Marie Curie

Jingzhi Yan

Patrice Le Calvez

M François Béguin

M Alain Chenciner

M John Guaschi

M Patrice Le Calvez

M Frédéric

Le Roux

M Marco Mazzucchelli

M Shigenori Matsumoto

Utilisation de feuilletages transverse à l'étude d'homéomorphismes préservant l'aire de surfaces

Keywords: homéomorphismes de surfaces, feuilletage transverse, orbite périodique, ensemble de rotation local, torsion faible, théorème de Poincaré-Birkhoff homeomorphisms of surfaces, transverse foliation, periodic orbits, local rotation set, torsion-low isotopy, Poincaré-Birkhoff theorem

Cette thèse est dédiée à mes parents et mon fiancé.

J'aimerais tout d'abord remercier mon directeur de thèse Patrice Le Calvez pour toutes ses aides. Cette thèse n'aurait jamais vu le jour sans ses suggestions pertinentes, sa générosité à partager ses connaissances et idées mathématiques, sa patience pendant les discussions et ses annotations de plusieurs versions de cette thèse. C'est une grande chance et un honneur de pouvoir faire mes etudes et travaux de recherche en mathématique avec lui.

Je remercie ensuite François Béguin et Shigenori Matsumoto d'avoir accepté d'être rapporteurs de cette thèse. Leurs annotations m'ont été précieuses. Je remercie aussi Sylvain Crovisier et Frédéric Le Roux pour ses plusieurs commentaires et m'expliquer ses résultats récents. Je remercie sincèrement Alain Chenciner, John Guaschi et Marco Mazzucchelli de m'avoir fait l'honneur de faire partie du jury.

Mes remerciements vont également à toute l'équipe de l'Institute Mathématique de Jussieu-Paris Rive Gauche où j'ai effectué mon travail de recherche. Je remercie tous mes collègues et amis avec qui j'ai partagé mathématiques et bons moments.

Je tiens particulièrement à remercier Zhiying Wen pour ses encouragements. Enfin, je voudrais exprimer ma sincère gratitude à mes parents et mon fiancé pour leur soutien constant et le bonheur que l'on partage.

Table des matières

A time-dependent vector field (X t ) t∈R is called a Hamiltonian vector field if it is defined by the equation:

dH t = ω(X t , •),
where (M, ω) is a symplectic manifold, and H : R × M → R is a smooth function. The Hamiltonian vector field induces a flow (ϕ t ) t∈R on M , which is the solution of the following equation ∂ ∂t ϕ t (z) = X t (ϕ t (z)).

We say that a diffeomorphism f of M is a Hamiltonian diffeomorphism if it is the time-1 map of a Hamiltonian flow. Let us recall some classical problems in symplectic geometry about existence of fixed points or periodic points of Hamiltonian diffeomorphisms of compact symplectic manifolds, which correspond to the periodic solutions of the Hamiltonian equations. A famous conjecture corresponding to the lower bound of the number of fixed points was formulated in 1960s by Vladimir I. Arnold and was proved in different cases. In the case of tori, Arnold's conjecture was proved by Charles Conley and Eduard Zehnder [START_REF] Conley | The Birkhoff-Lewis fixed point theorem and a conjecture of V. I[END_REF]. The most important breakthrough was Andreas Floer's proof [START_REF] Floer | Symplectic fixed points and holomorphic spheres[END_REF] in the case of monotone symplectic manifold by introducing a notion of Floer homology. In the study of Arnold's conjecture, Conley conjectured the existence of infinitely many periodic points for Hamiltonian diffeomorphisms of tori. This conjecture was proved by Nancy Hingston [START_REF] Hingston | Subharmonic solutions of Hamiltonian equations on tori[END_REF], and was generalized to Hamiltonian diffeomorphisms of closed symplectically aspherical manifold by Viktor L. Ginzburg [START_REF] Viktor | The Conley conjecture[END_REF]. But in the non-degenerate case, where 1 is not an eigenvalue of the Jacobian matrix at every fixed point of the Hamiltonian diffeomorphism, Conley's conjecture has been proven much earlier (see [START_REF] Conley | A global fixed point theorem for symplectic maps and subharmonic solutions of Hamiltonian equations on tori[END_REF] for tori case, and [START_REF] Salamon | Morse theory for periodic solutions of Hamiltonian systems and the Maslov index[END_REF] for more general case). In the case where the symplectic manifold is a surface, the situation is a little bit special because we have particular tools (for example, the Poincaré-Birkhoff theorem), and Conley's conjecture has been proven earlier and is generalized to homeomorphism (see [START_REF] Franks | Periodic points of Hamiltonian surface diffeomorphisms[END_REF] and [START_REF] Le | Periodic orbits of Hamiltonian homeomorphisms of surfaces[END_REF]). Recall that the Poincaré-Birkhoff theorem states that every orientation and area preserving homeomorphism of an annulus that rotates the two boundaries in opposite directions has at least two fixed points. This theorem has a lot of Chapter 1. Introduction generalizations (see [START_REF] Birkhoff | An extension of Poincaré's last geometric theorem[END_REF], [START_REF] Carter | An improvement of the Poincaré-Birkhoff fixed point theorem[END_REF], [START_REF] Franks | Generalizations of the Poincaré-Birkhoff theorem[END_REF], and [START_REF] Guillou | Théorème de translation plane de Brouwer et généralisations du théorème de Poincaré-Birkhoff[END_REF] for example), which are efficient tools to ensures the existence of periodic points in the surface.

In particular, when the symplectic manifold is a closed surface with positive genus and the Hamiltonian diffeomorphism has no degenerate fixed point, Conley's conjecture can be easily deduced. Indeed, if f is a Hamiltonian diffeomorphism of a closed surface M with positive genus that has finitely many fixed points, John Franks [START_REF] Franks | Rotation vectors and fixed points of area preserving surface diffeomorphisms[END_REF] proved that f has a contractible fixed point z with Lefschetz index 1. Here contractible means that the trajectory of z along the Hamiltonian flow that induces f is homotopic to zero in M , and hence there is an identity isotopy of f that fixes z. If f does not have any degenerate fixed point, we lift f to the universal covering space M of M and get a homeomorphism f which fixes all the lifts of z. We take one lift z of z, and can blow-up M at z by replacing z with the unit circle S 1 . Moreover, D f ( z) induces a continuous extension of f to S 1 , and we get a homeomorphism of the annulus M \ {z} ∪ S 1 . We still denote by f this homeomorphism. By the non-degeneracy of z, the Poincaré's rotation number of f | S 1 is not equal to 0 ∈ R/Z. By a generalization of Poincaré-Birkhoff Theorem, one deduces that f has infinitely many periodic points that correspond to different periodic points of f .

In the degenerate case, the problem is much more difficult. As in the previous paragraph, there always exists a contractible fixed point z with Lefschetz index 1 for a Hamiltonian diffeomorphism of closed surface with finitely many fixed points. In chapter 3, we will study this case and prove that if z is an isolated degenerate contractible fixed point of f with Lefschetz index 1, then z is accumulated by periodic points of f . This gives a new explanation of Conley's Conjecture for Hamiltonian diffeomorphism of closed surface with positive genus.

More generally, we do not really need f to be a Hamiltonian diffeomorphism. Instead, our proof is purely topological. We can define the Hamiltonian homeomorphisms of surfaces to be area preserving homeomorphisms that are isotopic to the identity and have zero mean rotation vector. Here, area preserving means that f preserves a Borel measure without atom such that the measure of each open set is positive and that the measure of each compact set is finite. Due to Shigenori Matsumoto [START_REF] Matsumoto | Types of fixed points of index one of surface homeomorphisms[END_REF], Franks' result about the existence of a fixed point with Lefschez index 1 was generalized to the case of Hamiltonian homeomorphism. The rotation number can also be generalized in the case of homeomorphism to a notion of the local rotation set introduced by Frédéric Le Roux [START_REF] Le | L'ensemble de rotation autour d'un point fixe[END_REF]. By a similar but more complicate discussion, we can give a new explanation of Conley's conjecture for Hamiltonian homeomorphism of closed surface with positive genus.

More precisely, we will study the case where f : M → M is an area preserving homeomorphism of an oriented surface M , I is an isotopy from the identity to f , and z 0 is an isolated fixed point of f with a Lefschetz index 1 which is also fixed by I and satisfies ρ s (I, z 0 ) = {k}, where ρ s (I, z 0 ) is the local rotation set (see Section 1.2.1). In particular, when f is a diffeomorphism, the condition ρ s (I, z 0 ) = {k} means that Df (z 0 ) has a positive real eigenvalue. Under these assumptions, Le Roux [START_REF] Le | L'ensemble de rotation autour d'un point fixe[END_REF] conjectured that z 0 is accumulated by periodic orbits. We will approach his conjecture by proving that if the total area of M is finite, then z 0 is accumulated not only by periodic points, but also by periodic orbits in the measure sense. More precisely, the Dirac measure at z 0 is the limit in the weak-star topology of a sequence of invariant probability measures supported on periodic orbits.

The motives

Searching for transverse foliations

In the proof of the first problem, we will use transverse foliations. More precisely, suppose that there exist (non-singular) oriented topological foliations on M , and fix such a foliation F. We say that a path γ : [0, 1] → M is positively transverse to F if it locally meets transversely every leaf from the left to the right. We say that F is a transverse foliation of an isotopy I, if for every z ∈ M , there exists a path that is homotopic to the trajectory of z along I and is positively transverse to F. Denote by Fix(I) the set of fixed points of f that is also fixed along I. When Fix(I) is not empty, we call F a (singular) transverse foliation of I, if Fix(I) is the set of singularities of F and if the restriction of F to M \ Fix(I) is a transverse foliation of the restriction of I to M \ Fix(I).

Transverse foliations are fruitful tools in the study of homeomorphisms of surfaces. For example, one can prove the existence of periodic orbits in several cases [START_REF] Le | Une version feuilletée équivariante du théorème de translation de Brouwer[END_REF], [START_REF] Le | Periodic orbits of Hamiltonian homeomorphisms of surfaces[END_REF]; one can give precise descriptions of the dynamics of some homeomorphisms of the torus R 2 /Z 2 [Dáv13], [START_REF] Koropecki | Strictly toral dynamics[END_REF]; . . . It is a natural question whether we can get a more efficient tool by choosing suitable maximal identity isotopies.

Of course the existence of a transverse foliation prohibits the existence of fixed points of I but also contractible fixed points of f associated to I. Patrice Le Calvez [START_REF] Le | Une version feuilletée équivariante du théorème de translation de Brouwer[END_REF] proved that if f does not have any contractible fixed point associated to I, there exists a transverse foliation of I. Later, Olivier Jaulent [START_REF] Jaulent | Existence d'un feuilletage positivement transverseá un homéomorphisme de surface[END_REF] generalized this result to the case where there exist contractible fixed points, and obtained singular foliations. Denote by Fix(f ) the set of fixed points of f . Jaulent proved that there exist a closed subset X ⊂ Fix(f ) and an identity isotopy I X on M \ X such that f | M \X does not have any contractible fixed point associated to I X . It means that there exists a singular foliation on M whose set of singularities is X and whose restriction to M \ X is transverse to I X . Recently, François Béguin, Sylvain Crovisier, and Le Roux [BCLR] generalized Jaulent's result, and proved that there exists an identity isotopy I of f such that f | M \Fix(I) does not have any contractible fixed point associated to I| M \Fix(I) . Then, there exists a singular foliation on M whose set of singularities is the set of fixed points of I and whose restriction to M \ Fix(I) is transverse to I| M \Fix(I) . We call such an identity isotopy I a maximal identity isotopy.

The primary idea is to choose a maximal isotopy that fixes as many fixed points as possible. When f : M → M is an orientation preserving diffeomorphism, and I is an identity isotopy of f fixing z 0 , we can give a natural blow-up at z 0 and extend f continuously to the circle added as in the previous section. Moreover, we can define a blow-up rotation number ρ(I, z 0 ) ∈ R, that is a representative of the Poincaré's rotation number on the circle added (see Section 2.10). Moreover, if the diffeomorphism f is area preserving, and if there exists a fixed point z 0 ∈ Fix(I) such that |ρ(I, z 0 )| > 1 and that the connected component M 0 of M \ (Fix(I) \ {z 0 }) containing z 0 is not homeomorphic to a sphere or a plane, by lift f | M \(Fix(I)\{z 0 }) to the universal cover of M \ (Fix(I) \ {z 0 }), we can find another fixed point of f that is not a fixed point of I as a corollary of a generalized version of Poincaré-Birkhoff theorem. Moreover, if Fix(I) is finite, by a technical construction, one can find another identity isotopy that fixes Fix(I) \ {z 0 } and has no less (probably more) fixed points than I (see Section 4.2). Then, it is reasonable to think that a maximal identity isotopy I such that -1 ≤ ρ(I, z) ≤ 1 for all z ∈ Fix(I), fixes more fixed points than a usual one. In chapter 4, we will study a more general case, and prove the existence of such an isotopy as a corollary.

Chapter 1. Introduction

More precisely, we will study area preserving homeomorphisms of an oriented surface isotopic to the identity, and prove the existence of a special kind of maximal identity isotopies: the torsion-low maximal identity isotopies. In this case, we also have more information about its transverse foliation: we can deduce the local dynamics of a transverse foliation near any isolated singularity.

Generalized local Poincaré-Birkhoff theorem to the case where there exist fixed points at the boundary

We are also interested in the generalization of local Poincaré-Birkhoff theorem. More precisely, let f be an orientation preserving homeomorphism of the annulus T 1 × [0, +∞). Let C 0 = T 1 × {0}, C 1 be an essential loop in T 1 × (0, +∞) that projects injectively to the first factor, and C 2 = f (C 1 ). When f twists the boundaries and satisfies an intersection condition, it has been proven that f has a fixed point in the annulus between C 0 and C 1 (see [START_REF] Birkhoff | An extension of Poincaré's last geometric theorem[END_REF] and [START_REF] Guillou | Théorème de translation plane de Brouwer et généralisations du théorème de Poincaré-Birkhoff[END_REF] for example). But what would happen if there exists a fixed point of f in C 0 ? We will give an answer in Chapter 5. Moreover, this result can be used to give a more delicate description of the dynamics of some local homeomorphisms.

Statements of the precise results of the thesis 1.2.1 The results of Chapter 3

Let f : M → M be an area preserving homeomorphism of an oriented surface M , z 0 be an isolated fixed point of f , and I be an identity isotopy of f fixing z 0 . For two small neighborhoods V ⊂ U of z 0 , we will write

ρ U,V (I) = ∩ m≥1 ∪ n≥m {ρ n (z) : z ∈ ∩ 0≤j≤n f -j (U ) \ (V ∪ f -n (V ))},
where ρ n (z) is the average change of angular coordinate along the trajectory of z. We define the local rotation set to be ρ s (I, z 0 ) = ∩ U ∪ V ρ U,V (I).

Let us say that a contractible q-periodic orbit has type (p, q) associated to I at z 0 if its trajectory along I is homotopic to pΓ in M \ Fix(I), where Γ is the boundary of a sufficiently small Jordan domain containing z 0 . We will prove the following result which is the main result of Chapter 3.

Theorem 1.1. Suppose that the Lefschetz index i(f, z 0 ) is equal to 1, and that the rotation set ρ s (I, z 0 ) is reduced to an integer. If one of the following situations occurs, i) the surface M is a plane, f has exactly one fixed point and has a periodic orbit besides z 0 ; ii) the total area of M is finite, then z 0 is accumulated by periodic points. More precisely, the following property holds: P): There exists ε > 0, such that either for all irreducible p/q ∈ (k, k + ε), or for all irreducible p/q ∈ (k -ε, k), there exists a contractible periodic orbit O p/q of type (p, q), such that µ O p/q → δ z 0 as p/q → k, in the weak-star topology, where µ O p/q is the invariant probability measure supported on O p/q , The rotation set of a local isotopy at a degenerate fixed point of an orientation and area preserving diffeomorphism is reduced to an integer. So, given an area preserving diffeomorphism f isotopic to the identity on a surface M with finite area, if z 0 is a degenerate 1.2. Statements of the precise results of the thesis fixed point whose Lefschetz index is equal to 1, the assumptions of the previous theorem are satisfied, and hence z 0 is accumulated by contractible periodic points. Formally, we have the following corollary: Corollary 1.2. Let f be an area preserving diffeomorphism of an oriented surface M with finite total area, and z 0 be a degenerate isolated fixed point such that i(f, z 0 ) = 1. If f is isotopic to the identity by an isotopy I that fixes z 0 , then z 0 is accumulated by contractible periodic points. Moreover, the property P) holds.

Let f be a C 1 diffeomorphism of R 2 . A function g : R 2 → R of class C 2 is called a generating function of f if ∂ 2 12 g < 1, and

f (x, y) = (X, Y ) ⇔ X -x = ∂ 2 g(X, y), Y -y = -∂ 1 g(X, y).
We know that the previous diffeomorphism f is orientation and area preserving by a direct computation.

Generating functions are usual objects in symplectic geometry. We will give the following version of our result whose conditions are described by generating functions.

Corollary 1.3. Let f be an area preserving diffeomorphism of an oriented surface M with finite area that is isotopic to the identity by an isotopy fixing z 0 ∈ M . Suppose that in a neighborhood of z 0 , f is conjugate to a local diffeomorphism at 0 that is generated by a generating function g, that 0 is a local extremum of g, and that the Hessian matrix of g at 0 is degenerate. Then z 0 is accumulated by periodic points, and the property P) holds.

In particular, a Hamiltonian diffeomorphism f of the torus T 2 that is close to the identity in C 1 topology can be lifted to the plane R 2 , and the lifted diffeomorphism can be defined by a generating function g. If z 0 is a local extremum of g, and if the Hessian of g at z 0 is degenerate, we are in the case of the previous corollary, and the image of z 0 in T 2 is a fixed point of f that is accumulated by contractible periodic points.

We will also give a version of our result whose assumptions are described by symplectically degenerate extremum that will be defined in the section 3.2.2. Marco Mazzucchelli noticed that the existence of a symplectically degenerate extremum implies the existence of infinitely many other periodic points, and asked whether a symplectically degenerate extremum actually corresponds to a fixed point accumulated by periodic points, in his paper [START_REF] Mazzucchelli | Symplectically degenerate maxima via generating functions[END_REF] which gave a simpler proof of Conley's conjecture of tori. In chapter 3, we will give a positive answer of Mazzucchelli's question in the case where the torus is the surface T 2 . More precisely, we have the following result.

Theorem 1.4. Let f : T 2 → T 2 be a Hamiltonian diffeomorphism, and z 0 be a symplectically degenerate extremum. Then z 0 is accumulated by periodic points, and the property P) holds.

The results of Chapter 4

We write f : (W, 0) → (W , 0) for an orientation preserving homeomorphism between two neighborhoods W and W of 0 in R 2 such that f (0) = 0. We say that f is an orientation preserving local homeomorphism at 0. More generally, we write f : (W, z 0 ) → (W , z 0 ) for an orientation preserving local homeomorphism between two neighborhoods W and W of z 0 in any oriented surface M such that f (z 0 ) = z 0 .

Chapter 1. Introduction Let f : (W, z 0 ) → (W , z 0 ) be an orientation preserving local homeomorphism at z 0 . A local isotopy I of f is a continuous family of local homeomorphisms (f t ) t∈[0,1] fixing z 0 . Suppose that f is not conjugate to a contraction or an expansion. We can give a preorder on the space of local isotopies such that for two local isotopies I and I , one has I I if and only if there exists k ≥ 0 such that I is locally homotopic to J k z 0 I, where J z 0 = (R 2πt ) t∈[0,1] is the local isotopy of the identity such that each R 2πt is the counter-clockwise rotation through an angle 2πt about the center z 0 . We will give the formal definitions in Section 2.3.

Let F be a singular oriented foliation on M . We say that F is locally transverse to a local isotopy I = (f t ) t∈[0,1] at z 0 , if there exists a neighborhood U 0 of z 0 such that F| U 0 has exactly one singularity z 0 , and if for every sufficiently small neighborhood U of z 0 , there exists a neighborhood V ⊂ U such that for all z ∈ V \ {z 0 }, there exists a path in U \ {z 0 } that is homotopic in U \ {z 0 } to the trajectory t → f t (z) of z along I and is positively transverse to F.

We will generalize the definitions of "positive type" and "negative type" by Matsumoto [START_REF] Matsumoto | Types of fixed points of index one of surface homeomorphisms[END_REF]. We say that I has a positive (resp. negative) rotation type at z 0 if there exists a foliation F locally transverse to I such that z 0 is a sink (resp. source) of F. We say that I has a zero rotation type at z 0 if there exists a foliation F locally transverse to I such that z 0 is an isolated singularity of F and is neither a sink nor a source of F. Two local isotopies I and I have the same rotation type if they are equivalent.

When z 0 is an isolated fixed point of f , a local isotopy of f has at least one of the previous rotation types. It is possible that a local isotopy of f has two rotation types. Let us say that f is locally non-wandering if there exists a neighborhood of z 0 that contains neither the positive nor the negative orbit of any wandering open set. If we assume that f is area preserving (or more generally f is locally non-wandering), we will show in Section 4.1 that a local isotopy of f has exactly one of the three rotation types. We say that I is torsion-low at z 0 if -every local isotopy I > I has a positive rotation type; -every local isotopy I < I has a negative rotation type. Under the previous assumptions, we will prove in Section 4.1 the existence of a torsion-low local isotopy I of f . Formally, we have the following result: Theorem 1.5. Let f : (W, z 0 ) → (W , z 0 ) be an orientation preserving local homeomorphism at an isolated fixed point z 0 . If f is locally non-wandering, then -a local isotopy of f has exactly one of the three kinds of rotation types; -there exists a local isotopy I 0 that is torsion-low at z 0 . Moreover, I 0 has a zero rotation type if the Lefschetz index i(f, z 0 ) is different from 1, and has either a positive or a negative rotation type if the Lefschetz index i(f, z 0 ) is equal to 1.

A torsion-low local isotopy has the following properties:

Proposition 1.6. Let f : (W, z 0 ) → (W , z 0 ) be an orientation preserving and locally non-wandering homeomorphism at an isolated fixed point z 0 , and I be a local isotopy of f . If I is torsion-low at z 0 , then

ρ s (I, z 0 ) ⊂ [-1, 1].
In particular, if f can be blown-up at z 0 , the rotation set is reduced to a real number in

[-1, 1]. Moreover, if f is a diffeomorphism in a neighborhood of z 0 , the blow-up rotation number satisfies -1 ≤ ρ(I, z 0 ) ≤ 1,
and the inequalities are both strict when z 0 is not degenerate.

When z 0 is not an isolated fixed point and f is area preserving, we will generalize the definition of torsion-low isotopy by considering the local rotation set. We say a local isotopy I of an orientation and area preserving local homeomorphism f at a non-isolated fixed point z 0 is torsion-low at z 0 if ρ s (I, z 0 ) ∩ [-1, 1] = ∅. One may fail to find a torsionlow local isotopy in some particular cases. In fact, there exists an orientation and area preserving local homeomorphism whose local rotation set is reduced to ∞, and hence there does not exist any torsion-low isotopy of this local homeomorphism. We will give such an example in Section 4.3.

However, if f is an area preserving homeomorphism of an oriented surface M that is isotopic to the identity, we can find a maximal identity isotopy I that is torsion-low as a local isotopy at each fixed point of I. Formally, we will prove the following theorem in Section 4.5, which is the main result of the Chapter 4.

Theorem 1.7. Let f be an area preserving homeomorphism of an oriented surface M that is isotopic to the identity. Then, there exists a maximal identity isotopy I such that I is torsion-low at z for every z ∈ Fix(I).

Remark 1.8. The area preserving condition is necessary for the result of this theorem. Even if f has only finitely many fixed points and is area preserving near each fixed point, one may still fail to find a maximal isotopy I that is torsion-low at every z ∈ Fix(I). We will give such an example in Section 4.3.

We say that an identity isotopy is torsion-low if it is torsion-low at each of its fixed points. A torsion-low maximal isotopy gives more information than a usual one. We have the following three results: Proposition 1.9. Let f be an area preserving homeomorphism of an oriented surface M that is isotopic to the identity and has finitely many fixed points. Let n = max{#Fix(I) : I is an identity isotopy of f }.

Then, there exists a torsion-low identity isotopy of f with n fixed points. Proposition 1.10. Let f be an area preserving homeomorphism of an oriented surface M that is isotopic to the identity, I be a maximal identity isotopy that is torsion-low at z ∈ Fix(I), and F be a transverse foliation of I. If z is isolated in the set of singularities of F, then we have the following results:

-if z is an isolated fixed point of f such that i(f, z) = 1, then z is a saddle 1 of F and i(F, z) = i(f, z); -if z is an isolated fixed point such that i(f, z) = 1, or if z is not isolated in Fix(f ),
then z is a sink or a source of F.

Proposition 1.11. Let f be an area preserving diffeomorphism of an oriented surface M that is isotopic to the identity. Then, there exists a maximal identity isotopy I, such that for all z ∈ Fix(I), -1 ≤ ρ(I, z) ≤ 1.

Moreover, the inequalities are both strict when z is not degenerate.

The results of Chapter 5

Let f be an orientation preserving homeomorphism of the annulus T 1 × [0, +∞). An essential loop in the annulus is a loop that is not homotopic to zero. Let C 0 = T 1 × {0}, C 1 be an essential loop in T 1 × (0, +∞) that projects injectively to the first factor, and C 2 = f (C 1 ). We denote by A i the closed annulus bounded by C 0 and C i . Let π : R×[0, ∞) → T 1 ×[0, ∞) be the universal cover, and f be a lift of f . Let p 1 : R×[0, ∞) → R be the projection on the first factor. We denote by C i the pre-images of C i for i = 0, 1, 2. We denote by Fix * (f ) the set of fixed points of f lifted to fixed points of f . We will prove the following results in Chapter 5:

Theorem 1.12. If Fix * (f ) ∩ C 0 = {z 0 }, and if p 1 ( f ( z) -z)p 1 ( f ( z ) -z ) < 0 for all z ∈ C 0 \ π -1 (z 0 ) and z ∈ C 1 , then
( f ( z) -z)p 1 ( f ( z ) -z ) < 0 for all z ∈ C 1 ,
then there exists a fixed point of f in the interior of A 1 lifted to fixed points of f .

We will also give a description of the dynamics of some local homeomorphisms as a corollary of our main result. Suppose that f : (W, z 0 ) → (W, z 0 ) is an orientation preserving local homeomorphism at z 0 that is locally non-wandering, and that I is a local isotopy of f that has a positive (resp. negative) rotation type. We know that if f can be blown-up at z 0 , the rotation number is included in [0, ∞] (resp. [-∞, 0]) (see [START_REF] Le | L'ensemble de rotation autour d'un point fixe[END_REF] or Section 2.10). We know also that if z 0 is a non-accumulated indifferent point, then there exist small invariant continuums at z 0 . Choosing a sufficiently small invariant continuum K, we can blown-up f | W \K , and the rotation number ρ(I, K) is included in [0, ∞] (resp. [-∞, 0]) (see Section 2.9 and 2.10). We say that a homeomorphism h of the circle is right semi-stable (resp. left semi-stable) if its lift h to R satisfies h(x) ≥ x (resp. h(x) ≤ x) for all x ∈ R, and the equality holds at some points. We have the following result: Corollary 1.14. In the previous two cases, if ρ(I, z 0 ) = 0, the dynamics on the circle added when blowing-up is right semi-stable (resp. left semi-stable).

Remark 1.15. In the proof of the first statement of Theorem 1.5, we can also using this corollary instead of Guillou's generalization of Poincaré-Birkhoff theorem.

Organization of the thesis

Now we give a plan of thesis. In Chapter 2, we will introduce many definitions and will recall previous results that will be essential in the proofs of our results.

In Chapter 3, we will study the dynamics near an degenerate fixed point with Lefschetz index one. We will first prove our main result of this part: Theorem 1.1; then, we will study a particular case where f is a diffeomorphism, and will proof several version of our results whose conditions are described in different ways: Corollary 1.3 and Theorem 1.4.

In Chapter 4, we will give the existence of torsion-low maximal isotopies and describe its properties. We will first study the local rotation types at an isolated fixed point of an orientation preserving homeomorphism, and will prove Theorem 1.5 and Proposition 1.3. Organization of the thesis 19 1.6; then we will also study the dynamics in global sense and prove the existence of a global torsion-low maximal identity isotopy: Theorem 1.7 in two cases, and will study its properties: Proposition 1.9, 1.10 and 1.11; next, we will give some explicit examples to get the optimality of our results.

In Chapter 5, we will prove two locally generalization of Poincaré-Birkhoff theorem: Theorem 1.12 and Corollary 1.13. We will also using our result to give a description of the dynamics of some local homeomorphisms: Corollary 1.14.

Chapter 2 Preliminaries 2.1 A classification of isolated fixed points

In this section, we will give a classification of isolated fixed points. More details can be found in [START_REF] Le | Dynamique des homéomorphismes du plan au voisinage d'un point fixe[END_REF].

Let f : (W, z 0 ) → (W , z 0 ) be a local homeomorphism with an isolated fixed point z 0 . We say that z 0 is an accumulated point if every neighborhood of z 0 contains a periodic orbit besides z 0 . Otherwise, we say that z 0 is a non-accumulated point.

We define a Jordan domain to be a bounded domain whose boundary is a simple closed curve. We say that z 0 is indifferent if there exists a neighborhood V ⊂ V ⊂ W of z 0 such that for every Jordan domain U ⊂ V containing z 0 , the connected component of ∩ k∈Z f -k (U ) containing z 0 intersects the boundary of U .

We say that z 0 is dissipative if there exists a fundamental system {U α } α∈J of the neighborhood of z 0 such that each U α is a Jordan domain and that f (∂U α ) ∩ ∂U α = ∅.

We say that z 0 is a saddle point if it is neither indifferent nor dissipative. Note that if f is area preserving, an isolated fixed point of f is either an indifferent point or a saddle point.

Lefschetz index

Let f : (W, 0) → (W , 0) be an orientation preserving local homeomorphism at an isolated fixed point 0 ∈ R 2 . Denote by S 1 the unit circle. If C ⊂ W is a simple closed curve which contains no fixed point of f , then we can define the index i(f, C) of f along the curve C to be the Brouwer degree of the map

ϕ : S 1 → S 1 t → f (γ(t)) -γ(t) ||f (γ(t)) -γ(t)|| ,
where γ : S 1 → C is a parametrization compatible with the orientation, and • is the usual Euclidean norm. Let U be a Jordan domain containing 0 and contained in a sufficiently small neighborhood of 0. We define the Lefschetz index of f at 0 to be i(f, ∂U ), which is independent of the choice of U . We denote it by i(f, 0). More generally, if f : (W, z 0 ) → (W , z 0 ) is an orientation preserving local homeomorphism at a fixed point z 0 on a surface M , we can conjugate it topologically to an orientation preserving local homeomorphism g at 0 and define the Lefschetz index of f at z 0 to be i(g, 0), which is independent of the choice of the conjugation. We denote it by i(f, z 0 ).

