Local density of diffeomorphisms with large centralizers

Densité locale des difféomorphismes ayant un gros centralisateur

C. Bonatti, S. Crovisier, G.M. Vago and A. Wilkinson

May 15, 2008

Abstract

Given any compact manifold M, we construct a non-empty open subset O of the space Diff^{1}(M) of C^{1}-diffeomorphisms and a dense subset D ⊂ O such that the centralizer of every diffeomorphism in D is uncountable, hence non-trivial.

Résumé : Pour toute variété M compacte, de dimension quelconque, nous construisons une partie O ⊂ Diff^{1}(M) non vide, ouverte dans l’espace Diff^{1}(M) des C^{1}-difféomorphismes de M, et un sousensemble D ⊂ O dense en O, constitué de difféomorphismes dont le centralisateur est non dénombrable, donc non trivial.

Key words: Trivial centralizer, trivial symmetries, Mather invariant.

Mots clés : Centralisateur trivial, symétries triviales, invariant de Mather.

MSC 2000: 37C85 - 37C80 - 37D45 - 37D15 - 37E30.

Introduction

The centralizer of a C^{r} diffeomorphism f ∈ Diff^{r}(M) is the group of diffeomorphisms commuting with f:

C(f) := {g ∈ Diff^{r}(M) : fg = gf}.

The centralizer C(f) always contains the group < f > of all the powers of f. For this reason, we say that f has a trivial centralizer if C(f) =< f >. If f is the time one map of a C^{r} vector field X, then C(f) contains the flow of X and hence contains a subgroup diffeomorphic to ℝ (or S^{1} = ℝ/ℤ if f is periodic).

The elements of C(f) are transformations of M which preserve the dynamics of f: in that sense they are the symmetries of f. How large is, in general, this symmetry group?

• On one hand, the structure on M given by a diffeomorphism is very flexible, so that one might expect that any symmetry could be broken by a small perturbation of the diffeomorphism.

• On the other hand, the symmetries are sought in the very large group Diff^{r}(M), which makes the problem harder. For example, one can easily show that the group C^{0}(f) of homeomorphisms commuting with a Morse-Smale diffeomorphism f is always uncountable.

Nevertheless it is natural to guess that general diffeomorphisms have no nontrivial smooth symmetries. Making this intuition explicit, Smale asked the following:

Question I ([S_{1}, S_{2}]). Let T^{r}(M) ⊂ Diff ^{r}(M), r ≥ 1 denote the set of C^{r} diffeomorphisms of a compact manifold M with trivial centralizer.

1. Is T^{r}(M) dense in Diff ^{r}(M)?

2. Is T^{r}(M) residual in Diff ^{r}(M)? That is, does it contain the intersection of countably many dense open subsets?

3. Is T^{r}(M) a dense open subset of Diff ^{r}(M)?

We think it’s natural to reformulate the third part of Smale’s question as:

Question II. Does T^{r}(M) contain a dense and open subset of Diff ^{r}(M)?

This question has many parameters, the most obvious being the regularity r of the diffeomorphisms and the dimension dim(M) of the manifold. The question has been answered in varying degrees of generality for specific parameters. For instance, Kopell [K] proved that Diff^{r}(S^{1}), r ≥ 2, contains a dense and open subset of diffeomorphisms with trivial centralizers. Many authors subsequently gave partial answers in higher dimension (see [BCW_{1}] for an attempt to list these partial results).

The present paper and [BCW_{2}] together give a complete answer to Smale’s problem for r = 1. More precisely:

• [BCW_{2}] proves that C^{1}-generic diffeomorphisms have a trivial centralizer^{1}, giving a positive answer to the first two parts of Smale question. [BCW_{3}] shows that C^{1}-generic conservative (volume preserving or symplectic) diffeomorphisms have a trivial centralizer.

• In this paper, we answer in the negative (for r = 1) the third part of Smale’s question (and to Question II) on any compact manifold.

Main theorem. Given any compact manifold M, there is a non-empty open subset O ⊂ Diff^{1}(M) and a dense subset D ⊂ O such that every f ∈ D is C^{∞} and its C^{∞}-centralizer C^{∞}(f) is uncountable (hence not trivial).

We will see below (see Theorem 5) that this statement also holds for symplectic diffeomorphisms on a symplectic manifold.

The uniform presentation of this result (Given any compact manifold,…) hides very different situations, arguments and results according to the dimension: namely, whether dim(M) < 3 or dim(M) ≥ 3. We discuss this breakdown of the results below.

Our paper also deals with the question of how large is the class of diffeomorphisms that can be embedded in a flow. This is a natural question, since the studies of the dynamical systems defined either by diffeomorphisms or by vector fields are in fact closely related. In the paper [P] titled Vector fields generate few diffeomorphisms, Palis proved that C^{1}-generic diffeomorphisms cannot be embedded in a flow. Our results somehow counterbalance Palis’ statement: diffeomorphisms that are the time one map of a flow are C^{1}-locally dense in dimension 1 and 2.

Theorem 1. There is a dense subset D ⊂ Diff ^{1}(S^{1}) such that every f ∈ D commutes with the flow of a C^{∞} Morse-Smale vector field X. More precisely, f is Morse-Smale and f^{q} is the time one map of the flow of X, where q = 2, if f is orientation reversing, and q is the period of the periodic orbits of f otherwise. Furthermore, C(f) is isomorphic to ℝ × (ℤ/qℤ).

(Section 1.6 presents open questions on centralizers of diffeomorphisms in Diff^{1}(S^{1}).)

Among compact surfaces, the case of the sphere is special because of the existence north pole-south pole diffeomorphisms. The symmetries of these dynamics allow us to get a centralizer isomorphic to S^{1} × ℝ and this is one of the reasons why we present this case separately.

Another specific feature of the north-south diffeomorphisms of the sphere is that for these maps it is possible to generalize the so-called Mather invariant, introduced by Mather in the one-dimensional case. Such an invariant plays a fundamental role in our constructions: the Mather invariant of a diffeomorphism f is trivial if and only if f can be perturbed to become the time one map of a vector field.

