Frédérique Bec

† Mélika

Ben Salem

Dornsbush revisited from an asymmetrical perspective: Evidence from G20 nominal effective exchange rates

Keywords: nominal exchange rate, asymmetrical overshooting correction. JEL Codes: C22, F31, F41

This paper develops an asymmetrical overshooting correction autoregressive model to capture excessive nominal exchange rate variation. It is based on the widely accepted perception that open economies might prefer under-evaluation to over-evaluation of their currency so as to foster their net exports. Our approach departs from existing works by allowing the strength of the overshooting correction mechanism to differ between over-depreciations and over-appreciations. It turns out that most of monthly effective exchange rates for the G20 countries are in fact well characterized by an overshooting correction after an over-appreciation only.

Introduction

Excessive exchange rate variation could be explained by the overshooting effect first identified by [START_REF] Dornbusch | Expectations and Exchange Rate Dynamics[END_REF]. In a small open economy, due to nominal price stickiness, a permanent expansionary (respectively restrictive) monetary shock would provoke a depreciation (resp. appreciation) of the nominal exchange rate which would go beyond the new long term exchange rate equilibrium level. Hence, this depreciation (resp. appreciation) should be followed by a few periods of appreciation (resp. depreciation) to reach the new equilibrium. Even though more sophisticated versions of this model have been proposed, this monetary description of the exchange rate behavior has received little empirical support so far (see [START_REF] Rogoff | Dornbusch's overshooting model after twenty-five years international monetary fund's second annual research conference mundell-fleming lecture[END_REF]). More recently, in another branch of literature, bounce-back augmented models have been found to be useful to describe transitory epochs of high growth rate GDP recovery following a recession and preceding a normal growth rate regime (see [START_REF] Kim | Nonlinearity and the permanent effects of recessions[END_REF] or [START_REF] Bec | The way out of recessions: A forecasting analysis for some euro area countries[END_REF]). This mechanism is similar to an exchange rate overshooting correction after a depreciation. The originality of this paper is to bring these two strands of research together to shed new light on the exchange rate overshooting by allowing it to be asymmetrical across appreciation and depreciation regimes. To our knowledge, this has not been investigated so far. The idea grounding this asymmetry is that countries should be more prompt in correcting over-appreciation than over-depreciation of their currency so as to foster their net exports. Our main contribution is to develop an asymmetrical overshooting correction (AOC hereafter) autoregression, extended to allow for a GARCH effect, to capture this behavior. Using G20 effective nominal exchange rate data since January 1994, evidence of overshooting correction is found in 15 cases out of 20: 13 correct the overshooting after appreciation only, one does so after appreciation and depreciation and a last one does it after depreciation only. The asymmetry revealed by our empirical results might explain the lack of overshooting evidence from previous studies.

The asymmetrical overshooting correction autoregression

The AOC function used in the subsequent empirical investigation is based on the bounceback function inspired by Friedman's view1 . This author claimed that "a large contraction in output ends to be followed on by a large business expansion; a mild contraction, by a mild expansion." This kind of dynamics, allowing for a correction mechanism which depends on the depth of the initial change, is a good candidate to capture the exchange rate overshooting. Eventually, the latter is likely to be more than proportional to the monetary shock by which it is triggered, due to price stickiness. More precisely, we propose to represent the nominal exchange rate first difference, ∆e t , by the AOC autoregression given by:

∆e t = µ + λ 1 m j=0 ∆e t-j-1 s t-j + λ 2 m j=0 ∆e t-j-1 (1 -s t-j ) + p k=1 ρ k ∆e t-k + ν t √ ε t , (1)
where ν t is a Gaussian strong white noise N (0,1), independent of ε t . The latter is allowed to exhibit GARCH(1,1) such that:

ε t = α 0 + α∆e 2 t-1 + βε t-1 , (2)
with α 0 > 0, and α, β ≥ 0. s t is an indicator function governing the appreciation/depreciation regime switching:

s t = 1 if ∆e t-1 ≤ 0 and 0 otherwise. ( 3
)
The originality of our approach compared to the empirical "overshooting" literature relies on the first two terms after the intercept µ on the right-hand side of Equation ( 1). Indeed, they allow the overshooting correction strength to depend on the sign and the size of the overshooting. If λ 1 < 0, the term λ 1 m j=0 ∆e t-j-1 s t-j will increase ∆e t during m + 1 periods after a shock as long as s t-j = 1, for j = 0, • • • , m, i.e. as long as the indicator function points to a corresponding decrease in the lagged exchange rate. Consequently, m + 1 reflects the duration of the overshooting correction. The estimate of λ 1 should be significantly different from zero in presence of overshooting after a depreciation2 . The top panel of Figure 1 plots a simulated path of the level of the nominal exchange rate after an expansionary monetary shock. The first decrease in e t brings the latter too low at time 2, which corresponds to the overshooting phenomenon. Consequently, it needs to bounce-back for two periods3 in order to reach its new equilibrium level from time 4 on: this is the overshooting correction mechanism. The bottom panel of Figure 1 plots the dynamics implied by this shock for the first difference of e t , which is the dependent variable of our AOC autoregression. There, periods 3 and 4 illustrate the role of the term λ 1 m j=0 ∆e t-j-1 s t-j in Equation ( 1). Note that this correction is proportional to the size of the past ∆e's. The next term λ 2 m j=0 ∆e t-j-1 (1 -s t-j ) of Equation ( 1) is a function mirroring the one just described above : there will be evidence of overshooting after an appreciation if λ 2 < 0.

Data and estimation results

Monthly data of broad effective exchange rates4 for the G20 members come from the Fed of St.Louis Federal Reserve Economic Data base (FRED). Table 1

reports results

for the 15 countries where an overshooting correction mechanism has been found. On top of the countries listed in the first column of this Table, Canada, Saudi Arabia, South Africa, the United Kingdom and Indonesia have also been studied. No evidence of overshooting correction has been found in the first four ones.5 Regarding Indonesian data, no autoregressive lag p, overshooting correction length m or GARCH lag orders made it possible to remove severe serial correlation in the model's estimated residuals. e(t)-e(t-1)

Figure 1: Simulated impact of λ 1 m j=0 ∆e t-j-1 s t-j on e t and ∆e t after a depreciation at time t = 2, which triggers two periods of overshooting correction before the new exchange rate equilibrium value is reached (λ 1 = -0.2 and m = 1).

Consequently, this exchange rate was excluded from our analysis. Unless otherwise mentioned, the series start in January 1994 and end in December 2018.6 ∆e t in Equation (1) denotes the demeaned first difference of these effective exchange rate series. The model given by Equations ( 1) and ( 2) is estimated by maximum likelihood method and results are reported7 in Table 1. For all the series, one autoregressive lag is enough to remove residuals serial correlation, so that p is at most 1 in Equation ( 1). The correction duration parameter is chosen among m = 0, 1, • • • , 6 as the one which maximizes the log-likelihood of the estimated model over the period 1994:9-2018:12.8 Evidence of overshooting correction is found in these 15 cases -at the 5%-level for 7 countries and the 10%-level for 8 countries -which represents three quarters of the series under study.