Chapter 2. Preliminaries

Local isotopies and the index of local isotopies

Let f : (W, z 0 ) → (W , z 0 ) be an orientation preserving local homeomorphism at z 0 ∈ M . A local isotopy I of f at z 0 is a family of homeomorphisms (f t ) t∈[0,1] such that -every f t is a homeomorphism between the neighborhoods V t ⊂ W and V t ⊂ W of z 0 , and

f 0 = Id V 0 , f 1 = f | V 1 ; -for all t, one has f t (z 0 ) = z 0 ; -the sets {(z, t) ∈ M × [0, 1] : z ∈ V t } and {(z, t) ∈ M × [0, 1] : z ∈ V t } are both open in M × [0, 1]; -the maps (z, t) → f t (z) and (z, t) → f -1
t (z) are both continuous. Let us introduce the index of a local isotopy which was defined by Le Roux [START_REF] Le | L'ensemble de rotation autour d'un point fixe[END_REF] and Le Calvez [START_REF] Le | Pourquoi les points périodiques des homéomorphismes du plan tournent-ils autour de certains points fixes?[END_REF].

Let f : (W, 0) → (W , 0) be an orientation preserving local homeomorphism at 0 ∈ R 2 , and I = (f t ) i∈[0,1] be a local isotopy of f . We denote by D r the disk with radius r and centered at 0. Then the isotopy f t is well defined in the disk D r if r is sufficiently small. Let

π : R × (-∞, 0) → C \ {0} R 2 \ {0} (θ, y) → -ye i2πθ ,
be the universal covering projection, and

I = ( f t ) t∈[0,1] be the lift of I| Dr\{0} to R × (-r, 0) such that f 0 is the identity. Let γ : [0, 1] → R × (-r, 0) be a path from z ∈ R × (-r, 0) to z + (1, 0). The map t → f 1 ( γ(t)) -γ(t) || f 1 ( γ(t)) -γ(t)||
takes the same value at both 0 and 1, and hence descends to a continuous map ϕ : [0, 1]/ 0∼1 → S 1 . We define the index of the isotopy I at 0 to be the Brouwer degree of ϕ, which does not depend on the choice of γ when r is sufficiently small. We denote it by i(I, 0). For two local isotopies (resp. isotopies) I = (f t ) t∈[0,1] and I = (g t ) t∈[0,1] , we denote by I -1 the local isotopy (resp. isotopy) (f -1 t ) t∈[0,1] , by I I the local isotopy (resp. isotopy)

(ϕ t ) t∈[0,1] such that ϕ t = f 2t for t ∈ [0, 1 2 ], g 2t-1 • f for t ∈ [ 1 2 , 0],
and by I n the local isotopy (resp. isotopy)

I • • • I n times for every n ≥ 1.
Recall that π 1 (homeo 0 (R 2 , 0)) ∼ = Z, where homeo 0 (R 2 , 0) is the space of homeomorphism of R 2 fixing 0 and isotopic to the identity (see [START_REF] Mccarty | Homeotopy groups[END_REF] or [START_REF] Hamstrom | Homotopy groups of the space of homeomorphisms on a 2-manifold[END_REF]). Then, the space of homotopy classes of local isotopies of f is isomorphism to Z. Let J = (R 2πt ) t∈[0,1] be the isotopy such that each R 2πt is the counter-clockwise rotation through an angle 2πt about the center 0. We will give a preorder on the set of local isotopies of f such that I I if and only if I is homotopic to J q I for a q ≥ 0. Indeed, fix a sufficiently small r such that I and I are well defined on D r . Let I = ( f t ) t∈[0,1] be the lift of I | Dr\{0} to R × (-r, 0) such that f 0 is the identity. We write

I I if p 1 f 1 (θ, y) ≤ p 1 f 1 (θ, y) for all (θ, y) ∈ R × (-r, 0),

Brouwer plane translation theorem

23

where p 1 is the projection onto the first factor. Thus is a preorder, and I I and I I ⇐⇒ I is locally homotopic to I .

In this case, we will say that I and I are equivalent and write I ∼ I . More generally, we consider an orientation preserving local homeomorphism on an oriented surface. Let f : (W, z 0 ) → (W , z 0 ) be an orientation preserving local homeomorphism at a fixed point z 0 in a surface M . Let h : (U, z 0 ) → (U , 0) be a local homeomorphism. Then h

• I • h -1 = (h • f t • h -1 ) t∈[0,1]
is a local isotopy at 0, and we define the index of I at z 0 to be i(h • I • h -1 , 0), which is independent of the choice of h. We denote it by i(I, z 0 ). Similarly, we have a preorder on the set of local isotopies of f at z 0 .

The Lefschetz index at an isolated fixed point and the indices of the local isotopies are related. We have the following result:

Proposition 2.1. ([LC08][LR13]
) Let f : W → W be an orientation preserving homeomorphism with an isolated fixed point z. Then, we have the following results:

-if i(f, z) = 1, there exists a unique homotopy class of local isotopies such that i(I, z) = i(f, z) -1 for every local isotopy I in this class, and the indices of the other local isotopies are equal to 0; -if i(f, z) = 1, the indices of all the local isotopies are equal to 0.

Brouwer plane translation theorem

In this section, we will recall the Brouwer plane translation theorem. More details can be found in [START_REF] Brouwer | Beweis des ebenen translationssatzes[END_REF], [START_REF] Guillou | Théorème de translation plane de Brouwer et généralisations du théorème de Poincaré-Birkhoff[END_REF] and [START_REF] Franks | A new proof of the Brouwer plane translation theorem[END_REF].

Let f be an orientation preserving homeomorphism of R 2 . If f does not have any fixed point, the Brouwer plane translation theorem asserts that every z ∈ R 2 is contained in a translation domain for f , i.e. an open connected set of R 2 whose boundary is L ∪ f (L), where L is the image of a proper embedding of R in R 2 such that L separates f (L) and f -1 (L).

As an immediate corollary, one knows that if f is an orientation and area preserving homeomorphism of a plane 1 with finite area, it has at least one fixed point.

Transverse foliations and its index at an isolated end

In this section, we will introduce the index of a foliation at an isolated end. More details can be found in [START_REF] Le | Pourquoi les points périodiques des homéomorphismes du plan tournent-ils autour de certains points fixes?[END_REF].

Let M be an oriented surface and F be an oriented topological foliation on M . For every point z, there is a neighborhood V of z and a homeomorphism h : V → (0, 1) 2 preserving the orientation such that the images of the leaves of F| V are the vertical lines upward. We call V a trivialization neighborhood of z, and h a trivialization chart.

Let z 0 be an isolated end of M . We choose a small annulus U ⊂ M such that z 0 is an end of U . Let h : U → D \ {0} be a homeomorphism which sends z 0 to 0 and preserves the orientation. Let γ : T 1 → D \ {0} be a simple closed curve homotopic to ∂D. We can cover the curve by finite trivialization neighborhoods {V i } 1≤i≤n of the foliation F h , where F h is the image of F| U . For every z ∈ V i , we denote by φ + V i ,z the positive half leaf of the the leaf in V i containing z. Then we can construct a continuous map ψ from the curve γ to D \ {0}, such that ψ(z) ∈ φ + V i ,z for all 0 ≤ i ≤ n and for all z ∈ V i . We define the index i(F, z 0 ) of F at z 0 to be the Brouwer degree of the application

θ → ψ(γ(θ)) -γ(θ) ψ(γ(θ)) -γ(θ) ,
which depends neither on the choice of ψ, nor on the choice of V i , nor on the choice of γ, nor on the choice of h. We say that a path γ : [0, 1] → M is positively transverse to F, if for every t 0 ∈ [0, 1], there exists a trivialization neighborhood V of γ(t 0 ) and ε > 0 such that γ([t 0 -ε, t 0 + ε] ∩ [0, 1]) ⊂ V and h • γ| [t 0 -ε,t 0 +ε]∩[0,1] intersects the vertical lines from left to right, where h : V → (0, 1) 2 is the trivialization chart.

Let f be a homeomorphism on M isotopic to the identity, and I = (f t ) t∈[0,1] be an identity isotopy of f . We say that an oriented foliation F on M is a transverse foliation of I if for every z ∈ M , there is a path that is homotopic to the trajectory t → f t (z) of z along I and is positively transverse to F.

Suppose I = (f t ) t∈[0,1] is a local isotopy at z 0 , we say that F is locally transverse to I if for every sufficiently small neighborhood U of z 0 , there exists a neighborhood V ⊂ U such that for all z ∈ V \ {z 0 }, there exists a path in U \ {z 0 } that is homotopic to the trajectory t → f t (z) of z along I and is positively transverse to F. Proposition 2.2. [START_REF] Le | Pourquoi les points périodiques des homéomorphismes du plan tournent-ils autour de certains points fixes?[END_REF] Suppose that I is an identity isotopy on a surface M with an isolated end z and F is a transverse foliation of I. If M is not a plane, F is also locally transverse to the local isotopy I at z. Proposition 2.3. [START_REF] Le | Pourquoi les points périodiques des homéomorphismes du plan tournent-ils autour de certains points fixes?[END_REF] Let f : (W, z 0 ) → (W , z 0 ) be an orientation preserving local homeomorphism at an isolated fixed point z 0 , I be a local isotopy of f at z 0 , and F be a foliation that is locally transverse to I, then i(F, z 0 ) = i(I, z 0 ) + 1;

-i(f, z 0 ) = i(F, z 0 ) if i(F, z 0 ) = 1.

The existence of a transverse foliation and Jaulent's preorder

Let f be a homeomorphism of M isotopic to the identity, and I = (f t ) t∈[0,1] be an identity isotopy of f . A contractible fixed point z of f associated to I is a fixed point of f such that the trajectory of z along I, that is the path t → f t (z), is a loop homotopic to zero in M .

Theorem 2.4. [START_REF] Le | Une version feuilletée équivariante du théorème de translation de Brouwer[END_REF] If I = (f t ) t∈[0,1] is an identity isotopy of a homeomorphism f of M such that there exists no contractible fixed point of f associated to I, then there exists a transverse foliation F of I.

One can extend this result to the case where there exist contractible fixed points by defining the following preorder of Jaulent [START_REF] Jaulent | Existence d'un feuilletage positivement transverseá un homéomorphisme de surface[END_REF].

Let us denote by Fix(f ) the set of fixed points of f , and for every identity isotopy I = (f t ) t∈[0,1] of f , by Fix(I) = ∩ t∈[0,1] Fix(f t ) the set of fixed points of I. Let X be a closed subset of Fix(f ). We denote by (X, I X ) the couple that consists of a closed subset X ⊂ Fix(f ) such that f | M \X is isotopic to the identity and an identity isotopy

I X of f | M \X .
Let π X : M X → M \ X be the universal cover, and I X = ( f t ) t∈[0,1] be the identity isotopy that lifts I X . We say that f X = f 1 is the lift of f associated to I X . We say that a path γ : [0, 1] → M \ X from z to f (z) is associated to I X if there exists a path γ : [0, 1] → M X that is the lift of γ and satisfies f X ( γ(0)) = γ(1). We write (X,

I X ) (Y, I Y ), if -X ⊂ Y ⊂ (X ∪ π X (Fix( f X )))
; -all the paths in M \ Y associated to I Y are also associated to I X . The preorder is well defined. Moreover, if one has (X, I X ) (Y, I Y ) and (Y, I Y ) (X, I X ), then one knows X = Y and f X = f Y . In this case, we will write (X, I X ) ∼ (Y, I Y ). Jaulent proved that two couples (X, I X ) and (X, I X ) are always equivalent if M \ X is not homeomorphic to an annulus or a torus.

When the closed subset X ⊂ Fix(f ) is totally disconnected, an identity isotopy I X on M \ X can be extended to an identity isotopy on M that fixes every point in X; but when X is not totally disconnected, one may fail to find such an extension. A necessary condition for the existence of such an extension is that for every closed subset Y ⊂ X, there exists (Y, I Y ) that satisfies (Y, I Y ) (X, I X ). By a result (unpublished yet) due to Béguin, Crovisier and Le Roux, this condition is also sufficient to prove the existence of an identity isotopy I of f on M that fixes every point in X and satisfies (X, I X ) ∼ (X, I | M \X ) (here, we do not know whether I X can be extended). Formally, we denote by I the set of couples (X, I X ) such that for all closed subset Y ⊂ X, there exists (Y, I Y ) that satisfies (Y, I Y ) (X, I X ). Then, we have the following results: Proposition 2.5. [BCLR] 2 For (X, I X ) ∈ I, there exists an identity isotopy I of f on M that fixes every point in X and satisfies (X, I X ) ∼ (X, I | M \X ).

Proposition 2.6. [START_REF] Jaulent | Existence d'un feuilletage positivement transverseá un homéomorphisme de surface[END_REF] Let (X, I X ) ∈ I, and f X be the lift of f | M \X to M X associated to I X . If z ∈ Fix(f ) \ Fix(I) is a fixed point of f such that f X fixes all the points in π -1 X {z}, then there exists (X ∪ {z}, I X∪{z} ) ∈ I such that (X, I X ) (X ∪ {z}, I X∪{z} ). In particular, if (X, I X ) is maximal in (I, ), f | M \X has no contractible fixed point associated to I X . Proposition 2.7. [START_REF] Jaulent | Existence d'un feuilletage positivement transverseá un homéomorphisme de surface[END_REF] If {(X α , I Xα )} α∈J is a totally ordered chain in (I, ), then there exists (X ∞ , I X∞ ) ∈ I that is an upper bound of the this chain, where X ∞ = ∪ α∈J X α Theorem 2.8. [START_REF] Jaulent | Existence d'un feuilletage positivement transverseá un homéomorphisme de surface[END_REF] If I is an identity isotopy of a homeomorphism f on M , then there exists a maximal (X, I X ) ∈ I such that (Fix(I), I) (X, I X ). Moreover, f | M \X has no contractible fixed point associated to I X , and there exists a transverse foliation F of I X on M \ X. Remark 2.9. Here, we can also consider the previous foliation F to be a singular foliation on M whose singularities are the points in X. Moreover, F is also a transverse foliation of I X for all (X, I X ) ∼ (X, I X ). In particular, if I X is the restriction to M \ X of an identity isotopy I on M , we will say that F a transverse (singular) foliation of I .

Remark 2.10. In this article, we denote also by I X an identity isotopy on M that fixes all the points in X, when there is no ambiguity. Proposition 2.6, 2.7 and Theorem 2.8 are still valid if we replace the definition of I with the set of couples (X, I X ) of a closed subset X ⊂ Fix(f ) and an identity isotopy I X on M that fixes every point in X. When Fix(f ) is totally disconnected, it is obvious; when Fix(f ) is not totally disconnected, we should admit Proposition 2.5.

We call (Y, I Y ) ∈ I an extension of (X, I X ) if (X, I X ) (Y, I Y ); we call I an extension of (X, I X ) ∈ I if (X, I X ) (Fix(I ), I ); we call I an extension of I if I is an extension of (Fix(I), I). We say that I is a maximal extension if (Fix(I ), I ) is maximal in (I, ).

In particular, when M is a plane, Béguin, Crovisier and Le Roux proved the following result (unpublished yet).

Proposition 2.11. [BCLR] If f is an orientation preserving homeomorphism of the plane, and if X ⊂ Fix(f ) is a connected and closed subset, then there exists an identity isotopy I of f such that X ⊂ Fix(I).

Dynamics of an oriented foliation in a neighborhood of an isolated singularity

In this section, we consider singular foliations. A sink (resp. asource) of F is an isolated singular point of F such that there is a homeomorphism h : U → D which sends z 0 to 0 and sends the restricted foliation F| U \{z 0 } to the radial foliation of D \ {0} with the leaves toward (resp. backward) 0, where U is a neighborhood of z 0 and D is the unit disk. A petal of F is a closed topological disk whose boundary is the union of a leaf and a singularity. Let F 0 be the foliation on R 2 \ {0} whose leaves are the horizontal lines except the x-axis which is cut into two leaves. Let S 0 = {y ≥ 0 : x 2 + y 2 ≤ 1} be the half-disk. We call a closed topological disk S a hyperbolic sector if there exist -a closed set K ⊂ S such that K ∩ ∂S is reduced to a singularity z 0 and K \ {z 0 } is the union of the leaves of F that are contained in S, -and a continuous map φ : S → S 0 that maps K to 0 and the leaves of F| S\K to the leaves of F 0 | S 0 . Proposition 2.12. [START_REF] Le | L'ensemble de rotation autour d'un point fixe[END_REF] We have one of the following cases: i) (sink or source) there exists a neighborhood of z 0 that contains neither a closed leaf, nor a petal, nor a hyperbolic sector; ii) (cycle) every neighborhood of z 0 contains a closed leaf; iii) (petal) every neighborhood of z 0 contains a petal, and does not contain any hyperbolic sector; iv) (saddle) every neighborhood of z 0 contains a hyperbolic sector, and does not contain any petal; v) (mixed) every neighborhood of z 0 contains both a petal and a hyperbolic sector. Moreover, i(F, z 0 ) is equal to 1 in the first two cases, is strictly bigger than 1 in the petal case, and is strictly smaller than 1 in the saddle case. Remark 2.13. In particular, let f : (W, z 0 ) → (W , z 0 ) be an orientation preserving local homeomorphism at z 0 , I be a local isotopy of f , F be an oriented foliation that is locally transverse to I, and z 0 be an isolated singularity of F. If P is a petal in a small neighborhood of z 0 and φ is the leaf in ∂P , then φ ∪ {z 0 } divides M into two parts. We denote by L(φ) the one to the left and R(φ) the one to the right. By definition, P contains the positive orbit of R(φ) ∩ L(f (φ)) or the negative orbit of L(φ) ∩ R(f -1 (φ)). Then, a petal in a small neighborhood of z 0 contains the positive or the negative orbit of a wandering open set. So does the topological disk whose boundary is a closed leaf in a small neighborhood of z 0 . So, if f is area preserving, or if there exists a neighborhood of z 0 that contains neither the positive nor the negative orbit of any wandering open set, then z 0 is either a sink, a source, or a saddle of F.

In some particular cases, the local dynamics of a transverse foliation can be easily deduced. We have the following result: Proposition 2.14. [START_REF] Le | Pourquoi les points périodiques des homéomorphismes du plan tournent-ils autour de certains points fixes?[END_REF] Let I be a local isotopy at z 0 such that i(I, z 0 ) = 0. If I is another local isotopy at z 0 and if F is an oriented foliation that is locally transverse to I . Then, -the indices i(I , z 0 ) and i(I, z 0 ) are equal if

I ∼ I; -z 0 is a sink of F if I > I; -z 0 is a source of F if I < I.

The local rotation type of a local isotopy

In this section, suppose that f : (W, z 0 ) → (W , z 0 ) is an orientation and area preserving local homeomorphism at an isolated fixed point z 0 , and that I is a local isotopy of f . We say that I has a positive rotation type (resp. negative rotation type) if there exists a locally transverse foliation F of I such that z 0 is a sink (resp. source) of F. Matsumoto [START_REF] Matsumoto | Types of fixed points of index one of surface homeomorphisms[END_REF] proved the following result: Proposition 2.15. [START_REF] Matsumoto | Types of fixed points of index one of surface homeomorphisms[END_REF] If i(f, z 0 ) is equal to 1, I has only one of the two kinds of rotation types.

Remark 2.16. By considering the index of foliation, one deduces the following corollary: if F and F are two locally transverse foliations of I, and if 0 is a sink (resp. source) of F, then 0 is a sink (resp. a source) of F .

Prime-ends compactification and rotation number

In this section, we first recall some facts and definitions from Carathéodory's primeends theory, and then give the definition of the prime-ends rotation number. More details can be found in [START_REF] Milnor | Dynamics in one complex variable[END_REF] and [START_REF] Koropecki | Prime ends rotation numbers and periodic points[END_REF].

Let U R 2 be a simply connected domain, then there exists a natural compactification of U by adding a circle, that can be defined in different ways. One explanation is the following: we can identify R 2 with C and consider a conformal diffeomorphism

h : U → D,
where D is the unit disk. We endow U S 1 with the topology of the pre-image of the natural topology of D by the application

h : U S 1 → D,
Chapter 2. Preliminaries whose restriction is h on U and the identity on S 1 .

Any arc in U which lands at a point z of ∂U corresponds, under h, to an arc in D which lands at a point of S 1 , and arcs which land at distinct points of ∂U necessarily correspond to arcs which land at distinct points of S 1 . We define an end-cut to be the image of a simple arc γ : [0, 1) → U with a limit point in ∂U . Its image by h has a limit point in S 1 . We say that two end-cuts are equivalent if their images have the same limit point in S 1 . We say that a point z ∈ ∂U is accessible if there is an end-cut that lands at z. Then the set of points of S 1 that are limit points of an end-cut is dense in S 1 , and accessible points of ∂U are dense in ∂U . We define a cross-cut by the image of a simple arc γ : (0, 1) → U which extends to an arc γ : [0, 1] → U joining two points of ∂U and such that each of the two components of U \ γ has a boundary point in ∂U \ γ.

Let f be an orientation preserving homeomorphism of U . We can extend f to a homeomorphism of the prime-ends compactification U S 1 , and denote it by f . In fact, for a point z ∈ S 1 which is a limit point of an end-cut γ, we can naturally define f (z) to be the limit point of f • γ. Then we can define the prime-ends rotation number ρ(f, U ) ∈ T 1 = R/Z to be the Poincaré's rotation number of f | S 1 . In particular, if f fixes every point in ∂U , ρ(f, U ) = 0.

Let K ⊂ R 2 be a continuum, and U K be the unbounded component of R 2 \ K. Then, U K is an annulus and becomes a simply connected domain of the Riemann sphere if we identify R 2 with C and add a point at infinity. The prime-ends compactification also gives us a compactification of the end of U K corresponding to K by adding the circle of prime-ends. We can define end-cuts and cross-cuts similarly.

Let f : (W, 0) → (W , 0) be an orientation preserving local homeomorphism at 0 ∈ R 2 , and K ⊂ W be an invariant continuum containing 0. Similarly, we can naturally extend

f | U K ∩W to a homeomorphism f K : U K ∩ W ∪ S 1 → U K ∩ W ∪ S 1
, and define the rotation number ρ(f, K) ∈ R/Z to be the Poincaré's rotation number of f K | S 1 .

Furthermore, if I = (f t ) t∈[0,1] is a local isotopy of f at 0, we consider the universal covering projections

π : R × (-∞, 0) → C \ {0} R 2 \ {0} (θ, y) → -ye i2πθ and π : R → S 1 θ → e i2πθ . Let U K = π -1 (U K ), W = π -1 (W ), and W = π -1 (W ). Let π K : U K R → U K T 1
be the map such that π K = π in U K and π K = π on R. We endow the topology on

U K R such that π K is a universal cover. Let I = ( f t ) t∈[0,1] be the lift of (f t | V \{0} ) t∈[0,1] such that f 0 is the identity, where V is a small neighborhood of 0. Let f : W → W be the lift of f | W \{0} such that f = f 1 in π -1 (V ), we call it the lift of f associated to I. Let f K : ( W ∩ U K ) R → ( W ∩ U K ) R be the lift of f K such that f K = f in W ∩ U K , we call
it the lift of f K associated to I. We define the rotation number ρ(I, K) = lim n→∞ f n K (θ) -θ n which is a real number that does not depend on the choice of θ. We know that ρ(I, K) is a representative of ρ(f, K) in R.

We have the following property:

2.10. The local rotation set 29 Proposition 2.17. [START_REF] Le | Dynamique des homéomorphismes du plan au voisinage d'un point fixe[END_REF] Let f : (W, 0) → (W , 0) be an orientation preserving local homeomorphism at a non-accumulated indifferent point 0. Let U ⊂ U ⊂ W be a Jordan domain such that U does not contain any periodic orbit except 0, and that for all V ⊂ U , the connected component of ∩ n∈Z f -n (V ) containing 0 intersects the boundary of V . Let K 0 be the connected component of ∩ n∈Z f -n (U ) containing 0. Then for every local isotopy I of f , and for every invariant continuum K ⊂ U containing 0, one has ρ(I, K) = ρ(I, K 0 ).

This proposition implies that if f : (W, 0) → (W , 0) is an orientation preserving local homeomorphism at a non-accumulated indifferent point 0, we can define the rotation number ρ(I, 0) for every local isotopy I of f at 0, by writing ρ(I, 0) = ρ(I, K) where K is a non-trivial invariant continuum sufficiently close to 0.

More generally, if f : (W, z 0 ) → (W , z 0 ) is an orientation preserving local homeomorphism at a non-accumulated indifferent point z 0 ∈ M , we can conjugate it to a local homeomorphism at 0, and get the previous definitions and results similarly.

The local rotation set

In this section, we will give a definition of the local rotation set and will describe the relations between the rotation set and the rotation number. More details can be found in [START_REF] Le | L'ensemble de rotation autour d'un point fixe[END_REF].

Let f : (W, 0) → (W , 0) be an orientation preserving local homeomorphism at 0 ∈ R 2 , and I = (f t ) t∈[0,1] be a local isotopy of f . Given two neighborhoods V ⊂ U of 0 and an integer n ≥ 1, we define

E(U, V, n) = {z ∈ U : z / ∈ V, f n (z) / ∈ V, f i (z) ∈ U for all 1 ≤ i ≤ n}.
We define the rotation set of I relative to U and V by

ρ U,V (I) = ∩ m≥1 ∪ n≥m {ρ n (z), z ∈ E(U, V, n)} ⊂ [-∞, ∞],
where ρ n (z) is the average change of angular coordinate along the trajectory of z. More precisely, let

π : R × (-∞, 0) → C \ {0} R 2 \ {0} (θ, y) → -ye i2πθ
be the universal covering projection, f : π -1 (W ) → π -1 (W ) be the lift of f associated to I, and p 1 : R × (-∞, 0) → R be the projection onto the first factor. We define

ρ n (z) = p 1 ( f n ( z) -z) n ,
where z is any point in π -1 {z}.

We define the local rotation set of I to be

ρ s (I, 0) = ∩ U ∪ V ρ U,V (I) ⊂ [-∞, ∞],
where V ⊂ U ⊂ W are neighborhoods of 0. We say that f can be blown-up at 0 if there exists an orientation preserving homeomorphism Φ : R 2 \ {0} → T 1 × (-∞, 0), such that Φf Φ -1 can be extended continuously to T 1 × {0}. We denote this extension by h. Suppose that f is not conjugate the contraction z → z 2 or the expansion z → 2z. We define the blow-up rotation number ρ(f, 0) of f at Chapter 2. Preliminaries 0 to be the Poincaré rotation number of h|

T 1 . Let I = (f t ) t∈[0,1] be a local isotopy of f , ( h t ) be the natural lift of Φ| T 1 ×(0,r) • f t | Dr\{0} • Φ -1 | T 1 ×(0,r)
, where D r is a sufficiently small disk with radius r and centered at 0, and h be the lift of h such that h = h 1 in a neighborhood of R × {0}. We define the blow-up rotation number ρ(I, 0) of I at 0 to be the rotation number of h| T 1 associated to the lift h| R×{0} , which is a representative of ρ(f, 0) on R. J-M Gambaudo, Le Calvez and E. Pécou [START_REF] Gambaudo | Une généralisation d'un théorème de Naishul[END_REF] proved that neither ρ(f, 0) nor ρ(I, 0) depend on the choice of Φ, which generalizes a previous result of Naȋshul [START_REF] Naȋshul | Topological invariants of analytic and area-preserving mappings and their application to analytic differential equations in C 2 and CP 2[END_REF].

In particular, if f is a diffeomorphism, f can be blown-up at 0 and the extension of f on

T 1 is induced by the map v → Df (0)v Df (0)v
on the space of unit tangent vectors. More generally, if f : (W, z 0 ) → (W , z 0 ) is an orientation preserving local homeomorphism at z 0 that is not conjugate to the contraction or the expansion, we can give the previous definitions for f by conjugate it to an orientation preserving local homeomorphism at 0 ∈ R 2 .

The local rotation set can be empty. However, due to Le Roux [LR08], we know that the rotation set is not empty if f is area preserving. More precisely, we have the following result:

Proposition 2.18. [START_REF] Le | L'ensemble de rotation autour d'un point fixe[END_REF] Let f : (W, z 0 ) → (W , z 0 ) be an orientation preserving local homeomorphism at z 0 , and

I = (f t ) t∈[0,1] be a local isotopy of f . Then ρ s (I, z 0 ) is empty if and only if f is conjugate to one of the following maps -the contraction z → z 2 ; -the expansion z → 2z; -a holomorphic function z → e i2π p q z(1 + z qr ) with q, r ∈ N and p ∈ Z.
Remark 2.19. In the three cases, f can be blown-up at z 0 . But ρ(f, z 0 ) is defined only in the third case. More precisely, ρ(f, z 0 ) is equal to p q + Z. Moreover, if I is conjugate to z → z i2πt p q (1 + tz qr ), then ρ(I, z 0 ) is equal to p q .

We say that z is a contractible fixed point of f associated to the local isotopy

I = (f t ) t∈[0,1] if the trajectory t → f t (z) of z along I is a loop homotopic to zero in W \ {z 0 }.
We say that f satisfies the local intersection condition, if there exists a neighborhood of z 0 that does not contain any simple closed curve which is the boundary of a Jordan domain containing z 0 and does not intersect its image by f . In particular, if f is area preserving or if there exists a neighborhood of z 0 that contains neither the positive nor the negative orbit of any wandering open set, f satisfies the local intersection condition.

The local rotation set satisfies the following properties:

Proposition 2.20. [START_REF] Le | L'ensemble de rotation autour d'un point fixe[END_REF] Let f : (W, z 0 ) → (W , z 0 ) be an orientation preserving local homeomorphism at z 0 , and I be a local isotopy of f at z 0 . One has the following results: i) for all integer p, q, ρ s (J p z 0 I q , z 0 ) = qρ s (I, z 0 ) + p; ii) if z 0 is accumulated by contractible fixed points of f associated to I, then 0 ∈ ρ s (I, z 0 ); iii) if f satisfies the local intersection condition and if 0 is an interior point of the convex hull of ρ s (I, z 0 ), then z 0 is accumulated by contractible fixed points of f associated to I; iv) if I has a positive (resp. negative) rotation type, then ρ s (I,

z 0 ) ⊂ [0, +∞] (resp. ρ s (I, z 0 ) ⊂ [-∞, 0]); 2.11. Some generalizations of Poincaré-Birkhoff theorem v) if ρ s (I, z 0 ) is a non-empty set that is contained in (0, +∞] (resp. [-∞, 0)), then I has a positive (resp.negative) rotation type; vi) if ρ s (I, z 0 ) is a non-empty set that is contained in [0, ∞] (resp. [-∞, 0]
) and is not reduced to 0, and if z 0 is not accumulated by contractible fixed points of f associated to I, then I has a positive (resp. negative) rotation type; vii) if f can be blown-up at z 0 , and if ρ s (I, z 0 ) is not empty, then ρ s (I, z 0 ) is reduced to the single real number ρ(I, z 0 ). viii) if z 0 is a non-accumulated indifferent point, ρ s (I, z 0 ) is reduced to ρ(I, z 0 ) (the rotation number defined in Section 2.9).

Remark 2.21. When f satisfies the local intersection condition, one can deduce that ρ s (I, z 0 ) is a closed interval as a corollary of the assertion i), ii), iii) of the proposition.

In general case, due to an unpublished work of Jonathan Conejeros, ρ s (I, z 0 ) is always a closed interval.

Remark 2.22. Le Roux also gives several criteria implying that f can be blown-up at z 0 . The one we need in this article is due to Béguin, Crovisier and Le Roux [START_REF] Le | L'ensemble de rotation autour d'un point fixe[END_REF]:

If there exists an arc γ at z 0 whose germ is disjoint with the germs of f n (γ) for all n = 0, then f can be blown-up at z 0 .