1[BCW_{1}] is an announcement which gives the structure of the detailed proof written in [BCW_{2}].

Theorem 2. Let O ⊂ Diff ^{1}(S^{2}) denote the (open) subset of Morse-Smale diffeomorphisms f such that the nonwandering set Ω(f) consists of two fixed points, one source N_{f} and one sink S_{f}, such that the derivatives D_{N_{f}} f and D_{S_{f}} f have each a complex (non-real) eigenvalue.

Then there is a dense subset D ⊂ O such that every f ∈ D is the time one map of a Morse-Smale C^{∞}-vector field. Furthermore C(f) is isomorphic to ℝ × S^{1}.

Theorem 2 is a bridge between the one-dimensional case and the general two-dimensional case. One the one hand, north pole-south pole dynamics on the sphere and Morse-Smale dynamics on the circle share the Mather invariant; on the other hand, other features of these dynamics on the sphere occur in simple dynamics on a general compact surface. The general case is solved by a combination of the arguments used for the sphere in a neighborhood of the sinks and the sources, together with an analysis of the local situation in a neighborhood of the saddles.

Theorem 3. Let S be a connected closed surface. Let O ⊂ Diff ^{1}(S) be the set of Morse-Smale diffeomorphisms f such that:

• any periodic point is a (hyperbolic) fixed point,

• f has at least one hyperbolic saddle point,

• for any hyperbolic saddle x, every eigenvalue of Df(x) is positive,

• for any sink or source x, the derivative Df(x) has a complex (non-real) eigenvalue,

• there are no heteroclinic orbits: if x ≠ y are saddle points then W^{s}(x) ∩ W^{u}(y) = ∅.

Then O is a non-empty open subset of Diff ^{1}(S) and there is a dense subset D ⊂ O such that every f ∈ D is the time one map of a Morse-Smale C^{∞}-vector field. Furthermore, C(f) is the flow of this vector field, hence isomorphic to ℝ.

An important hypothesis in Theorems 2 and 3 (which holds trivially in Theorem 1) is that the derivative at each sink and source is conjugate to the composition of a homothety with a (non-trivial) rotation. This condition is open in dimension 2, but is nowhere dense in higher dimension. This explains why we are not able to obtain the local density of the embeddability in a flow in higher dimension, and naturally leads to the following question:

Question III. Let M be a compact manifold of dimension d ≥ 3. Denote by O the C^{1}-interior of the C^{1}-closure of the set of diffeomorphisms which are the time one map of a flow. Is O empty?

In low dimension we find large centralizers among the simplest dynamical systems (the Morse-Smale systems). By contrast, in higher dimension we will use C^{1}-open subsets of wild diffeomorphisms to obtain periodic islands where the return map is the identity map. The resulting large centralizers for these wild diffeomorphisms are completely different. In low dimension, we embed the diffeomorphisms in a flow, and the centralizer is precisely the flow; hence all the diffeomorphisms in the centralizer have the same dynamics. In higher dimension, the diffeomorphisms we exhibit in the centralizer will be equal to the identity map everywhere but in the islands, in restriction to which they can be anything. This explains our result:

Theorem 4. Given any compact manifold M of dimension d ≥ 3, there is a non-empty open subset O ⊂ Diff ^{1}(M) and a dense subset D ⊂ O such that every f ∈ D has non-trivial centralizer.

More precisely, for f ∈ D the centralizer C(f) contains a subgroup isomorphic to the group Diff ^{1}(ℝ^{d}, ℝ^{d} \ 𝔻^{d}) of diffeomorphisms of ℝ^{d} which are equal to the identity map outside the unit disc 𝔻^{d}.

The large centralizer we build for a diffeomorphism in Theorem 4 consists of diffeomorphisms which have a very small support, and which are therefore C^{0}-close to the identity. It would be interesting to know if this is always the case. Let us formalize this question:

Question IV. Let M be a compact manifold with dim(M) ≥ 3 and ε > 0. Let O_{ε} ⊂ Diff^{1}(M) be the set of diffeomorphisms f such that, for every g ∈ C(f) there exists n ∈ ℤ such that g ∘ f^{n} is ε-close to the identity map for the C^{0}- distance. Does O_{ε} contain a dense open subset of Diff^{1}(M) for every ε?

If for non-conservative diffeomorphisms the existence of periodic islands depends on wild dynamics, the same islands appear in a more natural way for symplectic diffeomorphisms in a neighborhood of totally elliptic points. In order to state precisely this last result we need some notations. Let M be a compact manifold with even dimension dim(M) = 2d. If M carries a symplectic form ω, then we denote by Symp_{ω}^{1}(M) the space of C^{1} diffeomorphisms of M that preserve ω (these diffeomorphisms are called symplectomorphisms).

Recall that a periodic point x of period n of a symplectomorphism f is called totally elliptic if all the eigenvalues of Df^{n}(x) have modulus equal to 1. If e^{iα} is an eigenvalue of x then e^{−iα} is also an eigenvalue. Assume that 0 < α_{1} < ⋯ < α_{d} < π are the absolute values of the argument of the eigenvalues of x. Then x is C^{1}-robustly totally elliptic: every symplectomorphism g that is C^{1}-close to f has a totally elliptic point x_{g} of period n close to x.^{2}

Theorem 5. Let (M, ω) be a symplectic manifold and let O ⊂ Symp_{ω}^{1}(M) denote the non-empty open subset of the symplectomorphisms having a C^{1}- robust totally elliptic periodic point.

Then there is a dense subset D ⊂ O such that such that every f ∈ D has a non-trivial centralizer. More precisely, for f ∈ D the symplectic centralizer C_{ω}(f) contains a subgroup isomorphic to the group Symp_{ω}^{1}(ℝ^{2d}, ℝ^{2d} \ 𝔻^{2d}) of symplectomorphisms of ℝ^{2d} equal to the identity map on the complement of the unit disc 𝔻^{2d}.