Actually, 13 overshoot after appreciation only, while the remaining two overshoot either after depreciation only or after appreciation and depreciation. The overshooting correction duration is short: m = 1 in most of the cases, which amounts to 2 months. The overshooting correction is found to last longer in Japan, India, Argentina, South Korea and Mexico. Japan is a special case in that the Yen is a "safe haven" currency, and this country's inflation rate has remained close to zero since the early nineties. Consequently, the nominal price is likely to be sluggish and the adjustment to the purchasing power parity by the nominal exchange rate might be slow. The size of the overshooting correction coefficients, λ1 and λ2 , ranges from -0.04 to -0.32. These values imply that on average, the correction is a little smaller than 20% of the overshooting magnitude during the few months following the over-adjustment. Interestingly, most countries seem to implement a "leaning-against-the-wind" foreign exchange policy in order to mitigate the initial appreciation and hence avoid large negative impacts on their current account.

Italy is the only case where overshooting correction occurs both after depreciation or appreciation. With λ1 = -0.08 and λ2 = -0.14, the symmetry of the overshooting correction mechanism cannot be excluded. This might be explained by the Maastricht agreement constraints imposed to Euro area applicants during the nineties, especially regarding inflation and interest rates which de facto excluded competitive devaluation.

Nevertheless, overshooting is supported by these results. Finally, Mexico foreign exchange policy appears to be concerned by over-depreciations only. Actually, after the 1995 financial crisis, this country's guidelines for exchange rate policy aimed to increase the institutional credibility of the central bank, while implementing institutional reforms and policies to strengthen Mexico's position in the foreign exchange market.

Conclusion

Overall, our empirical results support the exchange rate overshooting feature put forward by monetary approaches of exchange rate dynamics. Indeed, most of the series

Table 1 :

 1 Maximum likelihood estimates of the AOC autoregression

	Country	Sample	m	λ1	λ2	ρ1	α	β	Q(6) AIC
	Argentina	03:6-18:12 6 -0.00 -0.07 0.48 0.12 0.69 4.64 3.953
				(0.95)) (0.03) (0.00) (0.07) (0.00) (0.59)
	Australia	94:9-18:12 1 -0.10 -0.28 0.48 0.09 0.84 0.67 4.028
				(0.25)) (0.00) (0.00) (0.00) (0.00) (0.99)
	Brazil	95:4-18:12 1 -0.12 -0.32 0.71 0.86 0.29 8.23 4.846
				(0.17)) (0.00) (0.00) (0.00) (0.00) (0.22)
	China	94:9-18:12 1 -0.04 -0.18 0.53 0.12 0.58 6.36 2.997
				(0.66)) (0.01) (0.00) (0.08) (0.04) (0.38)
	Euro Area	94:9-18:12 1 -0.07 -0.17 0.42 0.06 0.85 0.93 3.242
				(0.37)) (0.07) (0.00) (0.12) (0.00) (0.99)
	France	94:9-18:12 1 -0.09 -0.14 0.38 0.10 0.73 1.05 1.941
				(0.24)	(0.08) (0.00) (0.13) (0.00) (0.98)
	Germany	94:9-18:12 1 -0.09 -0.16 0.44 0.04 0.87 0.97 2.310
				(0.20)	(0.08) (0.00) (0.24) (0.00) (0.99)
	India	94:9-18:12 3 -0.02 -0.13 0.29 0.11 0.83 4.17 3.692
				(0.73)) (0.04) (0.00) (0.00) (0.00) (0.65)
	Italy	94:9-18:12 1 -0.08 -0.14 0.40 -0.03 1.01 4.28 2.132
				(0.07)	(0.06) (0.00) (0.00) (0.00) (0.64)
	Japan	94:9-18:12 5 -0.03 -0.04 0.30 0.30 0.00 4.51 4.175
				(0.30)	(0.06) (0.00) (0.00) (0.99) (0.61)
	Mexico	95:9-18:12 5 -0.06 -0.05 0.26 0.30 0.46 4.21 4.741
				(0.06)) (0.19) (0.00) (0.00) (0.00) (0.65)
	Russia	99:7-18:12 1 -0.27 -0.21 0.68 0.32 0.69 5.59 4.099
				(0.77)	(0.00) (0.00) (0.00) (0.00) (0.47)
	South Korea 98:9-18:12 3 -0.05 -0.12 0.39 0.17 0.69 4.11 4.095
				(0.36)) (0.04) (0.00) (0.00) (0.00) (0.66)
	Turkey	94:12-18:12 1	0.31 -0.10 0.49 0.04 0.91 3.21 6.166
				(0.05)) (0.10) (0.00) (0.00) (0.00) (0.78)
	USA	94:9-18:12 1 -0.08 -0.16 0.56 0.07 0.76 2.92 3.285
				(0.44)	(0.07) (0.00) (0.16) (0.00) (0.82)
	Notes: Figures in bold denote significant overshooting correction coefficients at the 10%-level maximum.
	p-values are in ( ). Q(6) is the Ljung-Box test of no serial correlation up to order 6.

See Friedman (1993), who refers to his work published in the 44th NBER Annual Report in 1964.

The nominal exchange rate data used below is the number of foreign currency units per domestic currency unit, so that a decrease corresponds to a depreciation.

In this illustration, m = 1 so that the overshooting correction lasts two periods.

Effective exchange rates are studied as in e.g.[START_REF] Adolfson | Evaluating an estimated new keynesian small open economy model[END_REF] or[START_REF] Bjornland | Monetary policy and exchange rate overshooting: Dornbusch was right after all[END_REF] due to the small open economy assumption underlying the overshooting effect.

The results for these countries are available upon request.

Due to unavailability or clear structural change in the exchange rate policy, data for Russia, Brazil, Mexico, South Korea, and Argentina start later, as indicated in the second column of Table1.

The estimates of constant terms µ and α 0 are not reported to save space.

Note that it amounts to minimize the AIC as all the models contain exactly the same number of parameters.

considered here show overshooting correction evidence, especially after an appreciation.

The asymmetry revealed by our empirical results might explain the lack of overshooting evidence from previous studies. Its theoretical modelling is on our research agenda.


Jérémie Bettinelli
email: jeremie.bettinelli@normalesup.org.

A bijection for nonorientable general maps

Keywords: . map, graph, bijection, nonorientable surface, Brownian surface

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Introduction

Motivation

The study of maps has seen tremendous developments in the past few decades. One of the reasons is that they provide natural discrete versions of a given surface. In particular, when taken according to a wellchosen natural probability distribution, it has been shown for several models that a random map converges (after scaling, in a certain sense) toward a limiting object. This limiting object is a random metric space and has (almost surely) the same topology as the surface on which the considered maps are drawn. It is called the Brownian map when the surface is the sphere, and the Brownian S for a general orientable surface S.

In the most-studied case of the sphere, it has been shown [START_REF] Gall | Uniqueness and universality of the Brownian map[END_REF][START_REF] Miermont | The Brownian map is the scaling limit of uniform random plane quadrangulations[END_REF] for example that a uniform quadrangulation (map with only faces of degree 4) with n faces converges to the Brownian map as n → ∞. The case of a more general compact orientable surface (with a boundary allowed) has been studied, mostly in the context of uniform quadrangulations: partial convergence has been established in a series of papers ending with [START_REF] Bettinelli | Geodesics in Brownian surfaces (Brownian maps)[END_REF] and a full convergence is under investigation [START_REF] Bettinelli | Compact Brownian surfaces II. The general case[END_REF]. The full convergence in the particular case of the disk has recently been shown in [START_REF] Bettinelli | Compact Brownian surfaces I. Brownian disks[END_REF], where many more models are also considered.