In particular, if there exists a leaf γ + from z 0 and a leaf γ -toward z 0 (we are in this case if z 0 is a petal, a saddle, or a mixed singularity of F), we can choose a sector U as in the picture. Let V be a small neighborhood of z 0 . There exists a neighborhood W ⊂ V

γ - γ + z0 f (γ -) f (γ + ) U V W of z 0 such that f (U ∩ W ) ⊂ (Int(U ) ∩ V ) ∪ {z 0 }.
So, the germs at z 0 of f n (γ + ) are pairwise disjoint, and hence f can be blown-up at z 0 . Moreover, ρ(I, z 0 ) is equal to 0 in this case.

Le Roux also studied the dynamics near a non-accumulated saddle point, and proved the following result: Proposition 2.23. [START_REF] Le | L'ensemble de rotation autour d'un point fixe[END_REF] If z 0 is a non-accumulated saddle point, then f can be blownup at z 0 and ρ s (I, z 0 ) is reduced to a rational number. Moreover, if i(f, z 0 ) is equal to 1, this rational number is not an integer.

Some generalizations of Poincaré-Birkhoff theorem

In this section, we will introduce several generalizations of Poincaré-Birkhoff theorem. An essential loop in the annulus is a loop that is not homotopic to zero.

We first consider the homeomorphisms of closed annuli. Let f be a homeomorphism of T 1 × [0, 1] isotopic to the identity, I = (f t ) t∈[0,1] be an identity isotopy of f . Let π : R × [0, 1] → T 1 × R be the universal cover, I = ( f t ) t∈[0,1] be the identity isotopy that Chapter 2. Preliminaries lifts I, f = f 1 be the lift of f associated to I, and p 1 : R 2 → R be the projection on the first factor. The limits

lim n→∞ p 1 • f n (x, 0) -x n and lim n→∞ p 1 • f n (x, 1) -x n
exists for all x ∈ R, and do not depend on the choice of x. We define the rotation number of f on each boundary to be the corresponding limits. We define the rotation number of a point z ∈ T 1 × [0, 1] associated to I to be the limit

lim n→+∞ p 1 ( f n ( z) -z) n ,
if this limit exists. We say that f satisfies the intersection property if f • γ intersects γ, for every simple essential loop γ ⊂ T 1 × (0, 1). We have the following generalizations of Poincaré-Birkhoff theorem:

Proposition 2.24. [START_REF] Birkhoff | An extension of Poincaré's last geometric theorem[END_REF] Let f be a homeomorphism of T 1 ×[0, 1] isotopic to the identity and satisfying the intersection property. If the rotation number of f on the two boundaries are different, then there exists a q-periodic orbit of rotation number p/q for all irreducible rational p/q ∈ (ρ 1 , ρ 2 ), where ρ 1 and ρ 2 are the rotation numbers of f on the boundaries.

We also consider homeomorphisms of open annuli. Let f : T 1 × R → T 1 × R be a homeomorphism isotopic to the identity, and I = (f t ) t∈[0,1] be an identity isotopy of f . Let π : R × R → T 1 × R be the universal cover, I = ( f t ) t∈[0,1] be the identity isotopy that lifts I, f = f 1 be the lift of f associated to I, and p 1 : R 2 → R be the projection on the first factor. Similarly, we define the rotation number of a recurrent point z ∈ T 1 × R associated to I to be the limit

lim n→+∞ p 1 ( f n ( z) -z) n ∈ [-∞, ∞],
if this limit exists. We say that f satisfies the intersection property if f • γ intersects γ, for every simple essential loop γ ⊂ T 1 × R. Then, we have the following generalization of Poincaré-Birkhoff theorem: ]). Let f : T 1 ×R → T 1 ×R be a homeomorphism isotopic to the identity and satisfying the intersection property. If there exist two recurrent points of rotation numbers ρ 1 , ρ 2 ∈ [-∞, +∞] respectively such that ρ 1 < ρ 2 , then there exists a q-periodic orbit of rotation number p/q for all irreducible rational p/q ∈ (ρ 1 , ρ 2 ).

Proposition 2.25 ([Fra88], [ LC05
Remark 2.26. The result is also true for area preserving homeomorphisms of the closed or half closed annulus by considering a symmetry.

We will also need the following generalization due to Lucien Guillou, in which case the boundary of the annulus is not fixed:

Proposition 2.27. [Gui94] Let f : T 1 × [-a, a] → T 1 × [-b, b],
where 0 < a < b, be an embedding homotopic to the inclusion, and

f : R × [-a, a] → R × [-b, b] be the lift of f . If f does

not have any fixed point, and if f satisfies

(p 1 ( f (x, a)) -x)(p 1 ( f (x , -a)) -x ) < 0, for all x, x ∈ R, then there exists an essential loop γ in T 1 × [-a, a] such that f (γ) ∩ γ = ∅.
2.12. Topologically monotone periodic orbits for annulus homeomorphisms 33

Topologically monotone periodic orbits for annulus homeomorphisms

In this section, we will recall the braid type of a periodic orbit and the existence of the topologically monotone periodic orbits for annulus homeomorphisms under some conditions. More details can be found in [START_REF] Boyland | Rotation sets and monotone periodic orbits for annulus homeomorphisms[END_REF].

Denote by A the closed annulus T 1 × [0, 1]. Let f be a homeomorphism of A that preserves the orientation and each boundary circle, and f be a lift of f to the universal cover A = R × [0, 1]. Given z ∈ A, we define its rotation number under f as

ρ( z, f ) = lim n→∞ p 1 ( f n ( z)) -p 1 ( z) n ,
if this limit exists, where p 1 is the projection onto the first factor. We define the rotation set of f to be

ρ( f ) = {ρ( z, f ), z ∈ A}.
In particular, if I is an identity isotopy of f and f is the lift of f associated to I, this definition of the rotation number coincides with the definition of the rotation number in Section 2.11. Fix a copy of the closed annulus minus n interior points, and denote it by A n . Let G n be the group of isotopy classes of orientation preserving homeomorphism of A n . If O is an n-periodic orbit of f in the interior of A, then there is an orientation preserving homeomorphism h : Let p/q be an irreducible positive rational, and T p/q be the homeomorphism of A defined by (x, y) → (x + p/q, y). It descends to a homeomorphism T p/q of A. We denote by α p/q the braid type bt(O, T p/q ), where O is any periodic orbit of T p/q . We say that a q-periodic orbit O of f is a (p, q)-periodic orbit if ρ( z, f ) = p/q for any z in the lift of O. We say that a (p, q)-periodic orbit O is topologically monotone if bt(O, f ) = α p/q . We define the Farey interval I(p/q) of p/q to be the closed interval [max{r/s : r/s < p/q, 0 < s < q, and (r, s) = 1}, min{r/s : r/s > p/q, 0 < s < q, and (r, s) = 1}].

A \ O → A n . Philip Boyland defined the braid type bt(O, f ) to be the conjugacy class of [h • f | A\O • h -1 ] in G n , this isotopy class is independent of the choice of h. If O is an n-periodic orbit of f contained
In particular, the Farey interval of 1/q is equal to [0, 1/(q -1)].

Boyland proved the following result:

Proposition 2.28 [START_REF] Boyland | Rotation sets and monotone periodic orbits for annulus homeomorphisms[END_REF]). If f is an orientation and boundary preserving homeomorphism of the closed annulus, and p/q ∈ ρ( f ) is an irreducible positive rational, then f has a (p, q)-topologically monotone periodic orbit. If f has a (p, q)-orbit that is not topologically monotone, then I(p/q) ⊂ ρ( f ).

Annulus covering projection

Let M be an oriented surface, X 0 ⊂ M be a closed set, and

z 0 ∈ M \ X 0 . Denote by M 0 the connected component of M \ X 0 containing z 0 . Let V ⊂ U ⊂ M 0 be two small Jordan domains containing z 0 . Write U = U \ {z 0 } and V = V \ {z 0 }. Fix z 1 ∈ V . Let γ ⊂ V be a simple loop at z 1 such that the homotopic class [γ] of γ in V generates π 1 ( V , z 1 ). Let i : U → M 0 \ {z 0 } be the inclusion, then i * π 1 ( U , z 1 ) is a subgroup of π 1 (M 0 \{z 0 }, z 1 ).
Then, there exists a covering projection π : ( M , z 1 ) → (M 0 \{z 0 }, z 1 ) such that π * π 1 ( M , z 1 ) = i * π 1 ( U , z 1 ) by Theorem 2.13 in [START_REF] Spanier | Algebraic topology[END_REF]. Moreover, the fundamental group of M is isomorphic to Z, so M is an annulus.

Let U be the component of π -1 ( U ) containing z 1 . Then π * π 1 ( U , z 1 ) = π 1 ( U , z 1 ) and the restriction of π to U is a homeomorphism between U and U by Theorem 2.9 in [START_REF] Spanier | Algebraic topology[END_REF]. Consider the ideal-ends compactification of M , and denote by the end in U . Then π| U can be extended continuously to a homeomorphism between U ∪ { } and U . We denote it by h.

If f is an orientation preserving homeomorphism of M 0 , and z 0 is a fixed point of f . By choosing sufficiently small V , we can suppose that f (V ) ⊂ U . We know that

(f • π) * π 1 ( M , z 1 ) = i * π 1 ( U , f (z 1 )) = π * π 1 ( M , h -1 (f (z 1 ))), then we deduce by Theorem 2.5 of [Spa66] that there is a lift f of f to M that sends z 1 to h -1 (f (z 1 )). This map f is a homeomorphism because f * π 1 ( M , z 1 ) = π 1 ( M , h -1 (f (z 1 )) (see Corollary 2.7 in [Spa66]).
Moreover, f can be extend continuously to a homeomorphism of M ∪ { } that fixes .

In particular, if f is isotopic to the identity, and if [START_REF] Spanier | Algebraic topology[END_REF]. We have indeed lifted I to an identity isotopy I = ( f t ) t∈[0,1] . Moreover, f t can be extended continuously to a homeomorphism of M ∪ { } that fixes , and we get an isotopy on M ∪ { } that fixes . We still denote by f t the homeomorphism of M ∪ { } and by I the identity isotopy on M ∪ { } when there is no ambiguity. We call I the natural lift of I to M ∪ { }, and f = f 1 the lift of f to M ∪ { } associated to I.

I = (f t ) t∈[0,1] is an identity isotopy of f fixing z 0 , then there exists a lift f • (•) : I × M → M of the continuous map (t, z) → f t (π( z)) such that f 0 is equal to the identity, because π is a covering projection. Moreover, by choosing V small enough, we know that f t | V is conjugate to f t | V for t ∈ [0, 1], where V is the component of π -1 ( V ) containing z 1 . Then ( f t ) * π 1 ( M , z 1 ) = π 1 ( M , h -1 (f t (z 1 )), therefore f t is a homeomorphism by Corollary 2.7 in
Moreover, if I is a maximal isotopy, f has no contractible fixed point associated to I on M and I is also a maximal isotopy. Recall that π * π 1 ( M , z 1 ) = i * π 1 ( U , z 1 ). So, π(O) is a periodic orbit of type (p, q) associated to I at z 0 for all periodic orbit O of type (p, q) associated to I at , where p q ∈ Q is irreducible.

Let F be an oriented foliation on M 0 such that z 0 is a sink (resp. source). Then there exists a lift F of F| M 0 \{z 0 } to M , and is a sink (resp. source) of F. Denote by W the attracting (resp. repelling) basin of z 0 for F, and by W the attracting (resp. repelling) basin of for

F. Write Ẇ = W \ {z 0 }, ˙ W = W \ { }. Let z 1 ∈ ˙ W be a point sufficient close to . Then (π| ˙ W ) * π 1 ( ˙ W , z 1 ) = π 1 ( Ẇ , π(z 1 )), and hence π| ˙ W is a homeomorphism between ˙
W and Ẇ by Corollary 2.7 in [START_REF] Spanier | Algebraic topology[END_REF], and can be extended continuously to a homeomorphism between W and W .

Extend lifts of a homeomorphism to the boundary

In this section, let M be a plane, f be an orientation preserving homeomorphism of M , and X be an invariant, discrete, closed subset of M with at least 2 points.

We consider the Poincaré's disk model for the hyperbolic plane H, in which model, H is identified with the interior of the unit disk and the geodesics are segments of Euclidean 2.15. The linking number circles and straight lines that meet the boundary perpendicularly. A choice of hyperbolic structure on M \ X provides an identification of the universal cover of M \ X with H. A detailed description of the hyperbolic structures can be found in [START_REF] Casson | Automorphisms of surfaces after Nielsen and Thurston[END_REF]. The compactification of the interior of the unit disk by the unit circle induces a compactification of H by the circle S ∞ . Let π : H → M \ X be the universal cover. Then, f | M \X can be lifted to homeomorphisms of H. Moreover, we have the following result:

Proposition 2.29. [Han99] Each lift f of f | M \X extends uniquely to a homeomorphism of H ∪ S ∞ .
Remark 2.30. When X has infinitely many points, Michael Handel gave a proof in Section 3 of [START_REF] Handel | A fixed-point theorem for planar homeomorphisms[END_REF]; when X has finitely many points, the situation is easier and Handel's proof still works.

In particular, suppose that z 0 is an isolated point in X and is a fixed point of f . Let γ be a sufficiently small circle near z 0 whose lifts to H are horocycles. Fix one lift γ of γ. Denote by P the end point of γ in S ∞ . Fix z 1 ∈ γ and a lift z 1 of z 1 in γ. Let Γ be the group of parabolic covering translations that fix γ, and T be the parabolic covering translations that generates Γ. Then, π descends to an annulus cover π : (H/Γ, z 1 ) → (M \ X, z 1 ), where

z 1 = {T n ( z 1 ) : n ∈ Z}. Also, z → {T n ( z) : n ∈ Z} defines a universal cover π : H → H/Γ. M z 0 γ V H γ P V u T (u)
Let V be the disk containing z 0 and bounded by γ, V be the disk bounded by γ which is a component of π -1 (V \ {z 0 }). We know that π ( V ) is an annulus with π ( γ) as one of its boundary. We add a point at the other end, and get a disk V = π ( V ) ∪ { }. As in the previous section, π | π ( V ) extends continuously to a homeomorphism between V and V , and f can be lifted to a homeomorphism f of H/Γ ∪ { } fixing . Let f be a lift of f | H/Γ to H, it is also a lift of f | M \X and satisfies f • T = T • f . Moreover, both f and T extend continuously to homeomorphisms of H ∪ S ∞ fixing P . We denote still by f and T the two extensions respectively. The formula

f • T = T • f is still satisfied. So, f | H∪S∞\{P } descends to a homeomorphism of (H ∪S ∞ \{P })/Γ. Because (H ∪S ∞ \{P })/Γ is homeomorphic to a compactification of H/Γ ∪ { } by adding a circle at infinity S ∞ , one knows that f extends continuously to a homeomorphism of H/Γ ∪ { } ∪ S ∞ .

The linking number

Let f be an orientation preserving homeomorphism on R 2 , and I = (f t ) t∈[0,1] be an identity isotopy of f . If z 0 , z 1 are two fixed points of f , the map

t → f t (z 0 ) -f t (z 1 ) f t (z 0 ) -f t (z 1 )
descends to a continuous map from [0, 1]/ 0∼1 to S 1 . We define the linking number between z 0 and z 1 associated to I to be the Brouwer degree of this map, and denote it by L(I, z 0 , z 1 ). We say that z 0 and z 1 are linked (relatively to I) if the linking number is not zero. Suppose that I and I are identity isotopies of f , and that z 0 , z 1 are two fixed points of f . Note the following facts:

-if I and I fixes z 0 and satisfies I ∼ J k z 0 I as local isotopies at z 0 , then one can deduce L(I , z 0 , z 1 ) = L(I, z 0 , z 1 ) + k;

-if both I and I can be viewed as local isotopies at ∞, and if I is equivalent to I as local isotopies at ∞, then one can deduce

L(I , z 0 , z 1 ) = L(I, z 0 , z 1 ).
Chapter 3

Dynamics near an isolated fixed points with index one and zero rotation

Proof of the main theorem

Let M be an oriented surface, f : M → M be an area preserving homeomorphism of M isotopic to the identity, and z 0 be an isolated fixed point of f such that i(f, z 0 ) = 1. Let I be an identity isotopy of f fixing z 0 such that its rotation set, which was defined in section 2.10, is reduced to an integer k.

We say that the property P) holds for (f, I, z 0 ) if there exists ε > 0, such that either for all irreducible p/q ∈ (k, k + ε), or for all irreducible p/q ∈ (k -ε, k), there exists a contractible periodic orbit O p/q of type (p, q) associated to I at z 0 , such that µ O p/q → δ z 0 as p/q → k, in the weak-star topology, where µ O p/q is the invariant probability measure supported on O p/q , Our aim of this section is to prove the following result:

Theorem 3.1 (Theorem 1.1). Under the previous assumptions, if one of the following situation occurs, a) M is a plane, f has only one fixed point z 0 and has a periodic orbit besides z 0 ; b) the total area of M is finite, then the property P) holds for (f, I, z 0 ). Remark 3.2. Let I be a local isotopy of f at z 0 such that ρ s (I , z 0 ) is reduced to 0. Since f is area preserving and i(f, z 0 ) = 1, by Proposition 2.15, I has either a positive or a negative rotation type. Let F be a locally transverse foliation of I . If I has a positive rotation type, then z 0 is a sink of F and the interval in Property P) is (k, k + ε); if I has a negative rotation type, then z 0 is a source and the interval in Property P) is (k -ε, k).

We assume that I has a positive rotation type in this section, the other case can be treated similarly.

Remark 3.3. If z 0 is not accumulated by periodic orbits, since the rotation set is reduced to an integer and i(f, z 0 ) = 1, z 0 is an indifferent fixed point by Proposition 2.23. Then, by the assertion viii) of Proposition 2.20, one deduces that ρ(I, z 0 ) is equal to this integer.

We will prove the theorem in several cases.

Chapter 3. Dynamics near an isolated fixed points with index one and zero rotation

The case where M is a plane

In this section, we suppose that M is a plane, and that I is a maximal identity isotopy of f such that Fix(I) is reduced to z 0 . We will prove the following result in this section and get the proof of the first part of Theorem 1.1 as a corollary.

Theorem 3.4. Under the previous assumption, if ρ s (I, z 0 ) is reduced to 0, and if f has another periodic orbit besides z 0 , then the property P) holds for (f, I, z 0 ). This result is an important one in the proof of Theorem 1.1. In the latter cases, we will always reduce the problem to this case and get the result as a corollary. Before proving this result, we first prove the first case of Theorem 1.1 as a corollary.

Proof of the first case of Theorem 1.1. If k is equal to 0, the result follows directly from Theorem 3.4. So, we only deal with the case where ρ s (I, z 0 ) is reduced to a non-zero integer k. Let J be an identity isotopy of the identity fixing z 0 such that the blow-up rotation number ρ(J, z 0 ) is equal to 1. Write I = J -k I. By the first assertion of Proposition 2.20, ρ s (I , z 0 ) is reduced to 0. Since f has exactly one fixed point, I is maximal and the property P) holds for (f, I , z 0 ). A periodic orbit in the annulus M \ {z 0 } with rotation number p/q associated to I is a periodic orbit with rotation number k + p/q associated to I. Therefore, the property P) holds for (f, I, z 0 ). Now we begin the proof of Theorem 3.4 by some lemmas. Lemma 3.5. Let g be a homeomorphism of R 2 , I be a maximal identity isotopy of g that fixes z 0 , and F be a transverse foliation of I . Suppose that z 0 is an isolated fixed point of g and a sink of F . Let W be the attacting basin of z 0 for F . Suppose that either W is equal to R 2 or W is a proper subset of R 2 whose boundary is the union of some proper leaves of F . Let U be a Jordan domain containing z 0 that satisfies U ⊂ W and g(U ) ⊂ W .

If there exist a compact subset K ⊂ U and ε > 0 such that K contains a q-periodic orbit O p/q with rotation number p/q in the annulus R 2 \{z 0 } for all irreducible p/q ∈ (0, ε), then µ O p/q converges, in the weak-star topology, to the Dirac measure δ z 0 as p/q → 0, where µ O p/q is the invariant probability measure supported on O p/q .

Proof. We only need to prove that for every continuous function ϕ : W → R, for every η > 0, there exists δ > 0, such that for every q-periodic orbit O ⊂ K with irreducible rotation number p/q < δ, we have

| ϕdµ O -ϕ(z 0 )| < η,
where µ O is the invariant probability measure supported on O.

Let V be a neighborhood of z 0 such that |ϕ(z) -ϕ(z 0 )| < η/2 for all z ∈ V . Let π : R × (-∞, 0) → W \ {z 0 } be the universal cover which sends the vertical lines upward to the leaves of F , and p 1 : R × (-∞, 0) → R be the projection onto the first coordinate. Let U = π -1 (U \ {z 0 }), K = π -1 (K \ {z 0 }) and g be the lift of g to U associated to I . By the assumptions about W , we know that any arc that is positively transverse to F cannot come back into W once it leaves W . So

p 1 ( g(z)) -p 1 (z) > 0, for all z ∈ K.
Therefore there exists η 1 > 0 such that for all z ∈ π -1 (K \ V ), one has

p 1 ( g(z)) -p 1 (z) > η 1 .
3.1. Proof of the main theorem 39 One deduces that for all q-periodic orbit O ⊂ K with irreducible rotation number p/q <

δ = ηη 1 4| sup K ϕ| , #(O \ V )η 1 q < p q , hence, | ϕdµ O -ϕ(z 0 )| < η/2 + 2 sup K |ϕ| #(O \ V ) q < η.
Remark 3.6. In this lemma, the homeomorphism g do not need to be area preserving.

The assumptions about W prohibit the following bad situation:

z 0 W z f (z)
Lemma 3.7. If ρ s (I, z 0 ) is reduced to 0, and if f can be blown-up at ∞ such that the blow-up rotation number at ∞, that is defined in Section 2.10, is different from 0, then the property P) holds for (f, I, z 0 ).

In order to to prove this lemma, we need the following sublemma: Sublemma 3.8. Under the conditions of the previous Lemma, ρ(I, ∞) is negative, and there exists ε > 0 such that for all irreducible p/q ∈ (0, ε), there exists a q-periodic orbit with rotation number p/q in the annulus M \ {z 0 }.

Proof. We consider a transverse foliation F of I. It has exactly two singularities z 0 and ∞. Since f is area preserving, f | M \{z 0 } satisfies the intersection property, and using the remark that follows Proposition 2.12, one can deduce that F does not have any closed leaf. Because f can be blown-up at ∞ and the blow-up rotation number ρ(I, ∞) is different from 0, we deduce that ∞ is either a sink or a source. By the assumption in Remark 3.2, z 0 is a sink of F, so ∞ is a source of F, and hence ρ(I, ∞) is smaller than 0. Write ρ = -ρ(I, ∞). We denote by S ∞ the circle added at ∞ when blowing-up f at ∞, and still by f the extension of f to M S ∞ . One has to consider the following two cases: -Suppose that z 0 is accumulated by periodic orbits. Let z 1 be a periodic point of f in the annulus M \ {z 0 }. Its rotation number is strictly positive. We denote by ε this number. Because the rotation set ρ s (I, z 0 ) is equal to 0, the rotation number of a periodic orbit tends to 0 as the periodic orbit tend to z 0 . Hence for all irreducible p/q ∈ (0, ε), there exists a periodic orbit near z 0 with rotation number r/s ∈ (0, p/q). The restriction of the homeomorphism f to the annulus M \ {z 0 } satisfies the intersection property, then by Proposition 2.25, there exists a q-periodic orbit with rotation number p/q in the annulus for all irreducible p/q ∈ (0, ε). -Suppose that z 0 is not accumulated by periodic orbits. Then, by Proposition 2.23 z 0 is an indifferent fixed point, and by the assertion viii) of Proposition 2.20, ρ(I, z 0 ), which was defined in Section 2.9, is equal to 0. Let K 0 be a small enough invariant continuum at z 0 such that ρ(I, K 0 ) = 0 ( see Section 2.9). We denote by Chapter 3. Dynamics near an isolated fixed points with index one and zero rotation (M \K 0 ) T 1 S ∞ the prime-ends compactification at the ends K 0 and the compactification at ∞, which is a closed annulus. We can extend f to both boundaries and get a homeomorphism of the closed annulus satisfying the intersection condition. Moreover, the rotation number of f on the upper boundary T 1 is equal to 0, and on the lower boundary S ∞ is equal to ρ. So, by Proposition 2.24, for all irreducible p/q between 0 and ρ, there exists a periodic orbit in the annulus with rotation number p/q. Remark 3.9. In the first case of the proof, it is natural to think that we can prove by a generalization of Poincaré-Birkhoff theorem that there exists a periodic orbit in the annulus with rotation number p/q for all irreducible p/q between 0 and ρ. But in fact, the annulus in this case is half-open, and we do not know whether there exists such a generalization of Poincaré-Birkhoff theorem. So, we choose another periodic orbit to avoid treating the half-open annulus.

Proof of Lemma 3.7. Let us blow-up f at ∞ by adding a circle S ∞ and paste two copies of the closed disk by S ∞ . We get a sphere S and a homeomorphism f that equals to f on each copy and has two fixed points z 0 and σ(z 0 ), where σ is the natural involution. Let I be an identity isotopy that fixes z 0 and σ(z 0 ) and satisfies ρ s (I , z 0 ) = {0}. Because I is a maximal isotopy, f | M \{z 0 } has no contractible fixed point associated to I. Because the blow-up rotation number ρ(I, ∞) is different from 0, the extension of f to S ∞ does not have any fixed point with rotation number 0 (associated to I). So, f | S\{z 0 ,σ(z 0 )} has no contractible fixed point associated to I | S\{z 0 ,σ(z 0 )} . Therefore, I is a maximal isotopy, and one knows Fix(I ) = {z 0 , σ(z 0 )}. Let F be a transverse foliation of I . Then F has exactly two singularities z 0 and σ(z 0 ). By the assumption in Remark 3.2, z 0 is a sink of F . Since the involution σ is orientation reversing, ρ(I , σ(z 0 )) = 0 and I has a negative rotation type at σ(z 0 ). So σ(z 0 ) is a source of F , and hence F does not have any petal. One has to consider the following two cases:

-Suppose that all the leaves of F are curves from σ(z 0 ) to z 0 . Sublemma 3.8 implies that the compact set M S ∞ satisfies the conditions of Lemma 3.5, and we can deduce the result. -Suppose that there exists a closed leaf in F . Since f is area preserving, similarly to the remark that follows Proposition 2.12, one can deduce that there does not exist any closed leaf in M or in σ(M ). So, each closed leaf intersects S ∞ . Let W be the basin of z 0 for F . Then ∂W is a closed leaf, and hence intersects S ∞ . Denote this leaf by φ. We suppose that z 0 is to the right of φ, the other case can be treated similarly. Denote by R(φ

) (resp. L(φ)) the component of S \ φ to S∞ z 0 φ f (φ) the right (resp. left) of φ. Since f (φ) is included in R(φ), we know that both R(φ) ∩ M and R(φ) \ (M ∪ S ∞ ) are not empty. Choose a suitable essential curve Γ ⊂ (R(φ) ∩ L(f (φ)
)) ⊂ W that transversely intersects S ∞ at only finitely many points (see the gray curve between φ and f (φ) in the picture). Then, (L(Γ) ∩ M ) has finitely many connected components, and so does (L(Γ)

∩ (M ∪ S ∞ )). Moreover, each component of (L(Γ) ∩ (M ∪ S ∞ )) contains a segment of S ∞ . Since both M and S ∞ are invariant by f , one knows that f -1 (L(Γ) ∩ (M ∪ S ∞ )) is included in L(Γ)∩(M ∪S ∞ ). So, if V is a component of (L(Γ)∩(M ∪S ∞ )), there exists n > 0 such that f -n (V ) ⊂ V . Moreover, one knows that f -n (V ∩ S ∞ ) ⊂ V ∩ S ∞
and that the rotation number of each point in S ∞ is equal to ρ, so there exists m > 0 such that ρ = m/n and the rotation number of every periodic point of f in V is equal to ρ. Therefore, the rotation number of every periodic point z ∈ (L(Γ) ∩ M ) of f is equal to the rotation number of S ∞ . So, all the periodic orbits in the annulus M \{z 0 } with rotation number in (0, ρ) is contained in R(Γ)∩M . We find a compact set R(Γ) ∩ M that satisfies the conditions of Lemma 3.5, and can deduce the result.

Lemma 3.10. If ρ s (I, z 0 ) is reduced to 0, and if O ⊂ M \ {z 0 } is a periodic orbit of f , then the rotation number of O (associated to I) is positive.

Proof. Let F be a transverse foliation of I. Then F has only one singularity z 0 , and z 0 is a sink of F by the assumption in Remark 3.2. Since f is area preserving, by the remark that follows 2.12 one knows that F does not have any closed leaf. Let W be the attracting basin of z 0 for F. It is either M or a proper subset of M whose boundary is the union of some proper leaves. In the first case, any periodic orbit of f | M \{z 0 } has a positive rotation number associated to I, and the proof is finished. In the second case, note that each connected component of M \ W is a disk foliated by proper leaves, and hence does not contain any loop that is transverse to F. Moreover, any loop transverse to F can not meet a boundary leaf of W , and hence is contained in W . One deduces that every periodic orbit of f distinct from {z 0 } is contained in W , and its trajectory along the isotopy is homotopic to a transverse loop in W . So, its rotation number is positive.

