >0 χ (p 1 ) (f ) * χ (p 2 ) (f ) * • • • * χ (p k ) (f ). (4.4)
The computation of the coefficients χ (n) (f ) for the first five values of n is displayed in Appendix A below. They have been obtained by hand by the recursive formula (4.4), using (2.5) repeatedly.

Comparing with the computations at the end of Section 3, one observes that, up to order n = 4, the coefficient χ (n) (f ) coincides with the coefficient χ (n) λ (f ) of the BCH-recursion in the weight λ = 1 case. We shall prove this fact at any order in Theorem 5.3 below. Remark 4.1. Formula (4.3) defines the post-Lie Magnus expansion in any complete filtered post-Lie algebra L. If the underlying Lie algebra is Abelian, then the post-Lie Magnus expansion is reduced to the so-called pre-Lie Magnus expansion. The latter already appears in [START_REF] Agrachev | Chronological algebras and nonstationary vector fields[END_REF] and encompasses classical Magnus expansion [START_REF]On the exponential solution of differential equations for a linear operator[END_REF]. Remark 4.2. We may deduce a Magnus-type differential equation similar to (4.1) for the post-Lie Magnus expansion (4.3), by differentiating exp * χ(f t) = exp(f t) with respect to t. This results in

χ(f t) = dexp * -1 -χ(f t) exp * (-χ(f t)) f , χ(0) = 0.

Inverse post-Lie Magnus expansion

The inverse post-Lie Magnus expansion θ is the bijective map from L T pl E onto itself given by the following formula

θ(f ) = log(exp * (f )) (4.5)
or exp(θ(f )) = exp * (f ).

The homogeneous component θ

(n) = θ (n) (f ) of degree n of the expansion θ(f t) = n≥1 θ (n) (f )t n
is given by θ (1) (f ) = f and the following recursive formula [START_REF] Ebrahimi-Fard | On the Lie enveloping algebra of a post-Lie algebra[END_REF] θ

(n) (f ) = 1 n n-1 j=1 1 j! k 1 +•••+k j =n-1 k i >0 θ (k 1 ) θ (k 2 ) • • • θ (k j ) f + n-1 j=1 B j j! k 1 +•••+k j =n-1 k i >0 ad θ (k 1 ) • • • ad θ (k j ) f + n-1 j=2 j-1 q=1 B q q! k 1 +•••+kq=j-1 k i >0 ad θ (k 1 ) • • • ad θ (kq ) × n-j p=1 1 p! k 1 +•••+kp=n-j k i >0 θ (k 1 ) θ (k 2 ) • • • θ (kp) f , (4.6)
where ad θ (i) (f ) := θ (i) , f , and the B i 's are the Bernoulli numbers. The computation of the first θ (n) 's is given in Appendix B.

Remark 4.3. The same fact, described in Remark 4.1, will be repeated again in the case of the inverse post-Lie Magnus expansion. In other words, the formula in (4.6) for the inverse post-Lie Magnus expansion is reduced, in the case of commutative post-Lie algebras, to the inverse pre-Lie Magnus expansion formula described below (see also [START_REF] Ebrahimi-Fard | A Magnus-and Fer-type formula in dendriform algebras[END_REF][START_REF] Manchon | A short survey on pre-Lie algebras[END_REF])

W (x) := e L [x] -1 L [x] (x) = ∞ n=0 1 (n + 1)! L (n) [x](x).
Modulo removal of a fictitious unit, W is also known as the pre-Lie exponential [START_REF] Agrachev | Chronological algebras and nonstationary vector fields[END_REF].

Remark 4.4. Similar to Remark 4.2, we may deduce a Magnus-type differential equation similar to (4.1) for the inverse post-Lie Magnus expansion [START_REF] Ebrahimi-Fard | On the Lie enveloping algebra of a post-Lie algebra[END_REF][START_REF] Ebrahimi-Fard | Post-Lie algebras and factorization theorems[END_REF] θ(f t) = dexp -1 -θ(f t) exp(θ(f t)) f , θ(0) = 0.

Post-Lie Magnus expansion and BCH-recursion

We now show that the Baker-Campbell-Hausdorff recursion driven by a weight λ = 1 Rota-Baxter operator identifies with the Magnus expansion relative to the post-Lie structure naturally associated to the corresponding Rota-Baxter algebra. χ (4) (f ) = -

1 12 (f f ) (f f ) + (f (f f )) f + ((f f ) f ) f + 1 24 [f, f (f f )] + [f, (f f ) f ] , χ (5) (f ) = - 1 720 f f f (f f ) + 1 144 (f f ) f (f f ) -f ((f f ) f ) f -f (f (f f )) f -f f ((f f ) f ) + 5 f (f f ) (f f ) + 5 (f f ) f (f f ) + 6 (f f ) (f f ) f + 3 (f (f f ) f ) f + 3 f (f (f f )) f + 3 f ((f f ) f ) f + 3(f f ) (f f ) f + 3 ((f f ) f ) f f + 1 180 f, [f, f (f f )] -f (f (f f )) - 1 120 [f f, f (f f )] - 1 36 [f, (f f ) (f f )] - 1 72 f, f (f f ) f + f (f f ) f + (f f ) f f - 1 360 f f, [f, f f ] + 1 720 f, f, [f, f f ] .