All the previously mentioned results strongly rely on powerful bijective encodings of the considered maps. It turns out that quadrangulations are particularly well behaved with respect to these bijective encodings and this is the main reason why they are usually the first to be studied. However, if one wishes to study other models and, in particular, surfaces with a boundary, one needs more general bijective encodings. In the case of compact orientable surfaces, the so-called Schaeffer-like bijections [CV81, Sch98, BDG04, CMS09, AB13] allow one to conduct most studies.

Until very recently, no such bijections were known in the case of a nonorientable surface. In [START_REF] Chapuy | A bijection for rooted maps on general surfaces[END_REF], Chapuy and Dołęga took the first step by exhibiting a bijection allowing to encode nonorientable bipartite quadrangulations. In this work, we give an alternate description of their bijection, which provides an explicit construction for pointed quadrangulations (i) and we show how to generalize it to nonorientable general maps. These works lay the bases for the future study of nonorientable Brownian surfaces [START_REF] Bettinelli | Nonorientable Brownian surfaces[END_REF].

Another cause of interest for maps is their remarkable enumerative properties. In fact, although maps are intricate objects by nature, many classes of them possess a quite simple enumerative structure. Thanks to different involved enumeration techniques (generating functions, matrix integrals, algebraic combinatorics), many classes of maps have been enumerated and map enumeration has become over the years a full-fledged research domain.

In the case of the sphere, Tutte [START_REF] Tutte | A census of planar maps[END_REF] gave a very simple closed formula for the number of rooted maps with a given number of edges. A bijective proof of this formula was given by Cori and Vauquelin [START_REF] Vauquelin | Planar maps are well labeled trees[END_REF] and later popularized by Schaeffer [START_REF] Schaeffer | Conjugaison d'arbres et cartes combinatoires aléatoires[END_REF]. It relies on their so-called Cori-Vauquelin-Schaeffer bijection encoding quadrangulations of the sphere with trees whose vertices carry integer labels satisfying local constraints. For more general surfaces, Bender and Canfield [START_REF] Bender | The asymptotic number of rooted maps on a surface[END_REF] showed that the number of rooted maps with n edges on a given surface (orientable or not) is asymptotically equal to a constant times n 5(h-1)/2 12 n , where h is the type of the considered surface and the constant depends on the surface. Extending the Cori-Vauquelin-Schaeffer bijection, a combinatorial interpretation of this fact in the orientable case was given by Chapuy, Marcus and Schaeffer [START_REF] Chapuy | A bijection for rooted maps on orientable surfaces[END_REF]. Their approach rely on a bijection between bipartite quadrangulations (it is a classical simple fact that bipartite quadrangulations are in bijection with general maps) and one-face maps of the same surface, whose vertices carry integer labels satisfying some local constraints.

In parallel, Bouttier, Di Francesco and Guitter [START_REF] Bouttier | Planar maps as labeled mobiles[END_REF] extended the original Cori-Vauquelin-Schaeffer bijection to encode maps of the sphere with an arbitrary face degree distribution. Unifying both aforementioned extensions, Chapuy [START_REF] Chapuy | Asymptotic enumeration of constellations and related families of maps on orientable surfaces[END_REF] proved similar asymptotic enumeration results for more families of maps on an orientable surface.

Nonorientability does not causes too much difficulties for generating function approaches. In addition to Bender and Canfield's results, we may cite the work of Gao, who showed [START_REF] Gao | The number of rooted triangular maps on a surface[END_REF] that the number of rooted triangulations with n edges on a given surface (orientable or not) is asymptotically equal to a constant (depending on the surface) times n 5(h-1)/2 (12 √ 3) n . He also studied the algebraicity of the generating function of rooted maps on a given surface with face degree constraints [START_REF] Gao | The number of degree restricted maps on general surfaces[END_REF].

In their recent work [START_REF] Chapuy | A bijection for rooted maps on general surfaces[END_REF], Chapuy and Dołęga extended the construction of [START_REF] Chapuy | A bijection for rooted maps on orientable surfaces[END_REF] to bipartite nonorientable quadrangulations. In this paper, we give a different construction of their bijection, and extend it by an approach reminiscent of [START_REF] Bouttier | Planar maps as labeled mobiles[END_REF]. In order to achieve this goal, we somehow fix a local orientation of the surface via a global process, in the sense that the process uses the information of the whole map. As a result, the constraints satisfied by the labels of the encoding objects are also global and this provides us from giving a simple characterization of these objects. In the very particular case of quadrangulations, the global constraints can be expressed as local constraints and the encoding objects (so called well-labeled unicellular maps) take a simple form and this allowed Chapuy and Dołęga [START_REF] Chapuy | A bijection for rooted maps on general surfaces[END_REF] to give a combinatorial interpretation of Bender and Canfield's asymptotic formula. In the particular case of triangulations, the same miracle occurs and we are able to give a combinatorial interpretation of the results of [START_REF] Gao | The number of rooted triangular maps on a surface[END_REF].

First definitions

From now on and until the end of the paper, we fix a compact surface without boundary S, orientable or not. A map is a cellular embedding of a finite graph (possibly with multiple edges and loops) into S, considered up to homeomorphisms. Cellular means that the faces of the map (the connected components of the complement of edges) are homeomorphic to 2-dimensional open disks. A corner is an angular sector determined by two consecutive half-edges incident to the same vertex and to the same face. The degree of a face is its number of corners. All the maps we consider are rooted, that is, given with a distinguished oriented corner called the root. The (unoriented) corner corresponding to the root will be called the root corner, the vertex incident to the root will be called the root vertex, the face containing the root will be called the root face and the edge incident to the root face that follows the root will be called the root edge. See Figure 1.

A bijection for nonorientable general maps 3 degree constraints [START_REF] Gao | The number of degree restricted maps on general surfaces[END_REF].

In their recent work [START_REF] Chapuy | A bijection for rooted maps on general surfaces[END_REF], Chapuy and Dołęga extended the construction of [START_REF] Chapuy | A bijection for rooted maps on orientable surfaces[END_REF] to bipartite nonorientable quadrangulations. In this paper, we give a different construction of their bijection, and extend it by an approach reminiscent of [START_REF] Bouttier | Planar maps as labeled mobiles[END_REF]. In order to achieve this goal, we somehow fix a local orientation of the surface via a global process, in the sense that the process uses the information of the whole map. As a result, the constraints satisfied by the labels of the encoding objects are also global and this provides us from giving a simple characterization of these objects. In the very particular case of quadrangulations, the global constraints can be expressed as local constraints and the encoding objects (so called well-labeled unicellular maps) take a simple form and this allowed Chapuy and Dołęga [START_REF] Chapuy | A bijection for rooted maps on general surfaces[END_REF] to give a combinatorial interpretation of Bender and Canfield's asymptotic formula. In the particular case of triangulations, the same miracle occurs and we are able to give a combinatorial interpretation of the results of [START_REF] Gao | The number of rooted triangular maps on a surface[END_REF].