Lemma 3.11. If ρ s (I, z 0 ) is reduced to 0, and if f has another periodic orbit besides z 0 , then there exist an interger q > 1 and a q-periodic orbit O with rotation number 1/q (associated to

I) such that f | M \(O∪{z 0 }) is isotopic to a homeomorphism R 1/q satisfying R q 1/q = Id.
Proof. Let O 0 be a periodic orbit of f distinct from {z 0 }. By the previous lemma, the rotation number ρ of O 0 in the annulus M \ {z 0 } associated to I is positive. Similarly to the proof of Sublemma 3.8, there exists a q-periodic orbit with rotation number p/q in the annulus M \ {z 0 } for all irreducible p/q ∈ (0, ρ). Let F be a transverse foliation of I. One knows that z 0 is a sink of F by the assumption in Remark 3.2. Let W be the attracting basin of z 0 for F. One has to consider the following three cases:

i) Suppose that W is equal to M . Let T : (x, y) → (x + 1, y) be the translation of R 2 . It induces a universal covering map π : R 2 → R 2 /T T 1 × R. Let h : M \ {z 0 } → T 1 × R be an orientation preserving map that maps the leaves of F to the vertical lines {π({x} × R) : x ∈ R} upward. Write I = (h • f t • h -1 ) t∈[0,1] , and f = h • f • h -1 .
We will prove that there exists a positive integer q, and a q-periodic orbit O of f with rotation number 1/q (associated to I ) such that f | (T 1 ×R)\O is isotopic to a homeomorphism R 1/q satisfying that R q 1/q = Id, and hence h -1 (O) is a q-periodic orbit of f with rotation number 1/q (associated to

I) such that f | M \(h -1 (O)∪{z 0 }) is isotopic to h -1 • R 1/q • h.
Chapter 3. Dynamics near an isolated fixed points with index one and zero rotation

Fix a q-periodic orbit O of f with rotation number 1/q in the annulus M \ {z 0 } for 1/q ∈ (0, ρ). Choose 0 and (

< M 1 < M 2 such that O ⊂ T 1 × (-M 1 , M 1 ),
t∈[0,1] f t (T 1 × [-M 1 , M 1 ])) ⊂ T 1 × (-M 2 , M 2 ).
Let f be the lift of f associated to I . One knows that

p 1 ( f ( z)) -p 1 ( z) > 0 for all z ∈ R 2 ,
where p 1 is the projection to the first factor. Let ϕ 1 be the homeomorphism of T 1 ×R whose lift to R 2 is defined by

ϕ 1 (x, y) = (x, y), for |y| ≤ M 2 , (x + |y| -M 2 , y), for |y| > M 2 .
We know that η(y) = sup x∈R |p 2 ( f (x, y))-y| is a continuous function, where p 2 is the projection onto the second factor. So, there exist M 3 > M 2 and a homeomorphism

ϕ 2 of T 1 × R whose lift ϕ 2 to R 2 satisfies p 1 • ϕ 2 = Id and ϕ 2 (x, y) = (x, y), for |y| ≤ M 2 , (x, y + sign(y)(η(y) + 1)), for |y| ≥ M 3 . Let f = ϕ 2 • ϕ 1 • f .
It is a contraction near each end and hence can be blown-up at each end by adding a circle. Moreover, by choosing suitable blow-up, the rotation numbers at the boundary can be any real number, and we get a homeomorphism f of closed annulus and a lift f of f such that O is a (1, q)-periodic orbit and ρ( f ) (see Section 2.12 for the definition) is a closed subset in (0, ∞). One deduces by Proposition 2.28 that O is topologically monotone (Otherwise I(1/q) = [0, 1/(q -1)] ⊂ ρ( f )). Therefore, f | (T 1 ×R)\O is isotopic to a homeomorphism R 1/q satisfying R q 1/q = Id, and so is f | (T 1 ×R)\O . The lemma is proved. ii) Suppose that W is a proper subset of M whose boundary is the union of some proper leaves, and that z 0 is not accumulated by periodic orbits.

In this case, one knows by Remark 2.22 that f can be blown-up at ∞, and that the blow-up rotation number ρ(I, ∞) is equal to 0. One knows by Remark 3.3 that z 0 is a non-accumulated indifferent point, and that ρ(I, z 0 ) is equal to 0.

Recall that there exists a q-periodic orbit with rotation number p/q in the annulus M \ {z 0 } for all irreducible p/q ∈ (0, ρ). We fix a q-periodic orbit O of f with rotation number 1/q in the annulus M \ {z 0 } for 1/q ∈ (0, ρ). Let γ 1 be a simple closed curve that separates O and z 0 . Denote by U -the component of M \ γ 1 containing O. We deduce by the assertions i) and ii) of Proposition 2.20 that there exists a neighborhood of z 0 that does not contain any q-periodic point of f with rotation number 1/q. So, by choosing γ 1 sufficiently close to z 0 , we can suppose that all the q-periodic points of f with rotation number 1/q are contained in U -. Let γ 2 be a simple closed curve that separate γ 1 and z 0 such that

∪ t∈[0,1] f t (U -) is in the component of M \γ 2 containing γ 1 . Denote by U the component of M \γ 2 containing z 0 . Let V ⊂ U be a small Jordan domain containing z 0 such that ∪ t∈[0,1] f t (V ) ⊂ U ,
and K ⊂ V be a sufficiently small invariant continuum at z 0 such that ρ(I, K) = 0. Let M \ K ∪ S ∞ ∪ S 1 be a compactification of M \ K, where S ∞ is the circle added when blowing f at ∞ and S 1 is the circle added when blowing f | M \K at the end K.

It is a closed annulus, and f | M \K extends continuously to a homeomorphism f of

M \ K ∪ S ∞ ∪ S 1 .
The homeomorphism f has a (1, q) periodic orbit O and hence by Proposition 2.28 has a (1, q) topologically monotone periodic orbit O (It could be equal to O or different from O). Since the rotation number of f at both boundary is equal to

0, O is included in M \ K and hence in U -. So, f | M \(K∪O ) is isotopic to a homeomorphism R 1/q satisfying that R q 1/q = Id. Let h : M \ K → M \ {z 0 } be a homeomorphism whose restriction to M \ U is equal to the identity. Then, f = h • f | M \K • h -1 is a homeomorphism of M \ {z 0 } which coincides f in M \U . The restriction of f •f -1 to M \U is equal to the identity, using Alexander's trick one deduces that f • f -1 | M \(O ∪{z 0 }) is isotopic to the identity. So, f | M \(O ∪{z 0 }) and f | M \(O ∪{z 0 }) are isotopic. Therefore, f | M \(O ∪{z 0 }) is isotopic to R 1/q = h| M \(O ∪{z 0 }) • R 1/q • h -1 | M \(O ∪{z 0 }) which satisfies of course R q
1/q = Id. iii) Suppose that W is a proper subset of M whose boundary is the union of some proper leaves, and that z 0 is accumulated by periodic orbits.

As in case ii), f can be blown-up at ∞ and the blow-up rotation number ρ(I, ∞) is equal to 0. Recall that there exists a q-periodic orbit with rotation number p/q in the annulus M \{z 0 } for all irreducible p/q ∈ (0, ρ). Fix two prime integers q 1 and q 2 such that 1/q 2 < 1/q 1 < ρ. Choose a q 1 -periodic orbit O 1 and a q 2 -periodic orbit O 2 in M \{z 0 } with rotation number (associated to I) 1/q 1 and 1/q 2 respectively. Recall that the rotation number of every periodic orbit in M \ {z 0 } is positive and that ρ s (I, z 0 ) is reduced to 0. One deduces by the assertion i) and ii) that for any given integer q > 1, there is a neighborhood of z 0 that does not contain any q-periodic point of f . So, there exists a Jordan domain U containing z 0 that does not contain any periodic point of f with period not bigger that q 2 except z 0 . Let γ 1 ⊂ U be a simple closed curve that separates z 0 and

O 1 ∪ O 2 . Denote by U - 1 the component of M \ γ 1 containing O 1 ∪ O 2 .
Let γ 2 be a simple closed curve that separates γ 1 and z 0 such that the trajectory of each z ∈ U - 1 along I q 2 is in the component U - 2 of M \ γ 2 containing γ 1 . Let γ 3 be a simple closed curve that separates γ 2 and z 0 such that the trajectory of each z ∈ U - 2 along I q 2 is in the component U - 3 of M \ γ 3 containing γ 2 . Since γ 3 ⊂ U , there does not exist any periodic points of f with periodic not bigger that q 2 in γ 3 . We can perturb f in M \ (U - 3 ∪{z 0 }) and get a homeomorphism f such that f has finitely many periodic points with periods not bigger than q 2 in M \ U - 3 . Let X be the union of periodic orbits of f with periodic not bigger than q 2 that intersects M \ U - 3 . It is a finite set containing z 0 . We consider the annulus covering π : M → M \ X such that the restriction of π to a sufficiently small annulus near one end is a homeomorphism between this annulus and a small annulus near ∞ in M \X. As in Section 2.13, we add a point at this end of M . Let U - i be the component of π -1 (U - i ) that has an end and O i be the lift of O i in U - 2 for i = 1, 2. Let f be the lift of f | M \X . It extends continuously to a homeomorphism of M ∪ { }, and the dynamics of f near is conjugate to the dynamics of f near ∞. So, f can be blown-up at , and by choosing a suitable isotopy I of f , the blow-up rotation number ρ( I , ) is equal to 0. Moreover, O i is a q i -periodic orbit of f with rotation number 1/q i (associated to I ), for i = 1, 2. Referring to Section 2.14, one knows that f can be blown-up at the other end. We blow-up f at both ends and get a homeomorphism f of a closed annulus. For i = 1, 2, the homeomorphism f has a (1, q i ) periodic orbit, so one can deduce by Proposition 2.28 that f has a (1, q i ) topologically monotone periodic orbit O i . The zero rotation circle we added at does not contain any periodic points with rotation number different from 0, so it does not contain O 1 or O 2 . The rotation number of f at the circle we added at the other end is different from 1/q 1 or 1/q 2 . Suppose that it is different from 1/q 1 , the other case can be treated similarly. Then, O 1 is included in M , and π( O 1 ) is a periodic orbit of f of period not bigger than q 1 . So, π( O 1 ) is included in U - 3 , and hence is a periodic orbit of f in U - 1 .

U -

1 U - 2 U - 1 U - 2
We will prove by contradiction that O 1 is included in U - 1 . Otherwise, suppose that there exists

z ∈ O 1 in another component U - 1 of π -1 (U - 1 ). Then, z is a fixed point of f q 1 . Let U - 2 be the component of π -1 (U - 2 ) containing U - 1 . Since f q 1 (U - 1 ) ⊂ U - 2 , one deduces that f q 1 ( U - 1 ) ⊂ U - 2 .
Recall that the rotation number of O 1 is 1/q 1 . So, the rotation number of f at the outer boundary is 1/q 1 , which contradicts our assumption. Let h : M → M \ {z 0 } be a homeomorphism whose restriction to U - 2 is equal to π. As in the end of case ii), we deduce that f | M \(π( O 1 )∪{z 0 }) is isotopic to a homeomorphism R 1/q 1 satisfying that R q 1 1/q 1 = Id. The lemma is proved.

Proof of Theorem 3.4. By the previous lemma, there exist an integer q > 1 and a qperiodic orbit O with rotation number 1/q (associated to I) such that f | M \(O∪{z 0 }) is isotopic to a homeomorphism R 1/q satisfying R q 1/q = Id. Let I = (ϕ t ) t∈[0,1] be an identity isotopy of f q that fixes every point in O ∪ {z 0 }. Since the rotation number of O associated to I is 1/q , each point in O is a fixed point of f q and its rotation number associated to I q is 1. Because I fixes O ∪ {z 0 }, I | M \{z 0 } is homotopic to J -1 z 0 I q | M \{z 0 } , where J z 0 is an identity isotopy of the identity fixing z 0 such that ρ(J z 0 , z 0 ) = 1. By the first assertion of Proposition 2.20, one knows that ρ s (I , z 0 ) is reduced to -1.

Let π : M → M \ O be the universal cover. Since M \ O is a surface of finite type, we can endow it a hyperbolic structure, and M can be viewed to be the hyperbolic plane. Fix z 0 ∈ π -1 (z 0 ). Let f be the lift of f | M \O that fixes z 0 . Then, f can be blown-up at ∞.

Let I = ( ϕ t ) t∈[0,1] be the identity isotopy of f q that lifts I . Then, ρ s ( I , z 0 ) is reduced to -1. On the other hand, ∞ is accumulated by the points of π -1 {z 0 } which are fixed points of I , so by the assertion ii) of Proposition 2.20, one knows that 0 is belong to ρ s ( I , ∞). But f can be blown-up at ∞, by the assertion vii) of Proposition 2.20, we know that ρ s ( I , ∞) is reduced to 0.

Let I 0 be an identity isotopy of f that fixes z 0 and satisfies ρ s ( I 0 , z 0 ) = {0}. Then ρ s ( I q 0 , z 0 ) is reduced to 0, and hence

I q 0 | M \{ z 0 } is homotopic to J z 0 I | M \{ z 0 } . So, ρ s ( I q 0 , ∞) is reduced to -1,
and by the assertion i) of Proposition 2.20, we deduce that ρ s ( I 0 , ∞) is reduced to -1/q . Since f can be blown-up at ∞, by the assertion vii) of Proposition 2.20, one knows that the blow-up rotation number ρ( I 0 , ∞) is equal to -1/q .

Every

z 0 ∈ π -1 {z 0 } \ { z 0 } is a contractible fixed point of f q | M \{ z 0 } associated to I | M \{ z 0 } , so it is not a contractible fixed point of f | M \{ z 0 } associated to I 0 | M \{ z 0 } .
Let O be a periodic orbit of f in the annulus M \ { z 0 } such that z 0 / ∈ π ( O ) and the rotation number of O associated to I 0 | M \{ z 0 } is p/q. Then O is a periodic orbit of f q in the annulus M \ { z 0 } and the rotation number associated to I | M \{ z 0 } is pq q -1. So, π ( O ) is a periodic orbit of f in the annulus M \ {z 0 }, the rotation number associated to I is pq q -1, the rotation number associated to I q is pq q , and the rotation number associated to

I is p/q. In particular, if z is a contractible fixed point of f | M \{ z 0 } associated to I 0 | M \{ z 0 } , π ( z ) is a contractible fixed point of f | M \{z 0 } associated to I| M \{z 0 } . So, f | M \{ z 0 } does not have any contractible fixed point associated to I 0 .
Moreover, if p/q is irreducible, and if O is a periodic orbit of f of type (p, q) associated to I 0 at z 0 such that z 0 / ∈ π ( O ), then π ( O ) is a periodic orbit of f of type (p, q) associated to I at z 0 .

By Lemma 3.7, the property P) holds for ( f , I 0 , z 0 ), and then holds for (f, I, z 0 ).

The case where the total area of M is finite

In this section, we assume that the area of M is finite. Recall that f is an area preserving homeomorphism of M , that z 0 is an isolated fixed point of f satisfying i(f, z 0 ) = 1, that I is an identity isotopy of f that fixes z 0 and satisfies ρ s (I, z 0 ) = {k}. Let (X, I X ) be a maximal extension of I that satisfies ρ s (I X , z 0 ) = ρ s (I, z 0 ). Write X 0 = X \ {z 0 }. Then, X 0 is a closed subset of Fix(f ), and I X can be extended to a maximal identity isotopy on M \ X 0 that fixes z 0 . To simplify the notation, we still denote by I X this extension. Moreover, by definition of Jaulent's preorder, we know that a periodic orbit of type (p, q) associated to I X at z 0 is a periodic orbit of type (p, q) associated to I at z 0 . Let M 0 be the connected component of M \ X 0 that contains z 0 . Of course the total area of M 0 is also finite. When M is a sphere, f | M \{z 0 } has at least one fixed point (see Section 2.4), and hence X 0 is not empty. So, M 0 is not a sphere. To simplify the notations, we denote by f 0 the restriction of f to M 0 , and by I 0 the restriction of I X to M 0 . If the property P) holds for (f 0 , I 0 , z 0 ), it holds for (f, I, z 0 ). So, we will prove the following proposition, and the second part of Theorem 1.1 is also proved. Proposition 3.12. Under the previous assumptions, the property P) holds for (f 0 , I 0 , z 0 ). We will prove this proposition in the following four cases: -the component M 0 is a plane and ρ s (I, z 0 ) is reduced to 0; -the component M 0 is neither a sphere nor a plane and ρ s (I, z 0 ) is reduced to 0; -the component M 0 is a plane and ρ s (I, z 0 ) is reduced to a non-zero integer k; -the component M 0 is neither a sphere nor a plane and ρ s (I, z 0 ) is reduced to an non-zero integer k. We will use some results that will be deduced in the first two cases to obtain the last two cases.

The case where M 0 is a plane and ρ s (I, z 0 ) is reduced to 0 In this case, I 0 is a maximal identity isotopy on the plane M 0 that fixes only one point z 0 and satisfies ρ s (I 0 , z 0 ) = {0}. The result of Proposition 3.12 is just a corollary of Chapter 3. Dynamics near an isolated fixed points with index one and zero rotation Theorem 3.4 and the following lemma:

Lemma 3.13. Under the previous assumptions, there exists a periodic orbit of f in the annulus M 0 \ {z 0 }.

Proof. Of course, we can assume that z 0 is not accumulated by periodic orbits. As in Remark 3.3, one knows that z 0 is an indifferent fixed point with rotation number ρ(I, z 0 ) = 0.

Let F be a transverse foliation of I 0 . One knows that F has a unique singularity z 0 and an end ∞. By the assumption in Remark 3.2, z 0 is a sink of F. Since f 0 is area preserving and the total area of M 0 is finite, ∞ is a source of F and all the leaves of F are lines from ∞ to z 0 . Let π : R × (0, 1) → M 0 \ {z 0 } be the universal cover such that the leaves of the lift F of F are the vertical lines oriented upward. Let f be the lift of f 0 associated to I 0 , and p 1 : R × (0, 1) → R be the projection onto the first factor. Then we know that p 1 ( f ( z) -z) > 0, for all z ∈ R × (0, 1).

Let V be a small Jordan domain in the annulus M 0 \ {z 0 } such that f (V ) ∩ V = ∅. Let V be one of the connected components of π -1 (V ). By choosing V small enough, one can suppose that

|p 1 ( z) -p 1 ( z )| < 1 2 for all z, z ∈ V .
Then, for every z ∈ V and z ∈ π -1 {z}, we know that

p 1 ( f n ( z) -z) n ≥ n k=1 χ V (f k (z)) -1 2 n .
We define U = ∪ k∈Z f k (V ). By Poincaré Recurrence Theorem, almost all points in U are recurrent. By Birkhoff-Khinchen Theorem, for almost all z ∈ U , and every z ∈ π -1 {z}, both of the two limits

lim n→∞ p 1 ( f n ( z) -z) n and lim n→∞ n k=1 χ V (f k (z)) n
exist, and there exists a non negative measurable function ϕ on U that satisfies ϕ • f = ϕ and

lim n→∞ n-1 k=0 χ V (f k (z)) n = ϕ(z) for almost all z ∈ U.
Moreover, by Lebesgue's dominated convergence theorem,

U ϕ = χ V = Area(V ) > 0.
Therefore, there exist a recurrent point z ∈ V and z ∈ π -1 {z} such that the limit lim n→∞ p 1 ( f n ( z) -z) n exists and is positive. So, the rotation number of z is positive. We denote it by ρ.

On the other hand, let K 0 be a small enough continuum at z 0 whose rotation number is 0. We denote by (M 0 \ K 0 ) T 1 the prime-ends compactification at the end K 0 , which is an annulus. We can extend f to T 1 and know that the rotation number on T 1 is 0. Then, there exists a fixed point on T 1 whose rotation number is 0.

By the remark that follows Proposition 2.25, there exists a q-periodic orbit of rotation number p/q in the annulus (M 0 \ K 0 ), for all irreducible p/q ∈ (0, ρ).

The case where M 0 is neither a sphere nor a plane and ρ s (I, z 0 ) is reduced to 0 Recall that f 0 is an area preserving homeomorphism of M 0 , that z 0 is an isolated fixed point of f 0 satisfying i(f 0 , z 0 ) = 1, that I 0 is a maximal identity isotopy that fixes only one point z 0 and satisfies ρ s (I 0 , z 0 ) = {0}.

As in Section 2.13, let π : M → M 0 \ {z 0 } be the annulus covering projection, I be the natural lift of I 0 to M ∪ { }, f be the lift of f 0 to M ∪ { } associated to I 0 . Then I is a maximal identity isotopy and Fix( I) is reduced to . For all irreducible p/q ∈ Q, if O is a periodic orbit of type (p, q) associated to I at , then π(O) is a periodic orbit of type (p, q) associated to I 0 at z 0 . So, if the property P) holds for ( f , I, ), then it holds for (f 0 , I 0 , z 0 ). The result of Proposition 3.12 is a corollary of Theorem 3.4 and the following Proposition 3.14, which is the most difficult part of this Chapter.

Proposition 3.14. There exists a periodic orbit of f besides .

The idea of the proof of the proposition is the following: we will first consider several simple situations such that there exists a periodic orbit of f besides , then we suppose that we are not in these situations and follow the idea of Le Calvez (see Section 11 of [START_REF] Le | Une version feuilletée équivariante du théorème de translation de Brouwer[END_REF]) to get a contradiction.

Let us begin with some necessary assumptions and lemmas. Of course, we can suppose that is not accumulated by periodic orbits of f . As in Remark 3.3, is an indifferent fixed point of f and the rotation number ρ( I, ) is equal to 0. Let F be a transverse foliation of I 0 , and F be the lift of F. By the assumption in Remark 3.2, z 0 is a sink of F, and is a sink of F. Denote by W the attracting basin of z 0 for F, and by W the attracting basin of for

F. Write Ẇ = W \ {z 0 } and ˙ W = W \ { }. Recall that π| ˙ W is a homeomorphism between ˙
W and Ẇ and can be extended continuously to a homeomorphism between W and W . The area on M 0 induces an area on M . So f is area preserving, and the area of W is finite. Lemma 3.15. Under the previous assumptions, if there exists an invariant continuum K ⊂ W with positive area, then there exists a periodic orbit besides .

Proof. The proof is similar to the proof of Lemma 3.13 except some small modifications when we try to find a recurrent point with positive rotation number. We will give a more precise description.

Since W is different from M ∪ { }, we can not get a lift of f as in the proof of Lemma 3.13. Instead, we will get a similar one by the following procedure. Let π : R 2 → ˙ W be a universal cover which sends the vertical lines upwards to the leaves of

F | ˙ W . Since K is an invariant subset of W , we can lift f | K\{ } to a homeomorphism f of π -1 (K \ { }) such that p 1 ( f ( z) -z) > 0, for all z ∈ π -1 (K \ { }),
where p 1 is the projection onto the first factor. Also, we should replace the small Jordan domain V in the proof of Lemma 3.13 with V ∩ K by choosing suitable V such that the area of V ∩ K is positive, that f (V ) ∩ V = ∅, and that for every component V of π -1 (V ), one has

|p 1 ( z) -p 1 ( z )| < 1/2 for all z, z ∈ V .
We can always find such a set because the area of K is positive.

Chapter 3. Dynamics near an isolated fixed points with index one and zero rotation Lemma 3.16. Under the previous assumptions, if there exists an invariant continuum K ⊂ W containing such that ρ( I, K) = 0, then there exists a periodic orbit in M .

Proof. Recall that π| ˙ W is a homeomorphism between ˙ W and Ẇ . So, W is a proper subset of M ∪ { }, and the boundary of W is the union of some proper leaves. By Remark 2.22, one knows that f can be blown-up at ∞ and the blow-up rotation number ρ( I, ∞) is equal to 0. We consider the prime-ends compactification of M \ K at the end K, and extend f continuously to a homeomorphism of ( M \ K) S 1 . We get a homeomorphism g of the closed annulus S ∞ ( M \ K) S 1 that coincides with f on M \ K, where S ∞ is the circle we added when blowing-up f at ∞.

Moreover, g satisfies the intersection property and has different rotation numbers at each boundary, then by Proposition 2.24, there exists a periodic orbit in M \ K, which is also a periodic orbit of f . Lemma 3.17. Suppose that there exists a closed disk D ⊂ W containing as an interior point such that the connected component of k∈Z f -k (D) containing is contained in the interior of D. Then f has another periodic orbit besides .

Proof. We will proof this lemma by contradiction. Suppose that f does not have any other periodic orbit. Let K be the connected component of k∈Z f -k (D) containing . We identify K as a point {K}, and still denote by f the reduced homeomorphism. The fixed point {K} is a non-accumulated saddle-point of f with index i( f , {K}) = i( f , K) = i( f , 0) = 1. By Proposition 2.23, f can be blown-up at {K} and ρ( f , {K}) is different from 0 ∈ R/Z. So, ρ( I, K) is different from 0. By the previous lemma, f has another periodic orbit besides , which is a contradiction. Now we begin the proof of Proposition 3.14.

Proof of Proposition 3.14. We will prove this proposition by contradiction. Suppose that there does not exist any other periodic orbit except . Let (D n ) n∈N be an increasing sequence of closed disks containing as an interior point such that D n is contained in the interior of D n+1 for all n ∈ N and ∪ n∈N D n = W . Let K n be the connected component of ∩ k∈Z f -k (D n ) containing . By Lemma 3.16, we know that ρ( I, K n ) is equal to 0 for every n ∈ N. By Lemma 3.17, each K n intersects the boundary of D n . Let K = ∪ n∈N K n ⊂ M ∪ { }. It is an invariant set of f . The boundary of W is the union of proper leaves, so for every point in ∂ W , either its image or its pre-image by f will leave W . Therefore, K can not touch the boundary of W , and is included in W . But each K n intersects the boundary of D n , so K intersects every neighborhood of ∞.

Lemma 3.18. There does not exist any connected component of

M \ K that is included in W .
Proof. We will give a proof by contradiction. Suppose that there exists a component U of M \ K such that U ⊂ W . Then ∂ U is a subset of K, and so is ∂( f n ( U )) for every n ∈ Z. So, f n ( U ) is a component of M \ K for every n ∈ Z. Recall that the area of W is finite, one deduces that the area of U is finite. So, for every n ∈ Z, the area of f n ( U ) is finite. Recall that K can not touch the boundary of W , one knows that a component of M \ K that is not included in W contains a proper leaf in ∂ W and hence has infinite area. So, f n ( U ) is included in W for every n ∈ Z, and hence ∪ n∈Z f n ( U ) is included in W . Therefore, there exists q ∈ N such that f q ( U ) = U . Moreover, is not an interior point of U , and U is homeomorphic to a disk. Then, one deduces by the Brouwer plane translation theorem (see Section 2.4) that f q has a fixed point in U . Therefore, f has a periodic point different from . We get a contradiction.

Let π : M → M be the universal cover, and T be a generator of the group of covering automorphisms. Let I = ( f t ) t∈[0,1] be the natural lift of I, and F be the lift of F. Write f = f 1 . It is the lift of f associated to I. Write K = π -1 (K \ { }), and W = π -1 ( ˙ W ). Because K is connected and is adherent to both ends of M , each connected component of M \K is simply connected. So, if U is one of the connected components of M \K, and if U is one of the components of π -1 ( U ), then U does not intersect T ( U ). Therefore, M \ K is not connected and has infinitely many components. By Lemma 3.18, each component of M \ K contains a proper leaf in ∂ W , and hence a disk bounded by this leaf. As in the following picture, this disk contains the image or the pre-image of this proper leaf by f .

U φ f ( φ) U φ f -1 ( φ) Figure 3.1: Each component U of M \ K is invariant by f So, every component of M \ K is invariant by f . Lemma 3.19. Each leaf in W is an arc from ∞ to .
Proof. Recall that the area of W is finite. So, there exist a leaf included in ∂ W such that W is to its right and a leaf included in ∂ W such that W is to its left. (Otherwise, W contains the positive or negative orbit of a wandering open set W \ f ( W ) or W \ f -1 ( W ) respectively.) Therefore, the following two situations can not occur, and each leaf in W is Proof. We can prove this lemma by contradiction. Suppose that Φ ⊂ K. Then a component of M \ K is either to the left or to the right of Φ. Moreover, if it is to the right (resp. left) of Φ, it is to the right (resp. left) of f ( Φ). Therefore, R( Φ) ∩ L( f ( Φ)) is included in K, and so the interior of K is not empty. We deduce that K is an invariant set of f with non-empty interior and finite area. By Lemma 3.15, there exists a periodic orbit of f in M , which is a contradiction.

Lemma 3.21. Let Φ be a leaf in W , t → Φ(t) be an oriented parametrization of Φ, and U be a component of M \ K. If Φ intersects U , then both the area of L( Φ) ∩ U and the area of R( Φ) ∩ U are infinite, and there exists t 0 such that Φ(t) ∈ U for all t ≤ t 0 .

Proof. We will first give a proof of the first statement by contradiction. We suppose that the area of L( Φ) ∩ U is finite, the other case can be treated similarly. Then,

L( Φ) ∩ R( f -1 ( Φ)) ∩ U is a wandering open set whose negative orbit is contained in L( Φ) ∩ U . It contradicts the fact that f is area preserving.
Let us prove the second statement. We know that both the area of L( Φ) ∩ U and the area of R( Φ) ∩ U are infinite. Since π | U is injective, both the area of π (L( Φ) ∩ U ) and the area of π (R( Φ) ∩ U ) are infinite. The area of W is finite, so both π (L( Φ) ∩ U ) and π (R( Φ) ∩ U ) intersect M \ W , and hence both L( Φ) ∩ U and R( Φ) ∩ U intersect M \ W . Therefore, there exists a proper leaf Φ 1 in L( Φ) ∩ U and a proper leaf Φ 2 in R( Φ) ∩ U . Fix a parametrization t → Φ 1 (t) of Φ 1 and a parametrization t → Φ 2 (t) of Φ 2 , and draw a path γ in U from a point of Φ 1 to a point of Φ 2 . Let s 1 = inf{t : Φ 1 (t) ∈ γ}, s 2 = sup{t : Φ 2 (t) ∈ γ}, and γ be the sub-path of γ connecting Φ 1 (s 1 ) to Φ 2 (s 2 ). Then, as in the following picture, Γ = Φ| (-∞,s 1 ] γ Φ 2 | [s 2 ,∞) is an oriented proper arc and satisfies

Φ U Φ1 Φ2 γ Φ U Φ1 Φ2 γ Φ U Φ1 Φ2 γ Φ U Φ1 Φ2 γ Figure 3
.2: Four possible cases in the proof of Lemma 3.21 R(Γ) ⊂ U . We know that Φ intersects γ . Let t 0 be a lower bound of the set {t : Φ(t) ∈ γ }.

We know that Φ| (-∞,t 0 ] ⊂ U .

Let δ : T 1 → W be an embedding that intersects F transversely, and δ : R → W be the lift of δ. Then δ intersects every leaf in W , and intersects each leaf at only one point. Moreover, if δ intersects Φ and Φ at δ(t) and δ(t ) respectively, and if t < t , then Φ is to the left of Φ , and Φ is to the right of Φ. We define a map h from R to the space of leaves of F in W by h(t) = Φ if δ(t) ∈ Φ.

Lemma 3.22. The set of points

t ∈ R such that h(t) ∩ U = ∅ is open for each component U of W \ K.
Proof. We fix a component U of M \ K, and will first prove that the set {t :

h(t) ∩ U = ∅} is open. Given a real number t such that h(t) intersects U and z ∈ h(t) ∩ U , there is a trivialization neighborhood V of z such that V ⊂ ( U ∩ W ). Moreover, h -1 (V ) is an open interval containing t. So, the set {t : h(t) ∩ U = ∅} is open. By Lemma 3.20, each leaf of F in W intersects at least a component of M \ K. By lemma 3.21, each leaf of F in W intersects at most one component of M \ K. So, each leaf of F in W intersects exactly one component of M \ K. Since M \ K has countable components, R = ∪ U {t : h(t) ∩ U = ∅}
is a disjoint union of countable many open sets. This is impossible.

The case where M 0 is a plane and ρ s (I, z 0 ) is reduced to a non-zero integer k Recall that f 0 is an area preserving homeomorphism of M 0 , that z 0 is an isolated fixed point of f 0 satisfying i(f 0 , z 0 ) = 1, and that I 0 is a maximal identity isotopy that fixes only one point z 0 and satisfies ρ s (I 0 , z 0 ) = {k}. In this case, one can easily deduce that the result of Proposition 3.12 is just a corollary of the result in the previous two cases. We will give a brief explanation. Let J be the identity isotopy of the identity map on M 0 fixing z 0 and satisfying ρ s (J, z 0 ) = 1. Write I 0 = J -k I 0 . It is an identity isotopy of f 0 that satisfies ρ s (I 0 , z 0 ) = {0}. By the result of Proposition 3.12 in the two cases we have already proved, the property P) holds for (f 0 , I 0 , z 0 ). A periodic orbit in M 0 of type (p, q) associated to I 0 at z 0 is a periodic orbit of type (kq + p, q) associated to I 0 at z 0 . So, the property P) holds for (f 0 , I 0 , z 0 ).

