¸ois Bacher
email: francois.bacher@univ-lille.fr.

FINITENESS OF HYPERBOLIC ENTROPY FOR HOLOMORPHIC FOLIATIONS WITH NON-DEGENERATE SINGULARITIES

Keywords: Singular holomorphic foliation, Leafwise Poincaré metric, Hyperbolic entropy, Non-degenerate singularities

INTRODUCTION

There has been a lot of progress in the dynamical theory of laminations by Riemann surfaces during the last two decades. More precisely, much of attention has been focused on building an ergodic theory when the leaves are hyperbolic. To have such a setup, the case of the projective spaces is very typical. Indeed, every polynomial vector field on C n can be compactified naturally into a holomorphic foliation on P n . This foliation is always singular. Let d, n P N with n ě 2, denote by F d pP n q the space of singular holomorphic foliations of degree d on P n . Lins Neto and Soares [START_REF] Neto | Algebraic solutions of one-dimensional foliations[END_REF], using a work of Jouanolou [START_REF] Jouanolou | Équations de Pfaff algébriques[END_REF], show that a generic foliation F P F d pP n q has only non-degenerate singularities. Moreover, by a result of Lins Neto [START_REF] Neto | Uniformization and the Poincaré metric on the leaves of a foliation by curves[END_REF] and Glutsyuk [START_REF] Glutsyuk | Hyperbolicity of the leaves of a generic one-dimensional holomorphic foliation on a nonsingular projective algebraic variety[END_REF], such a foliation is hyperbolic if d ě 2. It is even Brody-hyperbolic in the sense of [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF]. Loray and Rebelo [START_REF] Loray | Minimal, rigid foliations by curves on CP n[END_REF] also build a non-empty open subset of these foliations, the leaves of which are all dense in P n . When n " 2, Nguyên [START_REF] Nguyên | Singular holomorphic foliations by curves II: Negative Lyapunov exponent[END_REF] uses the integrability of the holonomy cocycle in [START_REF] Nguyên | Singular holomorphic foliations by curves I: integrability of holonomy cocycle in dimension 2[END_REF] to compute the Lyapunov exponent of a generic foliation F P F d pP 2 q. We recall briefly some recent studies and refer the reader to the survey articles [START_REF] Dinh | Some open problems on holomorphic foliation theory[END_REF][START_REF] Fornaess | Riemann surface laminations with singularities[END_REF][START_REF] Nguyên | Ergodic theory for Riemann surface laminations: a survey[END_REF][START_REF] Nguyên | Ergodic theorems for laminations and foliations: recent results and perspectives[END_REF] for a more detailed exposition.

By solving heat equations with respect to harmonic currents, Dinh, Nguyên and Sibony are able in [START_REF] Dinh | Heat equation and ergodic theorems for Riemann surface laminations[END_REF] to prove abstract ergodic theorems for laminations and foliations. This new approach enables them to develop an effective ergodic theory for laminations and foliations, and in particular, geometric versions of Birkhoff's theorem in this context. In two articles [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations I[END_REF][START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF], the three authors study a modulus of continuity for the leafwise Poincaré metric. More precisely, they show that it is Hölder in the case of a compact regular hyperbolic foliation, and Hölder with a logarithmic slope towards the origin in the case of linearizable singularities. Somehow, their work on the heat equation implicitly studies the dynamics of foliations in a canonical time, which is measured by the Poincaré distance in the universal covering. From this viewpoint, they introduce a canonical notion of hyperbolic entropy and prove the following finiteness results. (1) (Dinh-Nguyên-Sibony [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations I[END_REF]Theorem 3.10]) Let F " pX, L q be a smooth compact lamination by hyperbolic Riemann surfaces. Then, the hyperbolic entropy of F is finite.

(2) (Dinh-Nguyên-Sibony [6, Theorem 1.1]) Let F " pM, L , Eq be a Brody-hyperbolic singular holomorphic foliation on a compact complex surface. Suppose that all the singularities of F are linearizable. Then, the hyperbolic entropy of F is finite.

The finiteness of the hyperbolic entropy in this theorem is strongly dependent on their previous result on the modulus of continuity of the leafwise Poincaré metric (see [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations I[END_REF]Theorem 2.1] and [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF]Theorem 3.2]). In our previous work [START_REF] Bacher | Poincaré metric of holomorphic foliations with non-degenerate singularities[END_REF][START_REF] Bacher | Heat diffusions on holomorphic foliations with non-degenerate singularities[END_REF], we generalize this regularity result to foliations with non-degenerate singularities. In this article, we obtain the following generalization of Theorem 1.1 [START_REF] Bacher | Heat diffusions on holomorphic foliations with non-degenerate singularities[END_REF]. Theorem 1.2. Let F " pM, L , Eq be a Brody-hyperbolic singular holomorphic foliation on a compact complex surface. Suppose that all the singularities of F are non-degenerate. Then, the hyperbolic entropy of F is finite.

Let us explain briefly the method of our proof. We follow the general strategy of the three authors in [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF] for linearizable singularities. They are able to ensure that two points are at small Bowen distance by solving a Beltrami equation for a map that is obtained by gluing local orthogonal projections from a leaf to another. To control the Beltrami coefficient that could explode near the singularities, they need to correct this function and make it holomorphic when approaching the singular set. They use a small hyperbolic step and holonomy mappings to carry information from a transversal to another. By a crucial refinement lemma, they construct a covering such that they can define such an orthogonal projection for two points in the same cell, up to a hyperbolic time R. Of course, they need to estimate the cardinality of the covering all along the refinement process to show that the entropy is finite.

To adapt their proof, we use a classification of non-degenerate singularities in dimension 2 in three types.

' The linearizable singularities. ' The singularities with two separatrices and real negative characteristic number. ' The resonant singularities. By Poincaré linearization theorem and Briot-Bouquet theorem, every non-degenerate singularity is of one of these types. Moreover, Poincaré-Dulac theorem enables us to have an explicit form for resonant singularities. We show the same kind of estimates as [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF] for both cases with separatrices, and stronger ones for the resonant case. This gives us an initial covering on which we control the behaviour of close leaves in small hyperbolic time. To obtain such estimates, we use a generalization of Grönwall Lemma for non-linear differential equations, due to Lins Neto and Canille Martins [START_REF] Neto | Hermitian metrics inducing the Poincaré metric, in the leaves of a singular holomorphic foliation by curves[END_REF]. We also use their estimation of the Poincaré metric near the singularities to compare the flow time and the hyperbolic time. To correct the orthogonal projection in the neighbourhood of singularities, we fix a point in the initial leaf, and move in the second leaf with the same time of flow. This is what the three authors do, without naming it as such, since everything is explicit for linearizable singularities. It remains to check and adapt each technical element of [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF]. This is easily said, but everything is more difficult than it is in the linearizable case. Sometimes the resonant case and sometimes the two separatrices case need new arguments. Some of our techniques are slightly different from the three authors', but the main structure of our proof is very similar. Most of the time, our work on the linearizable case is just reproving what is already there in [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF] but in a slightly different setting. This enables us to clarify our work and statements in both other cases.

The article is organised as follows. In Section 2, we introduce the hyperbolic entropy following [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations I[END_REF]. Moreover, we recall our previous work on local orthogonal projections from a leaf to another and the generalization of the Grönwall Lemma. In Section 3, we study the flow in a small step of hyperbolic time for the three types of singularities. We obtain a first cell decomposition. In Section 4, we show a sufficient condition for the entropy to be finite. This criterion involves the orthogonal projections and their corrections near the singularities. In Section 5, we build a hyperbolically dense mesh of transversals and the initial covering that is refined later to obtain the Bowen cells. In Section 6, we study the holonomy mappings in small hyperbolic time to carry information during the refinement process. In Section 7, we consider trees that encode the dynamics on the universal cover D, and are compatible with our mesh of transversals. Section 8 ends the proof by exposing more precisely our refinement algorithm and building the orthogonal projection that is needed for our criterion.

Notations. Throughout this paper, we denote by D the unit disk of C, and rD the open disk of radius r P R ˚for the standard Euclidean metric of C. For R P R ˚, we denote by D R the open disk of hyperbolic radius R in D, so that D R " rD with r " e R ´1 e R `1 , or if r P r0, 1q, with R " ln 1`r 1´r . More generally, for ρ P R ˚and U a subset of a vector space with a marked point z 0 , ρU denotes the image of U by the homothety z Þ Ñ z 0 `ρpz ´z0 q. In particular, if D Ă C is a disk of radius r, ρD is the disk of same center and radius rρ.

We consider several distances on a complex manifold M . For g M a Hermitian metric on M , the distance induced by g M is denoted by d. Consider a singular holomorphic foliation F " pM, L , Eq. If L is a leaf of F , g M induces a distance on L that we denote by d L . If L is hyperbolic, then L is endowed with the Poincaré metric denoted g P and the induced distance denoted d P . We use the same notation for the Poincaré metric and distance on D. If x P M zE is such that the leaf through x, denoted L x , is hyperbolic, we note ϕ x : D Ñ L x a uniformization of L x such that ϕ x p0q " x. For u, v two functions from K to M , to a leaf or to D, we denote by

d K pu, vq " sup xPK dpupxq, vpxqq, d P,K pu, vq " sup xPK d P pupxq, vpxqq.
We try to make our notations different for different contexts. If z P C and r P R ˚, we note Dpz, rq the disk of center z and radius r. Inside a metric space, we denote by Bpx, rq the ball of center x and radius r. We try to keep this notation for ambiant Hermitian distances. Inside a leaf, we denote by L x rrs " ty P L x ; d Lx px, yq ă ru. Inside the Poincaré disk, if ξ P D and R P R `, we note D R pξq " tζ P D; d P pξ, ζq ă Ru.

The term "constant" means a real positive number that does not depend on a point x P M zE, nor on the hyperbolic radius R that will go to `8. Most of our proof relies on the fact that some constants h, h 1 , ℏ are sufficiently small, independently on R, given that it is sufficiently large. We may have forgotten to say it somewhere and the reader can suppose it is in the hypotheses of every statement. When we do not care about constants, we simply denote them by C, C 1 , C 2 , . . . When we want to keep track of them to clarify our arguments, we denote them by C 0 , C 1 , C 2 , . . . Finally, we denote by ras the smallest integer k such that k ě a, for a P R. We also denote ℜpaq (resp. ℑpaq) the real (resp. imaginary) part of a complex number a.

Acknowledgments. The author is supported by the Labex CEMPI (ANR-11-LABX-0007-01) and by the project QuaSiDy (ANR-21-CE40-0016).

PRELIMINARIES

2.1. Leafwise Poincaré metric. In all this section, we let F " pM, L , Eq be a singular holomorphic foliation on a complex manifold M . Suppose that M is endowed with a Hermitian metric g M and for x P M zE, consider

ηpxq " sup ! }α 1 p0q} g M ; α : D Ñ L x holomorphic such that αp0q " x ) .
Above, }v} g M is the norm of a vector v P T x L x with respect to the Hermitian metric g M . That is, }v} g M " pg M,x pv, vqq 1{2 . The function η was introduced by Verjovsky in [START_REF] Verjovsky | A uniformization theorem for holomorphic foliations. The Lefschetz centennial conference, Part III[END_REF]. It is designed to satisfy the following facts.

Proposition 2.1.

(1) For x P M zE, ηpxq ă `8 if and only if the leaf

L x is hyperbolic, that is, it is uniformized by the Poincaré disk D. (2) If L x is hyperbolic, we have ηpxq " }ϕ 1 p0q} g M , where ϕ : D Ñ L x is any uniformiza- tion of L x such that ϕp0q " x. (3) If L x is hyperbolic, then 4g M η 2 induces the Poincaré metric on L x .
In this article, we are interested in the case of hyperbolic leaves and we need to specify our global setting. We follow [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF] in our vocabulary. Definition 2.2. If all the leaves of F are hyperbolic, we say that F is hyperbolic. If moreover there exists a constant c 0 ą 0 such that ηpxq ă c 0 for all x P M zE, we say that F is Brody-hyperbolic.

From now on, we suppose that F is hyperbolic. We also need to define the type of singularities we deal with. Definition 2.3. Near a singularity a P E, there exists a vector field X defining F . In coordinates pz 1 , . . . , z n q centered at a, we can write

Xpzq " n ÿ j"1 F j pzq B Bz j .
The functions F j can be developed as a power series F j " ř αPN n c α,j z α . The 1-jet of X at a is defined in the chart pU, zq as X 1 " ř n j"1 ř |α|ď1 c α,j z α B Bz j . See [10, Chapter I] for more details. If the 1-jet of X has an isolated singularity at a, we say that a is a non-degenerate singularity of F .

We use the following estimate of η for non-degenerate singularities. It can be found in [1, Proposition 4.2] and its proof is basically the same as Dinh, Nguyên and Sibony's one [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF]Proposition 3.3], together with a local estimate that is due to Lins Neto and Canille Martins [START_REF] Neto | Hermitian metrics inducing the Poincaré metric, in the leaves of a singular holomorphic foliation by curves[END_REF]Theorem 2]. Proposition 2.4. Note d the distance induced by g M . Suppose that M is compact and that F is Brody-hyperbolic with only non-degenerate singularities. Then, there exists a constant C ě 1 such that C ´1dpx, Eq log ‹ dpx, Eq ď ηpxq ď Cdpx, Eq log ‹ dpx, Eq, x P M zE, where log ‹ " 1 `|log| is a log-type function.