First definitions

From now on and until the end of the paper, we fix a compact surface without boundary S, orientable or not. A map is a cellular embedding of a finite graph (possibly with multiple edges and loops) into S, considered up to homeomorphisms. Cellular means that the faces of the map (the connected components of the complement of edges) are homeomorphic to 2-dimensional open disks. A corner is an angular sector determined by two consecutive half-edges incident to the same vertex and to the same face. The degree of a face is its number of corners. All the maps we consider are rooted, that is, given with a distinguished oriented corner called the root. The (unoriented) corner corresponding to the root will be called the root corner, the vertex incident to the root will be called the root vertex, the face containing the root will be called the root face and the edge incident to the root face that follows the root will be called the root edge. See Figure 1.

ρ σ v f e c
Fig. 1: The (green) root ρ, the (purple) root corner c, the (blue) root vertex v, the (light pink) root face f and the (red) root edge e. The root of the root flipped map is σ (in orange).

We will use the following involution we call a root flip: from a map m, we define the root flipped map m by rerooting m at the only oriented corner incident to the other extremity of the root edge and defining the same root edge. We use the classical notation V (m) to denote the vertex set of a map m and we denote by d m the graph metric on V (m). A map is bipartite if its vertex set can be partitioned into two subsets such that every edge links a vertex from one subset to a vertex from the other subset.

Aim of the paper

In this work, we present a bijection between the set of pointed bipartite maps on S and pairs consisting of what we call a well-labeled unicellular mobile and a parameter ε ∈ {+, -}. A well-labeled unicellular mobile is a one-face map with green or white vertices whose white vertices carry positive integer labels satisfying some compatibility relations, which need more background to be properly stated (see Definition 2 for a rigorous definition).

Fig. 1: The (green) root ρ, the (purple) root corner c, the (blue) root vertex v, the (light pink) root face f and the (red) root edge e. The root of the root flipped map is σ (in orange).

We will use the following involution we call a root flip: from a map m, we define the root flipped map m by rerooting m at the only oriented corner incident to the other extremity of the root edge and defining the same root edge. We use the classical notation V (m) to denote the vertex set of a map m and we denote by d m the graph metric on V (m). A map is bipartite if its vertex set can be partitioned into two subsets such that every edge links a vertex from one subset to a vertex from the other subset.

Aim of the paper

In this work, we present a bijection between the set of pointed bipartite maps on S and pairs consisting of what we call a welllabeled unicellular mobile and a parameter ε ∈ {+, -}. A welllabeled unicellular mobile is a one-face map with green or white vertices whose white vertices carry positive integer labels satisfying some compatibility relations, which need more background to be properly stated (see Definition 2 for a rigorous definition).

If the surface S is orientable, we recover the famous Bouttier-Di Francesco-Guitter bijection [BDG04] and the following basic properties continue to hold in the nonorientable case:

Proposition 1. If (m, v •
) is a pointed bipartite map and ((u, l), ε) denotes the corresponding pair, then (i) V (m) \ {v • } corresponds to the white vertices of u and the label of a white vertex is given by its distance to v • in m; (ii) the faces of m correspond to the green vertices of u : moreover, the degree of a face of m is twice the degree of the corresponding green vertex; (iii) the maps m and u have the same number of edges.

Property (i) is absolutely crucial from a metric point of view, as the labeled unicellular map somehow captures part of the metric information of the map, namely all the distances to the distinguished vertex v • .

Our construction is based on a rule that gives an orientation to every corner of the map. We introduce what we call level loops; these can be thought of as contour lines in topography, where the height of a given vertex is its distance to the distinguished vertex v • . The orientation of the root gives a canonical orientation to all these level loops. Using these local orientations, we then apply similar rules as in the orientable case in order to complete the construction.

We will then see how to extend our bijection to general maps, which are not necessarily bipartite. In the case of triangulations, that is, maps with only faces of degree 3, the encoding mobiles happen to have a particularly simple structure. This allows us to recover the following enumeration result ([Gao91, Theorem 1]).

Proposition 2. The number of (rooted) triangulations of S with 2n faces (and thus 3n edges and n+2-2h vertices, by the Euler characteristic formula) is asymptotically equivalent to

c S n 5(h-1)/2 12 √ 3 n ,
where h is the type of S and c S is a constant that depends on S.

The values for h ≤ 1 of c S are given in the full version [START_REF] Bettinelli | A bijection for nonorientable general maps[END_REF] of this paper. For small values of h, the generating function of triangulations can also be computed: we recover [Gao91, Theorem 3] and we add the case of the Klein bottle.

Proposition 3. The generating function of triangulations counted with weight x per vertex is given by

1 2 (1 -2σ)(1 -σ + σ 2 ) - 1 2 1 -6σ + 6σ 2 if S is the projective plane and 3σ(1 -σ) 1 -6σ + 6σ 2 -2 7 -30σ + 30σ 2 -6(1 -2σ) 1 -6σ + 6σ 2
if S is the Klein bottle, where σ is an algebraic function of x given by

x = 1 2 σ(1 -σ)(1 -2σ) σ(0) = 0.
The analog of Proposition 2 was obtained by the same method for quadrangulations on an orientable surface in [START_REF] Chapuy | A bijection for rooted maps on orientable surfaces[END_REF], as well as on a nonorientable surface in [START_REF] Chapuy | A bijection for rooted maps on general surfaces[END_REF]. Unfortunately, the constraints on the labels of a well-labeled unicellular mobile in general are too intricate to derive similar enumeration results. In fact, the labels satisfy global constraints instead of just local constraints as soon as the maps we consider are neither triangulations nor quadrangulations.

2 From pointed bipartite maps to well-labeled unicellular mobiles

Let (m, v • ) be a pointed bipartite map. For every v ∈ V (m), we define its label by l(v) := d m (v • , v).
The fact that m is bipartite implies that the labels of two neighboring vertices differ by exactly 1. If the root vertex has a smaller label than the other extremity of the root edge, we set ε := + ; otherwise, we replace m by its root flipped version m and set ε := -. We construct the level loops as follows. At first, we only construct unoriented level loops: we will orient all the loops at the end. We consider the border of a face along some edge and we denote by i -1 and i the labels of the vertices incident to this edge. We travel on the border of the face along the edge, from the vertex labeled i -1 toward the vertex labeled i. When we reach the vertex labeled i, we turn around the vertex until we see an edge linking it to a vertex with label i -1; in the process, we cross a nonnegative number of edges linking the vertex to vertices labeled i + 1. We then travel along the edge toward the vertex labeled i -1. We keep traveling along the border of the face we are visiting until we either close the loop or reach a vertex labeled i + 1. In the later case, the vertex visited just before the vertex labeled i + 1 necessarily has label i; we turn around this vertex until we see an edge linking it to a vertex with label i -1. We iterate the process until we close the loop. We say that such a loop is at level i. We iterate the process until each side of every edge of type i -1 -i is visited by a loop at level i. We finally add around v • a single loop at level 0.