The case where M 0 is neither a sphere nor a plane and ρ s (I, z 0 ) is reduced to a non-zero integer k Recall that f 0 is an area preserving homeomorphism of M 0 , that z 0 is an isolated fixed point of f 0 satisfying i(f 0 , z 0 ) = 1, that I 0 is a maximal identity isotopy that fixes only one point z 0 and satisfies ρ s (I 0 , z 0 ) = {k}.

As in Section 2.13, let π : M → M 0 \ {z 0 } be the annulus covering projection, I be the natural lift of I 0 to M ∪ { }, and f be the lift of f 0 associated to I 0 to M ∪ { }. Then I is a maximal identity isotopy and Fix( I) is reduced to . As before, if the property P) holds for ( f , I, ), it holds for (f 0 , I 0 , z 0 ).

Let F be a transverse foliation of I 0 , and F be the lift of F. Since ρ s (I 0 , z 0 ) is reduced to a non-zero integer, by the assertion v) of Proposition 2.20, z 0 is a sink or a source of F and is a sink or a source of F. Let W be the attracting or repelling basin of z 0 for F, and W be the attracting or repelling basin of for F. Recall that π| W \{ } is a homeomorphism be tween W \ { } and W \ {z 0 }. So, W is a strict subset of M ∪ { }, and its boundary is the union of some proper leaves. By Remark 2.22, one knows that f can be blown-up at ∞ and ρ( I, ∞) is equal to 0.

Let J be the identity isotopy of the identity map of M ∪ { } fixing and satisfying ρ s (J, ) = 1. Write I = J -k I. We know that ρ s ( I , ) is reduced to 0, and that the blowup rotation number ρ( I , ∞) is equal to k. One deduces by the assertion ii) of Proposition 2.20 that there exists a neighborhood of ∞ that does not contain any contractible fixed points of f | M associated to I | M . Let (Y, I Y ) be a maximal extension of ({ }, I ) (see Section 2.6). One knows that Y is a closed subset of the union of { } and the set of contractible fixed points of f | M associated to I | M . So, there is a neighborhood of ∞ that does not intersect Y , and hence Y is a compact subset in M ∪ { }. One knows also that ρ s ( I Y , ) is reduced to 0, and that the blow-up rotation number ρ( I Y , ∞) is equal to k. As Chapter 3. Dynamics near an isolated fixed points with index one and zero rotation in the previous subsection, in order to prove the result of Proposition 3.12, we only need to prove that the property P) holds for ( f , I Y , ), which is the aim of this subsection.

Proposition 3.23. Under the previous assumptions, the property P) holds for ( f , I Y , ).

Proof. To get this result, one has to consider two cases: Y is reduced to a single point or it contains at least two points. In the first case, the proposition is a corollary of Lemma 3.7. Now, we will prove the proposition in the second case. Suppose that Y contains at least two points and write Y 0 = Y \ { }. Let M 0 be the connected component of M ∪ { } \ Y 0 containing . Recall that Y is a compact subset of M ∪ { }. So, one has to consider the following two cases:

-M 0 is a bounded plane, -M 0 is neither a sphere nor a plane. In the first case, the area of M 0 is finite, and the problem is reduced to the case in the first part of Section 3.1.2; while in the second case, we will prove the result like in the second part of Section 3.1.2. Now, we suppose that M 0 is neither a sphere nor a plane. Let π : M → M 0 be an annulus covering map, Ȋ be the natural lift of I Y | M 0 to M ∪ {˘ }, and f be the lift of

f | M 0 to M ∪ {˘ } associated to I Y | M 0 .
As before, if the Property P) holds for ( f , Ȋ, ˘ ), then it holds also for ( f , I Y , ). So, the proposition is a corollary of Theorem 3.4 and the following Lemma 3.24.

Lemma 3.24. There exists a periodic orbit of f besides ˘ .

Proof. The proof is similar to the proof of Proposition 3.14. Let F Y be a transverse foliation of I Y , and F be the lift of

F Y | M 0 to M ∪ {˘ }. Recall
the assumption in Remark 3.2, one knows that is a sink of F Y and that ˘ is a sink of F. Let W * be the attracting basin of for F Y and W be the attracting basin of ˘ for F. Recall that π | W \{˘ } is a homeomorphism between W \ {˘ } and W * \ { }.

When k ≥ 1, one deduces by Proposition 2.20 that the end ∞ is sink of F Y . In this case W * is a bounded subset of M ∪ { }, and hence the area of both W * and W are finite. We can repeat the proof of Proposition 3.14, and get the result. Now, we suppose that k ≤ -1. In this case, the end ∞ is a source of of F Y .

Sublemma 3.25. Each leaf in W is an arc from infinite to ˘ .

Proof. When the area of W is finite, we deduces the result as in Lemma 3.19. Now suppose that the area of W is infinite. We consider the compactification of M ∪ { } by adding a point ∞ at infinite, the added point ∞ is a source of F Y and is at the boundary of W * . So, there exists a leaf in W * from the singularity ∞ to whose lift in W is a leaf from infinite to ˘ . Therefore, each leaf in W is an arc from infinite to ˘ .

The difference between our case and the case of Proposition 3.14 is that the area of W may be infinite. But we did not use this condition except in the proof of Lemma 3.19 and Lemma 3.20. We have proven Sublemma 3.25 corresponding to Lemma 3.19. We will prove that the area of K is finite, so the result of Lemma 3.20 is still valid.

Formally, suppose that there does not exist any periodic orbits besides ˘ . Let (D n ) n∈N be an increasing sequence of closed disks containing ˘ such that D n is contained in the interior of D n+1 for all n ∈ N and

∪ n∈N D n = W . Let K n be the connected component of ∩ k∈Z f -k (D n ) containing ˘ and K = ∪ n∈N K n ⊂ M ∪ { }.
We will prove that the area of K is finite.

The case of diffeomorphisms

53 Let K n = π (K n ), and K = ∪ n K n ⊂ M ∪ { } ∪ S ∞
, where S ∞ is the circle we added when blowing-up f at ∞. As before, we can deduce that K ⊂ W . Recall that π | W \{˘ } is a homeomorphism between W \ {˘ } and W * \ { }. Therefore, we know that π (K) ⊂ K , and that the area of K is not bigger than the area of K . So, we only need to prove that the area of K is finite.

We will prove it by contradiction. Suppose that the area of K is infinite. One deduces that K ∩ S ∞ = ∅. As was proven in Section 3.1.2, one knows that ρ( Ȋ, K n ) = 0, and so ρ( I Y , K n ) = 0 for all n ∈ N. Since (Y, I Y ) is a maximal extension of ({ }, J -k I), one deduces that ρ(J -k I, K n ) is equal to 0 and that ρ( I, K n ) is equal to k, for all n ∈ N.

Since K ∩ S ∞ is invariant by f and the blow-up rotation number ρ( I, ∞) = 0, there exists a fixed point z 1 ∈ K ∩ S ∞ , and the rotation number of z 1 (associated to I) in the annulus M ∪ S ∞ is 0.

Let π : M → M ∪ S ∞ be the universal cover, T be a generator of the group of covering automorphism, and f the lift of f associated to I. Fix one z 1 ∈ π -1 ( z 1 ). It is a fixed point of f . Let U be a small neighborhood of z 1 such that T n (U ) ∩ U = ∅ for all n = 0. Let V ⊂ U be a neighborhood of z 1 such that f 2 (V ) ⊂ U . Fix n large enough such that K n ∩V = ∅, and choose an arc γ in V connecting z 1 and an accessible point of K n such that γ ∩ K n has exactly one point. By choosing a sub-arc of γ ∪ f -2 (γ), we get a cross-cut γ . On one hand, T (γ ) ∩ γ = ∅ because γ ⊂ V . On the other hand, we consider the primeends compactification of M ∪ S ∞ \ K n at the end K n , and denote by f K n the extension of f | M \K n . As was in Section 2.9, let π

K n : π -1 ( M ∪S ∞ \K n )∪R → ( M ∪S ∞ \K n )∪S 1 be the universal cover, and f K n the lift of f K n whose restriction to π -1 ( M ∪ S ∞ \ K n ) is equal to f . Recall that ρ( I, K n ) is equal to k ≤ -1. So, the end-cut f -2 K n (γ) > T -2k-1 (γ) ≥ T (γ), γ γ f -2 K n (γ)
T (γ)

T (γ )
which means γ ∩ T (γ ) = ∅. We get a contradiction.

The case of diffeomorphisms

The index at a degenerate fixed point that is an extremum of a generating function

Let f be a diffeomorphism of R 2 and g : R 2 → R be a C 2 function, we call g a generating function of f if ∂ 2 12 g < 1, and if

f (x, y) = (X, Y ) ⇔ X -x = ∂ 2 g(X, y), Y -y = -∂ 1 g(X, y). (3.1)
Every C 2 function g : R 2 → R satisfiying ∂ 2 12 g ≤ c < 1 defines a diffeomorphism f of R 2 by the previous equations. Moreover, the Jacobian matrix J f of f is equal to

1 1 -∂ 2 12 g(X, y) 1 ∂ 2 22 g(X, y) -∂ 2 11 g(X, y) -∂ 2 11 g(X, y)∂ 2 22 g(X, y) + (1 -∂ 2 12 g(X, y)) 2 .
Chapter 3. Dynamics near an isolated fixed points with index one and zero rotation Since det J f = 1, the diffeomorphism f is orientation and area preserving. On the other hand, every orientation and area preserving diffeomorphism

f of R 2 satisfying 0 < ε ≤ ∂ 1 (p 1 • f ) ≤ M < ∞ can
be generated by a generating function, where p 1 is the projection onto the first factor. Moreover, we can naturally define an identity isotopy I 0 = (f t ) t∈[0,1] of f such that f t is generated by tg. Precisely, the diffeomorphisms f t are defined by the following equations:

f t (x, y) = (X t , Y t ) ⇔ X t -x = t∂ 2 g(X t , y), Y t -y = -t∂ 1 g(X t , y). (3.2)
A point (x, y) is a fixed point of f if and only if it is a critical point of g. We say that a fixed point (x, y) of f is degenerate if 1 is an eigenvalue of J f (x, y). We will see later that a fixed point (x, y) of f is degenerate if and only if the Hessian matrix of g at (x, y) is degenerate.

We can also define a local generating function. Precisely, if (x, y) is a critical point of a C 2 function g such that ∂ 12 g(x, y) < 1, then one can define an orientation and area preserving local diffeomorphism f at (x, y) by the equations (3.1). On the other side, if (x, y) is a fixed point of an orientation and area preserving diffeomorphism f such that ∂ 1 (p 1 • f )(x, y) > 0, where p 1 is the projection to the first factor, then one can find a C 2 function g defined in a neighborhood of (x, y), that defines the germ of f at (x, y) by the equations (3.1). Moreover, in both cases, we can define a local isotopy of f at (x, y) by the equations (3.2), and will call it the local isotopy induced by g.

In this section, suppose that f : (W, 0) → (W , 0) is a local diffeomorphism at 0 ∈ R 2 , and that g is a local generating function of f . We will prove the following Proposition 3.26, and deduce Corollary 1.3 as an immediate consequence of Theorem 1.1 and Proposition 3.26. Proposition 3.26. If 0 is an isolated critical point of g and a local extremum of g, and if the Hessian matrix of g at 0 is degenerate, then i(f, 0) is equal to 1.

Proof. The idea is to compute the indices of the local isotopies, so that we can deduce the Lefschetz index by Proposition 2.1.

We denote by I 0 = (f t ) t∈[0,1] the local isotopy induced by g. We have the following lemma Lemma 3.27. The blow-up rotation number ρ(I 0 , 0) is equal to 0.

Proof. Since Hess(g)( 0) is degenerate, one deduces that 0 is an eigenvalue of Hess(g)(0). Let v be an eigenvector of Hess(g)(0) corresponding to the eigenvalue 0. We will prove that v is a common eigenvector of J ft (0) corresponding to the eigenvalue 1 for t ∈ [0, 1], and hence the blow-up rotation number ρ(I 0 , 0) is equal to 0. Write

Hess(g)(0) = σ σ τ and v = a b .
Then, one deduces that τ -σ 2 = 0, a + σb = 0, and σa + τ b = 0.

By a direct computation, one knows that for every t ∈ [0, 1],

J ft (0) = 1 1 -tσ 1 tτ -t -t 2 τ + (1 -tσ) 2 = 1 1 -tσ 1 tτ -t 1 -2tσ ,
and then

J ft (0)v = 1 1 -tσ a + tτ b -t a + b -2tσb = 1 1 -tσ a -tσa tσb + b -2tσb = a b .
Since f is area preserving, the rotation set at 0 is not empty. By the assertion vii) of Proposition 2.20, and the previous lemma, one can deduce that ρ s (I 0 , 0) is reduced to 0, and that for all local isotopy I of f that is not equivalent to I 0 , the rotation set ρ s (I, 0) is reduced to a non-zero integer.

Lemma 3.28. If I is a local isotopy of f that is not equivalent to I 0 , then i(I, 0) is equal to 0.

Proof. Let F be foliation locally transverse to I. Since ρ s (I, 0) is reduced to a non-zero integer, one can deduce by the assertion v) of Proposition 2.20 that 0 is either a sink or a source of F. By Proposition 2.3, one deduces that i(I, 0) = i(F, 0) -1 = 0.

In order to compute the index of I 0 , we will construct an isotopy I that is equivalent to I 0 , and prove that i(I , 0) = 0.

We define I = (f t ) t∈[0,1] in a neighborhood of 0 by

f t (x, y) = (x, y) + 2t(X -x, 0) for 0 ≤ t ≤ 1/2, (X, y) + (2t -1)(0, Y -y) for 1/2 ≤ t ≤ 1,
where (X, Y ) = f (x, y).

Lemma 3.29. The family I = (f t ) t∈[0,1] is a local isotopy of f .

Proof. For every fixed t ∈ [0, 1], We will prove that f t is a local diffeomorphism by computing the determinant of the Jacobian matrices, and then get the result. Indeed, one knows

∂ 1 X = 1/(1 -∂ 12 g) > 0.
Then for t ∈ [0, 1/2],

det

J f t = det 1 + 2t(∂ 1 X -1)) 2t∂ 2 X 0 1 = 2t∂ 1 X + (1 -2t) > 0; and for t ∈ [1/2, 1], det J f t = det ∂ 1 X ∂ 2 X (2t -1)∂ 1 Y (2 -2t) + (2t -1)∂ 2 Y = (2t -1) det J f + (2 -2t)∂ 1 X > 0.
Lemma 3.30. The blow-up rotation number ρ(I , 0) is equal to 0, and hence I is equivalent to I 0 .

Chapter 3. Dynamics near an isolated fixed points with index one and zero rotation

Proof. As in the proof of Lemma 3.27, we will prove that an eigenvector of Hess(g)(0) corresponding to the eigenvalue 0 is a common eigenvector of J f t (0) corresponding to the eigenvalue 1 for t ∈ [0, 1], and hence deduce the lemma. We keep the notations in the proof of Lemma 3.27, and recall that τ -σ 2 = 0, a + σb = 0, and σa + τ b = 0.

For t ∈ [0, 1/2],

J f t (0) = Id + 2t ∂ 1 X(0, 0) -1 ∂ 2 X(0, 0) 0 0 = Id + 2t 1 -σ σ τ 0 0 ,
and

J f t (0)v = v + 2t 1 -σ σa + τ b 0 = v. For t ∈ [1/2, 1], J f t (0) = J f (0) -(2 -2t) 0 0 ∂ 1 Y (0, 0) ∂ 2 Y (0, 0) -1 = J f (0) - 2 -2t 1 -σ 0 0 --σ ,
and

J f t (0)v = J f (0)v + 2 -2t 1 -σ 0 a + σb = v.
We have verified that v is a common eigenvector of J f t (0) corresponding to the eigenvalue 1 for t ∈ [0, 1].

To conclude, we will define a locally transverse foliation F 0 of I such that 0 is a sink or a source of F 0 , and then deduce by Proposition 2.3 that i(I , 0) = i(F 0 , 0) -1 = 0. Indeed, let F 0 be the foliation in a neighborhood of 0 whose leaves are the integral curves of the gradient vector field1 of g. One knows that 0 is a sink of F 0 if 0 is a local maximum of g, and is a source of F 0 if 0 is a minimum of g. We can finish our proof by the following lemma.

Lemma 3.31. The foliation F 0 is locally transverse to I . Proof. Let U be a sufficiently small Jordan domain containing 0 such that F 0 is well defined on U , and V ⊂ U be a sufficiently small neighborhood of 0 such that f t is well defined on V for t ∈ [0, 1], that f does not have any other fixed point in V except 0, and that ∪ t∈[0,1] f t (V ) ⊂ U . We will prove that for every z = (x, y) ∈ V \ {0}, the path γ z : t → f t (x, y) is positively transverse to F 0 , and then deduce the lemma.

Indeed, for t

∈ [0, 1/2], det 2(X -x) ∂ 1 g(f t (x, y)) 0 ∂ 2 g(f t (x, y)) = 2(X -x)∂ 2 g(f t (x, y)) = 2(X -x)∂ 2 g(2tX + (1 -2t)x, y) = 2(X -x)[∂ 2 g(X, y) + (2t -1)(X -x)∂ 2 12 g(ξ, y)] = 2(X -x) 2 [1 -(1 -2t)∂ 2
12 g(ξ, y)] ≥ 0 where ξ is a real number between x and X, and the inequality is strict if X = x.

For t ∈ [1/2, 1], det 0 ∂ 1 g(f t (x, y)) 2(Y -y) ∂ 2 g(f t (x, y)) = -2(Y -y)∂ 1 g(f t (x, y)) = -2(Y -y)∂ 1 g(X, (2 -2t)y + (2t -1)Y ) = -2(Y -y)[∂ 1 g(X, y) + (2t -1)(Y -y)∂ 2 12 g(X, η)] = 2(Y -y) 2 [1 -(2t -1)∂ 2
12 g(X, η)] ≥ 0 where η is a real number between y and Y , and the inequality is strict if Y = y.

Since z = (x, y) is not a fixed point, either X = x or Y = y. If both of the inequalities are satisfied, γ z is positively transverse to

F 0 ; if X = x and Y = y, γ z | t∈[0, 1 2 ] is positively transverse to F 0 , and γ z | t∈[ 1 2 ,1] is reduced to a point; if X = x and Y = y, γ z | t∈[0, 1 2
] is reduced to a point, and

γ z | t∈[ 1 2 ,1] is positively transverse to F 0 .
Remark 3.32. In the proof, we have indeed proven that F 0 is locally transverse to any local isotopy of f that is equivalent to I 0 .

Discrete symplectic actions and symplectically degenerate extrema

In this section, we will introduce symplectically degenerate extrema. More details can be found in [START_REF] Mazzucchelli | Symplectically degenerate maxima via generating functions[END_REF].

We say that a diffeomorphism F : T 2 → T 2 is Hamiltonian if it is area preserving and if there exists a lift f satisfying

f (z + k) = f (z) + k for all k ∈ Z 2 , and T 2 (f -Id)dxdy = 0.
Refering to [START_REF] Mcduff | Introduction to symplectic topology[END_REF], this definition coincides with the usual definition of a Hamiltonian diffeomorphism of a symplectic manifold. More precisely, we call a time-dependent vector field (X t ) t∈R a Hamiltonian vector field if it is defined by the equation:

dH t = ω(X t , •),
where (M, ω) is a symplectic manifold and H : R × M → R is a smooth function. The Hamiltonian vector field induces a Hamiltonian flow (ϕ t ) t∈R on M , which is the solution of the following equation

∂ ∂t ϕ t (z) = X t (ϕ t (z)).
Chapter 3. Dynamics near an isolated fixed points with index one and zero rotation

We say that a diffeomorphism F of M is a Hamiltonian diffeomorphism if it is the time-1 map of a Hamiltonian flow. So, for a Hamiltonian diffeomorphism, there exists a natural identity isotopy I which is defined by the Hamiltonian flow. We say that a fixed point of a Hamiltonian diffeomorphism is contractible if its trajectory along I is a loop homotopic to zero in M , and that a q-periodic point of a Hamiltonian diffeomorphism is contractible if it is a contractible fixed point of F q .

Let F : T 2 → T 2 be a Hamiltonian diffeomorphism. Then F is the time-1 map of a Hamiltonian flow, and we can factorize F by

F = F k-1 • • • • • F 0 ,
where F j is C 1 -close to the identity, for j = 0, • • • , k -1. For every j, let f j be the lift of F j that is C 1 -close to the identity, and g j be a generating function of f j . We define the discrete symplectic action

g : R 2k → R by g(z) := j∈Z k (< y j , x j -x j+1 > +g j (x j+1 , y j )),
where z = (z 0 , ..., z k-1 ) and z j = (x j , y j ). By a direct computation, we know that for every j ∈ Z k ,

∂ ∂x j g(z) = y j -y j-1 + ∂ 1 g j-1 (x j , y j-1 ), and ∂ ∂y j g(z) = x j -x j+1 + ∂ 2 g j (x j+1 , y j ).
So, z ∈ R 2k is a critical point of g if and only if z j+1 = f j (z j ) for every j ∈ Z k , and therefore if and only if

z 0 ∈ R 2 is a fixed point of f = f k-1 • • • • • f 0 .
In particular, each f j commutes with the integer translation, and so g is invariant by the diagonal action of Z 2 on R 2k and descends to a function

G : R 2k /Z 2 → R. Moreover, [z] ∈ R 2k /Z 2 is a critical point of G if
and only if z j+1 = f j (z j ) for every j ∈ Z k , and therefore if and only if [z 0 ] ∈ T 2 is a contractible fixed point of F , where

F = F k-1 •• • ••F 0 .
In particular, critical points of G one-to-one correspond to contractible fixed points of F . Moreover, for any period q ∈ N, contractible q-periodic points of F correspond to the equivalent classes in R 2kq /Z 2 of critical points of the discrete symplectic action g ×q : R 2kq /Z 2 → R defined by

g ×q (z) := j∈Z kq (< y j , x j -x j+1 > +g (j mod k) (x j+1 , y j )),
where z = (z 0 , ..., z kp-1 ) and z j = (x j , y j ).

Moreover, if [z 0 ] ∈ T 2 is a contractible fixed point of F , then by a suitable shift one can suppose that [z 0 ] is fixed along the Hamiltonian flow, and hence the factors F j fixes [z 0 ] for j = 0, • • • , k -1. So, z 0 is a fixed point of each f j and a critical point of each g j . We denote by C * (z ×kn 0 ) the graded group of relative homology Remark 3.34. In particular, a symplectically degenerate maximum satisfies the condition of the proposition, and hence is a degenerate fixed point of F . Moreover, the previous proposition is still valid if we replace C kn+1 (z ×n ) with C + kn+1 (z ×n ), and a symplectically degenerate minimum is also a degenerate fixed point of F .

H * ({g ×n < g ×n (z ×kn 0 )} ∪ {z ×kn 0 }, {g ×n < g ×n (z ×kn 0 )}).

The index at a symplectically degenerate extremum

As in the previous subsection, let F : T 2 → T 2 be a Hamiltonian diffeomorphism, and

F = F k-1 •• • ••F 0 be
a factorization by Hamiltonian diffeomorphisms F i which are C 1 -close to the identity. Let f j be the lift of F j to R 2 that is C 1 -close to the identity, and g j be a generating function of f j , for j = 0, • • • , k-1. As was recalled in the previous subsection, if z 0 is a symplectically degenerate extremum, then the blow-up rotation number ρ(I, [z 0 ]) is equal to 0 for any identity isotopy of F fixing [z 0 ]. We will prove the following Proposition 3.35, and then can deduce Theorem 1.4 as an immediate corollary of Theorem 1.1. Proposition 3.35. If z 0 is a symplectically degenerate extremum, then i(F, [z 0 ]) is equal to 1.

We will only deal with the case where 0 is a symplectically degenerated maximum, the other case can be treated similarly. Let us begin by some lemmas. Lemma 3.36. Suppose that g is a (local or global) generating function of a diffeomorphism f , and that 0 is a local maximum of g such that the Hessian matrix of g at 0 is degenerate. Let I = (f t ) be the identity isotopy of f induced by g as in Section 3.2.1, and θ(t) be a continuous function such that

J ft (0) cos θ(0) sin θ(0) J ft (0) cos θ(0) sin θ(0) = cos θ(t) sin θ(t) .
Then, one can deduce that θ(1) ≥ θ(0).

Chapter 3. Dynamics near an isolated fixed points with index one and zero rotation

Proof. As in Section 3.2.1, we denote the Hessian of g at 0 by Hess(g)(0) = σ σ τ .

Since 0 is a local maximal point of g, Hessg(0) is negative semi-definite. So, we knows that ≤ 0, τ ≤ 0, and τ -σ 2 = 0.

As was proved in Section 3.2.1, if (a, b) is a unit eigenvector of Hess(g)(0) corresponding to the eigenvalue 0 , then it is a common eigenvector of J ft (0) corresponding to the eigenvalue 1. Recall that a + σb = 0, and σa + τ b = 0. So,

J ft (0) -b a = 1 1 -tσ 1 tτ -t 1 -2tσ -b a = -b a + t( + τ ) 1 -tσ a b .
Therefore

J ft (0)Ω = Ω 1 t( +τ ) 1-tσ 0 1 , where Ω = a -b b a is a normal matrix. Since t( +τ ) 1-tσ ≤ 0, one can deduce that θ(1) ≥ θ(0). (a, b) (-b, a) J ft (0) (a, b) (-b, a) + t( +τ ) 1-tσ (a, b)
Lemma 3.37. If 0 is a symplectically degenerate maximum, then there exists a normal matrix Ω such that

Ω -1 J f j (0)Ω = 1 c j 0 1 for j = 0, • • • , k -1
, where c j are non-positive real numbers.

Proof. Let I j = (f j,t ) t∈[0,1] be the local isotopy of f j induced by g j as in section 3.2.1. Let F j be the foliation whose leaves are the integral curves of the gradient vector field of g j . As in Section 3.2.1, one can deduce that 0 is a sink of F j and that F j is locally transverse to I j . Therefore, one knows that ρ(I j , 0) ≥ 0, and that ρ(I j , 0) = 0 if and only if 0 is a degenerate fixed point of f j . Let θ : [0, k] → R be a continuous function such that

J f jt (0) cos θ(j) sin θ(j) J f jt (0) cos θ(j) sin θ(j) = cos θ(j + t) sin θ(j + t) .
One knows that θ(j + 1) > θ(j) if ρ(I j , 0) > 0, and θ(j + 1) ≥ θ(j) if ρ(I j , 0) = 0. But we know that ρ(I k-1 • • • I 0 , z 0 ) = ρ(I, [z 0 ]) = 0, so there exists θ(0) ∈ R and a 3.2. The case of diffeomorphisms 61 continuous function θ as above such that θ(k) = θ(0). Therefore, ρ(I j , z 0 ) = 0 for

j = 0, • • • , k -1 and cos θ(0) sin θ(0)
is a common eigenvector of J f j (0) corresponding to the eigenvalue 1. As in the proof of the previous lemma, we can prove this lemma by choosing Ω = cos θ(0) -sin θ(0) sin θ(0) cos θ(0) .

Lemma 3.38. Suppose that g is a (local or global) generating function of a diffeomorphism f , that 0 is a local maximum of g, and that the Hessian matrix of g at 0 is degenerate. If Ω is a normal matrix, and if f = Ω -1 f Ω is generated by g in a neighborhood of 0, then 0 is a local maximum of g and Hess(g )(0) is degenerate.

Proof. Since Hess(g)(0) is degenerate, 1 is an eigenvalue of J f (0) and hence an eigenvalue of J f (0). So, Hess(g )(0) is degenerate.

Let F be the foliation whose leaves are integral curves of the gradient vector field of g, and F be the foliation whose leaves are integral curves of the gradient vector field of g . Let I 0 be a local isotopy of f satisfies ρ(I 0 , 0) = 0, and I 0 be a local isotopy of f satisfies ρ(I 0 , 0) = 0. As was proved in Section 3.2.1, F is locally transverse to I 0 and F is locally transverse to I 0 . Therefore, Ω • F is locally transverse to I 0 . Since 0 is a maximal point of g, it is a sink of F. By the remark that follows Proposition 2.15, one deduces that 0 is a sink of Ω • F , and hence a sink of F . Therefore, 0 is a local maximum of g . Lemma 3.39. Let g 0 and g 1 be local generating functions of f 0 and f 1 respectively such that 0 is a local maximal point of both g 0 and g 1 , and that the Hessian matrices satisfy

Hess(g i )(0) = 0 0 0 c i ,
where c i ≤ 0 for i = 0, 1. Then there exists a function g which is a generating function of f = f 1 • f 0 in a neighborhood of 0. Moreover, 0 is a local maximal point of g, and the Hessian matrix satisfies Hess(g)(0) = 0 0 0 c 0 + c 1 .