We remark that, according to [ABC], C^{1}-generic symplectomorphisms of connected manifolds are transitive, and the manifold is a unique homoclinic class. In particular, there is a C^{1}-dense and open subset of symplectomorphisms having a hyperbolic periodic point with some transverse homoclinic intersection. Such a transverse homoclinic intersection prevents the diffeomorphism from being embedded in a flow: thus non-embeddability in a flow is satisfied on a C^{1}-open and dense subset of Symp_{ω}^{1}(M). The same argument holds for volume preserving diffeomorphisms, showing that, if μ is a smooth volume form on a manifold M of dimension dim(M) ≥ 2, then the non embeddability in a flow is satisfied on a C^{1}-open and dense subset of Diff_{μ}^{1}(M). However we have not been able to extend our main theorem for volume preserving diffeomorphisms:

Question V. Let M be a closed manifold endowed with a smooth volume form μ. Does there exist a non-empty open subset O ⊂ Diff_{μ}^{1}(M) and a dense subset D ⊂ O such that for every f ∈ D the centralizer C_{μ}(f) is not trivial ?

1 Local embeddability in a flow on the circle

1.1 Preparation of diffeomorphisms of the circle

The following proposition summarizes some very classical properties of diffeomorphisms of the circle.

Proposition 1.1. Let D_{0} ⊂ Diff ^{1}(S^{1}) be the set of diffeomorphisms f satisfying the following properties:

• f is a C^{∞} Morse-Smale diffeomorphism (i.e. the non wandering set consists of finitely many hyperbolic periodic points, alternately attracting or repelling); • for every periodic point x ∈ P er(f), there is a neighborhood U_{x} of x such

2This uses the fact that if λ is an eigenvalue of a symplectic matrix, then \overline{λ}, \frac{1}{λ}, \frac{1}{\overline{λ}} are also eigenvalues.

that the restriction f|_{U_{x}}: U_{x} → f(U_{x}) is an affine map (for the natural affine structure on S^{1} = ℝ/ℤ);

• if x, y ∈ P er(f) are periodic points with distinct orbits, then Df^{q}(x) ≠ Df^{q}(y), where q is the period of x.

Then D_{0} is dense in Diff^{1}(S^{1}).

For α > 1 and β ∈ (0, 1), we introduce the set D_{α,β} of orientation-preserving C^{∞} diffeomorphisms of the interval [0, 1] with the following properties:

• {0, 1} is the set of fixed points of f, and f(x) > x for x ∈ (0, 1);

• f(x) = αx for small x and f(x) = 1 + β(x − 1) for x close to 1.

Applying Proposition 1.1, we will prove Theorem 1 by working inside the space D_{α,β}.

1.2 The Mather invariant

We recall here a construction introduced by J. Mather [M] which associates to any diffeomorphism f ∈ D_{α,β} a class of diffeomorphisms of S^{1}.

Let us fix α > 1 and β ∈ (0, 1) and introduce a C^{∞} orientation preserving diffeomorphism φ: (0, 1) → R such that φ(x) = ln x/ln(α) for x small, and φ(x) = ln(1 − x)/ln(β) for x close to 1. Observe that there exists a constant K_{0} > 0 such that φ^{−1}(x) = exp(ln(α) x) for x < −K_{0} and φ^{−1}(x) = 1 − exp(ln(β) x), for x > K_{0}.

For any f ∈ D_{α,β} the conjugated diffeomorphism θ_{f} = φ ∘ f ∘ φ^{−1} of ℝ satisfies θ_{f}(x) > x for all x; furthermore, θ_{f}(x) agrees with x + 1 when |x| is larger than a constant K_{f} > K_{0}.

The space ℝ/θ_{f} of the orbits of θ_{f} is a smooth circle S_{f} which has two natural identifications with the (affine) circle S^{1} = ℝ/ℤ: two points x, y ∈ (−∞, −K_{f}] (resp. x, y ∈ [K_{f}, +∞)) are in the same orbit for θ_{f} if and only if they differ by an integer. This leads to two diffeomorphisms π_{+} : S_{f} → S^{1} and π_{−}: S_{f} → S^{1}, respectively. Let Δ_{f,φ} = π_{+} ∘ π_{−}^{−1} : S^{1} → S^{1}.

Proposition 1.2. The diffeomorphism f is the time one map of a C^{1}-vector field if and only if Δ_{f,φ} is a rotation.

Proof. Note that f ∈ D_{α,β} agrees with the time one map of the vector field X^{−} = ln(α) x \frac{∂}{∂x} in a neighborhood of 0 and with X^{+} = ln(β) (x − 1) \frac{∂}{∂x} in a neighborhood of 1. Furthermore, if f is the time one map of a C^{1}-vector field X on [0, 1], then X = X^{−} in a neighborhood of 0 and X = X^{+} in a neighborhood of 1. The hypothesis on φ implies φ_{∗}(X_{−}) = \frac{∂}{∂x} on some interval (−∞, L_{−}) and φ_{∗}(X_{+}) = \frac{∂}{∂x} on an interval (L_{+}, ∞).

Assume that Δ_{f,φ} is a rotation. Then we define a vector field Y on ℝ as follows: consider n > 0 such that θ_{f}^{n}(x) > K_{f}. Now let Y (x) = (D_{x}θ_{f}^{n})^{−1} (\frac{∂}{∂x}). This vector does not depend of n (because θ_{f} is the translation t ↦ t + 1 for t ≥ K_{f}).

Claim. if x < −K_{f} then Y (x) = \frac{∂}{∂x}.