We orient, give an origin and order the level loops as follows. We consider all the loops visiting the root corner: note that there exists at least one such loop. We orient all these loops according to the orientation given by the root, we set their origin at the root corner and we declare them to be the first loops, ordered by increasing level. We then start from the origin of the first loop and travel on it. Every time we encounter new loops, we declare them to be the next ones, ordered in increasing level. We set their origin at the location we are and we orient them by the orientation induced by the loop on which we are traveling. When we arrive back at the origin of our loop, we move to the next one and iterate the process until every loop has been oriented. See Figure 3. Note that this operation terminates as every loop at level i ≥ 1 touches a loop at level i -1.

2 From pointed bipartite maps to well-labeled unicellular mobiles Let (m, v • ) be a pointed bipartite map. For every v ∈ V (m), we define its label by l(v) := d m (v • , v). The fact that m is bipartite implies that the labels of two neighboring vertices differ by exactly 1. If the root vertex has a smaller label than the other extremity of the root edge, we set ε := + ; otherwise, we replace m by its root flipped version m and set ε := -. We construct the level loops as follows. At first, we only construct unoriented level loops: we will orient all the loops at the end. We consider the border of a face along some edge and we denote by i -1 and i the labels of the vertices incident to this edge. We travel on the border of the face along the edge, from the vertex labeled i -1 toward the vertex labeled i. When we reach the vertex labeled i, we turn around the vertex until we see an edge linking it to a vertex with label i -1; in the process, we cross a nonnegative number of edges linking the vertex to vertices labeled i + 1. We then travel along the edge toward the vertex labeled i -1. We keep traveling along the border of the face we are visiting until we either close the loop or reach a vertex labeled i + 1. In the later case, the vertex visited just before the vertex labeled i + 1 necessarily has label i; we turn around this vertex until we see an edge linking it to a vertex with label i -1. We iterate the process until we close the loop. We say that such a loop is at level i. We iterate the process until each side of every edge of type i -1i is visited by a loop at level i. We finally add around v • a single loop at level 0.

We orient, give an origin and order the level loops as follows. We consider all the loops visiting the root corner: note that there exists at least one such loop. We orient all these loops according to the orientation given by the root, we set their origin at the root corner and we declare them to be the first loops, ordered by increasing level. We then start from the origin of the first loop and travel on it. Every time we encounter new loops, we declare them to be the next ones, ordered in increasing level. We set their origin at the location we are and we orient them by the orientation induced by the loop on which we are traveling. When we arrive back at the origin of our loop, we move to the next one and iterate the process until every loop has been oriented. See Figure 2. Note that this operation terminates as every loop at level i ≥ 1 touches a loop at level i -1. The root of m is represented in green and the root of u is in purple. The existence of the green edge (of u) is ensured by the rules of the construction.

We consider a loop and denote by i its level. The loop visits one by one several corners of the map. Among these corners, we say that the loop selects those with label i that are immediately preceded by a corner labeled i -1. We add an extra vertex in the middle of every face: these vertices will be called green vertices in the following, in contrast with the original vertices, which we will call white vertices. Inside each face, we link in a noncrossing fashion by green edges all the corners that are selected by some loop to the green vertex of the face.

Finally, we consider the embedded graph u whose vertex set consists of the union of V • (u) := V (m) \ {v • } with the set V • (u) of green vertices and whose edge set is composed of the green edges. We root it with the convention depicted on Figure 3. The white vertices of u We consider a loop and denote by i its level. The loop visits one by one several corners of the map. Among these corners, we say that the loop selects those with label i that are immediately preceded by a corner labeled i -1. We add an extra vertex in the middle of every face: these vertices will be called green vertices in the following, in contrast with the original vertices, which we will call white vertices. Inside each face, we link in a noncrossing fashion by green edges all the corners that are selected by some loop to the green vertex of the face.

Finally, we consider the embedded graph u whose vertex set consists of the union of V • (u) := V (m) \ {v • } with the set V • (u) of green vertices and whose edge set is composed of the green edges. We root it with the convention depicted on Figure 2. The white vertices of u inherit the labels from the function l: we set Φ(m, v • ) := ((u, l), ε). See Figure 4.

3 Definition of the encoding object and inverse construction

Preliminaries

We now describe in more details the encoding object we obtained in Section 2. First, we say that a pair (u, l) is a labeled unicellular mobile if it satisfies the following conditions:

u is a rooted one-face map of S whose vertex set is partitioned into V • (u) V • (u) in such a way that every edge links a vertex from V • (u) to a vertex from V • (u); inherit the labels from the function l: we set Φ(m, v • ) := ((u, l), ε). See Figure 4.

3 Definition of the encoding object and inverse construction

Preliminaries

We now describe in more details the encoding object we obtained in Section 2. First, we say that a pair (u, l) is a labeled unicellular mobile if it satisfies the following conditions:

⋄ u is a rooted one-face map of S whose vertex set is partitioned into V • (u) ⊔ V • (u) in such a way that every edge links a vertex from V • (u) to a vertex from V • (u); ⋄ l : V • (u) → N is a function with minimum 1;

⋄ the root vertex belongs to V • (u).

As the face of u is homeomorphic to a disk by definition, and as the root of u gives a canonical orientation to this disk, the corners of u are naturally ordered in a cyclic way around its face. In what follows, we only consider the corners of u that are incident to vertices of V • (u). Definition 1. We call arc any contiguous interval of two or more subsequent corners such that the first and last corners have strictly smaller labels than the other corners. The first and last corners of the arc are called its extremities and the other corners are called its internal corners.

An arc whose extremities have labels i and j is called an {i, j}-arc and its level is defined as the number i ∨ j. An arc is said to be trivial if it contains only two corners.

The following lemma is quite straightforward so that, for space reasons, we give it without proof and refer the interested reader to the full version of the paper.

Lemma 4. We have the following properties. (i) Let A and B be two distinct nontrivial arcs whose intersection contains internal corners of A or B.

Then A ⊆ B or B ⊆ A and the level of the larger arc is strictly lower than that of the smaller arc.

Fig. 3:

The level loops.

Inside each face, the single loop at maximal level is not represented and not taken into account in the ordering of the loops.

The letters indicate the ranks of the loops in the order, the half-arrowheads their orientation and the dots their origin.

A bijection for nonorientable general maps 7 (ii) The following properties are equivalent: (a) For all i ≥ 1, every nontrivial arc at level i contains a corner with label i + 1.

v • 0 1 1 1 1 1 2 2 2 2
(b) For all i ≥ 1, every nontrivial arc at level i has a range of internal corner labels of the form {i + 1, i + 2, . . . , m} for some m. (c) For all i ≥ 2 and for every corner with label i, either the first subsequent corner with label strictly smaller than i has label i -1 or the last preceding corner with label strictly smaller than i has label i -1.

(iii) The properties of (ii) imply that, for all i ≥ 2, every arc at level i ≥ 2 and every corner labeled i ≥ 2 is included in a unique arc at level i -1, which is nontrivial.

Construction

In order to give the explicit conditions satisfied by a well-labeled unicellular mobile, we need to start performing the inverse construction. We suppose that (u, l) is a labeled unicellular mobile satisfying the properties of Lemma 4.(ii). We add inside the unique face of u a new vertex v • with label l(v • ) := 0. We connect all the corners with label 1 to v • in a noncrossing fashion and, for each nontrivial arc at level i ≥ 2, we add a temporary vertex to which we link all its corners that are labeled i + 1. Lemma 4.(i) ensures that this can be done in a noncrossing fashion.