Proof. Suppose that g 0 (0) = g 1 (0) = 0. Since 0 is a local maximal point of both g 0 and g 1 , it is a critical point of both g 0 and g 1 . So,

∂ 1 g 0 (0, 0) = ∂ 2 g 0 (0, 0) = ∂ 1 g 1 (0, 0) = ∂ 2 g 1 (0, 0) = 0. Write (x 1 , y 1 ) = f 0 (x 0 , y 0 ) and (x 2 , y 2 ) = f 1 (x 1 , y 1 )
. By definition of generating functions, one knows that

y 1 -y 0 + ∂ 1 g 0 (x 1 , y 0 ) = 0, and x 1 -x 2 + ∂ 2 g 1 (x 2 , y 1 ) = 0. (3.3) Note that det ∂ 2 11 g 0 (0, 0) 1 1 ∂ 2 22 g 1 (0, 0) = -1.
So, by implicit function theorem, there exists a C 1 diffeomorphism ϕ : W → W such that (x 1 , y 1 ) = ϕ(x 2 , y 0 ), where W and W are sufficiently small neighborhoods of 0 in R 2 . Moreover,

J ϕ (0, 0) = - ∂ 2 11 g 0 (0, 0) 1 1 ∂ 2 22 g 1 (0, 0) -1 0 ∂ 2 12 g 0 (0, 0) -1 ∂ 2 12 g 1 (0, 0) -1 0 = 1 -c 1 0 1 .
Chapter 3. Dynamics near an isolated fixed points with index one and zero rotation Let g(x 2 , y 0 ) = g 0 (x 1 , y 0 ) + g 1 (x 2 , y 1 ) + (x 2 -x 1 )(y 0 -y 1 ), where (x 1 , y 1 ) = ϕ(x 2 , y 0 ). We know that g(0, 0) = 0. In a neighborhood of 0, by a direct computation and equations (3.3), one knows that

∂ 1 g(x 2 , y 0 ) =∂ 1 g 0 (x 1 , y 0 )∂ 1 x 1 (x 2 , y 0 ) + ∂ 1 g 1 (x 2 , y 1 ) + ∂ 2 g 1 (x 2 , y 1 )∂ 1 y 1 (x 2 , y 0 ) + (1 -∂ 1 x 1 (x 2 , y 0 ))(y 0 -y 1 ) -∂ 1 y 1 (x 2 , y 0 )(x 2 -x 1 ) =∂ 1 g 0 (x 1 , y 0 ) + ∂ 1 g 1 (x 2 , y 1 ).
Similarly, one gets

∂ 2 g(x 2 , y 0 ) = ∂ 2 g 0 (x 1 , y 0 ) + ∂ 2 g 1 (x 2 , y 1 ).
So, g is a C 2 function near 0. Moreover,

∂ 2 12 g(0, 0) = ∂ 2 11 g 0 (0, 0)∂ 2 y 1 (0, 0) + ∂ 2 12 g 0 (0, 0) + ∂ 2 12 g 1 (0, 0)∂ 2 y 1 (0, 0) = 0.
Because g 0 and g 1 locally generate f 0 and f 1 respectively, one deduces

∂ 1 g(x 2 , y 0 ) = -(y 2 -y 0 ) and ∂ 2 g(x 2 , y 0 ) = x 2 -x 0 .
Therefore, g is a generating function of f in a neighborhood of 0. By a direct computation, one gets

∂ 2 11 g(0, 0) = ∂ 2 11 g 0 (0, 0)∂ 1 x 1 (0, 0) + ∂ 2 11 g 1 (0, 0) + ∂ 2 12 g 1 (0, 0)∂ 1 y 1 (0, 0) = 0,
and

∂ 2 22 g(0, 0) = ∂ 2 12 g 0 (0, 0)∂ 2 x 1 (0, 0) + ∂ 2 22 g 0 (0, 0) + ∂ 2 22 g 1 (0, 0) = c 0 + c 1 . So, Hess(g)(0) = 0 0 0 c 0 + c 1 .
We will conclude by proving that 0 is a locally maximum of g. Let ε > 0 be a small real number such that |εc 1 | < 1. We will prove that in a sufficiently small neighborhood of 0,

g(x 2 , y 0 ) ≤ g 0 (x 1 + 1 ε (y 0 -y 1 ), y 0 ) + g 1 (x 2 , y 1 + ε(x 2 -x 1 )) ≤ 0,
and hence 0 is a locally maximum of g because the second inequality is strict for (x 2 , y 0 ) = 0. Indeed, by Taylor's theorem and equations (3.3), one knows that in a sufficiently small neighborhood of 0,

g 0 (x 1 + 1 ε (y 0 -y 1 ), y 0 ) = g 0 (x 1 , y 0 ) + 1 ε ∂ 1 g 0 (x 1 , y 0 )(y 0 -y 1 ) + 1 2ε 2 ∂ 2 11 g 0 (ξ, y 0 )(y 0 -y 1 ) 2 = g 0 (x 1 , y 0 ) + 1 ε (y 0 -y 1 ) 2 + 1 2ε 2 ∂ 2 11 g 0 (ξ, y 0 )(y 0 -y 1 ) 2 ,
where ξ is a real number between x 1 and x 1 + 1 ε (y 0 -y 1 ). Similarly, one deduces that in sufficiently small neighborhood of 0,

g 1 (x 2 , y 1 + ε(x 2 -x 1 )) = g 1 (x 2 , y 1 ) + ε(x 2 -x 1 ) 2 + ε 2 2 ∂ 2 22 g 1 (x 2 , η)(x 2 -x 1 ) 2 ,
where η is a real number between y 1 and y 1 + ε(x 2 -x 1 ). So,

g(x 2 , y 0 ) =g 0 (x 1 + 1 ε (y 0 -y 1 ), y 0 ) + g 1 (x 2 , y 1 + ε(x 2 -x 1 )) - 1 2ε (y 0 -y 1 ) 2 - ε 2 (x 2 -x 1 ) 2 + (x 2 -x 1 )(y 0 -y 1 ) - 1 2ε (1 + 1 ε ∂ 11 g 0 (ξ, y 0 ))(y 0 -y 1 ) 2 - ε 2 (1 + ε∂ 22 g 1 (x 2 , η))(x 2 -x 1 ) 2 .
For (x 2 , y 0 ) = 0 that is in a sufficiently small neighborhood of 0, one can suppose that

| 1 ε 2 ∂ 11 g 0 (ξ, y 0 )| < 1, and |ε∂ 22 g 1 (x 2 , η)| < 1. So, g(x 2 , y 0 ) ≤ g 0 (x 1 + 1 ε (y 0 -y 1 ), y 0 ) + g 1 (x 2 , y 1 + ε(x 2 -x 1 )).
Now, we begin the proof of Proposition 3.35.

Proof of Propostion 3.35. Suppose that z 0 is a symplectically degenerated maximum. By Lemma 3.37, there exists a coordinate transformation such that in the new coordinate system the Jacobian of each f j at z 0 has the form

1 c j 0 1
where c j is a non-positive real number. We consider everything in the new coordinate system. Each f j can be locally generated by a generating function g j , and the Hessian of g j at z 0 has the form 0 0 0 c j .

By Lemma 3.38, z 0 is a local maximum of each g j . So, by Lemma 3.39, we can construct a generating function g such that z 0 is a local maximum of g , -Hess(g

)(z 0 ) is degenerate, -g generates f = f k-1 • • • f 0 in a neighborhood of z 0 . So, by Proposition 3.26, we know i(f, z 0 ) is equal to 1, and hence i(F, [z 0 ]) is equal to 1.
Chapter 4. Torsion-low isotopies Remark 4.2. In particular, a local homeomorphism satisfying the assumption of Theorem 1.5 also satisfies the condition of the previous lemma. But not all local isotopies can not have both a positive and a negative rotation type. As we can see in Section 4.3, there exist local isotopies that have both positive and negative rotation types. Now, we begin the proof of Theorem 1.5.

Proof of Theorem 1.5 . To simplify the notations, we suppose that the local homeomorphism is at 0 ∈ R 2 . One has to consider two cases: i(f, 0) is equal to 1 or not. a) Suppose that i(f, 0) = 1. By Proposition 2.1, there exists a unique homotopy class of local isotopies at 0 such that i(I 0 , 0) = i(f, 0) -1 = 0 for every local isotopy I 0 in this class. Let F be a locally transverse foliation of I 0 . Then i(F, 0) = i(I 0 , 0) + 1 = 1 by Proposition 2.3, and therefore 0 is neither a sink nor a source of F. This implies that I 0 has neither a positive nor a negative rotation type. So, I 0 has a zero rotation type at z 0 . For a local isotopy I at 0 that is not in the homotopy class of I 0 , by Proposition 2.14, it has only a positive rotation type if I > I 0 , and has only a negative rotation type if I < I 0 . Then, both statements of Theorem 1.5 are proved. b) Suppose that i(f, 0) = 1. Let I be a local isotopy of f , and F be an oriented foliation that is locally transverse to I. Since there exists a neighborhood U ⊂ W of 0 that contains neither the positive nor the negative orbit of any wandering open set, one knows (see the remark following Proposition 2.12) that 0 is either a sink, a source or a saddle of F. As recalled in Proposition 2.12, in the first two cases i(F, 0) is equal to 1, and in the last case i(F, 0) is not positive. By Proposition 2.3 one deduces that i(F, 0) = 1 because i(f, 0) = 1. So, 0 is a sink or a source. Therefore, I has exactly one of the three rotation types by Lemma 4.1.

Since there exists a neighborhood U ⊂ W of 0 that contains neither the positive nor the negative orbit of any wandering open set, one deduces by Proposition 2.18 that ρ s (I, 0) is not empty, and knows that f satisfies the local intersection condition. Moreover, 0 is an isolated fixed point, so one can deduce by the first three assertions of Proposition 2.20 that there exists k ∈ Z such that ρ s (I, 0) is a subset of [k, k + 1]. By the assertion i) of Proposition 2.20, there exists a local isotopy I 0 of f such that ρ s (I 0 , 0) is a nonempty subset of [0, 1] and is not reduced to 1. Then, as a corollary of the assertions iv)-vi) of Proposition 2.20, -I has a positive rotation type if I > I 0 , -I has a negative rotation type if I < I 0 .

Remark 4.3. It is easy to see that the condition that there exists a neighborhood U ⊂ W of 0 that contains neither the positive nor the negative orbit of any wandering open set is necessary for the first assertion of the theorem. Indeed, if we do not require this condition, even if f satisfies the local intersection condition, there still exists local isotopies that have both positive (resp. negative) and zero rotation types. We will give one such example in Section 4.3.

Remark 4.4. Matsumoto [START_REF] Matsumoto | Types of fixed points of index one of surface homeomorphisms[END_REF] defined a notion of positive and negative type for an orientation and area preserving local homeomorphism at an isolated fixed point with Lefschetz index 1. In this case, our definitions of "positive rotation type" (resp. "negative rotation type") is equivalent to his definition of "positive type" (resp. negative type"). Now, let us prove Proposition 1.6.

The existence of a global torsion-low isotopy 67

Proof of Proposition 1.6. The first statement is just a corollary of the definition of the torsion-low property and the assertions i), iv) of Proposition 2.20. Suppose now that f can be blown-up at z 0 . If f satisfies the hypothesis, ρ s (I, z 0 ) is not empty by Proposition 2.18. So, using the assertion vii) of Proposition 2.20, one deduces that ρ s (I, z 0 ) is reduced to a single point in [-1, 1]. Suppose now that f is a diffeomorphism in a neighborhood of z 0 . The first part of the third statement is just a special case of the second statement.

To conclude, let us prove the last part of the third statement. To simplify the notations, we suppose that z 0 = 0 ∈ R 2 . Since there exists a neighborhood of 0 that contains neither the positive nor the negative orbit of any wandering open set, Df (0) can not have two real eigenvalues such that the absolute values of both eigenvalues are strictly smaller (resp. bigger) than 1. Since 1 is not an eigenvalue of Df (0), one has to consider the following three cases:

-Suppose that Df (0) do not have any real eigenvalue. In this case, ρ(I, 0) is not an integer. -Suppose that Df (0) has two real eigenvalues λ 1 and λ 2 such that λ 1 < -1 < λ 2 < 0.

In this case, ρ(I, 0) is equal to 1 2 or -1 2 , and is not an integer. -Suppose that Df (0) has two real eigenvalues λ 1 and λ 2 such that 0 < λ 1 < 1 < λ 2 .

In this case, i(f, 0) = -1, and I has a zero rotation type at 0. So, ρ(I, 0) is equal to 0. Anyway, we know that ρ(I, 0) belongs to (-1, 1).

The existence of a global torsion-low isotopy

Let f be an orientation and area preserving homeomorphism of a connected oriented surface M that is isotopic to the identity. The main aim of this section is to prove the existence of a torsion-low maximal isotopy of f , i.e. Theorem 1.7.

When Fix(f ) = ∅, the theorem is trivial, and so we suppose that Fix(f ) = ∅ in the following part of this section. Recall that I is the set of couples (X, I X ) that consists of a closed subset X ⊂ Fix(f ) and an identity isotopy I X of f on M that fixes all the points in X. We denote by I 0 be the set of (X, I X ) ∈ I such that I X is torsion-low at every z ∈ X. Recall that is Jaulent's preorder defined in Section 2.6. Then, Theorem 1.7 is just an immediate corollary of the following theorem. Moreover, the proof do not need any other assumptions when Fix(f ) is totally disconnected, while we should admit the yet unpublished results of Béguin, Le Roux and Crovisier stated in Section 2.6 when Fix(f ) is not totally disconnected. Theorem 4.5. Given (X, I X ) ∈ I 0 , there exists a maximal extension (X , I X ) of (X, I X ) that belongs to I 0 . Remark 4.6. We will see that, except in the case where M is a sphere and X is reduced to a point, I X and I X are equivalent as local isotopies at z, for every z ∈ X. In the case where M is a sphere and X is reduced to one point, this is not necessary the case. We will give an example in Section 4.3. Remark 4.7. One may fail to find a torsion-low maximal identity isotopy I such that 0 ∈ ρ s (I, z) for every z ∈ Fix(I) that is not isolated in Fix(f ). We will give an example in Section 4.3. In particular, in this example, for every torsion-low maximal identity isotopy, there is a point that is isolated in Fix(I) but is not isolated in Fix(f ).

Before proving this theorem, we will first state some properties of a torsion-low maximal isotopy.

Chapter 4. Torsion-low isotopies Proposition (Proposition 1.10). Let f be an area preserving homeomorphism of M that is isotopic to the identity, I be a maximal identity isotopy that is torsion-low at z ∈ Fix(I), and F be a transverse foliation of I. If z is an isolated singularity of F, then z is a saddle of F and i(F, z

) = i(f, z), if z is an isolated fixed point of f such that i(f, z) = 1; -z is a sink or a source of F if z is an isolated fixed point such that i(f, z) = 1 or if z is not isolated in Fix(f ).
Proof. One has to consider two cases: z is isolated in Fix(f ) or not. i) Suppose that z is isolated in Fix(f ), then as a corollary of Theorem 1.5, z is neither a sink nor a source of

F if i(f, z) = 1; -z is a sink or a source of F if i(f, z) = 1.
Moreover, in the first case, z is a saddle of F and i(F, z) = i(f, z) by Proposition 2.3 and the remark that follows Proposition 2.12. ii) Suppose that z is not isolated in Fix(f ). Let D be a small closed disk containing z as an interior point such that D does not contain any other fixed point of I, and V ⊂ D be a neighborhood of z such that for every z ∈ V , the trajectory of z along I is contained in D. We define the rotation number of a fixed point z ∈ V \ {z} to be the integer k such that its trajectory along I is homotopic to k∂D in D \ {z}.

Then, by the maximality of I, the rotation number of a fixed point z ∈ V \ {z} is nonzero, and 0 is not an interior point of the convex hull of ρ s (I, z), as tells us the assertion iii) of Proposition 2.20. Since z is accumulated by fixed points of f , there exist k 0 ⊂ Z \ {0} ∪ {±∞} and a sequence of fixed points {z n } n∈N converging to z, such that their rotation numbers converge to k 0 . Then, k 0 belongs to ρ s (I, z).

When k 0 > 0, ρ s (I, z) is included in [0, +∞] and not reduced to 0. By the assertion v) of Proposition 2.20, one deduces that z is a sink of F. For the same reason, when k 0 < 0, we deduce that z is a source of F.

The following result is an immediate corollary of Theorem 1.7 and Proposition 1.6.

Corollary 4.8 (Proposition 1.11). Let f be an area preserving diffeomorphism of M that is isotopic to the identity. Then, there exists a maximal isotopy I, such that for all z ∈ Fix(I), the rotation number satisfies

-1 ≤ ρ(I, z) ≤ 1.
Moreover, the inequalities are both strict if z is not degenerate.

Remark 4.9. One may fail to get the strict inequalities without the assumption of nondegenerality. We will give an example in Section 4.3. Now, we begin the proof of Theorem 4.5. We first note the following fact which results immediately from the definition:

If (Y, I Y ) ∈ I and z ∈ Y is a point such that I Y is not torsion-low at z, then z is isolated in Y .
Then, given such a couple (Y, I Y ) ∈ I, we will try to find an extension (Y , I Y ) of (Y \ {z}, I Y ) and z ∈ Y \ (Y \ {z}) such that I Y is torsion-low at z .

We will divide the proof into two cases. Unlike the second case, the first case does not use the result of Béguin, Le Roux and Crovisier stated in Section 2.6, but only use Jaulent's results.

Proof of Theorem 4.5 when Fix(f ) is totally disconnected

We suppose that Fix(f ) is totally disconnected in this subsection. In this case, Theorem 4.5 is a corollary of Zorn's lemma and the following Propositions 4.10-4.13. We will explain first why the propositions imply the theorem, then we will prove the four propositions one by one. We will also give a proof of Proposition 1.9 at the end of this subsection. Proposition 4.12. When M is a plane and f admits a fixed point, (X, I X ) ∈ I 0 is not maximal in (I 0 , ) if X = ∅. Proposition 4.13. When M is a sphere, (X, I X ) ∈ I 0 is not maximal in (I 0 , ) if #X ≤ 1.

Remark 4.14. Proposition 4.12 and 4.13 deal with two special cases. The first is easy, while the second is more difficult. Indeed, to find an identity isotopy on a plane that is torsion-low at one point, we do not need to know the dynamics at infinity; but to find an identity isotopy on a sphere that is torsion-low at two points, we need check the properties of the isotopy near both points.

Proof of Theorem 4.5 when Fix(f ) is totally disconnected. Fix (X, I X ) ∈ I 0 . Let I * be the set of equivalent classes of the extensions (X , I X ) ∈ I 0 of (X, I X ). Then, the preorder induces a partial order over I * . To simplify the notations, we still denote by this partial order. By Proposition 4.10, (I * , ) is a partial ordered set satisfying the condition of Zorn's lemma, so (I * , ) contains at least one maximal element. Choose one representative (X , I X ) of a maximal element of (I * , ). It is an extension of (X, I X ) and is maximal in (I 0 , ).

Using Proposition 4.11-4.13, we will prove by contradiction that a maximal couple (X, I X ) ∈ (I 0 , ) is also maximal in (I, ). Suppose that there exists a couple (X, I X ) ∈ I 0 that is maximal in (I 0 , ) but is not maximal in (I, ). Fix a maximal extension (Y, I Y ) of (X, I X ) in (I, ), and z ∈ Y \ X. Then, I Y is not torsion-low at z, and so z is isolated in Y . Write Y 0 = Y \ {z}. By Proposition 4.12 and 4.13, M \ Y 0 is neither a sphere nor a plane. By Proposition 4.11, there exist a maximal extension (Y , I Y ) of (Y 0 , I Y ) and z ∈ Y , such that I Y is torsion-low at z . Then (X ∪ {z }, I Y ) ∈ I 0 is an extension of (X, I X ), which contradicts the maximality of (X, I X ) in (I 0 , ).

Proof of Proposition 4.10. By Proposition 2.7, we know that there exists an upper bound (X ∞ , I X∞ ) ∈ I of the chain, where X ∞ = ∪ α∈J X α . We only need to prove that (X ∞ , I X∞ ) ∈ I 0 .

When J is finite, the result is obvious. We suppose that J is infinite. Fix z ∈ X ∞ . Either it is a limit point of X ∞ , or there exists α 0 ∈ J such that z is an isolated point of X α for all α ∈ J satisfying (X α 0 , I Xα 0 ) (X α , I Xα ). In the first case, 0 ∈ ρ s (I X∞ , z); in the second case, I X∞ is locally homotopic to I Xα 0 at z. In both case, I X∞ is torsion-low at z. Before proving Proposition 4.11, we will first prove the following two lemmas (Lemma 4.15 and 4.16). We will use Lemma 4.15 when proving Lemma 4.16, and we will use Lemma 4.16 when proving Proposition 4.11. Proof. Fix a couple (Y, I Y ) maximal in (I, ) and z 0 ∈ Y satisfying the assumptions of this lemma. Then, z 0 is an isolated point of Y . Write Y 0 = Y \ {z 0 }. Then Y 0 is a closed subset and M \Y 0 is neither a sphere nor a plane. Due to Remark 2.21, one has to consider the following four cases: i) z 0 is an isolated fixed point of f and there exists a local isotopy I z 0 > I Y at z 0 which does not have a positive rotation type; ii) z 0 is not an isolated fixed point of f and ρ s (I Y , z 0 ) ⊂ [-∞, -1); iii) z 0 is an isolated fixed point of f and there exists a local isotopy I z 0 < I Y at z 0 which does not have a negative rotation type; iv) z 0 is not an isolated fixed point of f and ρ s (I Y , z 0 ) ⊂ (1, +∞]. We will study the first two cases, the other ones can be treated in a similar way.

Let F Y be a transverse foliation of I Y . In case i), by Theorem 1.5, there exists a local isotopy I 0 at z 0 that is torsion-low at z 0 , and we know that I Y < I z 0 I 0 , so I Y has a negative rotation type at z 0 ; in case ii), we know that I Y has a negative rotation type at z 0 by the assertion v) of Proposition 2.20 and the fact that ρ s (I Y , z 0 ) ⊂ [-∞, -1). Anyway, z 0 is a source of F Y . We denote by W the repelling basin of z 0 for F Y .

Let π Y 0 : M Y 0 → M \ Y 0 be the universal cover, I = ( f t ) t∈[0,1] be the identity isotopy that lifts I Y | M \Y 0 , f = f 1 be the induced lift of f | M \Y 0 , and F be the lift of F Y . Then, I fixes every point in π -1 Y 0 {z 0 }, and every point in π -1 Y 0 {z 0 } is a source of F. We fix one element z 0 in π -1 Y 0 {z 0 }, and denote by W the repelling basin of z 0 for F. Let J z 0 be an identity isotopy of the identity map of M Y 0 that fixes z 0 and satisfies ρ s (J z 0 , z 0 ) = {1}. Let I * be a maximal extension of ({ z 0 }, J z 0 I), and F * be a transverse foliation of I * .

Because M \ Y 0 is neither a sphere not a plane, π -1 Y 0 {z 0 } is not reduced to one point, and W is a proper subset of M Y 0 . Moreover, if we consider the end ∞ as a singularity, the disk bounded by the union of {∞} and a leaf of F in the boundary of W is a petal. Consequently, f can be blown-up at ∞ by the criteria in Section 2.10. On the other hand, ∞ is accumulated by the points of π -1 Y 0 {z 0 }, so 0 belongs to ρ s ( I, ∞). Therefore, ρ s ( I, ∞) is reduced to 0 by the assertion vii) of Proposition 2.20, and ρ s ( I * , ∞) is reduced to -1 by the first assertion of Proposition 2.20.

We can assert that I * has finitely many fixed points. We will prove it by contradiction. Suppose that I * fixes infinitely many points. Because ρ s ( I * , ∞) is reduced to -1, ∞ is not accumulated by fixed points of I * . Since I fixes each point in π -1 Y 0 {z 0 }, I * does not fix any point in π -1 Y 0 {z 0 } \ { z 0 }. Since I Y is not torsion-low at z 0 , z 0 is isolated in Fix( I * ) (otherwise, z 0 is accumulated by fixed points of f and -1 ∈ ρ s (I Y , z 0 )). Therefore, there exists a non-isolated point z in Fix( I * ) such that z = π Y 0 ( z ) = z 0 , and one knows that 0 belongs to ρ s ( I * , z ). Moreover, z is a non-isolated fixed point of f . By Proposition 2.6, there exists an extension (Y , I Y ) of (Y 0 , I Y ) that fixes z . Let I be the identity isotopy that lifts I Y | M \Y 0 . Since π -1 Y 0 (z ) is included in Fix( I ), we have ρ s ( I , ∞) = 0. Therefore, I and J -1 z I * are equivalent as local isotopies at ∞, where J z is an identity isotopy of the identity map of M Y 0 that fixes z and satisfies ρ s (J z , z ) = {1}. Recall that π 1 (homeo 0 (R 2 , 0)) ∼ = Z, so I and J -1 z I * are also equivalent as local isotopies at z (see Section 2.3), which means that -1 belongs to ρ s (I Y , z ). So, I Y is torsion-low at z , which contradicts the assumption of this lemma.

Since ρ s ( I * , ∞) is reduced to -1, the assertion v) of Proposition 2.20 tells us that ∞ is a source of F * . We can assert that z 0 is not a sink of F * . Indeed, in case i), one knows that I * I z 0 as a local isotopy at z 0 , and that I z 0 does not have a positive rotation type, so I * does not have a positive rotation type; in case ii), one knows that ρ s ( I * , z 0 ) = ρ s (J z 0 I, z 0 ) ⊂ [-∞, 0), and the result is a corollary of the assertion v) of Proposition 2.20.

In M Y 0 {∞}, there does not exist any closed leaf or oriented simple closed curve that consists of leaves and singularities of F * with the orientation inherited from the orientation of leaves. We can prove this assertion by contradiction. Let Γ be such a curve. Since ∞ is a source of F * , it does not belong to Γ. Let U be the bounded component of M Y 0 \ Γ, then U contains the positive or the negative orbit of a wandering open set in U \ f (U ) or U \ f -1 (U ) respectively. This contradicts the area preserving property of f . Then, we can give a partial order < over the set of singularities of F * such that z < z if there exists a leaf or a connection of leaves and singularities with the orientation inherited from the orientation of leaves from z to z. Since F * has only finitely many singularities, there exists a minimal singularity z 1 . Moreover, F * does not have any closed leaf or a leaf from z 1 , and hence z 1 is a sink of F * . Therefore, f fixes z 1 and hence there exists a maximal extension (Y

1 , I Y 1 ) of (Y 0 , I Y ) such that Y 0 ∪ {z 1 } ⊂ Y 1 , where z 1 = π Y 0 ( z 1 ).
Now, we will prove by contradiction that Y 1 \ Y 0 contains at least two points. Suppose that Y 1 = Y 0 {z 1 }. Let F Y 1 be a transverse foliation of I Y 1 , I 1 be the identity isotopy that lifts I Y 1 | M \Y 0 , and F 1 be the lift of andz 1 is an isolated singularity of F 1 , so it is a sink, or a source, or a saddle of F 1 by Remark 2.13. We know that ρ s ( I * , ∞) is reduced to -1 and that ρ s ( I 1 , ∞) is reduced to 0, so I * and J z 1 I 1 are equivalent as local isotopies at z 1 . By the assumption, I Y 1 is not torsion-low at z 1 , so z 1 is a sink of F 1 , and z 1 is a sink of F Y 1 . Let W 1 be the attracting basin of z 1 for F 1 . A leaf in ∂ W 1 is a proper leaf. For every fixed point z of f , there exists a loop δ that is homotopic to its trajectory along

F Y 1 to M Y 0 . We know that (Y 0 , I Y ) ∼ (Y 0 , I Y 1 ), so the lift of f | M \Y 0 to M Y 0 associated to I Y 1 is also f . The set of singularities of F 1 is π -1 Y 0 {z 1 },
I 1 in M Y 0 \ π -1 Y 0 {z 1 } (so in M Y 0 \ {z 1 })
and is transverse to F 1 . The linking number L( I 1 , z, z 1 ) is the index of the trajectory of z along I 1 relatively to z 1 , so it is equal to the index of δ relatively to z 1 . When z is in W 1 , the loop δ is included in W 1 and is transverse to F 1 , so L( I 1 , z, z 1 ) is positive. When z is not in W 1 , it is in one of the connected component of M Y 0 \ W 1 , and so is δ, therefore L( I 1 , z, z 1 ) is equal to 0. Since I * fixes z 0 and z 1 , the linking number L( I * , z 0 , z 1 ) is equal to 0. Referring to Section 2.15, we know that

L( I 1 , z 0 , z 1 ) = L( I * , z 0 , z 1 ) -1 = -1,
and find a contradiction.

The following lemma is a consequence of the previous one.

, I Y 1 ) of (Y \ {z}, I Y ) such that #(Y 1 \ (Y \ {z})) > 1. If #(Y 1 \ (Y \ {z})) = ∞, the proof is finished; if #(Y 1 \ (Y \ {z}) < ∞, we fix a point z 1 ∈ Y 1 \ (Y \ {z}).
By hypothesis, I Y 1 is not torsion-low at z 1 and M \ (Y 1 \ {z 1 }) is neither a sphere nor a plane. Since a maximal extension of (Y 1 \{z 1 }, I Y 1 ) is also a maximal extension of (Y \{z}, I Y ), the couple (Y 1 , I Y 1 ) and z 1 ∈ Y 1 satisfies the assumptions of the previous lemma. We apply the previous lemma, and deduce that there exists a maximal extension (Y

2 , I Y 2 ) ∈ I of (Y 1 \ {z 1 }, I Y 1 ) such that #(Y 2 \ (Y 1 \ {z 1 })) > 1. If #(Y 2 \ (Y 1 \ {z 1 })) = ∞, the proof is finished; if #(Y 2 \ (Y 1 \ {z 1 })) < ∞, we continue the construction. . .
Then, either we end the proof in finitely many steps, or we can construct a strictly increasing sequence Y 0 {z} such that z 0 and z are linked relatively to I.

(Y \ {z}, I Y ) ≺ (Y 1 \ {z 1 }, I Y 1 ) ≺ (Y 2 \ {z 2 }, I Y 2 ) ≺ (Y 3 \ {z 3 }, I Y 3 ) • • • By Proposition 2.
Proof. Let F be a transverse foliation of I Y , and F be the lift of

F| M \Y 0 to M Y 0 . Fix z ∈ Y \ Y 0 and z ∈ π -1
Y 0 {z}. Since I Y is a maximal identity isotopy, the trajectory of z along I is a loop that is not homotopic to zero in M Y 0 \ π -1 Y 0 {z 0 }. Let δ be a loop that is transverse to F, and is homotopic to the trajectory of z along I in M Y 0 \ π -1 Y 0 {z 0 }. By choosing suitable δ, we can suppose that δ intersects itself at most finitely many times, that each intersection point is a double point, and that the intersections are transverse. So, M Y 0 \ δ has finitely many components, and we can define a locally constant function Λ :

M Y 0 \ δ → Z such that -Λ is equal to 0 in the component of M Y 0 \ δ that is not relatively compact; -Λ( z ) -Λ( z )
is equal to the (algebraic) intersection number of δ and any arc from z to z . This function is not constant, and we have either max Λ > 0 or min Λ < 0. Suppose that we are in the first case (the other case can be treated similarly). Let U be a component of M Y 0 \ δ such that Λ is equal to max Λ > 0 in U . As in the picture, the boundary of U is a sub-curve of δ with the orientation such that U is to the left of its boundary, and is also transverse to F. So, there exists a singularity of F in U . Note the fact that the set of singularities of F is Fix( I) = π -1 Y 0 {z 0 }. So, there exists an automorphism T of the universal cover space such that T ( z 0 ) belongs to U , and the index of δ relatively to T ( z 0 ) U δ δ δ is positive. Note also that the linking number L( I, z, T ( z 0 )) is equal to the index of δ relatively to T ( z 0 ), and hence is equal to Λ(T ( z 0 )) by definition of Λ. So, T ( z 0 ) and z are linked relatively to I. Consequently, z 0 and T -1 ( z) are linked relatively to I.

As in the proof of Lemma 4.15, we know that f can be blown-up at ∞. Since ∞ is accumulated by both the points in π -1 Y 0 {z 0 } and the points in π -1 Y 0 (Y \ Y 0 ), both ρ s ( I, ∞) and ρ s ( I , ∞) contain 0. Then, both ρ s ( I, ∞) and ρ s ( I , ∞) are reduced to 0, so I and I are equivalent as local isotopies at ∞. Therefore, for every point z ∈ Y \ Y 0 , there exists z ∈ π -1 Y 0 {z} such that z 0 and z are linked relatively to I . Let us denote by L the set of points z ∈ π -1 Y 0 (Y \ Y 0 ) such that z and z 0 are linked relatively to I . It contains infinitely many points.

Let γ be the trajectory of z 0 along the isotopy I , and V be the connected component of M Y 0 \ γ containing ∞. Then K = M Y 0 \ V is a compact set that contains all the fixed points of I that are linked with z 0 relatively to I . In particular, L ⊂ K. Then, there exists z ∈ K that is accumulated by points of L. We know that Fix( I ) is a closed set. So, z belongs to Fix( I ) = π -1 (Y \ Y 0 ). We find a point z that is not isolated in π -1 Y 0 (Y \ Y 0 ), and a point z = π Y ( z ) that is not isolated in Y . This means that I Y is torsion-low at z . We get a contradiction.

Proof of Proposition 4.12. We only need to prove that there exists (X, I X ) ∈ I 0 such that X = ∅, because one knows (∅, I) (X, I X ) for all (X, I X ) ∈ I when M is a plane.