2.2. Hyperbolic entropy. For x P M zE, denote by ϕ x : D Ñ L x a uniformization of L x such that ϕ x p0q " x. To unify notations, set also ϕ a pζq " a, for a P E and ζ P D. The idea of Dinh, Nguyên and Sibony [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations I[END_REF] is to consider the Poincaré distance in D to be a canonical time. More precisely, for R ě 0, consider the Bowen distance

d R px, yq " inf θPR sup ξPD R dpϕ x pξq, ϕ y pe iθ ξqq, x, y P M.
It measures the distance between the orbits of x and y up to time R. It is clear that it is independent on the choice of ϕ x . This enables us to define the entropy of F . For x P M , R, ε ą 0, denote by B R px, εq " ty P M ; d R px, yq ă εu the Bowen ball of radius ε and center x up to time R.

For Y Ă M , R, ε ą 0 and F Ă Y , we say that F is pR, εq-dense in Y if Y Ă Y xPF B R px, εq.
Denote by N pY, R, εq the minimal cardinality of an pR, εq-dense subset in Y . The hyperbolic entropy of Y is defined as

hpY q " sup εą0 lim sup RÑ`8 1 R log N pY, R, εq.
For Y " M , we denote it by hpF q. If M is compact, then it does not depend on the choice of g M . A similar and equivalent definition can be made with maximal pR, εq-separated sets, but we do not need it. The interested reader can see [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations I[END_REF] for more details.

Local orthogonal projection.

In order to show Theorem 1.2, we need to build cells in sufficiently small cardinality, such that two points x, y P M in the same cell are close up to time R. To ensure such a proximity, we build a smooth map ψ : D R Ñ L y close to ϕ x , take its lifting Ψ : D R Ñ D via ϕ y , slightly correct it into a close holomorphic map v : D R Ñ D, and finally correct v into a close rotation r θ : ξ Þ Ñ e iθ ξ. That way, we are able to show that ϕ x and ϕ y ˝rθ are close up to large time. Actually, the last two steps of this proof are hidden behind a result of Dinh, Nguyên and Sibony [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF]Proposition 3.6].

For the first step of this construction, we need to recall our previous work [START_REF] Bacher | Poincaré metric of holomorphic foliations with non-degenerate singularities[END_REF] on local orthogonal projections from a leaf to another. For x P M zE, the metric g M can be restricted to L x and induces a distance d Lx on it. For x P M zE and r ą 0, denote by L x rrs " tx 1 P L x ; d Lx px, x 1 q ă ru. Suppose that M is compact and that all the singularities of F are non-degenerate.

Lemma 2.5 ([1, Lemma 4.3]

). There exist constants ε 0 , ε 1 , k and K such that for two points x, y P M zE, if dpx, yq ď ε 1 dpx, Eq, then there exists a local orthogonal projection (4) If x 1 P L x rε 0 dpx, Eqs, y 1 P L y rkε 0 dpx, Eqs and dpx 1 , y 1 q ď ε 1 dpx 1 , Eq, then Φ x 1 y 1 " Φ xy on the intersection of their domains of definition.

Φ xy : L x rε 0 dpx,
More precisely, we build the local orthogonal projection by solving an implicit equation on the flow. That way, in singular charts, we are able to estimate the flow time that is needed to join y and Φ xy pxq, with the notations of the previous lemma.

2.4.

Variations on the Grönwall Lemma. In order to obtain thorough estimates, we use several generalizations of the Grönwall Lemma, including some non-linear cases. First, let us state it in a form that contains these various versions and then do some remarks. Proposition 2.7 (Lins Neto-Canille Martins [START_REF] Neto | Hermitian metrics inducing the Poincaré metric, in the leaves of a singular holomorphic foliation by curves[END_REF]Proposition 6]). Let F : R `ˆR `Ñ R be a continuous function such that F pt, xq ď F pt, yq if x ď y. Suppose that for any x 0 P R `, the Cauchy problem (2.1)

x 1 ptq " F pt, xptqq, xp0q " x 0 , has a unique maximal solution in a neighbourhood of t " 0. Let x : r0, r x q Ñ R `be a continuous function that satisfies xptq ď x 0 `ż t 0 F ps, xpsqqds, t P r0, r x q , and y : r0, r y q Ñ R `be the unique maximal solution of (2.1) starting at x 0 . Then, for t P r0, minpr x , r y qq, xptq ď yptq.

Remark 2.8. Actually, Lins Neto and Canille Martins prove a stronger estimate for partial orders on R n `, but we only need it for n " 1. We use this result in two contexts. The first one is of an autonomous system, sometimes in the non-linear case and sometimes in the linear case (i.e. F pt, xq " Cx, which gives the classical Grönwall Lemma). The second one is a linear but non-autonomous system. More precisely, for F pt, xq " Cx `f ptq. In that case, note that we have yptq " ˆx0 `ż t 0 f psqe ´Cs ds ˙eCt . We consider F " pM, L , Eq a Brody-hyperbolic singular holomorphic foliation on a compact complex surface M (that is dim C M " 2) with only non-degenerate singularities. Take a P E and U a » D 2 a neighbourhood of a on which F is generated by a vector field X. We suppose that the coordinates of U a and X extend to a neighbourhood of D 2 . We need to clarify our vocabulary.

Definition 3.1. For z P 1 2 D 2 zt0u, note φ z the flow of the vector field X starting at z, defined on a maximal open subset of C such that φ z ptq stays in D 2 .

A flow path for z and X is a C 1 map γ : r0, T s Ñ C, where T P R `, γp0q " 0 and φ z pγptqq is well defined for all t P r0, T s and belongs to 3 4 D 2 . Most of the time, T is implicit. The length of γ as a path in C will be denoted by ℓpγq. The Poincaré length ℓ P pγq of γ is by definition the Poincaré length of φ z ˝γ in L z . The notation ℓ P pγq is used only if there is no confusion possible for the point z and the vector field X.

Let δ : r0, T s Ñ L z X 3 4 D 2 be a C 1 map such that δp0q " z. A flow path γ : r0, T s Ñ C for z and X is said to correspond to δ if δptq " φ z pγptqq. Fix a uniformization ϕ z of L z such that ϕ z p0q " z. Let ξ P D be such that ϕ z pr0, ξsq Ă 3 4 D 2 . We say that a flow path γ : r0, 1s Ñ L z represents ξ if φ z pγptqq " ϕ z ptξq. Since the flow is a local biholomorphism, it is clear that for such a ξ, there is a unique flow path representing it. Similarly, for such a δ, there is a unique flow path corresponding to it.

Let z, w P 1 2 D 2 zt0u and R, δ ą 0. We say that z and w are pR, δq-relatively close following the flow of X if for all ξ P D R , ϕ z pξq P 3 4 D 2 and for γ the flow path for z and X representing ξ, φ w pγptqq belongs to D 2 and for all t P r0, 1s, }φ z pγptqq ´φw pγptqq} 1 ď δ }φ z pγptqq} 1 ; and if we also have the same properties when switching the roles of z and w. Here, we have denoted by }z} 1 " maxp|z 1 | , |z 2 |q for z " pz 1 , z 2 q P D 2 .

We need a classification of non-degenerate singularities in dimension 2. Theorem 3.2 (Briot-Bouquet, Poincaré-Dulac). Let a be a non-degenerate singularity of a foliation F on a compact complex surface. There exist local coordinates pz 1 , z 2 q P D 2 centered at a, such that F is generated on D 2 by one of the vector fields

X 1 " z 1 B Bz 1 `λz 2 B Bz 2 , λ P C ˚; X 2 " z 1 B Bz 1 `pmz 2 `µz m 1 q B Bz 2 , m P N ˚, µ P C ˚, |µ| ă 1 2 ; X 3 " z 1 B Bz 1 ´αz 2 `1 `z1 z q`1 2 f pz 1 , z 2 q ˘B Bz 2 , q P N, }f } 8 ă 1, α P p0, 1s X " pq `1q ´1, q ´1˘.
In X 3 , if q " 0, then we just have α " 1.

Proof. Let β be a characteristic number of the singularity.

If β P C ˚z `R´Y N ˚Y 1 N ˚˘,
then Poincaré linearization theorem implies that the singularity is linearizable and F is generated by some X 1 . If β P N ˚Y 1 N ˚, then by Poincaré-Dulac theorem (see for both [10, Chapter I, Section 5]) gives either X 1 or X 2 , whether the singularity is linearizable or not. Finally, if α P R ´, taking α ´1 if necessary, we can suppose that α P p0, 1s X rpq `1q ´1, q ´1q for some q P N. By Briot-Bouquet theorem and a refinement by Camacho-Kuiper-Palis [3, Lemma 7], we have the form X 3 . The estimates on µ and f can be obtained by homothety or transformations of the type pz 1 , z 2 q Þ Ñ pAz 1 , z 2 q for some A P C ˚.

□

In what follows, we suppose that F is generated on a neighbourhood of D 2 by one of the vector fields X " X j , for j P t1, 2, 3u. If j " 1, we talk about the linearizable case, if j " 2 about the Poincaré-Dulac case and if j " 3 about the Briot-Bouquet case. For X 3 , we also consider

p X 3 " ´1 α z 1 `1 `z1 z q`1 2 f pz 1 , z 2 q ˘´1 B Bz 1 `z2 B Bz 2 " ´1 α z 1 `1 `z1 z q`1 2 gpz 1 , z 2 q ˘B Bz 1 `z2 B Bz 2 .
Making an other homothety, we can still suppose that }g} 8 ă 1. Exchanging z 1 and z 2 if necessary, note that both X 3 and p X 3 are of the form

(3.1) r X 3 " z 1 B Bz 1 ´αz 2 `1 `zk 1 z 2 f pz 1 , z 2 q ˘B Bz 2 ,
with α P R `, k P N ˚and k ě α. We often use only this hypothesis. In that case, we are able to make a unified proof of our estimates for X 3 and p X 3 . The Briot-Bouquet case is basically the only one for which we specify that flow and flow paths are for X 3 , p X 3 or r X 3 (that is, for any of X 3 or p X 3 ). Now, we want to establish some first useful results that we use uniformly in all three cases and throughout our proof. Since 0 is a non-degenerate singularity of the vector field X, we know that there exist constants

C 0 , C 1 , C 2 ą 0 such that (3.2) C ´1 0 }z} 1 ď }Xpzq} 1 ď C 0 }z} 1 , z P D 2 ; (3.3) C ´1 1 }z ´w} 1 ď }Xpzq ´Xpwq} 1 ď C 1 }z ´w} 1 , z, w P D 2 ; (3.4) C ´1 2 }z} 1 |ln }z} 1 | ď ηpzq ď C 2 }z} 1 |ln }z} 1 | , z P 3 4 D 2 .
The last inequality is a consequence of Proposition 2.4.

Lemma 3.3.

Let z be a point in 1 2 D 2 zt0u and γ be a flow path for z. We suppose that ℓ P pγq ď R. There exists a constant C 3 ą 0 such that for all t,

|γptq| ď C ´1 0 |ln }z} 1 | `eC 3 R ´1˘. Proof. Consider a reparametrization r γ : r0, T s Ñ C of γ such that |r γ 1 ptq| " 1 for all t P r0, T s. It is clear that T ě sup uPr0,T s |r γpuq| ě |γptq| for all t. Let us translate the bound ℓ P pr γq ď R in terms of an integral. R ě 2 ż T 0 }Xpφ z pr γptqqq} ηpφ z pr γptqqq dt ě C ż T 0 dt |ln }φ z pr γptqq} 1 | ,
where we used first the relation g P " 4g M η 2 and the fact that |r γ 1 ptq| " 1, and second the equivalence of Hermitian metrics, (3.2) and (3.4). Now, by (3.2), we have

› › pφ z ˝r γq 1 ptq › › 1 ď C 0 }pφ z ˝r γq ptq}.
Then, Grönwall Lemma gives }φ z pr γptqq} 1 ď }z} 1 e C 0 t and the same argument on the reverse path ensures that }φ z pr γptqq} 1 ě }z} 1 e ´C0 t . Hence,

R ě C ż T 0 dt |ln }z} 1 | `C0 t " C C 0 ln ˆ1 `C0 T |ln }z} 1 | ˙. Thus, |γptq| ď T ď C ´1 0 |ln }z} 1 | ´eC 0 C ´1R ´1¯. □
The next result describes the Bowen ball of a singularity. It is close to [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF]Lemma 2.5].

Lemma 3.4. Let R ą 0 and ε P `0, 1 2 ˘. If 0 ă }z} 1 ă exp `lnpεqe C 3 R ˘, then ϕ z pD R q Ă εD 2 . Proof. The proof is by contradiction. Take z 1 P ϕ z pD R q such that 1 2 ą }z 1 } 1 ą ε. Let γ : r0, 1s Ñ C be a flow path with respect to z 1 , of Poincaré length less than R and such that φ z 1 pγp1qq " z. By Lemma 3.3, |γp1q| ď C ´1 0 |ln }z 1 } 1 | `eC 3 R ´1˘.
On the other hand, using the same arguments as the previous lemma, we have the contradiction

}z} 1 ě }z 1 } 1 e ´C0 |γp1q| ě }z 1 } 1 exp `ln }z 1 } 1 `eC 3 R ´1˘˘ě exp `lnpεqe C 3 R ˘. □
Fix ε P `0, 1 2 ˘and denote by r sing pRq " exp `ln `ε 4 ˘eC 3 R ˘and U sing pRq " r sing pRqD 2 , for R sufficiently large. The next paragraphs are devoted to prove our cell decomposition. We need to distinguish three cases for the three vector fields X j , for j P t1, 2, 3u.

Linearizable case.

What we show in this subsection is quite easy, but it clarifies our wishes, methods and notations for the two following subsections. Indeed, our idea for the Briot-Bouquet and Poincaré-Dulac cases is to compare them to the corresponding linearizable cases with λ " ´α or λ " m.

Take the vector field X " X 1 , with the notations of Theorem 3.2. For z, w P 1 2 D 2 zt0u, denote by zptq " pz 1 ptq, z 2 ptqq " φ z ptq and wptq " pw 1 ptq, w 2 ptqq " φ w ptq the coordinates of the flow trajectories. Set λ 1 " 1 and λ 2 " λ. We have z j ptq " z j e λ j t , w j ptq " w j e λ j t .