We now construct the unoriented level loops. Notice that, by Lemma 4.(iii), every corner of u is linked either to v • or to a temporary vertex. We call black edges these links. We select a corner of u and one incident edge of u. We denote by i the label of the selected corner. Starting from the

: V • (u) → N is a function with minimum 1; the root vertex belongs to V • (u).
As the face of u is homeomorphic to a disk by definition, and as the root of u gives a canonical orientation to this disk, the corners of u are naturally ordered in a cyclic way around its face. In what follows, we only consider corners of u that are incident to vertices of V • (u).

Definition 1. We call arc any contiguous interval of two or more subsequent corners such that the first and last corners have strictly smaller labels than the other corners. The first and last corners of the arc are called its extremities and the other corners are called its internal corners.

An arc whose extremities have labels i and j is called an {i, j}-arc and its level is defined as the number i ∨ j. An arc is said to be trivial if it contains only two corners.

The following lemma is quite straightforward so that, for space reasons, we give it without proof and refer the interested reader to the full version of the paper.

Lemma 4. We have the following properties.

(i) Let A and B be two distinct nontrivial arcs whose intersection contains internal corners of A or B.

Then A ⊆ B or B ⊆ A and the level of the larger arc is strictly lower than that of the smaller arc.

(ii) The following properties are equivalent: (a) For all i ≥ 1, every nontrivial arc at level i contains a corner with label i + 1.

(b) For all i ≥ 1, every nontrivial arc at level i has a range of internal corner labels of the form {i + 1, i + 2, . . . , m} for some m.

(c) For all i ≥ 2 and for every corner with label i, either the first subsequent corner with label strictly smaller than i has label i -1 or the last preceding corner with label strictly smaller than i has label i -1.

(iii) The properties of (ii) imply that, for all i ≥ 2, every arc at level i ≥ 2 and every corner labeled i ≥ 2 is included in a unique arc at level i -1, which is nontrivial.

Construction

In order to give the explicit conditions satisfied by a well-labeled unicellular mobile, we need to start performing the inverse construction. We suppose that (u, l) is a labeled unicellular mobile satisfying the properties of Lemma 4.(ii). We add inside the unique face of u a new vertex v • with label l(v • ) := 0. We connect all the corners with label 1 to v • in a noncrossing fashion and, for each nontrivial arc at level i ≥ 2, we add a temporary vertex to which we link all its corners that are labeled i + 1. Lemma 4.(i) ensures that this can be done in a noncrossing fashion.

We now construct the unoriented level loops. Notice that, by Lemma 4.(iii), every corner of u is linked either to v • or to a temporary vertex. We call black edges these links. We select a corner of u and one incident edge of u. We denote by i the label of the selected corner. Starting from the selected corner, we explore the boundary of the face of u, in the orientation given by the selected edge. We search the first subsequent corner with label smaller than or equal to i. We then begin to draw a loop from the selected corner to the second one, without crossing black edges; this can be done as two corners of u are linked to the same temporary vertex if and only if they have the same label and all the corners in one of the two intervals they delimit have labels larger than or equal to this common label. We then cross the edge of u at the second corner and iterate the process (always with the same i) until we close loop. We thus create a loop and define its level to be i. We iterate the process until every side of every black edge starting from a corner labeled i is visited by a loop at level i. We add a single loop at level 0 around v • . See Figure 5.

By construction, every portion of loop included in the face of u delimits an arc of (u, l). More precisely, every arc delimited by a portion of loop at level i is either trivial at level less than or equal to i or nontrivial at level i. Moreover, every nontrivial arc is delimited by exactly one loop and every trivial arc is delimited by at least one loop. We say that a level loop at level i is well oriented if every nontrivial arc it delimits is such that the extremity first visited by the portion of loop inside the face of u has label i. Jérémie Bettinelli or equal to this common label. We then cross the edge of u at the second corner and iterate the process (always with the same i) until we close the loop. We thus create a loop and define its level to be i. We iterate the process until every side of every black edge starting from a corner labeled i is visited by a loop at level i. We add a single loop at level 0 around v • . See Figure 5. Here again, we did not represent the loops at maximal level, that is, the loops circling around a single green vertex.

By construction, every portion of loop included in the face of u delimits an arc of (u, l). More precisely, every arc delimited by a portion of loop at level i is either trivial at level less than or equal to i or nontrivial at level i. Moreover, every nontrivial arc is delimited by exactly one loop and every trivial arc is delimited by at least one loop. We say that a level loop at level i is well oriented if every nontrivial arc it delimits is such that the extremity first visited by the portion of loop inside the face of u has label i.

Let A be an arc at level i ≥ 2 and let B be the arc at level i -1 defined by Lemma 4.(iii). As B is nontrivial, there is a unique loop that delimits it. We say that this loop overflies the portions of loops delimiting A (there may be more than one portion if A is trivial). We also say that the portions of loops delimiting the arcs at level 1 are overflown by the single loop at level 0. We similarly define the loop overflying a corner. We define a future root as shown on Figure 6. We will orient, give an origin and order a few level loops. Every time we orient a loop, we do the following. If the loop is not well oriented, we stop the process. If the loop is well oriented, for each nontrivial arc it delimits, we identify the corresponding temporary vertex with the extremity of the arc that is first visited by the portion of loop inside the face of u. By Lemma 4.(i), this operation can be done in such a way that the black edges do not cross each other. See Figure 7. Here again, we did not represent the loops at maximal level, that is, the loops circling around a single green vertex.

Let A be an arc at level i ≥ 2 and let B be the arc at level i -1 defined by Lemma 4.(iii). As B is nontrivial, there is a unique loop that delimits it. We say that this loop overflies the portions of loops delimiting A (there may be more than one portion if A is trivial). We also say that the portions of loops delimiting the arcs at level 1 are overflown by the single loop at level 0. We similarly define the loop overflying a corner.

or equal to this common label. We then cross the edge of u at the second corner and iterate the process (always with the same i) until we close the loop. We thus create a loop and define its level to be i. We iterate the process until every side of every black edge starting from a corner labeled i is visited by a loop at level i. We add a single loop at level 0 around v • . See Figure 5. Here again, we did not represent the loops at maximal level, that is, the loops circling around a single green vertex.

By construction, every portion of loop included in the face of u delimits an arc of (u, l). More precisely, every arc delimited by a portion of loop at level i is either trivial at level less than or equal to i or nontrivial at level i. Moreover, every nontrivial arc is delimited by exactly one loop and every trivial arc is delimited by at least one loop. We say that a level loop at level i is well oriented if every nontrivial arc it delimits is such that the extremity first visited by the portion of loop inside the face of u has label i.