Proof. Consider the natural projection π_{f}: ℝ → S_{f} that maps each point to its orbit for θ_{f}. Since Y is invariant under θ_{f}, the vector field (π_{f})_{∗}(Y) is welldefined. Since on (K_{f}, +∞) the vector Y (x) is equal to \frac{∂}{∂x}, the map π_{+} ∘ π_{f} agrees with the natural projection ℝ → S^{1}, and we have (π_{+}∘π_{f})_{∗}(Y (x)) = \frac{∂}{∂x}. As Δ_{f,φ} is a rotation, and as the rotations preserve the vector field \frac{∂}{∂x}, we obtain that (π_{−} ∘ π_{f})_{∗}(Y (x)) = (Δ_{f,φ}^{−1} ∘ π_{+} ∘ π_{f})_{∗}(Y (x)) = \frac{∂}{∂x}. As θ_{f} agrees with the translation t ↦ t + 1 on (−∞, −K_{f}], the projection π_{−} ∘ π_{f} agrees on (−∞, −K_{f}] with the natural projection ℝ ↦ ℝ/ℤ. Hence (π_{−} ∘ π_{f})_{∗}(Y (x)) = \frac{∂}{∂x} implies Y (x) = \frac{∂}{∂x}.

Notice that, by construction, the vector field Y is invariant under θ_{f}; furthermore, θ_{f} is the time one map of Y: this is true on a neighborhood of ±∞, and extends on ℝ because Y is θ_{f}-invariant.

Now, the vector field X = φ_{∗}^{−1}(Y), defined on (0, 1), agrees with X_{−} and X_{+} in a neighborhood of 0 and 1, respectively, hence induces a smooth vector field on [0, 1]. Finally, f is the time one map of X.

Conversely, if f is the time one map of a C^{1}-vector field X on [0, 1] then θ_{f} is the time one map of the vector field Y = φ_{∗}(X), which agrees with ∂/∂x in the neighborhood of ±∞ (because X agrees with X_{−} and X_{+} in a neighborhood of 0 and 1, respectively). Hence the projections (π_{−} ∘ π_{f})_{∗}(Y) and (π_{+} ∘ π_{f})_{∗}(Y) are both equal to the vector field ∂/∂x on S^{1}. This implies that (Δ_{f,φ})_{∗}(∂/∂x) = ∂/∂x, which implies that Δ_{f,φ} is a rotation.

Remark 1.3. The function Δ_{f,φ} defined here seems to depend on the choice of φ. There is a more intrinsic way to define the diffeomorphism Δ_{f,φ} “up to composition with rotations:”

The vector fields X_{−} and X_{+} defined in a neighborhood of 0 and 1, respectively, are the unique vector fields such that f is the time one map of the corresponding flows, in the neighborhood of 0 and 1, respectively. Each of these vector fields induces a parametrization of the orbit space (0, 1)/f = S_{f}, that is, up to the choice of an origin, a diffeomorphism π_{f}^{±}: S_{f} → S^{1}. The change of parametrization π_{f}^{+} ∘ (π_{f}^{−})^{−1} is well-defined, up to the choice of an origin of the circle, i.e. up to composition, at the right and at the left, by rotations. This class of maps is called the Mather invariant of f.

1.3 Forcing the Mather invariant to vanish

In this subsection we prove the following result.

Proposition 1.4. Let f be a diffeomorphism in D_{α,β}. Then any C^{1}- neighborhood U of f in Diff^{1}([0, 1]) contains a diffeomorphism g such that g = f in a neighborhood of {0, 1} and g is the time one map of a C^{∞}-vector field on [0, 1].

We retain the notation from the previous subsection. Fix f ∈ D_{α,β} and K_{f} > 0 such that θ_{f} = φ ∘ f ∘ φ^{−1} agrees with x ↦ x + 1 on (−∞, −K_{f}] ∪ [K_{f}, +∞).

Given a diffeomorphism h: ℝ → ℝ, the support of h, denoted by supp(h) is the closure of the set of points x such that h(x) ≠ x.

Lemma 1.5. Consider a number a > K_{f} and a diffeomorphism ψ̃: ℝ → ℝ whose support is contained in (a, a + 1). Let h denote the diffeomorphism φ^{−1} ∘ ψ̃ ∘ φ, and let ψ denote the diffeomorphism of S^{1} ≃ [a, a + 1]/a ∼ a + 1 induced by ψ̃.

Then the diffeomorphism g = f ∘ h belongs to D_{α,β}, and Δ_{g,φ} = ψ ∘ Δ_{f,φ}.

Proof. The diffeomorphism g agrees with f in neighborhoods of 0 and 1, which proves that g ∈ D_{α,β}. Furthermore, by construction, one may choose K_{g} = a + 1.

For x < −a, there is a (unique) integer such that θ_{f}^{n}(x) = θ_{g}^{n}(x) ∈ [a, a+1), and by construction of Δ_{f,φ}, the projection of θ_{f}^{n}(x) on S^{1} is Δ_{f,φ(x)}. Now the projection on S^{1} of θ_{g}^{n+1}(x) = θ_{f} ∘ ψ̃ ∘ θ_{f}^{n}(x) is ψ ∘ Δ_{f,φ}(x), by construction. As θ_{g} = θ_{f} = y ↦ y + 1 for y ≥ a + 1, one gets that the projection on S^{1} of θ_{g}^{n+k}(x) is ψ ∘ Δ_{f,φ}(x), for all k > 0; hence Δ_{g,φ} = ψ ∘ Δ_{f,φ}.

Iterating the process described in Lemma 1.5, we obtain:

Corollary 1.6. Consider a finite sequence of numbers a_{i} > K_{f}, i ∈ {1, …, ℓ}, such that a_{i+1} > a_{i} + 1 for all i ∈ {1, …, ℓ − 1}. For every i ∈ {1, …, ℓ}, fix a diffeomorphism ψ̃_{i}: ℝ → ℝ whose support is contained in (a_{i}, a_{i} + 1). Let h_{i} denote the diffeomorphism φ^{−1} ∘ ψ̃_{i} ∘ φ, and let ψ_{i} denote the diffeomorphism of S^{1} induced by ψ̃_{i}. (Note that the diffeomorphisms h_{i} have disjoint support, so that they are pairwise commuting.)