The cell decomposition is a consequence of the following estimate, together with analogous ones in both non-linearizable cases. In that case, it is somehow a weaker version of [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF]Proposition 2.7]. However, for the other two cases, we would not have something as strong as what Dinh, Nguyên and Sibony obtain in the linearizable case.

Lemma 3.5. Let h, δ P p0, 1q be sufficiently small. For z, w P 1 2 D 2 z `1 2 U sing pRq ˘, if for each j P t1, 2u we are in one of the following configurations, (C1) |z j | , |w j | ď δr sing pRq 2 , (C2) w j , z j ‰ 0, ˇˇ1 ´zj w j ˇˇď δ and ˇˇ1 ´wj z j ˇˇď δ, then z and w are ph, δq-relatively close following the flow.

Proof. Since our hypotheses are symmetric in z and w, it is sufficient to prove only the assertions about flow paths with respect to z. Fix ξ P D h , γ a flow path representing ξ and j P t1, 2u. If z j and w j are in configuration (C2), then

|z j pγptqq ´wj pγptqq| " ˇˇˇ1 ´wj z j ˇˇˇ| z j pγptqq| ď δ }zpγptqq} 1 .
Next, suppose that z j and w j are in configuration (C1). By Lemma 3.3 and since h is small, we have |γptq| ď Ch |ln }z} 1 |. It follows by Grönwall Lemma and (3.2) that }zpγptqq} 1 ě r sing pRq 1`Ch . On the other hand, if h is sufficiently small, |z j pγptqq ´wj pγptqq| ď p|z j | `|w j |q e |λ j ||γptq| ď 2δr sing pRq 2´Ch ď δ }zpγptqq} 1 . □ 3.3. Poincaré-Dulac case. Now, suppose that X " X 2 , with the notations of Theorem 3.2. With the same notations as before, we have the explicit flow z 1 ptq " z 1 e t , z 2 ptq " pz 2 `µtz m 1 q e mt , w 1 ptq " w 1 e t , w 2 ptq " pw 2 `µtw m 1 q e mt . The analogous of Lemma 3.5 is the following. Lemma 3.6. Let h, δ P p0, 1q be sufficiently small. For z, w 1 q| e mℜpγptqq . First, focus on the second term in the right hand side. If z 1 and w 1 are in configuration (C1.1), the same arguments as in the linearizable case show that it is bounded above by δ 2 }zpγptqq} 1 . On the other hand, if z 1 and w 1 are in configuration (C1.2), we get

P 1 2 D 2 z `1 2 U sing pRq ˘, if (C1.1) |z 1 | , |w 1 | ď δr sing pRq 2 or (C1.2) w 1 , z 1 ‰ 0, ˇˇ1 ´z1 w 1 ˇˇď |ln }w} 1 | ´1 δ 4 and ˇˇ1 ´w1 z 1 ˇˇď |ln }z} 1 | ´1 δ 4 , and (C2.1) |z 2 ´w2 | ď δ 2 max p|z 1 | m , |w 1 | m q or (C2.2) z 2 , w 2 ‰ 0, ˇˇ1 ´z2 w 2 ˇˇď |ln }w} 1 | ´1 δ 4 and ˇˇ1 ´w2 z 2 ˇˇď |ln }z} 1 | ´1 δ 4 ,
|µγptq| |z m 1 ´wm 1 | e mℜpγptqq ď Ch |ln }z} 1 | ˇˇˇ1 ´wm 1 z m 1 ˇˇˇ| z 1 pγptqq| m ď |ln }z} 1 | ˇˇˇ1 ´w1 z 1 ˇˇˇ| z 1 pγptqq| ,
if h is sufficiently small. In any case, the second term of (3.5) is bounded above by δ 2 }zpγptqq} 1 . Now, consider the first term. If z 2 and w 2 are in configuration (C2.1), then

|z 2 ´w2 | e mℜpγptqq ď |z 2 ´w2 | |z 1 | m |z 1 pγptqq| m ď δ 2 }zpγptqq} 1 .
Next, suppose that z 2 and w 2 are in configuration (C2.2). We distinguish two cases.

(i) |z 2 `µγptqz m 1 | ď |z 2 | 2 . In particular, z 1 ‰ 0 and |µγptqz m 1 | ě |z 2 | 2 . Hence, |z 2 ´w2 | e mℜpγptqq " |z 2 ´w2 | |z 1 pγptqq| m |z 1 | m ď 2 ˇˇˇ1 ´w2 z 2 ˇˇˇ| µγptq| |z 1 pγptqq| m ď δ 2 |z 1 pγptqq| ,
if h is sufficiently small, using Lemma 3.3 and

|z 1 pγptqq| ď 3 4 . (ii) |z 2 `µγptqz m 1 | ą |z 2 | 2 .
In particular, |z 2 pγptqq| ě |z 2 | 2 e mℜpγptqq . Thus,

|z 2 ´w2 | e mℜpγptqq ď 2 ˇˇˇ1 ´w2 z 2 ˇˇˇ| z 2 pγptqq| .
In any case, we get |z 2 pγptqq ´w2 pγptqq| ď δ }zpγptqq} 1 . □ 3.4. Briot-Bouquet case. This is the most delicate one. With the notations of (3.1), consider X " r X 3 , i.e. both X 3 and p X 3 . Note zptq " pz 1 ptq, z 2 ptqq " φ z ptq, and also r z 2 ptq " z 2 ptqe αt to compare it with the corresponding linearizable case, where we would have r z 2 ptq being constant equal to z 2 . Take also the same notations for w. We begin by some study of z alone.

|r z 2 ptq| ď |z 2 | `C ż |t| 0 |z 1 | α ˇˇˇr z 2 ˆs t |t| ˙ˇˇˇ2 ds.
We want to apply Proposition 2.7. With its notations, F ps, xq " C |z 1 | α x 2 and the unique solution ypsq is given by

x 0 1´C|z 1 | α x 0 s . Hence, |r z 2 pγptqq| ď |z 2 | 1 ´C |z 1 | α |z 2 | |γptq| ď |z 2 | 1 ´Ch |ln }z} 1 | |z 1 | α |z 2 | . Since x Þ Ñ x ln x is bounded on " 0, 3 4
‰
, we get |r z 2 pγptqq| ď 2 |z 2 | for h small enough. We argue the same on the reverse path from z 2 pγptqq to z 2 to obtain the other inequality. □

We can control the distance between z 2 pγptqq and w 2 pγptqq in small hyperbolic time. Lemma 3.9. Let h ą 0 be sufficiently small and z, w P 1 2 D 2 zt0u be such that }z} 1 ď 2 }w} 1 . Let γ : r0, T s Ñ C be a flow path for z with ℓ P pγq ď h. Then, ˇˇˇ`δ 2 ď δ. □ 3.5. Cell decomposition. Now, we wish to cover the bidisk by smaller bidisks such that we have a good control of the divergence of the flow in small hyperbolic time of two points in a cell. Let us take any j P t1, 2, 3u and X " X j if j ‰ 3, X " r X 3 if j " 3. Fix ε ą 0 and let R ą 0 be sufficiently large (depending on ε). Let C 4 be a strictly positive constant that is specified by our further computations. Name r 0 " e ´exppC 4 Rq , r n " r 0 e n expp´C 4 Rq , for n P 1, N , N " P e 2C 4 R T , and θ k " 2kπ N , for k P 1, N 1 , and Let us suppose by symmetry that z R U sing pRq. It is clear that if C 4 ą C 3 `2 and R is sufficiently large, then w R 1 2 U sing pRq. Similarly, if C 4 ą C 3 and D pjq " D 0 , z j and w j are in configuration (C1) of Lemmas 3.5, 3.6 and 3.10. On the other hand, if D pjq is D nk , then ˇˇ1 ´zj w j ˇˇ, ˇˇ1 ´wj z j ˇˇď Ce ´C4 R . Since z, w R 1 2 U sing pRq, z j and w j are in configuration (C2) of the three lemmas. We conclude by applying them. □ Remark 3.12. Actually, configuration (C2.1) in the Poincaré-Dulac case is most of the time far weaker than the configurations corresponding to separatrices in any other cases.

|r z 2 pγptqq ´r w 2 pγptqq| ď 2 |z 2 ´w2 | `|z 2 | sup uPr0,T
N 1 " P 4πe C 4 R T . Let D 0 " r 0 D and D nk " D `rn´1 e iθ k ,
On some sublevel t|z 2 | ď C |z 1 | m u, we can replace the disks in the second coordinate by disks of radius e ´2R |z 1 | m . We do so later.

GENERAL STRATEGY AND REDUCTIONS

4.1. Geometric setup. First, let us describe the general geometric assumptions we make to simplify our arguments. Since the entropy hpF q does not depend on the choice of the Hermitian metric g M , we build one that satisfies some suitable conditions. Let pU r , U a q rPR,aPE be a finite open covering of M by ' Singular flow boxes U a » D 2 such that F is generated on a neighbourhood of U a by one of the vector fields X j , for j P t1, 2, 3u. We also suppose that the Hermitian metric g M is given on U a by }dz} 2 . ' Regular flow boxes U r » D ˆTr such that 2U r is still a flow box and 2U r X E " H.

We often identify T r with t0u ˆTr . We also suppose that the regular flow boxes cover M z pY aPE ρU a q. Here, ρ ą 0 is fixed below. For this section, we need ρ ă 1 4 . 4.2. Reduction to studying orthogonal projections. We want to do some reductions to a criterion involving an orthogonal projection. Let us begin by the following. Whereas the three authors prove it in the setup of linearizable singularities, it is implicit in [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF] that it is enough to prove the following lemma. Lemma 4.2. Let R, ε ą 0 and x P M zE be such that ϕ x pD 2R q Ă 1 2 U a . If R is sufficiently large, then ϕ x pD R q Ă ε 2 U a . Proof. We prove that for ε ą 0, there exists K ε ą 0 such that for any z P U a with }z} 1 ě ε, there is

z 1 P `Ua z 1 2 U a ˘X L z with d P pz, z 1 q ď K ε .
It is easy to see that it implies the lemma. We have to distinguish the vector field we are dealing with.

Briot-Bouquet or linearizable case. Note z " pz 1 , z 2 q. Let j P t1, 2u be the coordinate such that |z j | " }z} 1 . By symmetry, considering p X 3 if necessary, we can suppose that j " 1. Then, φ z ptq, for t P R `, stays in L z . On the other hand, it reaches U a z 1 2 U a in flow time less than ln

6]

). There exists a constant C ą 0 such that if R is large enough and x, y P T are pR, e ´2R q-conformally close, then they are `R{3, Ce ´R{3 ˘-close, i.e. d R{3 px, yq ď Ce ´R{3 . Now, let us state the criterion we apply to show the finiteness of the entropy. Its proof occupies most of the end of the section. Proposition 4.5. Let h 1 ą 0 be sufficiently small, R ą 0 be sufficiently large and pV i q iPI be a covering of T such that cardpIq ď e gR and satisfying the following. Let i P I and x, y P V i .

(1) There exists a subset

F Ă D with D R Ă D h 1 pF q " Y ξPF D h 1 pξq, (2
) There exists a map ψ : D h 1 pF q Ñ L y without critical points, that is locally near ξ P D h 1 pF q given as the orthogonal projection from L x near ϕ x pξq to L y near ψpyq, (3) If ξ P F is such that ϕ x pξq P 2ρU a , then ϕ x pξq and ψpξq are `3h 1 , e ´2R ˘-relatively close following the flow, for r X 3 in the Briot-Bouquet case. Then, hpTq ď 3g.

Take α ą 0 and an open covering pV i q iPI that satisfies the hypotheses of Proposition 4.5 for Rp1 `αq. It is enough to show that for ε ą 0, if x, y P V i for some i P I, and R is sufficiently large, then d R{3 px, yq ď ε. By Lemma 4.4, it is even enough to show that x, y are pR, e ´2R q-conformally close. We fix x, y P V i and R ą 0. Let us also take F and ψ given by the hypotheses of Proposition 4.5. We need to control paths in singular flow boxes and therefore need to meet F often enough. We use the two following lemmas. Lemma 4.6. Let a P E and V a be a connected component of D Rp1`αq X ϕ ´1 x pρU a q. For ζ 1 , ζ 2 P V a , there exist paths λ 1 , . . . , λ N : r0, 1s Ñ D h 1 pF q X ϕ ´1 x p2ρU a q with λ 1 p0q " ζ 1 , λ N p1q " ζ 2 , λ j p1q " λ j`1 p0q P F , for j P 1, N ´1 and ℓ P pλ j q ď 3h 1 , j P 1, N .

Proof. Since F is Brody-hyperbolic and h 1 is sufficiently small, if ϕ x pξq P ρU a , then ϕ x pD h 1 pξqq Ă 2ρU a . Since V a is connected, there exists a path connecting ζ 1 and ζ 2 . Since F is h 1 -dense, one can correct this path into λ 1 , . . . , λ N asked by the lemma. □ Lemma 4.7. Keep the notations of Lemma 4.6. Let λ be the concatenation of λ 1 , . . . , λ N , z " ϕ x pζ 1 q and w " ψpζ 1 q. There is a flow path γ (resp. δ) for z (resp. w) such that ϕ x pλptqq " φ z pγptqq (resp. ψpλptqq " φ w pδptqq) and |γptq ´δptq| ď e ´p2`αqR .