Let A be an arc at level i ≥ 2 and let B be the arc at level i -1 defined by Lemma 4.(iii). As B is nontrivial, there is a unique loop that delimits it. We say that this loop overflies the portions of loops delimiting A (there may be more than one portion if A is trivial). We also say that the portions of loops delimiting the arcs at level 1 are overflown by the single loop at level 0. We similarly define the loop overflying a corner. We define a future root as shown on Figure 6. We will orient, give an origin and order a few level loops. Every time we orient a loop, we do the following. If the loop is not well oriented, we stop the process. If the loop is well oriented, for each nontrivial arc it delimits, we identify the corresponding temporary vertex with the extremity of the arc that is first visited by the portion of loop inside the face of u. By Lemma 4.(i), this operation can be done in such a way that the black edges do not cross each other. See Figure 7. We define a future root as shown on Figure 6. We will orient, give an origin and order a few level loops. Every time we orient a we do the following. If the loop is not well oriented, we stop the process. If the loop is well oriented, for each nontrivial arc it delimits, we identify the corresponding temporary vertex with the extremity of the arc that is first visited by the portion of loop inside the face of u. By Lemma 4.(i), this operation can be done in such a way that the black edges do not cross each other. See Figure 7.

The first loop in our order is the loop overflying the root corner of u. We orient it by the orientation prescribed by the future root and do the aforementioned identifications (provided that it is well oriented). After these identifications, the first loop passes through the future root corner. We set its origin at this location. We then travel on the first loop, starting at its origin. Every time we see an unoriented loop, we orient it in the same orientation as the orientation of the loop on which we are traveling. We declare this loop to be the next one and we set its origin at the location we are. If we see several new loops at the same time, we order them by increasing level. When we arrive back at the origin of our loop, we move to the next one and iterate the process until either we create a loop that is not well oriented or every loop has been oriented. Observe that, as every portion of loop sees the loop that overflies it, this process terminates. Definition 2. A well-labeled unicellular mobile is a labeled unicellular mobile that satisfies the properties of Lemma 4.(ii) and such that the previous process only creates well oriented level loops. Fig. 7: Identification of a temporary vertex. The red loop delimits the {i, j}-arc (i ≥ j) and the temporary vertex corresponding to this arc is identified with the first extremity visited by the portion of loop. When traveling on the red loop, we "see" the other loops in the order brown, purple, green, yellow, orange, blue, then the overflying loop (not represented) and finally light green and light purple.

The first loop in our order is the loop overflying the root corner of u. We orient it by the orientation prescribed by the future root and do the aforementioned identifications (provided that it is well oriented). After these identifications, the first loop passes through the future root corner. We set its origin at this location. We then travel on the first loop, starting at its origin. Every time we see an unoriented loop, we orient it in the same orientation as the orientation of the loop on which we are traveling. We declare this loop to be the next one and we set its origin at the location we are. If we see several new loops at the same time, we order them by increasing level. When we arrive back at the origin of our loop, we move to the next one and iterate the process until either we create a loop that is not well oriented or every loop has been oriented. Observe that, as every portion of loop sees the loop that overflies it, this process terminates. Definition 2. A well-labeled unicellular mobile is a labeled unicellular mobile that satisfies the properties of Lemma 4.(ii) and such that the previous process only creates well oriented level loops.

We now suppose that (u, l) is a well-labeled unicellular mobile and we perform the previous construction. We denote by m the map whose vertex set is V • (u) ∪ {v • }, whose edges are the black edges and whose root is the future root. We also denote by m the root flipped version of m and set Ψ((u, l), +) := (m, v • ) and Ψ((u, l), -) := ( m, v • ). See Figure 8.

The previous mappings are inverse one from another

We denote by B • the set of pointed bipartite maps of the surface S we consider and by U the set of well-labeled unicellular mobiles (defined in Section 3). The proof of the following theorem takes a few pages and can be found in the full version.

Theorem 5. The mappings Φ : B • → U × {+, -} and Ψ : U × {+, -} → B • are inverse bijections. Fig. 7: Identification of a temporary vertex. The red loop delimits the {i, j}-arc (i ≥ j) and the temporary vertex corresponding to this arc is identified with the first extremity visited by the portion of loop. When traveling on the red loop, we "see" the other loops in the order brown, purple, green, yellow, orange, blue, then the overflying loop (not represented) and finally light green and light purple.

We now suppose that (u, l) is a well-labeled unicellular mobile and we perform the previous construction. We denote by m the map whose vertex set is V • (u)∪{v • }, whose edges are the black edges and whose root is the future root. We also denote by m the root flipped version of m and set Ψ((u, l), +) := (m, v • ) and Ψ((u, l), -) := ( m, v • ). See Figure 8.

The previous mappings are inverse one from another

We denote by B • the set of pointed bipartite maps of the surface S we consider and by U the set of well-labeled unicellular mobiles (defined in Section 3). The proof of the following theorem takes a few pages and can be found in the full version.

Theorem 5. The mappings Φ : B • → U × {+, -} and Ψ : U × {+, -} → B • are inverse bijections.

Remark. If m is a quadrangulation, by Proposition 1.(ii), all the vertices in V • (u) have degree 2. We can remove them and merge the two incident edges into a single edge. Through this operation, (u, l) becomes a so-called well-labeled unicellular map and one recovers (a slight modification of) the Chapuy-Dołęga bijection [START_REF] Chapuy | A bijection for rooted maps on general surfaces[END_REF].

General maps

We now relax the hypothesis that the map is bipartite. We will slightly modify it in order to be able to apply our bijection from Section 2. We consider a general pointed map (m, v • ) and define the labeling function l : V (m) → Z + as before. There are now two kinds of edges: an edge will be called equilabeled if its extremities both have the same label. We then enlarge the map m by adding in the middle of each equilabeled edge an extra vertex splitting the edge into two new edges. We denote by m this enlarged Remark. If m is a quadrangulation, by Proposition 1.(ii), all the vertices in V • (u) have degree 2. We can remove them and merge the two incident edges into a single edge. Through this operation, (u, l) becomes a so-called well-labeled unicellular map and one recovers (a slight modification of) the Chapuy-Dołęga bijection [START_REF] Chapuy | A bijection for rooted maps on general surfaces[END_REF].

General maps

We now relax the hypothesis that the map is bipartite. We will slightly modify it in order to be able to apply our bijection from Section 2. We consider a general pointed map (m, v • ) and define the labeling function l : V (m) → Z + as before. There are now two kinds of edges: an edge will be called equilabeled if its extremities both have the same label. We then enlarge the map m by adding in the middle of each equilabeled edge an extra vertex splitting the edge into two new edges. We denote by m this enlarged map and we assign to each added vertex the common label of its two neighbors plus 1. This extends the definition of l to V ( m) and, clearly, for v ∈ V ( m), one has l

(v) = d m(v • , v).
The map m is bipartite, so we may apply the construction of Section 2: we set ((ũ, l), ε) := Φ( m, v • ). We slightly modify the encoding map as follows. Every vertex of V ( m) \ V (m) is by design of degree 2 and both corners incident to it are selected by the level loops, as they correspond to local maximums along the boundaries of the incident faces. As a result, it also has degree 2 in ũ; we suppress it and merge the two incident edges into a single edge. We call such a resulting edge a flagged edge and we assign to it the label of the suppressed vertex. We denote by (u, l) the resulting map, the function l being defined on a subset of the vertices and edges of u. Finally, if the root edge of m is equilabeled, then the root vertex of ũ is one of the added vertices of m. In this case, we transgress our usual definition of root and declare the root of u to be the edge resulting of the suppression of the root vertex of ũ, together with the side and local orientation induced by the root of ũ. Such a map will be called edge-rooted in what follows. See Figure 9. We extend the definition of Φ by setting Φ(m, v • ) := ((u, l), ε). The extended map and we assign to each added vertex the common label of its two neighbors plus 1. This extends the definition of l to V ( m) and, clearly, for v ∈ V ( m), one has l