One has to consider the following two cases: -Suppose that Fix(f ) is reduced to one point z 0 . In this case, similarly to the proof of Theorem 1.5, we can find an isotopy I 0 that fixes z 0 and is torsion-low at z 0 . Then, ({z 0 }, I 0 ) belongs to I 0 . -Suppose that Fix(f ) contains at least two points. In this case, there exists a maximal (Y, Proof of Proposition 4.13. One knows (X, I X ) (Y, I Y ) for all (Y, I Y ) ∈ I satisfying X ⊂ Y , when M is a sphere and #X ≤ 1. So, we only need to prove the following two facts: i) there exists (X, I X ) ∈ I 0 such that X = ∅; ii) given (X, I X ) ∈ I 0 such that #X = 1, there exists (X , I X ) ∈ I 0 such that X X . One has to consider the following two cases:

I Y ) ∈ I such that #Y ≥ 2. If I Y is torsion-low at a point in Y , the proof is finished; if I Y is not torsion-low
-Suppose that #Fix(f ) = 2. In this case, we will prove that there exists an identity isotopy that fixes both fixed points and is torsion-low at each fixed point, which implies both i) and ii).

Denote by N and S the two fixed points. Since both N and S are isolated fixed points, we can find an identity isotopy I that fixes both N and S and is torsion-low at S. We will prove that I is also torsion-low at N . Let J N (resp. J S ) be an identity isotopy of the identity map of the sphere that fixes both N and S and satisfies ρ s (J N , N ) = {1} (resp. ρ s (J S , S) = {1}). One knows that the restrictions to M \ {N, S} of J N and J -1 S are equivalent. For every k ≥ 1, since I is torsion-low at S, J -k S I has a negative rotation type as a local isotopy at S. Let F k be a transverse foliation of J -k S I. Then S is a source of F k . Since f is area preserving and F k has exactly two singularities, N is a sink of F k . Note the fact that the restrictions to M \ {S, N } of J k N I and J -k S I are homotopic. So, J k N I has a positive rotation type as a local isotopy at N . Similarly, for every k ≥ 1, J -k N I has a negative rotation type as a local isotopy at N . Therefore, I is torsion-low at N . -Suppose that #Fix(f ) ≥ 3.

In this case, there exists (Y, I Y ) ∈ I such that #Y ≥ 3. We can prove i) by a similar discussion to the second part of the proof of Proposition 4.12. We can also give the following direct proof. -N is an isolated fixed point of f and there exists a local isotopy I N > I Y at N which does not have a positive rotation type; -N is not an isolated fixed point of f and ρ s (I Y , N ) ⊂ [-∞, -1); -N is an isolated fixed point of f and there exists a local isotopy I N < I Y at N which does not have a negative rotation type; -N is not an isolated fixed point of f and ρ s (I Y , N ) ⊂ (1, +∞]. As before, we study the first two cases. Let F Y be a transverse foliation of I Y . As in the proof of Lemma 4.15, N is a source of F Y . Since f is area preserving and F Y has exactly two singularities, S is a sink of F Y . Let I be a maximal extension of (Y, J N I Y ). Since I Y is torsion-low at S and I is equivalent to J -1 S I Y as local isotopies at S, I has a negative rotation type at S. Moreover, as local isotopies at S, J k S I ∼ J k-1 S I Y has a positive rotation type at S for k ≥ 1, and has a negative rotation type for k ≤ -1. Therefore I is torsion-low at S. Let F be a transverse foliation of I . One knows that S is a source of F . Like in the proof of Lemma 4.15, we deduce that N is not a sink of F by our assumption at the beginning of this case. Therefore, F has another singularity, and hence one deduces that Fix(I ) ≥ 3. b) Suppose that S is not isolated in Fix(f ). We know that ρ s (I Y , S) ∩ [-1, 1] = ∅ by definition. We define the rotation number of a fixed point near S as in the proof of Proposition 1.10. By the maximality of I Y , the rotation number of a fixed point near S is not zero. Then, either there exists k ∈ Z \ {0} such that S is accumulated by fixed points of f with rotation number k, or ρ s (I Y , S) intersects {±∞}. In the second case, the interior of the convex hull of ρ s (I Y , S) contains a non-zero integer k , and hence 0 is in the interior of the convex hull of ρ s (J -k S I Y , S). So, S is accumulated by contractible fixed points of J -k S I Y by the assertion iii) of Proposition 2.20, and hence is accumulated by fixed points with rotation k (associate to I Y ). Anyway, there exists k ∈ Z \ {0} such that S is accumulated by fixed points of f with rotation number k. We fix one such k. Let I be a maximal extension of J -k S I Y . Then, I fixes at least 3 fixed points, and 0 belongs to ρ s (I , S). Therefore, I is torsion-low at S, and satisfies #Fix(I ) ≥ 3. Remark 4.18. In both case a) and case b), we construct an identity isotopy I that is torsion-low at S and has at least three fixed points. Even though ρ s (I X , S) and ρ s (I , S) are different, I is still an extension of (X, I X ) because M is a sphere and X is reduced to a single point. However, as was in Remark 4.6, for (X , I X ) ∈ I 0 that is a maximal extension of (X, I X ), I X and I X are not necessarily equivalent as local isotopies at S. Now, let us prove Proposition 1.9.

Proof of Proposition 1.9. Let f be an area preserving homeomorphism of M that is isotopic to the identity and has finitely many fixed points. When Fix(f ) is empty, the proposition is trivial. So, we suppose that Fix(f ) is not empty. Let n = max{#Fix(I) : I is an identity isotopy of f }.

One has to consider the following three cases:

-Suppose that M is a plane and f has exactly one fixed point. As in the first part of the proof of Proposition 4.12, there exists an identity isotopy that fixes this fixed point and is torsion-low at this fixed point. -Suppose that M is a sphere and f has exactly two fixed points. As in the first part of the proof of Proposition 4.13, there exists an identity isotopy that fixes these two fixed points and is torsion-low at each fixed point. -Suppose that we are not in the previous two cases. Let I be the set of identity isotopies of f with n fixed points. It is not empty. We can give a preorder over I such that I I if and only if #{z ∈ Fix(I), I is torsion-low at z} ≤ #{z ∈ Fix(I ), I is torsion-low at z}.

Since #{z ∈ Fix(I), I is torsion-low at z} is not bigger than n for all I ∈ I, I has a maximal element. Fix a maximal element I of I. We will prove by contradiction that I is torsion-low at every z ∈ Fix(I).

z 0 is an isolated fixed point of f and there exists a local isotopy I z 0 > I Y at z 0 which does not have a positive rotation type; z 0 is not an isolated fixed point of f and ρ s (I Y , z 0 ) ⊂ [-∞, -1); z 0 is an isolated fixed point of f and there exists a local isotopy I z 0 < I Y at z 0 which does not have a negative rotation type; z 0 is not an isolated fixed point of f and ρ s (I Y , z 0 ) ⊂ (1, +∞]. As before, we only study the first two cases.

Let F Y be a transverse foliation of I Y . As in the proof of Lemma 4.15, we know that z 0 is a source of F Y .

Since #∂M Y 0 > 1, the plane M Y 0 can be blown-up by prime-ends at infinity. Because I Y fixes ∂M Y 0 and z 0 , I Y | M Y 0 can be viewed as a local isotopy at ∞, and the blow-up rotation number ρ(I Y | M Y 0 , ∞), that was defined in Section 2.10, is equal to 0.

Let I * be a maximal extension of ({z 0 }, J z 0 I Y | M Y 0 ), and F * be a transverse foliation of I * . Note that I Y is not torsion-low at z 0 , by the same argument of the proof of Lemma 4.15, we know that z 0 is not a sink of F * .

We can assert that ∞ is a source of F * . Indeed, when the total area of

M Y 0 is finite, f | M Y 0 is area preserving as a local homeomorphism at ∞, so ρ s (I Y | M Y 0 , ∞)
is not empty by Proposition 2.18 and is reduced to 0 by the assertion vii) of Proposition 2.20. Then, by the assertion i) of Proposition 2.20, ρ s (I * , ∞) is reduced to -1, and by the assertion v) of Proposition 2.20, ∞ is a source of F * . However, the total area of M Y 0 may be infinite. In this case, we can not get the result that ρ s (I Y | M Y 0 , ∞) is not empty. But anyway, we can prove the assertion by considering the following two cases:

-Suppose that ρ s (I Y | M Y 0 , ∞) is not empty. As in the case where the total area of

M Y 0 is finite, ρ s (I Y | M Y 0 , ∞) is reduced to 0, and ρ s (I * , ∞) is reduced to -1. Therefore, ∞ is a source of F * by the assertion v) of Proposition 2.20. -Suppose that ρ s (I Y | M Y 0 , ∞) is empty. Since f | M Y 0 is area preserving, f | M Y 0 is not
conjugate to a contraction or an expansion at ∞. By Proposition 2.18, the germ of f | M Y 0 at ∞ is conjugate to a local homeomorphism z → e i2π p q z(1 + z qr ) at 0 with q, r ∈ N and p ∈ Z. Since ρ(I Y | M Y 0 , ∞) = 0, we can deduce that p ∈ qZ. Therefore, one has i(f | M Y 0 , ∞) > 1. Let I 0 = (g t ) t∈[0,1] be a local isotopy at 0 such that g t (z) = z(1 + tz qr ). Then, ρ(I 0 , 0) is equal to 0 and i(I 0 , 0) is positive. Since

ρ(I Y | M Y 0 , ∞) is equal to 0, I Y | M Y 0 is conjugate to a local isotopy that is in the same homotopy class of I 0 . So, i(I Y | M Y 0 , ∞) = i(I 0 , 0) is positive. Therefore, ∞
is a source of F * by Proposition 2.14. Then, like in the proof of Lemma 4.15, we deduce that I * fixes finitely many points, that there exists a sink z 1 of F * , and that there exists a maximal extension (Y , I Then, one has to consider two cases:

Y ) ∈ I of (Y 0 , I Y ) such that Y 0 ∪ {z 1 } ⊂ Y and #(Y \ Y 0 ) > 1.
-M Y 0 is neither a sphere nor a plane, -M Y 0 is a plane whose boundary contains more than two points. In the first case, we repeat the proof of Proposition 4.11 except that we should replace M \ Y 0 with M Y 0 . In the second case, the idea is similar, but we do not lift the isotopies to the universal cover because M Y 0 itself is a plane.

Proof of Proposition 4.21. As in the proof of Proposition 4.12, we only need to prove that there exists (X, I X ) ∈ I 0 such that X = ∅.

Since Fix(f ) is not totally disconnected, we can fix a connected component X of Fix(f ) that is not reduced to a point. By Proposition 2.11, there exists a maximal identity isotopy I of f that fixes all the points in X. So, 0 belongs to ρ s (I, z) for all z ∈ X, and hence (X, I) belongs to I 0 .

Proof of Proposition 4.22. As in the proof of Proposition 4.13, we only need to prove the following two facts: i) there exists (X, I X ) ∈ I 0 such that X = ∅; ii) given (X, I X ) ∈ I 0 such that #X = 1, there exists (X , I X ) ∈ I 0 such that X X . The proof of the first fact is the same to the proof of Proposition 4.21; while the proof of the second fact is similar to the proof of Proposition 4.13 in the case #Fix(f ) ≥ 3.

Examples

In this section, we will give some explicit examples to get the optimality of previous results.

Exemple 4.26. (A local isotopy that has both positive and negative rotation types)

Write f t for the homothety of factor 1 + t of a plane. One can note that f 1 has an isolated fixed point 0, and I = (f t ) t∈[0,1] has both positive and negative rotation types at 0. In fact, let

π : R × (-∞, 0) → C \ {0} R 2 \ {0} (θ, y) → -ye i2πθ
be the universal cover. Let F 1 be the foliation on R × (-∞, 0) whose leaves are the lines y = θ + c upward. It descends to an oriented foliation F 1 on C \ {0} that is locally transverse to I. Moreover, 0 is a sink of F 1 . Let F 2 be the foliation on R × (-∞, 0) whose leaves are the lines y = -θ + c downward. It descends to an oriented foliation F 2 on C \ {0} that is locally transverse to I. Moreover, 0 is a source of F 2 . We define a flow 2 on R 2 by

f t (x, y) =          x 2 +y 2
x 2 e -2t +y 2 e 2t (xe -t , ye t ) for x ≥ 0, y ≥ 0, (xe -t , ye -t ) for x ≤ 0, y ≥ 0, (xe -t , ye t ) for x ≤ 0, y ≤ 0, (xe t , ye t ) for x ≥ 0, y ≤ 0.

It is the flow of the (time-independent) continuous vector field V in the plane R 2 , where V is defined by

V (x, y) =          ( x(x 2 -3y 2 ) x 2 +y 2 , y(3x 2 -y 2 )
x 2 +y 2 ) for x > 0, y > 0, (-x, -y) for x ≤ 0, y ≥ 0, (-x, y) for x ≤ 0, y ≤ 0, (x, y) for x ≥ 0, y ≤ 0.

Then, f = f 1 has a unique fixed point 0, and I = (f t ) t∈[0,1] is an identity isotopy of f . We will prove that I has both positive and zero rotation types at 0 by constructing two transverse foliations F 1 , F 2 of I such that 0 is a sink of F 1 and is a mixed singularity of F 2 . We will construct F 1 by considering the integral curves of vector field. We define a 2. The flow on the first quadrant is just ft(z) = 1 ϕ t (1/z) , where z = x + iy is the complex coordinate and ϕ is the flow defined by ϕt(x, y) = (xe -t , ye t ).

Let f be a homeomorphism of C defined by f (z) = 0 for z = 0, ze i2π/|z| for z = 0.

It is area preserving and fix 0. Moreover, ρ s (I, 0) is reduced to ∞ for every isotopy I of f fixing 0. We will construct an orientation preserving diffeomorphism f of the sphere with 2 fixed points such that f is area preserving in a neighborhood of each fixed point but there does not exist any torsion-low maximal identity isotopy of f .

Let ϕ be a diffeomorphism of [0, 1] that satisfies ϕ(y) = y for y ∈ [0, 1/6] ∪ [5/6, 1], ϕ(y) < y for y ∈ (1/6, 5/6).

Let g be a diffeomorphism of R × [0, 1] that is defined by g(x, y) = (x + 3y, ϕ(y)).

We define an equivalent relation ∼ on R × [0, 1] such that

    
(x, y) ∼ (x + 1, y) for all (x, y) ∈ R × (0, 1) (x, 0) ∼ (x , 0) for all x, x ∈ R (x, 1) ∼ (x , 1) for all x, x ∈ R.

Then, R × [0, 1]/ ∼ is a sphere, and g descends to a diffeomorphism f of the sphere that has two fixed points and is area preserving near each fixed point. Note the facts that every maximal identity isotopy I fixes both fixed points of f , that the rotation number of I at each fixed point is an integer, and that the sum of the rotation numbers of I at both fixed point is 3. By Proposition 1.6, there does not exist any torsion-low maximal identity isotopy of f . Exemple 4.30. (Example of Remark 4.6)

In this example, we will construct an isotopy I * on the sphere such that I * is torsionlow at a fixed point z, but there does not exist any torsion-low maximal isotopy that is equivalent to I * as a local isotopy at z.

We will induce the isotopy by generating functions (see Appendix 4.4).

Let ϕ be a smooth 1-periodic function on R that satisfies Then, g is constant on R×(-∞, 0] and on R×[1, ∞) respectively, and satisfies g(x+1, y) = g(x, y). Moreover, one knows ∂ 2 12 g(x, y) = 0 for y ≤ 0 or y ≥ 1, πϕ(y) sin(2πx) for 0 < y < 1. So, ∂ 2 12 g ≤ 1 2 < 1. Therefore, g defines an identity isotopy I = (f t ) t∈[0,1] by the following equations: f t (x, y) = (X t , Y t ) ⇔ X t -x = t∂ 2 g(X t , y), Y t -y = -t∂ 1 g(X t , y),

For every t ∈ [0, 1], f t is the identity on R×(-∞, 0]∪R×[1, ∞), and satisfies f t (x+1, y) = f t (x, y). Moreover, for every t ∈ (0, 1], a point (x, y) is a fixed point of f t if and only if it is a critical point of g. Let F be the foliation whose leaves are the integral curves of the gradient vector field (x, y) → (∂ 1 g(x, y), ∂ 2 g(x, y)) of g. As will be proved in Appendix 4.4, F is a transverse foliation of I.

We know that ∂ 1 g(x, y) = 0 for y ≤ 0 or y ≥ 1, π sin(2πx) y 0 ϕ(s)ds for 0 < y < 1, and that ∂ 2 g(x, y) = 0 for y ≤ 0 or y ≥ 1, y sin 2 π y + ϕ(y) sin 2 (πx) for 0 < y < 1.

So, the set of critical points of g is The fixed point S is not isolated in Fix(I ), and so ρ s (I , S) is reduced to 0; N is isolated in Fix(f ) and is a sink of F ; and all the other fixed points of f are isolated in Fix(f ) and are saddles of F . Let I * be an identity isotopy of f fixing S such that ρ s (I * , S) is reduced to -1. Then, I * is torsion-low at S. We will prove that there does not exist any torsion-low maximal isotopy I such that ρ s (I , S) is reduced to -1.

C = {(n, 1
Indeed, a maximal identity isotopy of f fixes either all the fixed points of f (in which case, the isotopy is homotopic to I relatively to Fix(f )) or exactly two fixed points. If I is a maximal identity isotopy of f such that ρ s (I , S) is reduced to -1, then I fixes exactly two fixed points. Denote by {S, z 1 } the set of fixed points of I . One knows that z 1 is an isolated fixed point of f , and that J -1 z 1 I is equivalent to I as local isotopies at z 1 . Therefore, J -1 z 1 I does not have a negative rotation type at z 1 , and hence I is not torsion-low at z 1 .

Exemple 4.31. (Example of Remark 4.7)

In this example, we will construct an orientation and area preserving homeomorphism f of the sphere such that there does not exist any maximal identity isotopy I of f such that 0 ∈ ρ s (I, z) for every z ∈ Fix(I) that is not an isolated fixed point of f .

Let g be a homeomorphism on R × [0, 1] that is defined by

g(x, y) =     
(x, y) for 0 ≤ y ≤ 1 3 , (x + 3y -1, y) for 1 3 < y ≤ 2 3 , (x + 1, y) for 2 3 < y ≤ 1.

We define an equivalent relation ∼ on R × [0, 1] such that

    
(x, 0) ∼ (x , 0) for x, x ∈ R (x, y) ∼ (x + 1, y) for 0 < y < 1, (x, 1) ∼ (x , 1) for x, x ∈ R.

Then, R × [0, 1]/ ∼ is a sphere, g descends to an orientation and area preserving diffeomorphism f of the sphere that has infinitely many fixed points, and every fixed point of f is not isolated in Fix(f ). We will prove that there does not exist any maximal isotopy I such that for all z ∈ Fix(I), one has 0 ∈ ρ s (I, z). By definition of f , one knows that f can be blown-up at each fixed point, and hence for every identity isotopy I of f and every z ∈ Fix(I), the rotation set ρ s (I, z) is reduced to ρ(I, z). Then, we only need to prove that there does not exist any maximal identity isotopy I such that ρ(I, z) = 0 for every z ∈ Fix(I).

Denote by N and S the two components of Fix(f ) respectively. Note the following fact: for z 1 , z 2 ∈ Fix(f ) and every identity isotopy I fixing both z 1 and z 2 , one can deduce

ρ(I, z 1 ) + ρ(I, z 2 ) = 0 if z 1 , z 2 ∈ S, or if z 1 , z 2 ∈ N, 1 if z 1 ∈ S, z 2 ∈ N, or if z 1 ∈ N, z 2 ∈ S.
Let us conclude the proof by observing the properties of any maximal identity isotopy of f . Indeed, if I is a maximal identity isotopy of f , it satisfies one of the following properties:

-The set of fixed points of I is the union of N (resp. S) and a point z in S (resp. N ). In this case, ρ(I, z) = 1. -The set of fixed points of I is the union of a point z 1 in N (resp. S) and a point z 2 in S (resp. N ), and the rotation numbers satisfy ρ(I, z i ) = 0 for i = 1, 2.

Chapter 5

A generalization of the local Poincaré-Birkhoff theorem

Proof of the main results

Let f be an orientation preserving homeomorphism of the annulus T 1 × [0, +∞). Let C 0 = T 1 × {0}, C 1 be an essential loop in T 1 × (0, +∞) that projects injectively to the first factor, and C 2 = f (C 1 ). We denote by A i the closed annulus bounded by C 0 and C i , and write Int(A i ) = A i \ C i for i = 1, 2. Let π : R × [0, ∞) → T 1 × [0, ∞) be the universal cover, and f be a lift of f . Let p 1 : R × [0, ∞) → R be the projection onto the first factor. We denote by C i , A i and Int( A i ) the pre-images of C i for i = 0, 1, 2 , A i and Int(A i ) for i = 1, 2 respectively. We denote by Fix * (f ) the set of fixed points of f lifted to fixed points of f . We will prove the following two results in this section.

Theorem 5.1 (Theorem 1.12). If Fix * (f )∩C 0 = {z 0 }, and if p 1 ( f ( z)-z)p 1 ( f ( z )z ) < 0 for all z ∈ C 0 \ π -1 (z 0 ) and z ∈ C 1 , then we are in one of the following cases: i) there exists a fixed point of f in the interior of A 1 lifted to fixed points of f ; ii) there exists an essential loop γ in A 1 that does not intersect C 0 \ {z 0 } and satisfies γ ∩ f (γ) ⊂ {z 0 }.

Corollary 5.2 (Corollary 1.13). If A 1 does not contain the positive or the negative orbit of any wandering open set and if there exists z ∈ C 0 such that p 1 ( f ( z) -z)p 1 ( f ( z ) -z ) < 0 for all z ∈ C 1 , then there exists a fixed point of f in the interior of A 1 lifted to fixed points of f .

Before proving the theorem, we will first prove the corollary:

Proof of Corollary 1.13. One has to consider two cases:

-there exists z ∈ C 0 such that p 1 ( f ( z) -z) < 0, and p 1 ( f ( z ) -z ) > 0 for all z ∈ C 1 ; -there exists z ∈ C 0 such that p 1 ( f ( z) -z) > 0, and p 1 ( f ( z ) -z ) < 0 for all z ∈ C 1 ; We only study the first case, the second one can be treated similarly.

Let l ⊂ R × {0} be a maximal interval such that p 1 f (x, 0) < x for all x ∈ l. Shrinking the complement of the interval π(l) in C 0 to a point, we get an annulus Ȃ and a homeomorphism f of Ȃ. Let Ȃ1 ⊂ Ȃ be the quotient space of A 1 . Let ε > 0 be a small number such that f (C 1 ) ⊂ T 1 × (ε, ∞), and h be a homeomorphism between Ȃ and A such that h is equal the identity in T 1 × [ε, ∞). Let f = h • f • h -1 , and f be the lift of f that is equal to f in R × [ε, ∞). Then, f satisfies the condition of the previous theorem but does not satisfy ii). Consequently, there exists a fixed point of f in the interior of Ȃ1 lifted to 90 Chapter 5. A generalization of the local Poincaré-Birkhoff theorem fixed point of f , and so there there exists a fixed point of f in the interior of A 1 lifted to fixed points of f . Now we begin the proof of Theorem 1.12. One has to consider two cases: p 1 ( f ( z) -z) < 0 for all z ∈ C 0 \ π -1 (z 0 ), and p 1 ( f ( z ) -z ) > 0 for all z ∈ C 1 ; p 1 ( f ( z) -z) > 0 for all z ∈ C 0 \ π -1 (z 0 ), and p 1 ( f ( z ) -z ) < 0 for all z ∈ C 1 ; We only study the first case, and the second one can be treated similarly.

We construct an open annulus A = T 1 × R by pasting two copies of the annulus A at the lower boundary C 0 , and a homeomorphism on A that coincides with f on each copy. Let σ be the natural involution. To simplify the notations, we still denote by f this homeomorphism. Let π : R 2 → A be the universal cover, and T : (x, y) → (x + 1, y) be the translation on R 2 . We still denote by f the lift of f . Let A 1 = A 1 ∪ σ(A 1 ) and A 1 = π -1 (A 1 ). Let C * 0 = C 0 \ {z 0 } and C * 0 = π -1 (C * 0 ). We denote by γ -1 the path t → γ(1 -t) for any path γ defined on [0, 1].

Let g : W → R 2 be a continuous map defined on an open set W , and γ : [0, 1] → W be a path that does not intersect the fixed points set of g, we define the index of g along γ by i(g, γ) = γ dθ, where γ : t → f (γ(t)) -γ(t) is a path in R 2 \ {0}, and dθ = 1 2π xdy-ydx x 2 +y 2 is the polar form. Remark 5.3. In particular, if γ is a simple loop, and if g is defined on the domain D bounded by γ and has only finitely many fixed points in the domain D bounded by γ, then i(g, γ) = ± z∈Fix(g)∩D i(g, z),

where the sign depends on the orientation of γ.

We will prove the theorem by contradiction. Suppose that we are in neither of the two cases of Theorem 1.12. On one hand, Fix * (f ) ∩ A 1 is reduced to z 0 , and we can prove that the Lefschetz index i( f , z 0 ) is zero for z 0 ∈ π -1 (z 0 ). On the other hand, following the approach of [LCW10] which used the notion of "positive path" and was inspired by Birkhoff's paper [START_REF] Birkhoff | An extension of Poincaré's last geometric theorem[END_REF], we will construct a loop Γ ⊂ ( A 1 ) such that i( f , Γ) = -2. We will see that Theorem 1.12 is a corollary of the following Lemma 5.4 -Lemma 5.10. Lemma 5.4. If Fix * (f )∩Int(A 1 ) is reduced to z 0 , then i( f , z 0 ) = 0 for every z 0 ∈ π -1 (z 0 ), Proof. Let γ 0 be a path in A 1 joining σ(C 1 ) to C 1 that does not intersect {z 0 }. Lift γ 0 to an arc γ joining σ( C 1 ) to C 1 . Let γ be the segment of C 1 joining γ(1) to T (γ(1)), then Γ = γγ T (γ) -1 σ(γ ) -1 is a loop, and the domains bounded by Γ contains a unique lift z 0 of z 0 . We have i( f , z 0 ) = -i( f , Γ) = -i( f , γ) -0 + i( f , γ) + 0 = 0, because f commutes with T and σ.

Let G be a homeomorphism of a topological space X. We say that a path γ : [0, 1] → X is a positive path if for every t, t ∈ [0, 1],

t ≥ t ⇒ G(γ(t )) = γ(t).
Note that if γ is a positive path of G, then G(γ) -1 is a positive path of G -1 . Let A and B be two disjoint subset of X, we say that a path γ : [0, 1] → X joins A to B if γ(0) ∈ A, γ(1) ∈ B, and γ(t) / ∈ A ∪ B for all t ∈ (0, 1).

Proof of the main results

91

Lemma 5.5. If γ is a positive path of f joining C * 0 to C 1 such that f (γ(1)) / ∈ Int( A 1 ), if γ is a positive path of f joining C 1 to C * 0 such that γ (0) / ∈ Int( A 2 ) is at the right of γ(1), and if γ is the segment of C 1 joining γ(1) to γ (0), then i( f , Γ) = -2, where Γ is the loop γγ γ σ(γ ) -1 σ(γ ) -1 σ(γ) -1 . Proof. By a suitable conjugation that preserves the vertical lines, we can suppose

A 1 = T 1 × [0, 1].
Let us consider the simplex ∆ = {(t, t ) : 0 ≤ t ≤ t ≤ 1}. Let δ v , δ h and δ d be the paths along the edges of ∆ joining (0, 0) to (0,1), joining (0, 1) to (1, 1) and joining (0, 0) to (1, 1) respectively. The application Φ(t, t ) = f (γ(t )) -γ(t) does not vanish on ∆, and hence

Φ•δ d dθ = Φ•δv dθ + Φ•δ h dθ.
The first integral is just i( f , γ). The second coordinate of the curves Φ • δ v and Φ • δ h are both non-negative, the initial point Φ(0, 0) ⊂ (-∞, 0) × {0}, the end point φ(1, 1) ⊂ (0, ∞) × [0, ∞). One deduces that

i( f , γ) ∈ [- 1 2 , - 1 4
).

The image of the path t → f (γ (t)) -γ (t) is contained in (0, ∞) × R, so

i( f , γ ) ∈ (- 1 4 , 1 4
). Now, it remains to compute i( f , γ ), which is harder than the computations of the indices of the previous two paths. Let T r (x, y) = (x + r, y). We fix a large positive number r 0 such that p 1 ( f (T r 0 (γ (t ))) -γ (0)) > 0, for t ∈ [0, 1].

Let ε be a sufficiently small positive number such that f (T r 0 (γ (t ))) -γ (0) > r 0 ε.

We construct a continuous map Ψ(t, t , r) = f (T r (γ (t ))) -T tr(1+ε) γ (t) on ∆ × [0, r 0 ]. Let us prove that Ψ does not vanish on the closed surface (∆ × {0}) ∪ {t = 0} ∪ {t = 1}. Indeed, it does not vanish on ∆ × {0}, because γ is a positive path for f . Suppose that Proof of Corollary 1.14. Suppose that I has a positive rotation type, and that f can be blown-up at z 0 . The proofs in the other cases is similar. To simplify the notations, we can also suppose that z 0 = 0 ∈ R 2 . We will prove the proposition by contradiction. Let ϕ : R 2 \ {0} → T 1 × (-∞, 0) be an orientation preserving homeomorphism such that ϕ • f • ϕ -1 can be extended continuously to T 1 × {0}. Denote by h the extension homeomorphism on T 1 × {0}. We suppose that h is not right semi-stable, and will find a contradiction.

We will first show that 0 is an isolated fixed point of f by contradiction. Suppose that 0 is accumulated by fixed points of f . Since I has a positive rotation type, there exists a locally transverse foliation F of I such that 0 is a sink of F. So, for a fixed point z of f that is sufficiently close to 0, the change of angular coordinate along the trajectory of z is a positive integer. Therefore, by definition of the local rotation set, we know that ρ s (I, 0) contains a positive integer or +∞. On the other hand, we have ρ(I, z 0 ) = 0, which contradicts the assertion vii) of Proposition 2.20.

Let U ⊂ W be a small neighborhood of 0 that contains neither the positive nor the negative orbit of any wandering open set. By choosing U sufficiently small, we can also suppose that f does not have any fixed point in U except 0. Using the same technique of the proof of Lemma 4.1, we can find an orientation homeomorphism ϕ such that ϕ is equal to ϕ in sufficiently small neighborhood of 0 and that there exists a circle in ϕ (U \ {0}) such that ϕ • f • ϕ -1 maps each point in the circle to the right. More precisely, let π : R × (-∞, 0] → T 1 × (-∞, 0] be the universal covering map, and f the lift of the extension of ϕ • f • ϕ -1 (associated to the local isotopy (ϕ • f t • ϕ -1 ) t∈[0,1] ). There exists r > 0 such that ϕ -1 (T 1 × [-r, 0)) ⊂ U and that p 1 ( f (x, -r)) -x > 0, for all x ∈ R, where p 1 is the projection onto the first factor. But by our assumption, h is not right semi-stable, so there exists x 0 ∈ R such that p 1 ( f (x 0 , 0)) -x 0 < 0.

One deduces by Corollary 1.13 that ϕ • f • ϕ -1 has a fixed point in ϕ (U \ {0}), and hence there is a fixed point of f in U \ {0}. We get a contradiction.

  Conley's Conjecture to the dynamics near an isolated degenerate fixed point with Lefschetz index 1

  Figure 2.1: The hyperbolic sectors

  in a boundary circle of A, he extends f near this boundary and gets a homeomorphism f also on a closed annulus. Then O is in the interior of this new annulus. The braid type bt(O, f ) is independent of the choice of the extension, and Boyland defined bt(O, f ) = bt(O, f ).