Proof. The existence of the flow paths is a consequence of the fact that the flow is a local biholomorphism. We need to show the estimate |γptq ´δptq| ď e ´p2`αqR . Note that λ meets F at the ends of each λ j and that they satisfy ℓ P pλ j q ď 3h 1 . Hence, we can show by induction and using Lemma 2.6 that |γptq ´δptq| ď C }φ w pγptqq ´φz pγptqq} 1 }φ z pγptqq} ´1 1 ď Ce ´2Rp1`αq ď e ´p2`αqR , since ϕ x pξq and ψpξq are p3h 1 , e ´2pR`αq q-relatively close following the flow. Here, we prove by induction and use that the orthogonal projection from L x near φ z pγptqq onto L y near φ w pδptqq coincide with the one from L x near φ z pγptqq onto L y near φ w pγptqq. □

We need to control some monodromy phenomena for the flow by the next result.

Lemma 4.8. There exists ε 2 ą 0 such that if z, w P D 2 zt0u, for U a » D 2 a linearizable or Poincaré-Dulac singular flow box, are such that }z ´w} 1 ď 1 2 }z} 1 , then the following holds. If t 1 , t 2 , u 1 , u 2 satisfy φ z pt 1 q " φ z pt 2 q, φ w pu 1 q " φ w pu 2 q and |pt 1 ´t2 q ´pu 1 ´u2 q| ď ε 2 , then t 1 ´t2 " u 1 ´u2 .

For a Briot-Bouquet singularity, we have the same result for at least one of the vector fields X 3 or p X 3 , depending on z, w (but not on t 1 , t 2 , u 1 , u 2 ).

Proof. Name z " pz 1 , z 2 q and w " pw 1 , w 2 q. Note that the hypothesis }z ´w} 1 ď 1 2 }z} 1 implies that there is a coordinate j P t1, 2u with z j , w j ‰ 0. Without loss of generality, we suppose in the linearizable case that j " 1. Since we can choose X 3 or p X 3 , we do the same in the Briot-Bouquet case. The explicit form of the flow shows that t 1 ´t2 , u 1 ´u2 P 2iπZ. We conclude for ε 2 ă 2π. We argue similarly in the Poincaré-Dulac case. □ End of proof of Proposition 4.5. We correct the orthogonal projection ψ into a function r ψ, which coincides with ψ far from the singular set, is holomorphic near the singularities, and still satisfies (a) and (b) of being pR, e ´2R q-conformally close. Then, condition (c) is clear and it is sufficient to conclude. It is enough to construct this r ψ on V a such that it coincides with ψ on V a X ϕ ´1 x `ρU a z ρ 2 U a ˘and make this correction for each a and each connected component V a . We can also do so only if V a X ϕ ´1 x `ρ 2 U a ˘‰ H.

Fix any ξ P F X ϕ ´1 x pρU a q. Such a ξ exists because F is supposed to be Brodyhyperbolic, V a X ϕ ´1 x `ρ 2 U a ˘‰ H and h 1 is small. Set z " ϕ x pξq and w " ψpξq. Let ζ P V a and take paths λ : r0, 1s Ñ D given by Lemma 4.6 and γ, δ given by Lemma 4.7. For the Briot-Bouquet case, we choose flow paths for the vector field that satisfies Lemma 4.8 for z and w. Let χ : r0, 1s Ñ r0, 1s be a smooth function such that χ " 0 on " ρ 2 , 1

‰ and χ " 1 on " 0, ρ 4 ‰ . Define
r ψpζq " φ w pχ p}φ z pγp1qq}q pγp1q ´δp1qq `δp1qq .

Lemmas 4.8 and 4.7 imply that this definition does not depend on the choice of λ. The correction r ψ is holomorphic on V a Xϕ ´1 x `ρ 4 U a ˘and we conclude with the first remarks. □ Above, the correction process for ψ is close to [6, Lemma 2.12], but its proof is more difficult in our context, because it is less explicit. To ensure that a h 1 -dense subset F has its image in an appropriate cell, we use the following refinement result, due to Dinh, Nguyên and Sibony, on holonomy mappings. Lemma 4.9 (Dinh-Nguyên-Sibony [6, Lemma 4.4]). Let Ω be a subset of a finite union of copies of C. Let V 1 , . . . , V n be coverings of Ω by disks, all satisfying card V i ď K, for some uniform K ą 0. Then, there exists a covering V of Ω by disks, with card V ď 200 n K such that for any D P V, there are D i P V i , i P 1, n , satisfying 2D Ă 2D 1 X ¨¨¨X 2D n .

MESH OF TRANSVERSALS AND INITIAL COVERING

5.1. Mesh of transversals. At the end of our argument, we wish to apply Proposition 4.5. Hence, we need to build a covering of T, the union of some regular transversals. However, to build such a covering, we also need to control how leaves from these transversals behave when approaching the singular set. Therefore, we also build a mesh of transversals, covering almost all the manifold, which is in some sense hyperbolically dense in the foliation. That way, we are able to control at the beginning what happens to leaves in small hyperbolic time near all of these transversals, and then refine to control what happens all along the orbit in large hyperbolic time.

Let us fix some constant h 1 ą 0 which is sufficiently small to have all the results of Section 3 for h " 3h 1 . We also take a constant K P p0, 1q and name ℏ " Kh 1 . The constant K is fixed in the following sections. Note that since we are only concerned in what happens to points in T in time R, we let ε " dpT, Eq and r sing pRq " exp `ln `ε 4 ˘eC 3 R ˘. Hence, ϕ x pD R q X B pa, r sing pRqq " H, for a P E and x P T, by Lemma 3.4. First, let us state what we ask of our mesh of transversals. Proposition 5.1. There exists a constant K 1 ą 0 such that for all R ą 0 sufficiently large, there exists a mesh of transversals r T " pT i q iPI T satisfying (HT1) For x P M zE, if dpx, Eq ě r sing pRq, there is y P pY iPI T T i q X L x with d P px, yq ď ℏ; (HT2) For i P I T , there are at most K 1 elements j P I T such that there are x P T i , y P L x XT j with d P px, yq ď 2h 1 .

This is a difference in the structure of our argument and the one of the three authors of [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF]. They build transversals for each of the singular cells and have to consider some multiplicity of transversals. The property (HT2) enables us to avoid this because the refinement Lemma 4.9 applies to a bounded number of transversals.

In regular flow boxes U r » D ˆTr , the Poincaré metric is equivalent to the standard Hermitian metric. Then, we consider a lattice Λ " Bℏ pZ `iZq X D and T r,λ " tλu ˆTr . The union of these transversals clearly satisfy (HT1) and (HT2) if B is sufficiently small. So, the difficulty of Proposition 5.1 is for the singular flow boxes. We have to distinguish different cases for the different types of singularities. First, let a P E be a singularity of linearizable or Briot-Bouquet type. Define r T 0 " r sing pRq and for j P N, r T j " `rT 0 ˘p1´C 5 ℏq j , for some C 5 P `0, 1 ℏ ˘that our further computation specifies. Let also θ T 1 , . . . , θ T P be some angles that are C 5 ℏ-dense in r0, 2πs, for P "

Q π C 5 ℏ U . Define T a,j,k,u " " z " pz 1 , z 2 q P D 2 » U a ; z u " r T j e iθ T k , }z} 1 ď 3 2 r T j * ,
for j P 0, N a pRq , k P 1, P , u P t1, 2u and N a pRq " max ␣ j P N; r T j ď 2ρ

( . Here, we need ρ ă 1 4 so that the biggest transversal is still contained in 3 4 D 2 . Let us prove (HT1) and (HT2) for these transversals. This is what is done in the following statements. Lemma 5.2. Let a P E be a singularity as above. If C 5 is sufficiently small, then for any z P D 2 with }z} 1 P " r T j´1 , r T j ‰ , for some j P r1, N a pRqs, there exists z 1 P L z satisfying }z 1 } 1 " r T j and d P pz, z 1 q ď ℏ 2 .

Proof. Considering p X 3 if necessary, we may assume that }z} 1 " |z 1 | without loss of generality. For the flow φ z ptq starting from z in time t P R `, let

T " inf ␣ t P R `; }φ z ptq} 1 " r T j ( . It is clear that T ď ln r T j r T j´1
. Moreover, the path γ : r0, T s Ñ L z defined by γptq " φ z ptq satisfies }γptq} 1 ď r T j and }γpT q} 1 " r T j . Then, its Poincaré length is bounded by

ℓ P pγq " 2 ż T 0 }γ 1 ptq} ηpγptqq dt ď CT ˇˇln r T j ˇˇď C ˜ln r T j´1 ln r T j ´1¸" CC 5 ℏ 1 ´C5 ℏ ď ℏ 2 ,
by (3.2) and (3.4) and if C 5 is sufficiently small. □

Once we have reached the right norm, we need to turn around to a transversal angle.

Lemma 5.3. Let a P E be a singularity as above. If C 5 is sufficiently small, z P D 2 and j P 1, N a pRq are such that }z} 1 " r T j , then there exist k P 1, P , u P t1, 2u and a z 1 P L z X T a,j,k,u with d P pz, z 1 q ď ℏ 2 . Proof. Similarly to the previous lemma, we can assume that }z} 1 " |z 1 |. Note z 1 " r T j e iθ and let k P 1, P be such that ˇˇθ ´θT k ˇˇď C 5 ℏ. Consider the path γptq " φ z `it `θT k ´θ˘˘. We have γp0q " z and γp1q " ´rT j e iθ T k , z 1 2 ¯. By Grönwall Lemma, it is easy to see that |z 2 | ď 3 2 r T j . Thus, z 1 P T a,j,k,1 . On the other hand, if C 5 is sufficiently small,

ℓ P pγq ď C 1 ˇˇθ ´θT k ˇľ n r T j ˇˇ´C |θ ´θT k | ď ℏ 2 . □
For these singularities, the condition (HT1) is clear, and we need to prove (HT2).

Lemma 5.4. Let a P E be a singularity as above. There is a constant K 1 ą 0, independent on R such that for any j P 0, N a pRq , k P 1, P , u P t1, 2u, there are at most K 1 elements pj 1 , k 1 , u 1 q P 0, N a pRq ˆ 1, P ˆt1, 2u such that there exist x P T a,j,k,u , y P T a,j 1 ,k 1 ,u 1 X L x with d P px, yq ď 2h 1 .

Proof. Note that }x} 1 P " r T j , 3 2 r T j ‰ . On the other hand,

}y} 1 P " `rT j ˘1`Ch 1 , `3 2 r T j ˘1´Ch 1
ı by Grönwall Lemma. A direct computation implies that there is a bounded number of possible j 1 if j is given. The integers k 1 , u 1 are also in bounded quantity. □

We finish building the singular transversals, with a singularity a P E of Poincaré-Dulac type. We take the same radii r T j " pr sing pRqq p1´C 5 ℏq j , but this time for j P Z. We also keep the same angles θ T k , for k P 1, P . Consider the transversals

T a,j,k,1 " " pz 1 , z 2 q P D 2 » U a ; z 1 " r T j e iθ T k , |z 2 | ď 3 2 `rT j ˘m ˇˇln r T j ˇˇ* ; T a,j,k,2 " # pz 1 , z 2 q P D 2 » U a ; z 2 " `rT j ˘m ˇˇln r T j ˇˇe iθ T k , |z 1 | ď ˆ3 2 ˙1{m r T j + ;
for j P Z such that r sing pRq ď max `rT j , `rT j ˘m ˇˇln r T j ˇˇ˘a nd min `rT j , `rT j ˘m ˇˇln r T j ˇˇ˘ď 2ρ. We denote the maximal j by N a pRq and the minimal j by N 1 a pRq. For a unification purpose, denote by N 1 a pRq " 0 for the other singularities. Now, we need ρ so that 2 max `rT NapRq , `rT

NapRq ˘m ˇˇln r T NapRq ˇˇ˘ď 1.
That way, the biggest transversal is contained in 3 4 D 2 . The next results are analogous to Lemmas 5.2, 5.3 and 5.4. Lemma 5.5. Let a P E be as above and C 5 be small enough. If j P N 1 a pRq `1, N a pRq and z " pz 1 , z 2 q P D 2 are such that we are in one of the following configurations, then, there is z 1 " pz 1 1 , z 1 2 q P L z , with d P pz, z 1 q ď ℏ 2 and one of the following configurations. . We prove that t ě T by ensuring that |z 2 ptq| ě `rT j ˘m.

|z 2 ptq| ě |z 2 | ˆ1 ´|µtz m 1 | |z 2 | ˙˜r T j r T j´1 ¸2m ě `rT j ˘m ˇˇln r T j´1 ˇˇ˜1 ´ln `rT j {r T j´1 ln r T j´1 ˇˇ¸˜r T j r T j´1 ¸m .
Here, we have used both hypotheses of configuration [START_REF] Bacher | Heat diffusions on holomorphic foliations with non-degenerate singularities[END_REF]. Now, the right hand side is equal to `rT Lemma 5.6. Let a P E be as above. If C 5 is small enough and z P D 2 is in configuration (i) or (ii), then there are k P 1, P , j P t1, 2u and z 1 P L z X T a,j,k,u with d P pz, z 1 q ď ℏ 2 . Proof. In configuration (i), the proof is the same as for other singularities. Let us suppose we are in configuration (ii). Note z 2 " `rT j ˘m ˇˇln r T j ˇˇe iθ and k P 1, P with ˇˇθ ´θT k ˇˇď C 5 ℏ. We search a small t such that (5.1)

f ptq " ´1 `e´iθ µz m 1 `rT j ˘´m ˇˇln r T j ˇˇ´1 t ¯emt " e ipθ T k ´θq .
It is easy to see that |f 1 p0q| ě m ´|µ| ˇˇln r T j ˇˇ´1 ě 1 2 , shrinking again ρ if necessary. It is also clear that |f ptq ´f p0q ´tf 1 p0q| ď C |t| 2 for |t| ď 1 and some uniform constant C ą 0. Hence, f is injective on a small uniform disk and by Koebe 1 4 -Theorem, (5.1) admits a solution t " O pC 5 ℏq, if C 5 is small enough. We conclude the same as in Lemma 5.3. □ So, (HT1) is also true for this type of singularity. The following statement finishes the proof of Proposition 5.1, by showing (HT2) for Poincaré-Dulac singularities. Lemma 5.7. Let a P E be a singularity as above. There is a constant K 1 ą 0, independent on R such that for any j P 0, N a pRq , k P 1, P , u P t1, 2u, there are at most K 1 elements pj 1 , k 1 , u 1 q P 0, N a pRq ˆ 1, P ˆt1, 2u such that there exist x P T a,j,k,u , y P T a,j 1 ,k 1 ,u 1 X L x with d P px, yq ď 2h 1 .