Then the diffeomorphism g = f ∘ h_{1} ∘ h_{2} ∘ ⋯ ∘ h_{ℓ} belongs to D_{α,β}, and we have:

Δ_{g,φ} = ψ_{ℓ} ∘ ⋯ ∘ ψ_{1} ∘ Δ_{f,φ}.

Definition 1.7. Let a ∈ ℝ, and let \overline{a} be its projection on S^{1} = ℝ/ℤ. Given a diffeomorphism ψ: S^{1} → S^{1} with support in S^{1} \ \{\overline{a}\} we call the lift of ψ in (a, a + 1) the diffeomorphism ψ̃_{a}: ℝ → ℝ with support in (a, a + 1) such that for any x ∈ (a, a + 1) the image ψ_{a}(x) is the point of (a, a + 1) which projects to ψ(\overline{x}) where \overline{x} is the projection of x.

We denote by Θ_{a}(ψ) the diffeomorphism of [0, 1] whose expression in (0, 1) is Θ_{a}(ψ) = φ^{−1} ∘ ψ_{a} ∘ φ.

Lemma 1.8. For any C^{1}-neighborhood U of f there is a neighborhood V of Id_{S^{1}} ∈ Diff^{1}(S^{1}) with the following property:

Given any finite sequence a_{i} > K_{f}, i ∈ {1, …, ℓ}, such that a_{i+1} > a_{i} + 1 for all i ∈ \{1, …, ℓ − 1\}, we denote by \overline{a}_{i} the projection of a_{i} on S^{1}. For any i, let ψ_{i} ∈ V be a diffeomorphism of S^{1} with support in S^{1} \ {a_{i}}. Then the diffeomorphism g = f ∘ Θ_{a_{1}}(ψ_{1}) ∘ ⋯ Θ_{a_{ℓ}}(ψ_{ℓ}) belongs to U.

Proof. We fix a neighborhood U_{0} if the identity map of [0, 1] such that, if g_{1}, …, g_{n} ∈ U_{0} and if the support of the g_{i} are pairwise disjoint, then f ∘ g_{1} ∘ h_{2} ∘ ⋯ g_{n} belongs to U. Now the lemma is a direct consequence of Lemma 1.9 below.

Lemma 1.9. For any C^{1}-neighborhood U_{0} of f there is a neighborhood V of Id_{S^{1}} ∈ Diff^{1}(S^{1}) with the following property:

Consider any a > K_{f}, its projection \overline{a} on S^{1} and any diffeomorphism ψ ∈ V with support in S^{1} \ {a}. Then the diffeomorphism Θ_{a}(ψ) belongs to U_{0}.

Proof. Notice that there exists ε > 0 such that U_{0} contains any diffeomorphism h of [0, 1] with sup_{x∈[0,1]} |D_{x}h − 1| < ε.

Now consider a > K_{f} and an integer n > 0. Then for any diffeomorphism ψ of S^{1} with support in S^{1} \ \{\overline{a}\}, the lifts ψ_{a} and ψ_{a+n} are conjugated by the translation x ↦ x + n. As a consequence, Θ_{a+n}(ψ) is obtained from Θ_{a}(ψ) by the conjugacy by the homothety of ratio β^{n}. As a consequence we obtain that sup_{x∈[0,1]} |D_{x}Θ_{a+n}(ψ) − 1| = sup_{x∈[0,1]} |D_{x}Θ_{a}(ψ) − 1|.

Hence it suffices to prove the lemma for a ∈ [K_{f}, K_{f} + 1]. This is a direct consequece of the facts that the derivatives of φ and of φ^{−1} are bounded on φ^{−1}([K_{f}, K_{f} +2]) and [K_{f}, K_{f} +2] respectively, and that for any ψ with support in S^{1} \ {a}, one has:

sup_{x∈[a,a+1]} |D_{x}ψ_{a} − 1| = sup_{x∈S^{1}} |D_{x}ψ − 1|.

Let us now recall a classical result from [B] which is the key point of our proof.

Theorem 6 (Fragmentation lemma). Let M be a closed Riemannian manifold, let r > 0 and let U be a C^{1}-neighborhood of the identity map. Then for any smooth diffeomorphism f of M isotopic to the identity, there exist k ≥ 1 and g_{1}, …, g_{k} ∈ U such that g_{i} = id on the complement of a ball B(x_{i}, r), and

f = g_{1} ∘ ⋯ ∘ g_{k}.

Here we use Theorem 6 on the circle S^{1}, where it is an easy consequence of the result, by M. Herman, that any smooth diffeomorphism is the product of a rotation by a diffeomorphism smoothly conjugate to a rotation. In Section 2, we will also use Theorem 6 on the torus T^{2} and on compact surfaces.

Proof of Proposition 1.4. Given a C^{1}-neighborhood U of f, we choose a C^{1}- neighborhood V of the identity map of S^{1} given by Lemma 1.8. Using Theorem 6, we can write Δ_{f,φ} as a finite product Δ_{f,φ} = ψ_{1}^{−1} ∘ ⋯ ∘ ψ_{ℓ}^{−1} such that ψ_{i} ∈ V, and the support of ψ_{i} is contained in an interval of length \frac{1}{2} in S^{1} (and in particular is not all of S^{1}). Now we choose a finite sequence a_{i} > K_{k} such that a_{i+1} > a_{i} + 1, and such that the projection \overline{a}_{i} does not belong to the support of ψ_{i}. Let h_{i} = Θ_{a_{i}}(ψ_{i}).

Applying Lemma 1.8, we obtain that the diffeomorphism

g = f ∘ h_{1} ∘ h_{2} ∘ ⋯ ∘ h_{ℓ}

belongs to U; applying Corollary 1.6, we get that

Δ_{g,φ} = ψ_{ℓ} ∘ ⋯ ∘ ψ_{1} ∘ Δ_{f,φ} = Id_{S^{1}}.

1.4 Existence of an invariant vector field

In this section we prove the first part of Theorem 1.