B Computations on the inverse post-Lie Magnus expansion

Here, we calculate the first five inverse post-Lie Magnus elements:

θ (1) (f ) = f, θ (2) (f ) = 1 2 f f, θ (3) (f ) = 1 6 f (f f ) + 1 12 [f, f f ], θ (4) (f ) = 1 24 f f (f f ) + [f, f (f f )] ,
θ (5)

(f ) = 1 120 f f f (f f ) + 1 80 f, f f (f f ) + 1 720 f, [f, f (f f )] -f, f, [f, f f ] + 1 120 [f f, f (f f )] - 1 240 f f, [f, f f ] .

  2), one can see that L, [[• , •]] forms a Lie algebra, denoted L. In the case of an abelian post-Lie algebra, this amounts to Lie admissibility of pre-Lie algebras. The triple L, -[• , •], forms another post-Lie algebra [12, 28] sharing the same double Lie bracket, i.e., [[x, y]] = x y -y x + [x, y] = x y -y x -[x, y].

  ) since any element of L is primitive. The Grossman-Larson product (2.4) defines together with the coproduct ∆ another structure of Hopf algebra on U(L). The corresponding antipode will be denoted by S * . The Hopf algebras U(L), * , ∆ and U( L), ., ∆ are isomorphic [15, Section 3],[START_REF] Oudom | On the Lie enveloping algebra of a pre-Lie algebra[END_REF] Section 2].

  Both products on U L T pl E -the concatenation and the Grossman-Larson product (2.8)can also be extended to products on the completion U L T pl E . As a result, two different exponential functions can be defined on U L T pl E , namely, exp *

Theorem 5 . 1 .Proposition 5 . 2 .

 5152 Let A, R be a complete filtered Rota-Baxter algebra of weight λ = 1. We have the following equality in U(A) for any x ∈ A and t ∈ K:exp * (tx) = exp -t R(x) exp -tR(x) ,(5.1)where R = -id A -R, and * is the associative product defined in (2.4), using the post-Lie productx y = [R(x), y].The proof of this theorem will rely on the following proposition: In any complete filtered Rota-Baxter algebra A, R of weight λ = 1, we have the following identity in U(A):d n dt n exp -t R(x) exp -tR(x) = exp -t R(x) x * n exp -tR(x) . (5.2)Proof . The proof goes by induction. The base case k = 0 is trivial. For k = 1, we have in U(A):d dt exp -t R(x) exp -tR(x) = exp t(id A + R)(x) (id A + R)(x) exp -tR(x) -exp t(id A + R)(x) R(x) exp -tR(x) = exp -t R(x) x exp -tR(x) .Now, suppose that the statement is true in the case k = n -1, i.e., d n-1 dt n-1 exp -t R(x) exp -tR(x) = exp -t R(x) x * n-1 exp -tR(x) .

The remainder Baker-Campbell-Hausdorff series, BCH(x, y), is denoted by BCH(x, y) in[START_REF] Ebrahimi-Fard | Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion[END_REF]. We adopt here a more conventional notation.

The convention for the weight is with the opposite sign in[START_REF] Ebrahimi-Fard | Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion[END_REF].

Acknowledgments

The first author was funded by the Iraqi Ministry of Higher Education and Scientific Research. He would like to thank the Department of Mathematics at the University of Bergen, Norway, for warm hospitality during a visit in 2021, which was partially supported by the project Pure Mathematics in Norway, funded by Trond Mohn Foundation and Tromsø Research Foundation. He would also like to thank Mustansiriyah University, College of Science, Mathematics Department for their support in carrying out this work. The second author is supported by the Research Council of Norway through project 302831 "Computational Dynamics and Stochastics on Manifolds" (CODYSMA). The third author is supported by Agence Nationale de la Recherche, projet CARPLO ANR20-CE40-0007. We thank the anonymous referees for their remarks and suggestions which greatly helped us to improve the presentation of the present paper.

We then get

which means that (5.2) is true for k = n, and it is true for all n ≥ 0. This ends the proof.

Proof of Theorem 5.1. We have that

for all n ≥ 0.

One can therefore conclude that both members of (5.1) do coincide as infinite formal series.

Theorem 5.3. The post-Lie Magnus expansion χ, described in (4.3), coincides with the weighted BCH-recursion χ λ recursively given by (3.6), with weight λ = 1.

Proof . From equation (3.7), specialized to λ = 1, and by setting θ BCH := χ -1 1 , we obtain that

From (4.5), (5.1) and ( 5.3) we have exp θ BCH (tx) = exp * (tx) = exp θ(tx) .

(5.4)

Then the two θ's, namely the inverse BCH-recursion in (5.3) and the inverse post-Lie Magnus expansion (5.4), do coincide.

A Calculations on post-Lie Magnus expansion

The first five elements of the post-Lie Magnus expansion are