Proof. The arguments are similar to those of Lemma 5.4, using that

`rT j ˘m ˇˇln r T j ˇˇP " C ´1 `rT j ˘m , C `rT j ˘1 2 ı . □
Actually, we correct a bit the movement that makes us end on a transversal to control monodromy processes below. For u P t1, 2u, we consider a vector field X u j , for j P t1, 2, 3u. For the linearizable case, we put

X 1 1 " B Bz 1 `λz 2 B Bz 2 , X 2 1 " λ ´1z 1 B Bz 1 `z2 B Bz 2 .
For the Poincaré-Dulac case, we just put X 1 2 " X 2 2 " X 2 . For the Briot-Bouquet case, we put X 1 3 " X 3 and X 2 3 " p X 3 . When the j P t1, 2, 3u is implicitly determined by a singularity a, we simply denote the vector field by X u . This enables to have the following. We use it below to obtain Lemma 7.8. Somehow, this is a way to deal with the movement to reach a transversal. Since it is not so standard (compared with what we do in Section 7), we need to control the behaviour in both coordinates. Hence, we need it not to be too neutral, and avoid turning around inside a transversal level t|z u | " r T j u. Lemma 5.8. Let z P 2ρU a be such that dpx, Eq ě r sing pRq. There are a transversal T a,j,k,u and flow paths γ v : r0, 1s Ñ C for X v , v P t1, 2u, such that φ 1 z pγ 1 ptqq " φ 2 z pγ 2 ptqq, where φ v z denotes the flow of X v , φ v z pγ v p1qq P T a,j,k,u , ℓ P pγq ď ℏ and ' In the linearizable case, there is a constant c ą 0 such that

|ℑpγ v p1qq| ď c |ℜpγ v p1qq| `O pC 5 ℏq , v P t1, 2u.
' In the Briot-Bouquet and Poincaré-Dulac cases, |ℑpγ v p1qq| " OpC 5 ℏq.

Proof. We change slightly the processes of Lemmas 5.2, 5.3, 5.5 and 5.6. Note that they already give us a flow path γ, with γp1q " t 1 `t2 , where t 1 is given by Lemma 5.2 or 5.5, and t 2 by Lemma 5.3 or 5.6. In any case, |t 2 | " O pC 5 ℏq so it is enough to show that ' We can choose |ℜpt 1 q| ě c |ℑpt 1 q| and |ℜpλt 1 q| ě c |ℑpλt 1 q| in the linearizable case. ' ℑpt v 1 q " OpC 5 ℏq, where t v 1 is the flow time corresponding to X v and Lemma 5.3 for the Briot-Bouquet or Poincaré-Dulac case. For the Poincaré-Dulac case, t 1 is real. For the linearizable case, we consider a complex number ω of modulus 1 such that ℜpωq ą 0 and ℜpλωq ‰ 0. For φ z pωtq and with notations of Lemma 5.2, T ď 1 ℜpωq ln

r T j r T j´1
. We obtain our result if C 5 is small enough. For the Briot-Bouquet case, our process is in real flow time for X 1 . We have

φ 1 z pγ 1 ptqq " ´z1 e γ 1 ptq , z 2 pγ 1 ptqq ¯" φ 2 z pγ 2 ptqq " ´z1 pγ 2 ptqq, z 2 e γ 2 ptq
¯.

Note r z 2 ptq " z 2 ptqe αt and p z 2 ptq " r z 2 ptq ´z2 . Locally, it is clear that

γ 2 ptq " ´αγ 1 ptq `ln ˆ1 `p z 2 pγ 1 ptqq z 2 ˙.
Since p z 1 2 ptq " r z 1 2 ptq, by Lemmas 3.7 and 3.8, we get |p z 1

2 ptq| ď C |z 1 | α |z 2 | 2 . It follows that |p z 2 ptq| " O `C5 ℏ |z 2 | 2 ˘and since C 5 ℏ is small, that |ℑ pγ 1 ptqq| " O pC 5 ℏq. □ 5.
2. Initial covering. Now, we want to build an initial covering of r T on which we control the orthogonal projection and the flow in hyperbolic time 3h 1 . This is the covering that we progessively refine by Lemma 4.9 in order to apply Proposition 4.5. As we did for the transversals, let us begin by stating what we wish for our covering. Proposition 5.9. There exists a constant C 6 ą 0 such that for R sufficiently large we have the following. For each i P I T , there exists a covering by disks V i of T i such that (HD1) If T i is a regular transversal, D P V i and x, y P 2D, the orthogonal projection Φ xy exists on ϕ x pD 3h 1 q and satisfies dpx 1 , Φ xy px 1 qq ď e ´2R . (HD2) If T i is a singular transversal, D P V i and x, y P 2DzU sing pRq, then the orthogonal projection Φ xy exists in a neighbourhood of x and satisfies dpx, Φ xy pxqq ď e ´2R . Moreover, x and Φ xy pxq are p3h 1 , e ´2R q-relatively close following the flow. (HD3) max iPI T card V i ď e C 6 R .

In the regular case, we can simply cover T r,Λ by Ce 4R disks of radius Be ´2R , for B sufficiently small. The difficulty is again the singular case and we address particularly different types of singularities. For a linearizable or Briot-Bouquet singularity, we consider the covering D of Subsection 3.5 of a transversal T a,j,k,u . More precisely, we consider only the disks of D that intersect T a,j,k,u .

For a Poincaré-Dulac singularity, we consider the same covering if u " 2. On the other hand, if u " 1, we consider a covering of T a,j,k,1 by O `e2C 4 R ˘disks of radius `rT j ˘m ˇˇln r T j ˇˇe ´C4 R . Note that ˇˇln r T j ˇˇe ´C4 R ď e ´2R because C 4 ą C 3 `2. Actually, we need something stronger and need to enlarge again C 4 .

Proof of Proposition 5.9. Points (HD1) and (HD3) are clear. We have to prove (HD2). For a transversal T i " T a,j,k,u , D P V i and x " pz 1 , z 2 q, y " pw 1 , w 2 q P 2DzU sing pRq, we have z u " w u . In any case, it is then clear that

}z´w} 1 }z} 1 " O `|ln }z} 1 | e ´C4 R ˘" O `e´2R ˘.
Thus, the orthogonal projection exists and y 1 " pw 1 1 , w 1 2 q " Φ xy pxq " φ y ptq, for some t " Op|ln }z} 1 | e ´C4 R q. To show that x and y 1 are p3h 1 , e ´2R q-relatively close following the flow, we have to consider different cases. Note v P t1, 2u such that v ‰ u, so that the coordinate on the transversal is z v .

D " D 0 for a linearizable, Briot-Bouquet or Poincaré-Dulac singularity with u " 2. Note that dpx, y 1 q ď Cr 0 . Since C 4 ą C 3 and |z u | ě pr sing pRqq m , it is easy to check that z u and w 1 u are in configuration (C2) and z v , w 1 v in configuration (C1) of Lemmas 3.5, 3.6 and 3.10. D " D nk with the same singularities. Here, we have

}z´w} 1 }z} 1 " O `e´C 4 R ˘and t is a O `e´C 4 R ˘. Since tz v "
0u is a separatrix, Grönwall Lemma implies that z v and w 1 v are in configuration (C2). For a linearizable or Briot-Bouquet singularity, this is the same for z u and w 1 u . For the Poincaré-Dulac case,

z 2 ´w1 2 " z 2 ´pz 2 `µtw m 1 q e mt . The condition |w 1 | m ď C |z 2 | |ln }z} 1 | ´1 and t " O `e´C 4 R ˘give us that z 2 and w 1 2 are in configuration (C2.2) if C 4 ą C 3 `2. A Poincaré-Dulac singularity with u " 1. Since t " O `|ln }z} 1 | e ´C4 R ˘, z 1 and w 1 1 are in configuration (C1.2) if C 4 ą 2C 3 `2. On the other hand, z 2 ´w1 2 " z 2 ´pw 2 `µtz m 1 q e mt . This time, |z 2 ´w2 | " O `|z 1 | m |ln }z} 1 | e ´C4 R ˘and again t " O `e´2R ˘. Hence, |z 2 ´w1 2 | is a O `|z 1 | m e ´2R ˘and z 2 , w 1
2 are in configuration (C2.1). □

HOLONOMY AND FLOW IN HYPERBOLIC TIME

6.1. Comparison of three motion processes. We need to control the behaviour of an orthogonal projection. Since we work on a universal cover, it will be very convenient to have processes that are invariant under homotopy and that is why we study holonomy. Moreover, this motion enables us to carry information step by step from far away transversals to the regular transversals where we want to build orthogonal projections.

Still, we need it to be well defined on an initial disk and a way to keep disks in the process. Here, estimating the flow is crucial to guarantee these technical elements. The end of the proof relies deeply on comparing these three motions. We need some preparation.

Lemma 6.1. Let a P E be some singularity of F . There exists a constant ε 3 ą 0 such that for any z 0 P 3 4 U a , there exists a regular flow box U » D ˆT with B pz 0 , ε

3 }z 0 } 1 q Ă U . Proof. If z P U a satisfies }z ´z0 } 1 ď 1 2C 0 C 1 }z 0 } 1 , then }Xpzq ´Xpz 0 q} 1 ď 1 4 }Xpz 0 q} 1 by (3.2) and (3.3). Write X " X 1 B Bz 1 `X2 B Bz 2 .
Without loss of generality, we can suppose that }Xpz 0 q} 1 " |X 1 pz 0 q|. Considering the exponential map of X X 1 , we conclude. □

This preparation enables us to ensure that the holonomy maps we consider are not only germs, but are defined on the whole disks of the initial covering. Proposition 6.2. Let T i , T j P r T, z P T i and λ : r0, 1s Ñ L z be such that λp0q " z, λp1q P T j and ℓ P pλq ă 2h 1 . Let D P V i be a disk of the initial covering containing z. If R is sufficiently large, the holonomy map Hol λ along λ from T i to T j is well defined on 2D.

If w P 2D, there exists a unique map r Φ zw : D 2h 1 Ñ L w such that near ζ P D 2h 1 , if z 1 " ϕ z pζq and w 1 " r Φ zw pζq, then r Φ zw " Φ z 1 w 1 ˝ϕz , where Φ z 1 w 1 is the orthogonal projection from L z 1 near z 1 onto L w 1 near w 1 ; and moreover r Φ zw " Φ zw ˝ϕz in a neighbourhood of 0. Finally, if w 2 " Hol λ pwq and ζ P D 2h 1 is such that ϕ z pζq " z 1 " λp1q, then r Φ zw " Φ z 1 w 2 ˝ϕz in a neighbourhood of ζ.

Proof. Since F is Brody-hyperbolic and h 1 is small, the conclusions are clear for T i a regular transversal. If T i is a singular transversal, then note that z and w are p3h 1 , e ´2R qrelatively close following the flow by Proposition 5.9. Considering a flow path γ for z representing λ and using Lemma 6.1, we can cut γ in parts where λptq and φ w pγptqq are in the same flow box to ensure that the holonomy is well defined. The existence of r Φ zw and its link with Φ z 1 w 2 can be addressed similarly using Lemma 2.6. □ 6.2. Image of disks by holonomy. Now that we know the holonomy maps are well defined on the disks of the covering, we need a method to keep disks to apply Lemma 4.9.

We need the following notion, which is close to one that may be found in [6, p. 617].

Definition 6.3. Let U be an open subset of C and σ ą 1. We call U σ-quasi-round if there exists a disk D with σ ´1D Ă U Ă σD.

Given a σ-quasi-round open set, we use the following to make it round.

Lemma 6.4.