Proposition 1.10. Let D_{0}^{∗} be the set of diffeomorphisms f ∈ D_{0} which preserve a C^{∞} Morse-Smale vector field X and such that f^{q} is the time one map of X, where q is the period of the connected components of S^{1} \ P er(f). Then D_{0}^{∗} is dense in Diff^{1}(S^{1}).

Proof. By Proposition 1.1, it is enough to prove that D_{0}^{∗} is dense in D_{0}. Consider f ∈ D_{0}. The set P er(f) is finite. Let I be the set of segments joining two successive periodic points of f; in other words, every element I ∈ I is the closure of a connected component of S^{1} \ Per(f). Notice that f induces a permutation on I. Furthermore, all the elements of I have the same period denoted by q > 0, under this action (this period is equal to 2 if f reverses the orientation, and is equal to the period of the periodic orbits in the orientation preserving case).

Now consider a segment I ∈ I. The endpoints of I are the fixed points of the restriction f^{q}|_{I}; moreover, one endpoint (denoted by a) is a repeller and the other (denoted by b) is an attractor. Let h_{I}: I → [0, 1] be the affine map such that h_{I}(a) = 0 and h_{I}(b) = 1, and let φ_{I}: [0, 1] → [0, 1] denote the diffeomorphism h_{I} ∘ f^{q}|_{I} ∘ h_{I}^{−1}.

According to Proposition 1.4, there is a sequence (ψ_{I,n})_{n∈ℕ}, of diffeomorphisms converging to φ_{I} in the C^{1}-topology when n → +∞, and a sequence (Y_{I,n})_{n∈ℕ} of C^{∞} vector fields on [0, 1] such that ψ_{I,n} agrees with φ_{I} in a small neighborhood of {0, 1} and is time one map of Y_{I,n}. We define g_{I,n} = h_{I}^{−1} ∘ ψ_{I,n} ∘ h_{I}. Notice that each diffeomorphism g_{I,n} agrees with f^{q} in neighborhoods of the endpoints of I and converges to f^{q}|_{I} when n → ∞.

We now define a diffeomorphism f_{I,n} of S^{1} as follows:

f_{I,n} = \begin{cases}f & on S^{1} \ f^{q−1}(I) \\ g_{I,n} ∘ f^{−q+1} & on f^{q−1}(I). \end{cases}

This is a C^{∞} diffeomorphism since it agrees with f in a neighborhood of the periodic orbits. Moreover, (f_{I,n}) converges to f as n approaches +∞.

We denote by X_{I,n} the vector field, defined on the orbit ⋃_{0}^{q−1} f^{i}(I) of the segment I, as follows:

• X_{I,n} = (h_{I}^{−1})_{∗}(Y_{I,n}) on I;

• for all i ∈ {1, …, q − 1} and all x ∈ I:

X_{I,n}(f^{i}(x)) = f_{∗}^{i}(X_{I,n}(x)).

Finally, we fix a family I_{1}, …, I_{ℓ} ⊂ I such that for i ≠ j the segments I_{i} and I_{j} have distinct orbits, and conversely every orbit of segment in I contains one of the I_{i}.

We denote by f_{n} the diffeomorphism of S^{1} coinciding with f_{I_{i},n} on the orbit of I_{i} for all i ∈ {1, …, ℓ}. This diffeomorphism is well-defined because all the f_{I_{i},n} agree with f in a small neighborhood of the periodic points (the endpoints of the segments in I). We denote by X_{n} the vector field on S^{1} that agrees with X_{I_{i},n} on the orbit of I_{i}, for all i ∈ {1, …, ℓ}.

It is easy to see that X_{n} is a smooth vector field on S^{1}, invariant under f_{n}, and such that f_{n}^{q} is the time one map of X_{n}: the unique difficulty consists in checking the continuity and smoothness of the vector field X_{n} at the periodic points. As f_{n} is affine in the neighborhood of the periodic orbits, it follows that, at both sides of a periodic point x, the vector field X_{n} is the affine vector field vanishing at x and whose eigenvalue at x is ln D_{x}f. We hence have built a sequence (f_{n}) in D_{0}^{∗} converging to f in the C^{1}-topology, as required.

1.5 Centralizer of a diffeomorphism f ∈ D_{0}^{∗}

To complete the proof of Theorem 1, it remains to exhibit the centralizer C(f), for f ∈ D_{0}^{∗}. Let X be the smooth vector field such that f^{q} = X^{1}. Denote by {x_{i}} the set of periodic points of f indexed in an increasing way for a cyclic order (according to the natural orientation of the circle S^{1} = ℝ/ℤ). The flow \{X^{t}\}_{t∈ℝ} of X is contained in the centralizer C(f). Let h_{0} = f ∘X^{− \frac{1}{q}}; it belongs to C(f), it acts on P er(f) as f, and h_{0}^{q} = id. The group G_{f} generated by the flow of X and by h_{0} is isomorphic to ℝ × ℤ/qℤ and is contained in C(f). We will prove:

Proposition 1.11. With the notation above, C(f) is isomorphic to ℝ × ℤ/qℤ or ℝ × ℤ/qℤ × ℤ/2ℤ. More precisely, C(f) = G_{f} ≃ ℝ × ℤ/qℤ, except in the special case where f preserves the orientation and has exactly two periodic orbits; in this special case, either C(f) = G_{f} ≃ ℝ × ℤ/qℤ or C(f) = G_{f} × ℤ/2ℤ ≃ ℝ × ℤ/qℤ × ℤ/2ℤ.

Consider now g ∈ C(f). Let Y^{t} = g ∘ X^{t} ∘ g^{−1}. Then {Y^{t}}_{t∈ℝ} is a one parameter group of C^{1}-diffeomorphisms commuting with f, and f^{q} = Y^{1}. One easily deduces that Y^{t} fixes every (oriented) segment [x_{i}, x_{i+1}]. As f^{q} is an affine contraction or dilation in the neighborhood of x_{i} and x_{i+1}, one also easily deduces that Y^{t} = X^{t} for every t, so that g commutes with the flow of X. In particular, this shows that the group G_{f} is contained in the center of C(f) (i.e. every element of G_{f} commutes with every element of C(f)).