  ∞ to . Every leaf Φ ⊂ W divides M into two part. We denote by R( Φ) the component of M \ Φ to the right of Φ and by L( Φ) the component to the left. Lemma 3.20. There does not exist any leaf Φ ⊂ W such that Φ ⊂ K. Chapter 3. Dynamics near an isolated fixed points with index one and zero rotation

Figure 3 . 3 :

 33 Figure 3.3: The dynamics and foliation generated by g(x, y) = x 2 + y 2

3. 2 .)))} ∪ {z ×kn 0 }

 20 The case of diffeomorphisms 59 Then C j (z ×kn 0 ) are always trivial for j < mor(z ×kn 0 ) and j > mor(z ×kn 0 is the dimension of negative eigenvector space of Hessian matrix of g ×n at z ×kn 0 , and nul(z ×kn 0 ) is the dimension of the kernel of Hessian matrix of g ×n at z ×kn 0 . We say that z 0 is a symplectically degenerate maximum if z 0 is an isolated local maximum of the generating functions g 0 , • • • , g k-1 , and the local homology C kn+1 (z ×kn 0 ) is non-trivial for infinitely many n ∈ N. Similarly, we denote by C + * (z ×kn 0 ) the graded group of relative homology H * ({g ×n > g ×n (z ×kn 0 , {g ×n > g ×n (z ×kn 0 )}), and say that z 0 is a symplectically degenerate minimum if z 0 is an isolated local minimum of the generating functions g 0 , • • • , g k-1 , and the local homology C + kn+1 (z ×kn 0 ) is non-trivial for infinitely many n ∈ N. Proposition 3.33 ([Maz13][Rue85]). Let z = z ×k 0 be a critical point of g such that C kn+1 (z ×n ) is non-trivial for infinitely n ∈ N. Then 1 is the only eigenvalue of DF ([z 0 ]), and the blow-up rotation number ρ(I, [z 0 ]) is equal to 0 for any identity isotopy of F fixing [z 0 ].

Proposition 4. 10 .

 10 If {(X α , I Xα )} α∈J is a totally ordered chain in I 0 , then there exists an upper bound (X ∞ , I X∞ ) ∈ I 0 of this chain, where X ∞ = ∪ α∈J X α Proposition 4.11. For every maximal (Y, I Y ) ∈ I and z ∈ Y such that I Y is not torsionlow at z and M \ (Y \ {z}) is neither a sphere nor a plane, there exist a maximal extension (Y , I Y ) ∈ I of (Y \ {z}, I Y ) and z ∈ Y \ (Y \ {z}) such that I Y is torsion-low at z .

Lemma 4. 15 .

 15 Let us suppose that (Y, I Y ) is maximal in (I, ), that I Y is not torsion-low at z ∈ Y , and that M \ (Y \ {z}) is neither a sphere nor a plane. If for every maximal extension (Y , I Y ) of (Y \ {z}, I Y ) and every point z ∈ Y \ (Y \ {z}), I Y is not torsionlow at z , then there exists a maximal extension (Y , I Y ) ∈ I of (Y \ {z}, I Y ) such that #(Y \ (Y \ {z})) > 1.

Lemma 4. 16 .

 16 Let us suppose that (Y, I Y ) is maximal in (I, ), that I Y is not torsion-low at z ∈ Y , and that M \ (Y \ {z}) is neither a sphere nor a plane. If for every maximal extension (Y , I Y ) of (Y \ {z}, I Y ) and every point z ∈ Y \ (Y \ {z}), I Y is not torsionlow at z , then there exists a maximal extension (Y , I Y ) ∈ I of (Y \ {z}, I Y ) such that #(Y \ (Y \ {z})) = ∞. Proof. Fix a couple (Y, I Y ) maximal in (I, ) and z ∈ Y satisfying the assumptions of the lemma. By the previous lemma, there exists a maximal extension (Y 1

  7, there exists an upper bound (Y ∞ , I Y∞ ) ∈ I of this sequence, where Y ∞ = ∪ n≥1 (Y n \ {z n }). By Theorem 2.8, there exists a maximal extension (Y , I Y ) ∈ I of (Y ∞ , I Y∞ ). It is also a maximal extension of (Y \ {z}, I Y ), and satisfies #(Y \ (Y \ {z})) = ∞. Proof of Proposition 4.11. We will prove this proposition by contradiction. Fix a maximal element (Y, I Y ) ∈ I and z 0 ∈ Y such that I Y is not torsion-low at z 0 and M \ (Y \ {z 0 }) is neither a sphere nor a plane. Write Y 0 = Y \ {z 0 }, and suppose that for all maximal extension (Y , I Y ) of (Y 0 , I Y ) and z ∈ Y \Y 0 , I Y is not torsion-low at z . By the previous lemma, there exists a maximal extension (Y , I Y ) of (Y 0 , I Y ) such that #(Y \ Y 0 ) = ∞. Let π Y 0 : M Y 0 → M \ Y 0 be the universal cover, I be the identity isotopy that lifts I Y | M \Y 0 , I be the identity isotopy that lifts I Y | M \Y 0 , and f be the lift of f | M \Y 0 associated to I Y | M \Y 0 . Since both I Y and I Y are maximal and M \ Y 0 is neither a sphere nor a plane, the point z 0 does not belong to Y . Moreover, f is also the lift of f | M \Y 0 associated to I Y | M \Y 0 . In particular, f fixes every point in π -1 Y 0 ({z 0 } ∪ Y \ Y 0 ). Fix z 0 ∈ π -1 Y 0 {z 0 }. Sublemma 4.17. For every z ∈ Y \ Y 0 , there exists z ∈ π -1

  at every z ∈ Y , we fix z 0 ∈ Y and can find a maximal extension (Y , I Y ) of (Y \ {z 0 }, I Y ) and z ∈ Y \ (Y \ {z 0 }) such that I Y is torsion-low at z by Proposition 4.11. Consequently, ({z }, I Y ) belongs to I 0 .

Lemma 4. 25 .

 25 Suppose that (Y, I Y ) is maximal in I, that I Y is not torsion-low at z ∈ Y , and that the connected component of M \ (Y \ {z}) is neither a sphere nor a plane whose boundary is empty or reduced to one point. If for every maximal extension (Y , I Y ) of (Y \ {z}, I Y ) and every z ∈ Y \ (Y \ {z}), I Y is not torsion-low at z , then there exists a maximal extension (Y , I Y ) ∈ I of (Y \ {z}, I Y ) such that #(Y \ (Y \ {z})) = ∞. Proof. The proof is almost the same as the one of Lemma 4.16 except the following: every time we want to get a new couple, we should check that the previous couple satisfies the assumptions of Lemma 4.23 or Lemma 4.24 instead of the assumptions of Lemma 4.15. Now, we begin the proof of Proposition 4.20. The proof is similar to the one of Proposition 4.11. Chapter 4. Torsion-low isotopies Proof of Proposition 4.20. We will prove this proposition by contradiction. Fix a maximal (Y, I Y ) ∈ I and z 0 ∈ Y such that I Y is not torsion-low at z 0 and M \ (Y \ z 0 ) is neither a sphere nor a plane whose boundary is empty or reduced to a single point. Write Y 0 = Y \ {z 0 }, and denote by M Y 0 the connected component of M \ Y 0 containing z 0 . Suppose that for all maximal extension (Y , I Y ) of (Y \ {z}, I Y ) and z ∈ Y \ (Y \ {z}), I Y is not torsion-low at z . By the previous lemma, there exists a maximal extension (Y , I Y ) of (Y 0 , I Y ) such that #(Y \ Y 0 ) = ∞. Let us prove by contradiction that Y \Y 0 ⊂ M Y 0 . Suppose that there exists z 1 ∈ Y \Y 0 that is in another component of M \ Y 0 . Since I Y | M \Y 0 and I Y | M \Y 0 are homotopic, the trajectory of z 1 along I Y is homotopic to zero in M \ Y 0 . Moreover, because the trajectory of z 1 along I Y is in another component of M \ Y 0 , this trajectory is homotopic to zero in M \ Y , which contradicts the maximality of (Y, I Y ) by Proposition 2.6.

Figure 4 . 1 :

 41 Figure 4.1: The two foliations of Example 4.26

Figure 4 . 2 :

 42 Figure 4.2: The dynamics and two foliations of Example 4.27

Exemple 4 .

 4 29. (Example of Remark 1.8)

ϕ( 0 )

 0 = ϕ(3/4) = ϕ(1) = 0 and |ϕ| ≤ 1 2π , ϕ(s) > 0 for 0 < s < 3/4 ϕ(s) < 0 for 3/4 < s < 1 , y ≤ 0,y 0 s sin 2 π s + ϕ(s) sin 2 πxds for 0 < y < 1, 1 0 s sin 2 π s ds for y ≥ 1.

  m ) : n ∈ Z, m ∈ N} ∪ R × (-∞, 0] ∪ R × [1, ∞),and one deduces that ∂ 2 g(x, y) > 0 for (x, y) / ∈ C. We define an equivalent relation ∼ on R 2 by     (x, y) ∼ (x , y )for y, y ≤ 0, (x, y) ∼ (x + 1, y) for 0 < y < 1, (x, y) ∼ (x , y ) for y, y ≥ 1.Then, R 2 / ∼ is a sphere, f 1 descends to an area preserving homeomorphism f of the sphere, I descends to an identity isotopy I of f , and F descends to a transverse foliation F of I . Moreover, one knows that Fix(I ) = Fix(f ) = Sing(F ), where Sing(F ) is the set of singularities of F . We denote by S and N the two points R × (-∞, 0] and R × [1, ∞) in the sphere respectively.

Figure 4

 4 Figure 4.3: A sketch map of F

min z∈ C 1 p 1

 11 ( f (z) -z) > r 0 ε,

  we are in one of the following cases: i) there exists a fixed point of f in the interior of A 1 lifted to fixed points of f ; ii) there exists an essential loop γ in A 1 that does not intersect C 0 \ {z 0 } and satisfies γ ∩ f (γ) ⊂ {z 0 }.

Corollary 1.13. If A 1 does not contain the positive or the negative orbit of any wandering open set and if there exists z ∈ C 0 such that p 1

  Fix a maximal (Y, I Y ) ∈ I such that #Y ≥ 3. If Y is infinite, there exists a point z ∈ Y that is not isolated in Y , and hence I Y is torsion-low at z. If Y is finite, we consider a transverse foliation of I Y and know that there is a saddle singulary point z of F by the Poincaré-Hopf formula and Remark 2.13, and hence I Y is torsion-low at z. Anyway, there exists z ∈ Y such that ({z}, I

Y ) ∈ I 0 . To prove ii), we fix (X, I X ) ∈ I 0 such that X = {S}. For a maximal extension (Y, I Y ) ∈ I of (X, I X ) such that I Y is torsion-low at S, one knows that Y \ X is not empty. If I Y is torsion-low at another fixed point, the proof is finished; if I Y is not torsion-low at any other fixed points and if #(Y \ X) is bigger that 1, we get the result as a corollary of Proposition 4.11. Then, we only need to prove that there exists a maximal extension (Y, I Y ) ∈ I of (X, I X ) such that I Y is torsion-low at S and that satisfies one of the two conditions: I Y is torsion-low at another fixed point or #(Y \ X) > 1. Fix a maximal extension (Y, I Y ) ∈ I of (X, I X ) such that ρ s (I Y , S) = ρ s (I X , S). Of course, I Y is torsion-low at S. If I Y is torsion-low at another fixed point or if #(Y \ X) > 1, the proof is finished. Now, we suppose that Y = {S, N } and I Y is not torsion-low at N . One has to consider two cases: S is isolated in Fix(f ) or not. a) Suppose that S is isolated in Fix(f ). As in the proof of Lemma 4.15, one has two consider the following four cases:

It is a talk of Crovisier in the conference "Surfaces in Sao Paulo" in April, 2014.

It means the vector field: (x, y) → (∂1g(x, y), ∂2g(x, y)).

Remerciements

Chapter 4

Torsion-low isotopies

The rotation type at an isolated fixed point of an orientation preserving local homeomorphism

Let f : (W, 0) → (W , 0) be an orientation preserving local homeomorphism at the isolated fixed point 0 ∈ R. The main aim of this section is to detect the local rotation type of the local isotopies of f and prove Theorem 1.5.

Before proving the theorem, we will first prove the following lemma:

Lemma 4.1. If f satisfies the local intersection condition, then a local isotopy I = (f t ) t∈[0,1] of f can not have both a positive and a negative rotation type.

Proof. We will give a proof by contradiction. Suppose that F 1 and F 2 are two locally transverse foliations of I such that 0 is a sink of F 1 and a source of F 2 . Then, there exist two orientation preserving local homeomorphisms h 1 : (V 1 , 0) → (D, 0) and h 2 : (V 2 , 0) → (D, 0) such that h 1 (resp. h 2 ) sends the restricted foliation

) to the radial foliation on D with the orientation toward (resp. backward) 0, where D is the unit disk centered at 0, and V i ⊂ W is a small neighborhood of 0 such that f does not have any fixed point in V i except 0, and f (γ)∩γ = ∅ for all essential closed curve γ in V i \{0}, for i = 1, 2. We denote by D r the disk centered at 0 with radius r, and S r the boundary of D r . Choose 0 < r 2 < 1 such that for all z ∈ h -1 2 (S r 2 ), there exists an arc in V 2 \ {0} that is homotopic to t → f t (z) in V 2 \ {0} and is positively transverse to F 2 ; choose 0 < r 2 < r 2 such that

2 (S r 2 ) ⊂ D \ D r 2 ; choose 0 < r 1 < r 2 such that h -1 1 (D r 1 ) ⊂ h -1 2 (D r 2 ); and choose 0 < r 1 < r 1 such that h 1 • f • h -1 1 (D r 1 ) ⊂ D r 1 , and for all z ∈ h -1 1 (S r 1 ), there exists an arc in V 1 \ {0} that is homotopic to t → f t (z) in V 1 \ {0} and is positively transverse to F 1 . We consider a homeomorphism h : (V 2 , 0) → (D, 0) such that h| h -1

(θ, y) → -ye i2πθ be the universal covering projection, and f be the lift of h

where p 1 is the projection onto the first factor. Then, h • f • h -1 is a map satisfying the conditions of Proposition 2.27. But we know that h • f • h -1 (γ) ∩ γ = ∅ for all essential simple closed curve γ in D \ {0}, which is a contradiction.

Chapter 4. Torsion-low isotopies

Suppose that I is not torsion-low at z 0 ∈ Fix(I). Write Y 0 = Fix(I) \ {z 0 }. Since we are not in the previous two cases, M \ Y 0 is neither a plane nor a sphere. By Proposition 4.11, there exist a maximal extension I of (Y 0 , I) and z ∈ Fix(I ) \ Y 0 such that I is torsion-low at z . This contradicts the fact that I is maximal in (J, ).

Proof of Theorem 4.5 when Fix(f ) is not totally disconnected

We suppose that Fix(f ) is not totally disconnected in this subsection. In this case, the proof of Theorem 4.5 is similar to the one in the previous section except that we should consider more cases. More precisely, Theorem 4.5 is a corollary of Zorn's lemma and the following four similar propositions. The proof of Proposition 4.19 is just a copy of the one of Proposition 4.10; while the proofs of the others are the aim of this subsection.

Proposition 4.19. If {(X α , I Xα )} α∈J is a totally ordered chain in I 0 , then there exists an upper bound (X ∞ , I X∞ ) ∈ I 0 of the chain, where X ∞ = ∪ α∈J X α Proposition 4.20. For every maximal (Y, I Y ) ∈ I and z ∈ Y such that I Y is not torsionlow at z and M \ (Y \ {z}) is neither a sphere nor a plane 1 whose boundary is empty or reduced to one point, there exist a maximal extension

x 2 +y 2 , x(x 2 -3y 2 ) x 2 +y 2 ) for x > 0, y > 0, (y, -x) for x ≤ 0, y ≥ 0, (-y, -x) for x ≤ 0, y ≤ 0, (-y, x) for x ≥ 0, y ≤ 0.

One knows that ξ vanish at a unique point 0, is transverse to V , and satisfies det(V (x, y), ξ(x, y)) > 0 for all (x, y) = 0. So, the the foliation F 1 whose leaves are the integral curves of ξ is transverse to I. Moreover, by a direct computation, we can get the formulae of the integral curves of ξ and find that every integral curves go to 0 as in the picture. So, 0 is a sink of F 1 .

Let

be the universal cover, and ( f t ) t∈[0,1] be the identity isotopy that lifts I. We know that γ (x,y) :

), and is a vertical segment downward for every (x, y) ∈ [-1 4 , 0] × (-∞, 0). We define an oriented foliation F II on the domain ( 1 4 , 1 2 ) × (-∞, 0) whose leaves are the the restriction to

is the graph of y = log(4x -1) -c with the direction from right to left, for c ∈ (0, ∞). Then, γ (x,y) is positively transverse to F II for every (x, y) ∈ ( 1 4 , 1 2 ) × (-∞, 0). Similarly, we define an oriented foliation F IV on the domain (-1 4 , 0) × (-∞, 0) whose leaves are the restriction to (-1 4 , 0) × (-∞, 0) of the family of curves ( c ) c∈(-1,∞) such that c is the graph of y = log(-4x-c) with the direction from left to right, for c ∈ (-1, 0], c is the graph of y = log(-4x)-c with the direction from left to right, for c ∈ (0, ∞). Then, γ (x,y) is positively transverse to F IV for every (x, y) ∈ (-1 4 , 0) × (-∞, 0). Note the following facts:

) is equal to the restriction of F 1 to the same set. Moreover, one can deduce that 0 is a mixed singularity of F 2 .

Exemple 4.28. (An orientation and area preserving local homeomorphism whose local rotation set is reduced to ∞) Chapter 4. Torsion-low isotopies -The set of fixed points of I is a subset of N (resp. S) with exactly two points z 1 and z 2 , and the rotation numbers satisfies

Exemple 4.32. (Example of Remark 4.9) We will construct an orientation and area preserving diffeomorphism of the sphere such that there does not exist any maximal identity isotopy I satisfying -1 < ρ(I, z) < 1, for every z ∈ Fix(F ).

Let g be a diffeomorphism of R × [0, 1] that is defined by g(x, y) = (x + y, y).

We define an equivalent relation

Then R × [0, 1]/ ∼ is a sphere and g descends to an orientation and area preserving diffeomorphism f of the sphere that has exactly two fixed points. Note the facts that every maximal identity isotopy I fixes both fixed points of f , that the rotation number of I at each fixed point is an integer, and that the sum of the rotation numbers of I at both fixed point is 1. So, there does not exist any maximal isotopy I such that for all z ∈ Fix(I),

Appendix: Construct a transverse foliation from the generating function

Let f be a diffeomorphism of R 2 and g : R 2 → R be a C 2 function, we call g a generating function of f if ∂ 2 12 g < 1, and if

Every C 2 function g : R 2 → R satisfiying ∂ 2 12 g ≤ c < 1 defines a diffeomorphism f of R 2 by the previous equations, and every area preserving diffeomorphism

be generated by a generating function, where p 1 is the projection onto the first factor. Moreover, the Jacobian matrix J f of f is equal to

Since det J f = 1, the diffeomorphism f is orientation and area preserving. A point (x, y) is a fixed point of f if and only if it is a critical point of g. We can naturally define an identity isotopy I = (f t ) t∈[0,1] of f such that f t is generated by tg. Precisely, the diffeomorphisms f t are defined by the following equations:

4.4. Appendix: Construct a transverse foliation from the generating function 85

In this section, we suppose that f is a diffeomorphism of R 2 , and that g is a generating function of f . We will construct a transverse foliation of I. More precisely, denote by F the foliation whose leaves are the integral curves of the gradient vector field (x, y) → (∂ 1 g(x, y), ∂ 2 g(x, y)) of g, we will prove the following result:

Theorem 4.33. The foliation F is a transverse foliation of I.

Proof. We will prove the theorem by constructing an identity isotopy I of f that is homotopic to I relatively to Fix(f ) and satisfies that for every z ∈ R 2 \ Fix(f ), the trajectory of z along I is positively transverse to F.

We define I = (f t ) t∈[0,1] by the following equations:

where (X, Y ) = f (x, y).

Lemma 4.34. One can verify that I is an identity isotopy of f .

Proof. We know that ∂ 1 X(x, y) = 1/(1 -∂ 2 12 g(X, y)) > 0. By computing the determinant of the Jacobian matrix of f t , we know that det

To prove that I is an isotopy, we only need to check that f t is a bijection for every t ∈ (0, 1).

For t ∈ (0, 1 2 ), write f t (x, y) = (ϕ t,y (x), y). One deduces

So, f t is a surjection. Now, we will prove f t is an injection. Suppose that f t (x, y) = f t (x , y ), and write (X , Y ) = f (x , y ). One knows y = y and

and deduces

where ξ is a real number between X and X . So, one knows X = X . By definition of the generating function, one deduces (x, y) = (x , y ). Therefore, f t is injective.

Chapter 4. Torsion-low isotopies

So, f t is a surjection. Now, we will prove f t is an injection. Suppose that f t (x, y) = f t (x , y ), and write (X , Y ) = f (x , y ). One knows X = X and

So, one knows (y -y ) -(2t -1)(∂ 1 g(X, y) -∂ 1 g(X, y )) = 0, and then deduces (y -y ) -(2t -1)∂ 2 12 g(X, η)(y -y ), where η is a real number between y and y . So, one knows y = y , and then deduces (x, y) = (x , y ). We conclude that f t is an injection.

By definition, we know Fix(I 0 ) = Fix(I ) = Fix(f ). If Fix(f ) is empty or contains more than one point, we know that (Fix(f ), I 0 ) ∼ (Fix(f ), I ), and hence a transverse foliation of I is also a transverse foliation of I 0 ; if Fix(f ) is reduced to one point, we can deduce the same result by the following lemma and the fact that π 1 (homeo

Proof. Let θ : [0, 1] → R and θ : [0, 1] → R be the continuous functions that satisfies θ(0) = θ (0) = 0 and

To simplify the notations, we write Hess(g)(0) = , σ σ, τ .

One knows

We know 1 -tσ > 0 for all t ∈ [0, 1], so θ(t) belongs to (-π 2 , π 2 ) for all t ∈ [0, 1]. For t ∈ [0, 1 2 ], one knows

4.4. Appendix: Construct a transverse foliation from the generating function 87

We know ∂ 1 X(0, 0) > 0, so θ (t) belongs to (-π 2 , π 2 ) for all t ∈ [ 1 2 , 1]. Therefore, one deduces θ(1) = θ (1) ∈ (-π 2 , π 2 ), and hence ρ(I, 0) = ρ(I , 0) ∈ [-1, 1].

Lemma 4.36. For every z = (x, y) that is not a fixed point of f , the path γ z : t → f t (x, y) is positively transverse to F. Proof.

where ξ is a real number between x and X, and the inequality is strict if X = x.

where η is a real number between y and Y , and the inequality is strict if Y = y. Since z is not a fixed point, either X = x or Y = y. If both of the inequalities are satisfied, γ z intersects F positively transversely; if X = x and Y = y, γ z | t∈[0, 1 2 ] intersects F positively transversely, and

is reduced to a point, and γ z | t∈[ 1 2 ,1] intersects F positively transversely. t = t = 0, and note that Ψ(0, 0, r) = f (T r (γ (0))) -γ (0). Its image is contained in (0, ∞) × R, because the first coordinate satisfies

Suppose that t = 0, t ∈ (0, 1), and note that Ψ(0, t , r) = f (T r (γ (t ))) -γ (0) does not vanish, because γ (0) / ∈ Int( A 2 ), and f (T r (γ (t ))) ∈ Int( A 2 ) for t ∈ (0, 1). Suppose that t ∈ [0, 1), t = 1, and note that Ψ(t, 1, r) = f (T r (γ (1))) -T tr(1+ε) γ (t). Its image is contained in R × (-∞, 0). Suppose that t = t = 1, and note that Ψ

Let δ d be the segment joining (0, 0, 0) to (1, 1, 0), δ + be the segment joining (0, 0, 0) to (0, 0, r 0 ), δ v be the segment joining (0, 0, r 0 ) to (0, 1, r 0 ), δ h be the segment joining (0, 1, r 0 ) to (1, 1, r 0 ), and δ -be the segment joining (1, 1, r 0 ) to (1, 1, 0). So,

The images of both the initial point f (γ(0)) -γ(0) and the end point

Therefore i( f , γγ γ ) = -1,

Lemma 5.6. If there exists a positive path γ of f in A 1 starting from C * 0 such that γ| [0,1) ⊂ Int( A 1 ) and f (γ) Int( A 1 ), there exists a positive path γ of f joining C * 0 to C 1 such that f (γ (1)) / ∈ Int( A 1 ).

Proof of the main results

93

Proof. Taking a sub-path if necessary, we can suppose that γ(t) is in the interior of A 1 for all t ∈ (0, 1) and f (γ(1)) / ∈ Int( A 1 ). If γ(1) ∈ C 1 , the path γ satisfies the lemma. Suppose γ(1) / ∈ C 1 . Let δ be an arc joining f (γ(1)) to C 2 such that δ does not intersect

We define a free disk chain of f in A 1 to be a the sequence (σ i ) 0≤i≤n of closed topological disks σ i of A whose interiors are pairwise disjoint and such that f (σ i ) ∩ σ i = ∅ for all 0 ≤ i ≤ n, and f (σ i-1 ) ∩ σ i = ∅ for all 1 ≤ i ≤ n.

Lemma 5.7. If there exists a free disk chain (σ i ) 0≤i≤n of f in A 1 such that σ 0 ∩ C * 0 = ∅, and f (σ n )

Int( A 1 ), there exists a positive path γ of f starting from C * 0 such that γ| [0,1) ⊂ Int( A 1 ) and f (γ) Int( A 1 ).

Proof. Let m = min{n : there exists a free disk chain (σ i ) 0≤i≤n in A 1 such that σ 0 ∩ C * 0 = ∅, and f (σ n ) Int( A 1 )}, and (σ i ) 0≤i≤m be one of such free disk chain. We will show that

Otherwise, let i 0 = min{i : there exist j > i + 1, k ≥ 1, such that f k (σ i ) ∩ σ j = ∅}, j 0 = max{j : there exists k ≥ 1, such that f k (σ i 0 ) ∩ σ j = ∅}, and k 0 = min{k : f k (σ i 0 ) ∩ σ j 0 = ∅}. Then we get a free disk chain

and

This contradicts the minimality of m. The set ∪ 0≤i≤m f m-i (σ i ) is connected and intersects C * 0 . Moreover, f (σ m ) Int( A 1 ). Choose a point z 0 ∈ f m (σ 0 ) ∩ C * 0 , a point z m+1 ∈ σ m such that f (z m+1 ) / ∈ Int( A 1 ), and a point z i ∈ f m-i+1 (σ i-1 ) ∩ f m-i (σ i ) for 1 ≤ i ≤ m. Choose an arc γ i joining z i to z i+1 such that γ i \ {z i , z i+1 } ⊂ Int( f m-i (σ i )) for 0 ≤ i ≤ m. The path γ = γ 0 • • • γ m is a path starting from γ(0) ∈ C * 0 , and satisfying f (γ) Int( A 1 ). By choosing suitable γ i for i = 0, • • • , m, we can suppose that γ| [0,1) does not intersect C 1 .

We will prove that γ is a positive path. Indeed, we will prove that f (γ j ) ∩ γ i = ∅ for 0 ≤ i ≤ j ≤ m. By the construction, we know that γ i ⊂ f m-i (σ i ), and σ i is a free disk, so f

Replacing f with f -1 and C 2 with C 1 , we get similarly Lemma 5.8. If there exists a free disk chain

Lemma 5.9. If we are in neither of the two cases i) and ii) of Theorem 1.12, then there exists a free disk chain satisfying the conditions of Lemma 5.7.

Proof. Let

of W , such that the bricks are closed disks obtained as the closures of the connected components of the complement in W of a locally finite graph Σ whose vertices are locally the extremities of exactly 3 edges. Moreover, we can suppose that the bricks are free when lifted to A 1 by decomposing the bricks into smaller ones. Let J ⊂ I be the set of indices j such that there exists a sequence (σ i k ) 0≤k≤l with σ i 0 ∩ C * 0 = ∅, σ i l = σ j , and f (σ i k ) ∩ σ i k+1 = ∅. Then X = ∪ j∈J σ j is a subsurface with boundary of W that contains a neighborhood of C * 0 in W . We assert that there exists j ∈ J such that f (σ j ) Int(A 1 ). We prove it by contradiction. Suppose that f (σ j ) ⊂ Int(A 1 ) for all j ∈ J . Then f (X) ⊂ Int(A 1 ). We write

The boundary ∂X of X in A * does not intersect f (∂X), and a component γ of ∂X is either a simple loop, or becomes a simple loop when one adds z 0 . Since C * 0 and A * \ A 1 are in different component of A * \ ∂X, ∂X has either a component that is an essential loop or a component that becomes a simple loop when one adds z 0 . This means we are in the second case of Theorem 1.12.

We choose suitable lifts of the disks (σ j ) j∈J , and can find a free disk chain of f satisfying the conditions of Lemma 5.7.

Similarly, by replacing f with f -1 , we have the following lemma: Lemma 5.10. If we are in neither of the two cases i) and ii) of Theorem 1.12, then there exists a free disk chain satisfying the conditions of Lemma 5.8. Now we will give a proof of Theorem 1.12.

Proof of Theorem 1.12. We will prove the theorem by contradiction. Suppose that we are in neither of the two cases i) and ii) of Theorem 1.12. By Lemma 5.6, Lemma 5.7, and Lemma 5.9, one deduces that there exists a positive path γ of f joining C * 0 to C 1 such that f (γ(1)) / ∈ Int( A 1 ). By Lemma 5.8 and Lemma 5.10, one deduces that there exists a positive path γ of f joining C 1 to C * 0 such that γ (0) / ∈ Int( A 2 ). Moreover, by considering a translation of γ , we can suppose that γ (0) is at the right of γ(1). So, by Lemma 5.5, one deduces that there exists a loop Γ in A 1 such that i( f , Γ) = -2. But by Lemma 5.4, one knows that f does not have any fixed point with non-zero index, we get a contradiction.

An application to the local dynamics of homeomophisms

In this section, suppose that f : (W, z 0 ) → (W , z 0 ) is a local homeomorphism at z 0 such that there exists a neighborhood of z 0 that does not contain the positive or the negative orbit of any wandering open set, and that I = (f t ) t∈[0,1] is a local isotopy of f that has a positive (resp. negative) rotation type. We suppose also that either f can be blown-up at z 0 , or that z 0 is a non-accumulated indifferent point. Recall that in both case, we can define a rotation number ρ(I, z 0 ). We say that a homeomorphism h of the circle is right semi-stable (resp. left semi-stable) if its lift h to R satisfies h(x) ≥ x (resp. h(x) ≤ x) for all x ∈ R, and the equality holds at some points. The aim of this section is to prove the following corollary: Corollary 5.11 (Corollary 1.14). Under the previous assumptions, if ρ(I, z 0 ) = 0, the dynamics on the circle added when blowing-up is right semi-stable (resp. left semi-stable).

Remark 5.12. When z 0 is a non-accumulated indifferent point, this result have been proven by Le Calvez [START_REF] Le | Dynamique des homéomorphismes du plan au voisinage d'un point fixe[END_REF] in a different way.