There exists σ 0 ą 1 such that for σ P p1, σ 0 q and U σ-quasi-round, there exist four disks D 1 , D

D h 1 `ℏ Ă D h 1 Y p´1 ď k"0 D h 1 ´ℏpζ k q.
In particular, if ξ 0 , . . . , ξ p´1 P D h 1 `ℏ and satisfy d P pζ k , ξ k q ď ℏ, k P 0, p ´1 , then

D h 1 `ℏ Ă D h 1 Y p´1 ď k"0 D h 1 pξ k q.
Proof. The second statement is indeed a direct consequence of the first one. Note that by definition of our transversals, we have c ´1 ˇˇln r T j ˇˇď |ln }z} 1 | ď c ˇˇln r T j ˇˇ. Define C 8 " cC 7 . If h 1 is sufficiently small, Grönwall Lemma implies that the flow is defined on the whole disk C 8 h 1 ˇˇln r T j ˇˇD. Consider the paths λ k : r0, 1s Ñ L z defined by

λ k ptq " φ z ´C´1 8 th 1 ˇˇln r T j ˇˇe 2ikπ p ¯.
Take the lifting r λ k in D via ϕ z such that r λ k p0q " 0 and define ζ k " r λ k p1q. Thus, we have ℓpλ k q " C ´1 8 h 1 ˇˇln r T j ˇˇ, so ℓ P pλ k q ď h 1 by Lemma 7.1 and by definition of C 8 . Consequently,

ζ k P D h 1 . Let ξ P D h 1 `ℏ. If ξ P D h 1 ,
there is nothing to prove. If d P p0, ξq ě h 1 , define the radius r 1 " e h 1 ´1 e h 1 `1 and ξ 1 " r 1 ξ |ξ| . Let γ : r0, |ξ|s Ñ C be the flow path representing ξ and λptq " φ z ptγpr 1 qq, which is homotopic to the path φ z ˝γ r0,r 1 s . Hence, its lifting via ϕ z with r λp0q " 0 satisfies r λp1q " ξ 1 . By definition, we have ℓpλq " |γpr 1 q| and ℓ P pλq ě h 1 . Thus, |γpr 1 q| ě C ´1 8 h 1 ˇˇln r T j ˇˇand there exists r 1 1 P r0, r 1 s with |γpr 1

1 q| " C ´1 8 h 1 ˇˇln r T j ˇˇ. Note ξ 1 1 " r 1 1 ξ
|ξ| and the hyperbolic radius h 1 1 " ln

1`r 1 1 1´r 1 1
. Applying Lemma 7.1, we get

h 1 1 " ℓ P ´γ r0,r 1 1 s ¯ě C ´1 8 ˇˇln r T j ˇˇ´1 ℓ ´γ r0,r 1 1 s ¯ě C ´1 8 ˇˇln r T j ˇˇ´1 |γpr 1 1 q| " 3ℏ.
Hence, d P pξ, ξ 1 1 q ă h 1 ´2ℏ. It follows that it is enough to find k P 0, p ´1 with d P pξ 1 1 , ζ k q ď ℏ. It is clear that we can find k such that ˇˇγpr 1 1 q ´|γpr Denote by z k " φ k pzq, u 1 1 " γpr 1 1 q and t k " C ´1 8 h 1 ˇˇln r T j ˇˇe 2ikπ p . Consider the flow path which goes straight from 0 to u 1 1 and then travels the arc from u 1 1 to t k . It is homotopic to the straight line from 0 to t k . Thus, ϕ z pξ 1 1 q " φ z k pu 1 1 ´tk q. Grönwall Lemma implies that |ln }z k } 1 | ě 1 2 |ln }z} 1 | ě 1 2c ˇˇln r T j ˇˇif h 1 is sufficiently small. By Lemma 7.1,

d P pξ 1 1 , ζ k q ď π p C 7 C ´1 8 h 1 |ln }z k } 1 | ´1 ˇˇln r T j ˇˇď 2πh 1 p .
For p ě 2πh 1 ℏ " 6πC 2 8 , we have our result. □ Far from the singular set, we use a slightly different criterion.

Lemma 7.3. Note r 1 " e h 1 ´1 e h 1 `1 and ζ k " r 1 e 2ikπ p , for k P 0, p ´1 . If h 1 is sufficiently small,

D h 1 `ℏ Ă D h 1 Y p´1 ď k"0 D h 1 ´ℏpζ k q.
In particular, if ξ 0 , . . . , ξ p´1 P D h 1 `ℏ and satisfy d P pζ k , ξ k q ď ℏ, k P 0, p ´1 , then

D h 1 `ℏ Ă D h 1 Y p´1 ď k"0 D h 1 pξ k q.
Proof. Let ξ P D h 1 `ℏ. If d P p0, ξq ă h 1 , then ξ P D h 1 . Next, suppose that d P p0, ξq ě h 1 . Define ξ " r 1 |ξ| ξ so that d P pξ, ξ 1 q ă ℏ. Note that ℓ P pBD h 1 q " πpe h 1 ´e´h 1 q ď 4πh 1 if h 1 is sufficiently small. So, there exists k P 0, p ´1 with

d P pξ 1 , ζ k q ď 2πh 1 p ă ℏ. Since C 8 ą 1, d P pξ, ζ k q ă 2ℏ ď h 1 ´ℏ. □ 7.2.
Encoding the hyperbolic dynamics. Here, the notion we introduce is something that exists but is not precisely defined in [6, p. 619-623]. It encodes the transversals through which a leaf L x goes in time R, and ensures that we get a h 1 -dense subset.

Definition 7.4. For H P N, we define

A H " H ğ j"0 0, p ´1 j ,
where by convention 0, p ´1 0 " tHu. We see A H as a tree, the directed edges of which are pi 1 , . . . , i k q Ñ pi 1 , . . . , i k`1 q, for k P 0, H ´1 and i 1 , . . . , i k`1 P 0, p ´1 . A hyperbolic motion tree of deepness H is a map Θ : A H Ñ D with ΘpHq " 0 and for any vertex i 1 , . . . , i k P 0, p ´1 ,

D h 1 `ℏ pΘ pi 1 , . . . , i k qq Ă D h 1 pΘ pi 1 , . . . , i k qq Y p´1 ď i k`1 "0 D h 1 pΘ pi 1 , . . . , i k`1 qq ,
and Θpi 1 , . . . , i k`1 q P D h 1 `ℏpΘpi 1 , . . . , i k qq, i k`1 P 0, p ´1 . The cut-off definition tree of Θ is defined as the subset A Θ H Ă A H , made of elements pi 1 , . . . , i k q P A H , for k P 0, H such that for all j P 0, k , Θpi 1 , . . . , i j q P D 2h 1 `Hℏ . These trees are designed to get h 1 -covering subsets. Lemma 7.5. Let Θ : A H Ñ D be a hyperbolic motion tree of deepness H P N. Then,

D h 1 `Hℏ Ă ď SPA Θ H D h 1 pΘpSqq.
Proof. The proof is by induction on H. If H " 0, this is a definition. Suppose that the lemma is true for deepness up to H, let Θ be a hyperbolic motion tree of deepness H `1 and Ξ

" Θ A H . Note D h 1 pΘq " Y SPA Θ H`1 D h 1 pΘpSqq. Let ζ P D h 1 `pH`1qℏ . If ζ belongs to D h 1 `
Hℏ , then the induction hypothesis applies on Ξ and gives ζ P D h 1 pΘq. Next, suppose that d P p0, ζq ě h 1 `Hℏ, define r H " e h 1 `Hℏ ´1 e h 1 `Hℏ `1 , ζ " re iθ and ζ H " r H e iθ . The induction hypothesis on ζ H and the condition d P pζ, ζ H q ă ℏ give us ξ " Θpi 1 , . . . , i k q, with pi 1 , . . . , i k q P A Ξ H such that d P pζ, ξq ď h 1 `ℏ. By definition, there exists i k`1 P 0, p ´1 such that ξ 1 " Θpi 1 , . . . , i k`1 q satisfies d P pξ, ζq ă h 1 . Since pi 1 , . . . , i k q P A Ξ H and d P pξ 1 , 0q ď d P pζ, ξ 1 q `dP pζ, 0q ă 2h 1 `pH `1qℏ, pi 1 , . . . , i k`1 q P A Θ H`1 and the induction is finished. □ trees emerge from our work on the mesh of transversals.

Lemma 7.6. Let x P T and H " P R´h 1 ℏ T . If h 1 is small enough, then there is a hyperbolic motion tree Θ x : A H Ñ D such that for S P A Θx H , there exists T S P r T with ϕ x pΘ x pSqq P T S . Moreover, if ϕ x pSq P 2ρU a for S P A Θx H , then T S is a singular transversal T a,j,k,u . Proof. The proof is by induction on H. If T S is regular, then we apply Lemma 7.3 to first find ζ k for k P 0, p ´1 . If T S is singular, we find similar ζ k by Lemma 7.2. To find ξ k " Θ x pS ¨kq, we distinguish whether }ϕ x pζ k q} 1 ď 2ρ or not. In the first case, we apply Lemma 5.8. In the second case, we choose ξ k in a transversal such that }ϕ x pξ k q} 1 ą 2ρ. Note also that if d P p0, ζ k q ě 2h 1 `Hℏ, we can keep the ξ k " ζ k . In order to apply the lemmas, we need to check that }ϕ x pζ k q} 1 ě r sing pRq. If h 1 is sufficiently small, since we have d P p0, ζ k q ă 2h 1 `Hℏ ă R `2h 1 , we get this condition using Lemma 3.4. □

With the notations of the previous proof, we want to make the choice of ξ k in the singular case in some sense uniform on a disk of the initial covering. Let us introduce some notations. Let T i be a singular transversal, D P V i and z, w P 2D. Note z k " φ k pzq and ζ k,z the corresponding point in the disk that we obtain in Lemma 7.2. Suppose that z k P 2ρU a and follow the procedure of Lemma 5.8 to find a corresponding z 1 k P L z k and ξ k,z P D h 1 `ℏ, with z 1 k on a transversal and ϕ z pξ k,z q " z 1 k . Denote by r γ k,w its lifting via ϕ w such that r λ p1q k,w p0q " 0. We stop here with the strict analogy. Indeed, we wish to show that the construction for z also holds for w to have a uniform choice. Note w 1 k " Hol γ k,z pwq, which is well defined by Proposition 6.2. We want to find a path λ k,w ptq " φ w 2 k ptt w q, for t P r0, 1s and the notations of the proof of Proposition 6.5. Its endpoint is w 1 k by definition of t w . Since t w " Ope ´2R q and }z} 1 ´}w} 1 }z} 1

z ζ k,z z k ξ k,z z 1 k r γ p1q k,z r λ p1q k,z γ p1q k,z λ p1q k,z r γ p2q k,z r λ p2q k,z γ p2q k,z λ p2q k,z ϕ z ϕ z ϕ z 0 w ζ k,w w k ξ k,w " r λ p2q k,w p1q
w 1 k " Hol γ k,z pwq r γ p1q k,w r λ p1q k,w γ p1q k,w λ p1q k,w r γ p2q k,w r
" O `e´C 4 R ˘, the second part is of length smaller than ℏ 2 if R is sufficiently large. Moreover, using Lemma 7.1, we can compare the Poincaré length of the second part as a flow path for z k or w k in L z or L w . Choosing C 5 small enough, the four lemmas when we had not fixed ℏ show that (i) holds.

To prove (ii), note that λ k,w is the same flow path as the one for z, just completed by adding t w . Hence, we can cut λ k,z and λ k,w into parts on which they stay in the same flow box. This implies that Hol λ k,w " Hol λ k,z on 2D. Note that on the one hand γ k,z and λ k,z and on the other hand γ k,w and λ k,w have same starting and ending point in their lifting in D. Therefore, they are homotopic and we also have Hol γ k,z " Hol γ k,w on 2D. □

In practice, to determine a hyperbolic motion tree for a point x, we do not follow exactly the proof of Lemma 7.6. Instead, we fix a point in the cell (say the center), follow the proof for this point and apply Lemma 7.7 to all points in the same cell. Lemma 7.7 implies that we do not break any symmetry with this choice because we could have done the same with any point by (i), with the same holonomy map by (ii). We need to check the following. This plays a similar role to [6, Lemma 2.9] in our refinement process.

(3) Define J 1 D as the set of indices j P J D such that V H 1 i,j,D is a covering of D. Let J 1 " Y DPV 0 i J 1 D and V H 1 i,j " Y DPV 0 i V H 1 i,j,D . Let V H 1 `1 i be the covering of T i obtained by applying Lemma 4.9 to V H 1 i and `VH 1 i,j ˘jPJ 1 . Note that each V H 1 i,j

does not necessarily cover entirely T i , but we apply successively Lemma 4.9 on the subset each covers. II) If T i Ă ρ 2 U a , for some a P E, (1) For each D P V 0 i , fix one z P D. For k P 0, p ´1 , name by z k " φ k pzq and z 1 k P T k,D the point obtained by Lemma 5.8 if we have }z k } 1 ě r sing pRq. Consider the path γ k,D defined before Lemma 7.7 and note π k,D " Hol γ k,D , which is well defined on D. The holonomy map π ´1 k,D only depends on T i , T k,D and k, but not on D. Name J D " tk P 0, p ´1 ; }z k } 1 ě r sing pRqu.

(2) For all disk D 1 P V H 1 k,D such that π k,D pDq X D 1 ‰ H, π ´1 k,D pD 1 q is σ 1 -quasi-round and we cover it by four disks D 1 1 , . . . , D i,k ˘kP 0,p´1 . Note that V H 1 i,k covers at least the disks D for which }z k } 1 ě r sing pRq.

Lemma 8.1. This algorithm works well. That is, the peremptory assertions used to define it hold. Moreover, there exists a constant C 9 ą 0 such that max

iPI T card V H i ď e C 9 R .
Proof. Let us shortly list our assertions: in (I.1) and (II.1), the holonomy does not depend on x and x 1 or on D; in (I.1) and (II.1), the holonomy is well defined on D; in (I.2) and (II.2), the inverse images of D 1 are σ 1 -quasi-round; in (II.3), V H 1 i,k,D is a covering of D. Note that by Lemma 4.9 and by induction, we always have 2D 1 Ă 2D 0 , for some D 0 P V 0 i . By Propositions 6.2 and 6.5, it follows that the holonomies are well defined on 2D and that the inverse images of D 1 are σ 1 -quasi-round. The fact that the holonomy does not depend on x and x 1 in (I.1) follows from the following observation. Since in case (I), we have dpx, Eq ą c for some c ą 0 and all x P T i , we can cover T i by a finite number of flow boxes. Reducing h 1 if necessary, we can suppose that D 2h 1 is still contained in these flow boxes. The uniqueness of a point of a plaque belonging to a transversal gives us that the holonomy only depends on T i and T j . In (II.1), the similar result follows from Lemma 7.8. Finally, note that Lemmas 5.3 and 5.6 make us end far from the boundary of T k,D . Hence, π k,D is fully contained in T k,D and V H 1 i,k,D is a covering of D if R is sufficiently large. Therefore, the algorithm works well.