The diffeomorphism g preserves the (finite) set P er(f), so that P er(f) consists in periodic orbits of g. Let ℓ > 0 be the smallest positive integer such that g^{ℓ} preserves the orientation and has a fixed point. This implies that every x_{i} ∈ P er(f) is a fixed point of g^{ℓ}. As f^{q} is an affine contraction or dilation in a neighborhood of x_{i}, and g^{ℓ} commutes with f^{q}, it follows that g^{ℓ} is an affine map in the neighborhood x_{i}: in other words, g^{ℓ} agrees in the neighborhood of x_{i} with the time-t_{i} map X^{t_{i}} of the flow of X. As a consequence, g^{ℓ} agrees with X^{t_{i}} on the basin (stable or unstable manifold) of x_{i}. Since the basin of x_{i} meets the basin of x_{i+1}, an inductive argument shows that the time t_{i} does not depend on i. This proves that there exists a t such that g^{ℓ} = X^{t}. Let h_{g} = g ∘ X^{− \frac{t}{ℓ}}. Then h belongs to C(f), induces the same permutation of the periodic points of f as g, and (as g commutes with X^{− \frac{t}{ℓ}}), we have h_{g}^{ℓ} = id.

The proposition now follows from two remarks:

• Since the derivatives (at the period) of the periodic orbits f ∈ D_{0} are pairwise distinct, any diffeomorphism g ∈ C(f) preserves each periodic orbit of f.

• An orientation preserving diffeomorphism of S^{1} which is periodic and has a fixed point is the identity map.

We can now complete the proof: \underline{Case a:} f reverses the orientation. Then f has two fixed points. Assume g ∈ C(f) preserves the orientation; then so does h_{g}. By the first remark, h_{g} fixes the fixed points of f. By the second remark, h_{g} = id, and so g = X^{\frac{t}{ℓ}} ∈ G_{f}.

If g reverses the orientation, then f^{−1} ∘ g ∈ C(f) preserves the orientation: so there exists r ∈ ℝ such that g = X^{r} ∘ f ∈ G_{f}.

This shows that C(f) = G_{f} ≃ ℝ × ℤ/2ℤ. \underline{Case b:} f preserves the orientation and has at least 3 periodic orbits. Then the periodic orbits of f are cyclically ordered. Since g ∈ C(f) preserves every periodic orbit, it preserves this order, and hence g preserves the orientation.

Let x ∈ P er(f). Then h_{g}(x) belongs to the f-orbit of x, hence to the h_{0} orbit of x. So there exists r such that h_{g} ∘ h_{0}^{r} has a fixed point. As h_{0} belongs to the center of C(f), it commutes with h_{g}. As h_{0} and h_{g} are periodic we deduce that h_{g} ∘ h_{0}^{r} is periodic, hence is the identity map, because it is orientation-preserving and has a fixed point. So h_{g} = h_{0}^{r}. This proves that g ∈ G_{f}, and so C(f) = G_{f} ≃ ℝ × ℤ/qℤ. \underline{Case c:} f preserves the orientation and has exactly 2 periodic orbits. Notice that this case represents an open subset D_{1} ⊂ D_{0}^{∗}. We will prove that every f ∈ D_{1} satisfies C(f) = ℝ×ℤ/qℤ×ℤ/2ℤ or C(f) = G_{f} ≃ ℝ×ℤ/qℤ and that both behaviors are dense in D_{1}. The argument in the previous case shows that every g ∈ C(f) preserving the orientation belongs to G_{f}. The two possible behaviors depend on the existence of an orientation-reversing g ∈ C(f). We will exhibit an invariant (of differentiable conjugacy) that vanishes if and only if C(f) contains an orientation-reversing element.

The space of orbits of h_{0} is a circle S_{0} = S^{1}/h_{0}, and f induces on that circle a diffeomorphism f_{0} = f/h_{0} with exactly 2 fixed points: one attractor and one repellor; moreover, there are smooth coordinates on S_{0} ≃ ℝ/ℤ such that the fixed points of f_{0} are a source at 0 and a sink at \frac{1}{2}, and such that f_{0} is affine in the neighborhood of its fixed points. We fix ε > 0 such that f_{0} is affine on [−ε, ε] and on [\frac{1}{2} − ε, \frac{1}{2} + ε].

The vector field X induces on S_{0} a vector field Y, invariant by f_{0}, and f_{0}^{q} is the time 1 map of Y. It follows that f_{0} = Y ^{\frac{1}{q}}.

For r, s ∈ (0, ε] let t^{+}(r, s) and t^{−}(r, s) be the time length of the Y -orbit from r to \frac{1}{2} − s and from −r to \frac{1}{2} + s. One easily checks that t^{+}(r, s) − t^{−}(r, s) does not depend on r, s ∈ (0, ε]; let θ_{f} = t^{+}(r, s) − t^{−}(r, s).

Lemma 1.12. C(f) contains an orientation-reversing element if and only if θ_{f} = 0.

Proof. If g ∈ C(f) is orientation-reversing, then h_{g} projects on S_{0} to a diffeomorphism commuting with Y and agreeing with r ↦ −r on [−ε, ε) and with \frac{1}{2} − s ↦ \frac{1}{2} + s on [\frac{1}{2} − ε, \frac{1}{2} + ε]. This implies t^{−}(r, s) = t^{+}(r, s). Conversely, if θ_{f} = 0, then it is possible to build an orientation-reversing symmetry for the flow Y, agreeing with r ↦ −r on [−ε, ε) and with \frac{1}{2} −s ↦ \frac{1}{2} +s on [\frac{1}{2} −ε, \frac{1}{2} +ε]. This symmetry lifts on S^{1} to a element g ∈ C(f).