(v) = d m(v • , v).
The map m is bipartite, so we may apply the construction of Section 2: we set ((ũ, l), ε) := Φ( m, v • ). We slightly modify the encoding map as follows. Every vertex of V ( m) \ V (m) is by design of degree 2 and both corners incident to it are selected by the level loops, as they correspond to local maximums along the boundaries of the incident faces. As a result, it also has degree 2 in ũ; we suppress it and merge the two incident edges into a single edge. We call such a resulting edge a flagged edge and we assign to it the label of the suppressed vertex. We denote by (u, l) the resulting map, the function l being defined on a subset of the vertices and edges of u. Finally, if the root edge of m is equilabeled, then the root vertex of ũ is one of the added vertices of m. In this case, we transgress our usual definition of root and declare the root of u to be the edge resulting of the suppression of the root vertex of ũ, together with the side and local orientation induced by the root of ũ. Such a map will be called edge-rooted in what follows. See Figure 9. We extend the definition of Φ by setting Φ(m, v • ) := ((u, l), ε). The extended mapping Φ realizes a bijection between the set of pointed maps of S and its image. See the full version for the details.

Application to triangulations

In the case of triangulations, we can give a simpler characterization of the encoding objects and derive enumeration results. In fact, the green vertices of an encoding object can only be of the three types depicted on Figure 10.

Moreover, the label variations along two consecutive corners of a unicellular mobile whose green vertices are of such a type belong to {-1, 0, 1}. This entails that such a mobile satisfies the properties of Lemma 4.(ii), that every of its nontrivial arcs has the same label at both extremities, and finally that every orientation of any of its level loops is a good orientation. Moreover, the conditions on the labels entail that both extremities of every nontrivial arc are original white vertices (in the sense that they do not come from vertices added on the flagged edges) so that no temporary vertices will be identified with vertices added on mapping Φ realizes a bijection between the set of pointed maps of S and its image. See the full version for the details. Fig. 9: The bijection for a general map. The flagged edges of the mobile have been highlighted and their labels are represented by the red squares. On this example, the root edge is equilabeled; the root of the mobile is thus a flagged edge given with a side and local orientation.

Application to triangulations

In the case of triangulations, we can give a simpler characterization of the encoding objects and derive enumeration results. In fact, the green vertices of an encoding object can only be of the three types depicted on Figure 10. Moreover, the label variations along two consecutive corners of a unicellular mobile whose green vertices are of such a type belong to {-1, 0, 1}. This entails that such a mobile satisfies the properties of Lemma 4.(ii), that every of its nontrivial arcs has the same label at both extremities, and finally that every orientation of any of its level loops is a good orientation. Moreover, the conditions on the labels entail that both extremities of every nontrivial arc are original white mapping Φ realizes a bijection between the set of pointed maps of S and its image. See the full version for the details. Fig. 9: The bijection for a general map. The flagged edges of the mobile have been highlighted and their labels are represented by the red squares. On this example, the root edge is equilabeled; the root of the mobile is thus a flagged edge given with a side and local orientation.

Application to triangulations

In the case of triangulations, we can give a simpler characterization of the encoding objects and derive enumeration results. In fact, the green vertices of an encoding object can only be of the three types depicted on Figure 10. Moreover, the label variations along two consecutive corners of a unicellular mobile whose green vertices are of such a type belong to {-1, 0, 1}. This entails that such a mobile satisfies the properties of Lemma 4.(ii), that every of its nontrivial arcs has the same label at both extremities, and finally that every orientation of any of its level loops is a good orientation. Moreover, the conditions on the labels entail that both extremities of every nontrivial arc are original white the flagged edges. As a result, the image by Φ of the set of pointed triangulations is exactly the set of these objects. They are simple enough to describe and can be enumerated by generating function techniques. We refer the reader to the full version [START_REF] Bettinelli | A bijection for nonorientable general maps[END_REF] for this application and the proof of Propositions 2 and 3.

Fig. 3 :

 3 Fig. 3: Rooting u from the root of m.The root of m is represented in green and the root of u is in purple. The existence of the green edge (of u) is ensured by the rules of the construction.

Fig. 2 :

 2 Fig. 2: Rooting u from the root of m. The root of m is represented in green and the root of u is in purple. The existence of the green edge (of u) is ensured by the rules of the construction.

Fig. 2 :

 2 Fig. 2: The level loops. Inside each face, the single loop at maximal level is not represented and not taken into account in the ordering of the loops. The letters indicate the ranks of the loops in the order, the half-arrowheads their orientation and the dots their origin.

Fig. 4 :

 4 Fig. 4:The bijection, from a pointed bipartite map to a well-labeled unicellular mobile. The edges of the original map have been grayed out.

Fig. 4 :

 4 Fig.4: The bijection, from a pointed bipartite map to a well-labeled unicellular mobile. The edges of the original map have been grayed out.

  l

Fig. 5 :

 5 Fig. 5: The unoriented level loops.Here again, we did not represent the loops at maximal level, that is, the loops circling around a single green vertex.

  ℓ

Fig. 6 :

 6 Fig. 6:The future root of m (in green) from the root of u (in purple).

Fig. 5 :

 5 Fig. 5: The unoriented level loops.Here again, we did not represent the loops at maximal level, that is, the loops circling around a single green vertex.

Fig. 5 :

 5 Fig. 5:The unoriented level loops. Here again, we did not represent the loops at maximal level, that is, the loops circling around a single green vertex.

ℓFig. 6 :

 6 Fig. 6:The future root of m (in green) from the root of u (in purple).

Fig. 6 :

 6 Fig.6: The future root of m (in green) from the root of u (in purple).

Fig. 8 :

 8 Fig. 8: The bijection, from a well-labeled unicellular mobile to a pointed bipartite map.

Fig. 8 :

 8 Fig. 8: The bijection, from a well-labeled unicellular mobile to a pointed bipartite map.

Fig. 10 :

 10 Fig. 10: The three possible types of face and the corresponding green vertices.

Fig. 9 :

 9 Fig.9: The bijection for a general map. The flagged edges of the mobile have been highlighted and their labels are represented by the red squares. On this the root edge is equilabeled; the root of the mobile is thus a flagged edge given with a side and local orientation.

Fig. 10 :

 10 Fig. 10: The three possible types of face and the corresponding green vertices.

Fig. 10 :

 10 Fig. 10: The three possible types of face and the corresponding green vertices.

Acknowledgements

This work is partially supported by Grant ANR-14-CE25-0014 (GRAAL). The author also acknowledges partial support from the Isaac Newton Institute for Mathematical Sciences where part of this work was conducted. We thank Guillaume Chapuy and Maciej Dołęga for stimulating discussions and for sharing [CD15] with us, as well as Grégory Miermont for encouraging discussions, especially about further developments on nonorientable Brownian surfaces.