It remains to prove the control of the cardinality. Note K H 1 " max iPI T card V H 1 i . By Proposition 5.9, K 0 ď e C 6 R . Note that card J 1 ď K 1 , by (HT2). Since the holonomy maps only depend on the transversals, we have by construction card V H 1 i,j ď 4K H 1 and card V H 1 i,k ď 4K 1 K H 1 . By Lemma 4.9, we obtain for C " max `4 ˆ200 K 1 `1, 4K The construction of the covering V H i and of the hyperbolic motion tree clearly imply the following result. Lemma 8.2. Let T i P T be a regular transversal, D P V H i and x P 2D. Then, there exists a hyperbolic motion tree Θ x : A H Ñ D satisfying the conditions of Lemma 7.6 and the following. If S k " pi 1 , . . . , i k q P A Θx H , for j P 0, k ´1 , we denote by S j " pi 1 , . . . , i j q. Let ξ j " Θ x pS j q, γ j be the geodesic from ξ j to ξ j`1 , λ j " ϕ x ˝γj and λ S k " λ k´1 . . . λ 0 . Then, Hol λ S k is well defined on 2D, with image in 2D k , for some D k P V S k . This enables us to conclude the proof of Theorem 1.2 by checking our criterion. Proof. The cardinality condition is an immediate consequence of Lemma 8.1. Let D be in V H i with T i P T and x, y P D. Let Θ x : A H Ñ D be the hyperbolic motion tree of Lemma 8.2 and F " tΘ x pSq; S P A Θx H u. The set F is h 1 -dense by Lemma 7.5. For S P A Θx H , note x S " ϕ x pΘ x pSqq, y S " Hol λ S pyq, with the notations of Lemma 8.2. We build ψ by gluing all the r Φ x S y S of Proposition 6.2. If we can do so, Lemma 8.2, (HD1) and (HD2) imply points (2) and (3) of Proposition 4.5. By Proposition 6.2 and Lemma 8.2, these maps patch up well on a branch of Θ x . On the other hand, if given two S 1 and S 2 such that D h 1 pΘ x pS 1 qq X D h 1 pΘ x pS 2 qq ‰ H, we can build homotopic paths from x to the image of a point on the intersection because they have the same starting and ending point in D. The corresponding holonomies coincide as germs and coincide on 2D by analytic continuation. Hence, it is clear that r Φ x S 1 y S 1 and r Φ x S 2 y S 2 coincide. □

Lemma

  then z and w are ph, δq-relatively close following the flow. Proof. Similarly to Lemma 3.5, it is sufficient to prove the assertions concerning flow paths for z. Take also the same notations for ξ, γ. By the same arguments as in the linearizable case, we have |z 1 pγptqq ´w1 pγptqq| ď δ 4 }zpγptqq} 1 . Next, consider (3.5) |z 2 pγptqq ´w2 pγptqq| ď |z 2 ´w2 | e mℜpγptqq `|µγptq pz m 1 ´wm

Proposition 3 . 11 .

 311 r n ´rn´1 ˘, for n P 1, N and k P 1, N 1 . Define also the collection D " tD 0 u Y tD nk ; n P 1, N , k P 1, N 1 u. It is easy to see that D is a covering of the disk if R is sufficiently large. This is actually about the same covering as[6, p. 602]. Let h be sufficiently small and R sufficiently large. For D p1q , D p2q P D, let U " D p1q ˆDp2q and z, w P 2U X `1 2 D 2 zt0u ˘. If C 4 is well chosen,[START_REF] Bacher | Poincaré metric of holomorphic foliations with non-degenerate singularities[END_REF] If z and w belong to U sing pRq, then z and w are pR, εq-close;(2) If z or w does not belong to U sing pRq, then z and w are ph, e ´2R q-relatively close following the flow.Proof. Point (1) is a direct consequence of Lemma 3.4 and the definition of U sing pRq.

Proposition 4 . 1 (

 41 Dinh-Nguyên-Sibony [6, Proposition 4.1]). Denote by T " Y rPR T r . If hpTq ă 8, then hpF q ă 8.

1 2

 1 1 4ε and in hyperbolic time less than C ε , for some C ą 0. Poincaré-Dulac case. With the same notations, if |z 1 | ě |z 2 | 2 , then |z 1 | ě ε 2 2 and we argue similarly. If |z 2 | 2 ą |z 1 |, then 1 m |ln |z 2 || |µ| |z 1 | m ď 1 2 |z 2 |. In particular, φ z ptq escapes U a in positive real time 1 m |ln |z 2 || ď 1m |ln ε|. We conclude by the same observations. □ Let us recall a notion of[START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations I[END_REF] that clarifies our work on the orthogonal projection.

Definition 4 . 3 .

 43 Let R, δ ą 0 be such that δ ď e ´2R . Two points x, y P M zE are said to be pR, δq-conformally close if the following properties, and the same when exchanging the roles of x and y, are satisfied.(a) There exists a smooth function ψ : D R Ñ L y without critical points such that d Ly pψp0q, yq ď δ and d D R pψ, ϕ x q ď δ. (b) }dψ} 8 ď 2A for a constant A such that η ď A on M , and the norm is considered for the Poincaré metric at the source D R and g M on the goal L y . (c) Denote by y 1 " ψp0q. There exists a map Ψ : D R Ñ D such that Ψp0q " 0, ϕ y 1 ˝Ψ " ψ and the Beltrami coefficient µ Ψ satisfies }µ Ψ } C 1 ď δ. Recall that the Beltrami coefficient is defined to have BΨ Bt " µ ΨBΨBt . What interests us with this notion is the following lemma. What is hidden behind is that we correct Ψ into a close holomorphic map by solving a Beltrami equation, and then correct this holomorphic map into a rotation.

Lemma 4 . 4 (

 44 Dinh-Nguyên-Sibony [6, Proposition 3.

( 1 )

 1 Either |z 1 | P " r T j´1 , r T j ‰ and |z 2 | ď `rT j ˘m ˇˇln r T j ˇˇ, (2) Or |z 1 | ď r T j´1 and |z 2 | P

  (i) Either |z 1 | " r T j and |z 2 | ď `rT j ˘m ˇˇln r T j ˇˇ, (ii) Or |z 1 | ď r T j and |z 2 | " `rT j ˘m ˇˇln r T j ˇˇ. Proof. The situation is similar to Lemma 5.2, the proof of which we keep some notations. Consider the flow φ z ptq starting at z in positive real time. Note φ z ptq " pz 1 ptq, z 2 ptqq. Set T " inf ␣ t P R `; |z 1 ptq| " r T j or |z 2 ptq| " `rT j ˘m ˇˇln r T j ˇˇ( . Similarly, it is enough to show that T ď C ln r T j r T j´1 . Indeed, for t ď T , |z 2 ptq| ď `rT j ˘1 2 . In configuration (1), the same arguments still work. Let us suppose the setup of configuration (2). Denote by t " 2 ln

  the right radius found, we have to turn to the right angle.

  p1q k,z the geodesic from 0 to ζ k,z , r γ p2q k,z the geodesic from ζ k,z to ξ k,z and r γ k,z " r γ p2q k,z ¨r γ p1q k,z the concatenation of the two paths. For each of these, denote without tildas the projection on L z , that is, for ‹ P tp1q, p2q, Hu, γ ‹ k,z " ϕ z ˝r γ ‹ k,z . On the other hand, consider λ p1q k,z ptq " φ z `tC ´1 8 h 1 ˇˇln r T j ˇˇ˘, for t P r0, 1s. That way, we have λ p1q k,z p0q " z, λ p1q k,z p1q " z k . Since it is how we have built ζ k,z , γ p1q k,z and λ p1q k,z are homotopic. Consider then the lifting r λ p1q k,z of λ p1q k,z via ϕ z such that r λ p1q k,z p0q " 0. It follows that r λ p1q k,z p1q " ζ k,z . Finally, the procedure of Lemma 5.8 gives us a flow path and then a path λ p2q k,z on L z joining z k and z 1 k , by definition of ξ k,z . Its lifting r λ p2q k,z via ϕ z such that r λ p2q k,z p0q " ζ k,z clearly satisfies r λ p2q k,z p1q " ξ k,z . Let us also denote r λ k,z " r λ p2q k,z ¨r λ p1q k,z and λ k,z " λ p2q k,z ¨λp1q k,z the concatenation of these paths. We use analogous notations for w, with slight subtelty. To be more precise, let us denote by w k " φ k pwq, ζ k,w its preimage by ϕ w obtained by Lemma 7.2, r γ p1q k,w the geodesic joining 0 and ζ k,w , γ k,w " ϕ w ˝r γ p1q k,w , λ p1q k,w ptq " φ w `tC ´1 8 h 1 ˇˇln r T j ˇˇ˘a nd r λ p1q

p2q k,w joining w k and w 1 k

 1 with peculiar properties (see below). Once we have it, define r λ p2q k,w its lifting via ϕ w such that r λ p2q k,w p0q " ζ k,w and denote by ξ k,w its endpoint. Then, take analogous notations with r γ p2q k,w the geodesic joining ζ k,w with ξ k,w , γ p2q k,w " ϕ w ˝r γ p2q k,w and λ k,w , r λ k,w , γ k,w , r γ k,w the usual concatenations. See the diagramm below for a summary of all these notations.

  0

Lemma 7 . 7 .

 77 With the notations above, there exists a path λ p2q k,w with the following properties. (i) If C 5 is sufficiently small, ℓ P ´λp2q k,w ¯ď ℏ. (ii) Hol γ k,z " Hol γ k,w on 2D. Proof. Note δ k,z the flow path for z corresponding to λ k,z , that we cut in two parts δ pjq k,z : r0, 1s Ñ C, for j P t1, 2u corresponding to λ pjq k,z . Note that w k " φ w ´δp1q k,z p1q ¯. We build λ p2q k,w by concatenating the two following parts. (a) λ p2,1q k,w ptq " φ w k ´δp2q k,z ptq ¯, for t P r0, 1s. Note w 2 k its endpoint. (b) λ p2,2q

Proposition 8 . 3 .

 83 The covering `VH i ˘iPI T satisfies the hypotheses of Proposition 4.5.

  Eqs Ñ L y rkε 0 dpy, Eqs, Ly py, Φ xy pxqq ď kdpx, yq, (2) for x 1 , x 2 P L x rε 0 dpx, Eqs, d Ly pΦ xy px 1 q, Φ xy px 2 qq ď kd Lx px 1 , x 2 q.(3) Φ xy is smooth and in a finite set of charts,}Φxy ´id} 8 ď e K dpx, Φ xy pxqq, }Φ xy ´id} C 1 ď e K dpx, Φ xy pxqq dpx, Eq , }Φ xy ´id} C 2 ď e K dpx, Φ xy pxqq dpx, Eq 2 .

	satisfying
	(1) d

2.6 (

  [START_REF] Bacher | Poincaré metric of holomorphic foliations with non-degenerate singularities[END_REF] Lemma 3.1]). Consider X a holomorphic vector field on a neighbourhood of D2 with a non-degenerate singularity at the origin. Suppose that D 2 is endowed with the standard Hermitian metric on C 2 . Let x, y P 3 4 D 2 zt0u be such that Φ xy exists in the sense of Lemma 2.5. Denote by φ y the flow of X starting at y from an open neighbourhood of 0 in C to L y . Then, there exists t P C, with t " O `}x ´y} }x} ´1˘s uch that Φ xy pxq " φ y ptq.

  | α |r z 2 ptq| 2 . ptq " ´αr z 2 ptqz 1 ptq k z 2 ptqf pz 1 ptq, z 2 ptqq " ´αr z 2 ptq 2 z k 1 e pk´αqt f pz 1 ptq, z 2 ptqq. Now, since k ě α and 3 4 ě |z 1 ptq| " |z 1 | e ℜptq , we obtain our result. P pγq ď h, 1 2 |z 2 | ď |r z 2 pγptqq| ď 2 |z 2 |. Proof. Since h is small and by Lemma 3.3, |γptq| ď Ch |ln }z} 1 |. Integrating the inequality of Lemma 3.7 along a radius gives for t ‰ 0,

	Lemma 3.7. With the notations above, if |z 1 ptq| ď 3 4 ,
	|r z 1 2 ptq| ď α |z 1 Proof. By a simple computation, we get ˙k´α ˆ3 4
	(3.6)	r z 1 2

□

Lemma 3.8. If h ą 0 is sufficiently small, then for any z P 1 2 D 2 zt0u and any flow path γ such that ℓ

  s |z 1 pγpuqq ´w1 pγpuqq| . αe αℜptq ˇˇz 1 ptq k z 2 ptq 2 f pz 1 ptq, z 2 ptqq ´w1 ptq k w 2 ptq 2 f pw 1 ptq, w 2 ptqq ˇˇ. Name gpa, bq " a k b 2 f pa, bq. It is clear that there exists a constant C ą 0 such that ˇˇBg Ba ˇˇď C |a| k´1 |b| 2 and ˇˇBg Bb ˇˇď C |a| k |b|. Integrating along direction a and then b, we get |r z 1 2 ptq ´r w 1 2 ptq| ďCe αℜptq p|z 2 ptq| 2 max p|z 1 ptq| , |w 1 ptq|q k´1 |z 1 ptq ´w1 ptq| `|w 1 ptq| k max p|z 2 ptq| , |w 2 ptq|q |z 2 ptq ´w2 ptq|q, as in Lemma 3.7. Furthermore, by Grönwall Lemma, |z 2 ptq| ď |z 2 | 1 2 and by Lemma 3.8, |r z 2 ptq| ď 2 |z 2 | if h is sufficiently small. Thus, (3.7) |r z 1 2 ptq ´r w 1 2 ptq| ď C ´|z 2 | 3 2 |z 1 ptq ´w1 ptq| `|w 1 | α max p|z 2 | , |w 2 |q |r z 2 ptq ´r w 2 ptq| ¯. Applying the last inequality to a reparametrization r γ of γ such that |r γ 1 puq| " 1, integrating along r γ and applying Proposition 2.7, we obtain for β " C |w 1 | α max p|z 2 | , |w 2 |q, |r z 2 pγptqq ´r w 2 pγptqq| ď ˜|z 2 ´w2 | `C |z 2 | Now, since }w} 1 ď 2 }z} 1 , we have β |γptq| ď Ch |w 1 | α . Hence, if h is sufficiently small, e β|t| ď 2. For I " e β|γptq| ş |γptq| 0 e ´βs ds, we also deduce that I ď C |γptq|. Finally, it is quite clear that |z 2 | 2 are in configuration (C1). Now, suppose that z 2 and w 2 are in configuration (C2).

	Proof. Let us bound |r z 1 2 ptq ´r w 1 2 ptq| by (3.6). |r z 1 2 ptq ´r w 1 Lemmas 3.8 and 3.9, 2 ptq| " 3 2 sup γ ˇˇˇz 2 pγptqq ´w2 pγptqq z 2 pγptqq ˇˇˇ" ˇˇˇr z 2 pγptqq ´r w 2 pγptqq |z 1 p¨q ´w1 p¨q| r z 2 pγptqq ˇˇˇď 4 ˇˇˇ1 ´w2 ż |γptq| 0 z 2	e ´βs ds ¸eβ|γptq| .	By

ďC p|z 2 ptq| |r z 2 ptq| |z 1 ptq ´w1 ptq| `|w 1 | α max p|z 2 | , |w 2 |q |r z 2 ptq ´r w 2 ptq|q , if |t| ď Ch |ln }z} 1 | and h is sufficiently small. Here, we used first that φ z ptq, φ w ptq stay in 3 4 D 2 if h is sufficiently small, since ln }z} 1 and ln }w} 1 have bounded quotient; and the same trick 1 2 |γptq| ď Ch so we conclude by putting together all these observations. □ This enables us to prove a similar result to the other cases. Lemma 3.10. Let h, δ P p0, 1q be sufficiently small. For z, w P 1 2 D 2 z `1 2 U sing pRq ˘, if for each j P t1, 2u we are in one of the following configurations, (C1) |z j | , |w j | ď δr sing pRq 2 ,

(C2) w j , z j ‰ 0, ˇˇ1 ´zj w j ˇˇď δ 8 and ˇˇ1 ´wj z j ˇˇď δ 8 , then z and w are ph, δq-relatively close following the flow of r X 3 .

Proof. As usual, we only deal with flow paths for z. We also keep the same notations as Lemmas 3.5 and 3.6. By the same arguments as in the linearizable case, we still have |z 1 pγptqq ´w1 pγptqq| ď δ 4 }zpγptqq} 1 , and |z 2 pγptqq ´w2 pγptqq| ď δ }zpγptqq} 1 if z 2 and w

  2 , D 3 , D 4 with U Ă Y 4 k"1 D k and 2D k Ă 2U , for k P 1, 4 . Proof. Let D be a disk with σ ´1D Ă U Ă σD. By an affine transformation of C, one can suppose that D " D. Il is enough to build the D k satisfying 2D k Ă 2σ ´1D and σD Ă Y 4k"1 D k . Easy computations show that for σ ă σ 0 " Let z P 1 2 U a ztau, γ : r0, 1s Ñ C be a flow path for z and h 1 be sufficiently small. We denote by ℓ P pγq the Poincaré length of φ z ˝γ and ℓpγq the length of γ for the usual Hermitian metric of C. If ℓ P pγq ď h 1 , then there exists a constant C 7 ą 1 with C ´1 7 ℓpγq ď ℓ P pγq |ln }z} 1 | ď C 7 ℓpγq. and an integer p ą 0, for C 8 ą C 7 that our further computation specifies. For z P T a,j,k,u in a singular transversal, denote by These maps are standart flows. In what follows, we use criteria close to[START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF] Lemma 4.6]. Let z P T a,j,k,u be in a singular transversal. If p is sufficiently large, h 1 is sufficiently small and C 8 is well chosen, there exist ζ 0 , . . . , ζ p´1 P D h 1 with ϕ z pζ k q " φ k pzq for k P 0, p ´1 . Moreover,

	Lemma 7.1. Fix ℏ " 1 3C 2 8	h 1 φ k pzq " φ z	´C´1 8 h 1 ˇˇln r T j ˇˇe	2ikπ p ¯,	k P 0, p ´1 .
	Lemma 7.2.
	D k " D ´´4	? 3 3	´2¯σ ´1i k , ´2	´2? 3 3	¯σ´1 ¯, we have the result.	a ? 6 ´?2 « 1.02 and □

  1 4 using Lemma 6.4. We denote byV H 1 i,k,D " ␣ D 1 j ; j P 1, 4 , π k,D pDq X D 1 ‰ H ( and V H 1 i,k " Y DPV 0 i V H 1 i,k,D . (3) For each k P J D , V H 1 i,k,D is a covering of D. Note V H 1 `1 ithe covering of T i obtained by applying Lemma 4.9 to V H 1

i and `VH

1

Proof of the existence of an orthogonal projection.

  CK H 1 . By definition of H, we get K H ď C R{ℏ e C 6 R . □ 8.2.

		1 ˆ200 p`1	˘,
	card V H 1 `1 i	ď

Fix σ 1 P p0, σ 0 q. We want to prove the following. Note that this is where being in dimension 2 is crucial for the case of linearizable singularities [START_REF] Dinh | Entropy for hyperbolic Riemann surface laminations II[END_REF]. Indeed, for transversals of higher dimension, one would have polydisks of the form ∆ " r 1 D ˆ¨¨¨ˆr n D, with possibly very degenerate quotients r i r j and no refinement lemma would be available. Proposition 6.5. We keep the notations of Proposition 6.2. If h 1 is sufficiently small, R is sufficiently large and D 1 Ă 2D is a disk, then Hol λ pD 1 q is σ 1 -quasi-round.

Let us begin with the easiest cases. Proposition 6.5's beginning of proof. First, suppose that T i or T j is a regular transversal. Note that Hol λ is holomorphic without critical points. Hence, it is close to homotheties on small disks and we conclude. Here, we study a bounded number of maps, independently of R. Moreover, these arguments still work if we consider singular transversals that do not approach too much (independently of R) the singularities. Now, suppose that T i and T j are singular transversals, for a linearizable singularity. Let us introduce many notations, that we keep for the other types of singularities. Note z 0 " pz 0 1 , z 0 2 q the center of the disk D 1 , and z 1 " pz 1 1 , z 1 2 q " Hol λ pz 0 q. Let γ : r0, 1s Ñ L z 0 be a flow path corresponding to λ. Here, we can suppose that λ is a path on L z 0 , since we have a sequence of flow boxes that contain the image of λ and corresponding plaques for z 0 . Note w 0 " pw 0 1 , w 0 2 q P D 1 , w 1 " pw 1 1 , w 1 2 q " Hol λ pw 0 q, which is well defined by Proposition 6.2. Let w 1 1 " pw

2 q " φ w 0 pγp1qq. Finally, denote T i " T a,j 0 ,k 0 ,u 0 and T j " T a,j 1 ,k 1 ,u 1 . By definition, note that z 0 u 0 " w 0 u 0 and z 1 u 1 " w 1 u 1 . Let us go back to the linearizable case. For j P t1, 2u, λ 1 " 1 and λ 2 " λ, we have w 1 1 j ´z1 j " pw 0 j ´z0 j qe λ j γp1q . Hence, if u 0 " u 1 , Hol λ pD 1 q is actually a disk. On the other hand, if u 0 ‰ u 1 ,

Denote by t w " 1

. We keep this notation such that w 1 " φ w 1 1 pt w q in the other cases. Here, we have t w " 1

Finally, since u 0 ‰ u 1 , it is easy to see that we can not come from a separatrix disk, i.e. D ‰ D 0 . Therefore, ˇˇw

ˇˇď Ce ´C4 R and we conclude because w 0 u 1 runs through the disk D 1 of center z 0 u 1 . Next, consider a Poincaré-Dulac type singularity. We need to distinguish all four cases for starting and arrival transversals. u 0 " u 1 " 1. As in the linearizable case, w 1 1 " w 1 and w 1 2 ´z1

2 " pw 0 2 ´z0 2 qe mγp1q . It follows that Hol λ pD 1 q is a disk. u 0 " 2 and u 1 " 1. Here, we have t w " ln

and D ‰ D 0 . As before, it follows that

By the estimate on t w and since w 1 " φ w 1 1 pt w q, we get

because z 0 and w 0 are p3h 1 , e ´2R q-relatively close following the flow. Here, we used that since u 0 " 2, we still have

Putting together the last two equations, we conclude if R is sufficiently large by

Arguing as in Lemma 5.6, we get (6.1)

Hence, we conclude with the estimate

1 " e γp1q pz 0 1 ´w0 1 q. Moreover, the following bound and a similar for z 1m 1 ´w1 1m 1 show that (6.1) still holds.

Here, we have considered the case D ‰ D 0 but D " D 0 is even simpler. It follows that

We have our result for R sufficiently large. □ It leaves us with the Briot-Bouquet case, for which we still need some preparation. We need to ensharpen the estimates of Subsection 3.4. Let us reintroduce some of its notations. Consider the flow in r X 3 , for z P 1 2 D 2 , denote by φ z ptq " pz 1 ptq, z 2 ptqq, with z 1 ptq " z 1 e t . Denote by r z 2 ptq " z 2 ptqe αt and p z 2 ptq " r z 2 ptq ´z2 , as in Lemma 5.8. Lemma 6.6. Take the notations above for z 0 and w 0 in the proof of Proposition 6.5. If R is sufficiently large and h 1 is sufficiently small, then We just have to consider a Briot-Bouquet case and with our arguments in the regular case, we can suppose that }z 0 } 1 is sufficiently small (independently of R). Considering p X 3 if necessary, we can suppose that u 1 " 1. We have to distinguish whether u 0 " 1 or u 0 " 2.

Case u 0 " 1. In that case, t w " 0. It follows that

2 " e ´αγp1q `z0 2 ´w0 2 `p z 0 2 pγp1qq ´p w 0 2 pγp1qq ˘" e ´αγp1q `z0

For O p|z 0 2 |q ď σ 1 ´1, we have our result. Case u 0 " 2. As for other singularities, t w " ln

2 " e ´αγp1q pp z 0 2 pγp1qq ´p w 0 2 pγp1qqq. We get

2 " e ´αγp1q ˆp z 0 2 pγp1qq ´p w 0 2 pγp1qq ´αr z 1 2 p1 `z1

because z 0 2 and w 0 2 are p3h 1 , e ´2R q-relatively close following the flow. Here, we have denoted by r z 1 2 " r z 0 2 pγp1qq. By Lemma 6.6 and Lemma 3.8,

Choosing |z 0 1 | sufficiently small and R sufficiently large, we conclude. □

HYPERBOLIC MOTION TREES

7.1. Covering the Poincaré disk. We still face some serious problems. Whereas a transversal can only interact with a finite number of others by (HT2), it can do so in many ways. Indeed, a transversal that is close to the singularity has around e C 3 R monodromy paths of Poincaré length lower than 2h 1 . To tackle this, we consider some standard flow motion. We need a refinement of Lemma 3.3, the proof of which is the same.

Lemma 7.8. Let z, w P T a,j 0 ,k 0 ,u 0 and k P 0, p ´1 be such that z 1 k , w 1 k P T a,j 1 ,k 1 ,u 1 arrive on the same transversal, with the notations above. Here, we do not suppose that z, w are in the same disk, so w 1 k is indeed obtained by exactly the same process as z 1 k . Suppose that z 1 k , w 1 k P 2D P V a,j 1 ,k 1 ,u 1 belong to a same disk of the initial covering. Then, Hol ´1 λ k,z " Hol ´1 λ k,w on 2D, with λ k,z , λ k,w defined before.

Proof. Note that both holonomies are defined on 2D by Proposition 6.2. Let us introduce some notations. Denote by z 1 " Hol ´1 λ k,w pz 1 k q. Note δ z (resp. δ w ) the flow time for X u 0 such that z 1 k " φ z pδ z q (resp. w 1 k " φ w pδ w q). It is clear that z 1 " φ z pδ z ´δw `tq, with |t| " Ope ´2R q. Moreover, δ z and δ w can be written as a sum of two terms: δ z " t k `tz , δ w " t k `tw , where t k " C ´1 8 h 1 ˇˇln r T j ˇˇe 2ikπ p , and t z , t w are defined by Lemma 5.8. In the linearizable case, since z and z 1 belong to the same transversal, we have ℜpt z ´tw `tq " 0. It follows that |ℑpt z ´tw `tq| ă 2π and hence is 0. So, t z " t w ´t and the condition on t imply that we have the same holonomy, by the same argument as Proposition 6.2. For the Briot-Bouquet case or the Poincaré-Dulac case with u 0 " 1, we argue the same. Consider a Poincaré-Dulac singularity with u 0 " 2. Note z " pz 1 , z 2 q. It is enough to show that z 2 " pz 2 `µtz m 1 q e mt , with ℑptq " OpC 5 ℏq and C 5 ℏ small enough implies that t " 0. If ℜptq ě |ℑptq|, we have

Here, we have used that

T

. For H 1 P 0, H , we build a covering V H 1 i by disks that satisfies properties to ensure the existence of the orthogonal projection. What follows is the induction process, and contains some peremptory assertions to define the construction that we prove just after. Suppose that V H 1 i is already built. I) If T i X `M z Y aPE ρ 2 U a ˘‰ H, (1) For each D P V 0 i , we denote by J D the set of indices j P I T such that there exists x P D, x 1 P T j X L x with d P px, x 1 q ď 2h 1 . If h 1 is sufficiently small, the holonomy map along the geodesic from x to x 1 only depends on T i and T j and not on x and x 1 . We denote it by π ij , which is well defined on D.

(2) For all disk D 1 P V H 1 j such that π ij pDq X D 1 ‰ H, π ´1 ij pD 1 q is σ 1 -quasi-round, for σ 1 fixed below Lemma 6.4. We cover it by four disks D 1 1 , . . . , D