We conclude the proof of the proposition by proving

Lemma 1.13. The subsets {f ∈ D_{1}, θ_{f} = 0} and {f ∈ D_{1}, θ_{f} ≠ 0} are both dense in D_{1}.

The proof is very analoguous to the argument that forces the Mather invariant to vanish. One proves that C^{1}-small perturbations can give arbitrary values of θ_{f}.

If D_{1}^{∗} denotes the second set that appears in lemma 1.13, then Theorem 1 is now proved with the set

D = (D_{0}^{∗} \ D_{1}) ∪ D_{1}^{∗}.

1.6 Open questions on the circle

Theorem 1 presents a dense subset of Diff^{1}(S^{1}) of smooth diffeomorphisms whose centralizer is not trivial, and more precisely is isomorphic to a group ℝ × ℤ/qℤ, for some integer q. However, it is known that C^{1}-diffeomorphisms of S^{1} may have very different centralizers. For instance, [FF] presents faithful actions on S^{1} of any finitely generated nilpotent (non-abelian) group G. This nilpotent group G has a non-trivial center Z(G), and for every element f ∈ Z(G) of this center, the centralizer C(f) contains the whole group G, hence is not abelian.

Conjecture 1. The set of diffeomorphisms f ∈ Diff^{1}(S^{1}) with a non-abelian centralizer is dense in Diff^{1}(S^{1}).

It could be interesting to build examples of diffeomorphisms with irrational rotation number and having a non-abelian centralizer.

Theorem 1 uses the fact that C^{1}-small perturbations allow us to change arbitrarily the Mather invariant of any smooth diffeomorphism of the interval [0, 1] whose fixed points are precisely 0 and 1. This proves that, in the set D_{α,β} we defined, every class of smooth conjugacy is C^{1}-dense. This suggests another question. For α > 1 and 0 < β < 1, consider the set D̂_{α,β} ⊂ Diff^{1}([0, 1]) of diffeomorphisms f whose fixed points are precisely 0 and 1 and such that Df(0) = α and Df(1) = β (this set contains D_{α,β}, and it differs from D_{α,β} because we do not require f to be affine in the neighborhood of 0 and 1). Notice that D̂_{α,β} is invariant by conjugacy by orientation preserving diffeomorphisms of [0, 1].

Conjecture 2. Every C^{1}-conjugacy class is dense in D̂_{α,β}. In other words, given any two elements f, g ∈ D̂_{α,β}, there is a diffeomorphism arbitrarily C^{1}- close to g that is conjugated to f by a diffeomorphism of [0, 1].

A positive answer to this conjecture would allow us to show that every pathological behavior (in particular of the centralizer) built on a Morse-Smale example would appear densely in Diff^{1}(S^{1}). It would be interesting to understand the same question for diffeomorphisms with irrational rotation number:

Question VI. Is every C^{1}-conjugacy class dense in the set of diffeomorphisms having a given rotation number α ∈ ℝ \ ℚ?

2 Local embeddability in a flow on S^{2}

As in the one-dimensional case, the idea here is to measure how far certain diffeomorphisms of S^{2} are from the time one map of a vector field. One obtains in this way a generalization of the Mather invariant, which in this setting is a diffeomorphism of 𝕋^{2}. Such an invariant has already been constructed^{3} in [AY] by V. Afraimovich and T. Young, and we now have to show that by a C^{1}-small perturbation of the dynamics, this invariant vanishes.

2.1 Preparation of diffeomorphisms in O

Let O be as in Theorem 2. Let S^{2} be the unit sphere in ℝ^{3} endowed with the coordinates (x, y, z). We denote by N = (0, 0, 1) and S = (0, 0, −1) the north and the south poles of S^{2}. Notice that the coordinates x, y define local coordinates of S^{2} in local charts U_{N} and U_{S} in neighborhoods of N and S.

The following straightforward lemma asserts that one may assume that the fixed points of any diffeomorphism f in the open set O are N and S and that the derivative at these points are conformal maps.

Proposition 2.1. Consider a diffeomorphism f ∈ O. Then there is a smooth diffeomorphism h: S^{2} → S^{2} such that h(N_{f}) = N, h(S_{f}) = S are the fixed points of g = hfh^{−1}; furthermore, the derivatives D_{N}g and D_{S}g are conformal linear maps, i.e., each of them is a composition of a rotation with a homothety of ratio α > 1 and β < 1, respectively.

Finally, any C^{1}-neighborhood of g contains a diffeomorphism g̃ such that there are neighborhoods V_{N} ⊂ U_{N} and V_{S} ⊂ U_{S} of N and S, respectively, such that the expression of g̃ in the coordinates (x, y) is g̃(x, y) = D_{N}g(x, y) for (x, y) ∈ V_{N} and g̃(x, y) = D_{S}g(x, y) for (x, y) ∈ V_{S}.

2.2 Space of orbits of a conformal linear map

Let A ∈ GL(ℝ, 2) be a conformal matrix of norm α ≠ 1. There exists a ∈ [0, 2π) such that A = R_{a} ∘ h_{α} where R_{a} is the rotation of angle a and b and h_{α} is the homothety of ratio α. Notice that, for all n ∈ ℤ, the linear map A is the time one map of the vector field

X_{A,n} = ln α.(x\frac{∂}{∂x} + y \frac{∂}{∂y}) + (a + 2πn).(x\frac{∂}{∂y} − y \frac{∂}{∂x}).

The orbit space T_{A} = ℝ^{2} \ {0}/A (of the action of A on ℝ^{2} \ {0}) is a torus (diffeomorphic to T^{2} = ℝ^{2}/ℤ^{2}); we denote by π_{A} the canonical projection from ℝ^{2} \ {0} onto T_{A}. Moreover, the vector fields

3In [AY], the authors write that the Mather invariant for a diffeomorphism of S^{2} is always isotopic to the identity, but this is not correct (their Proposition 1 contains an error). For this reason, we choose here to build in detail the construction of this invariant on the sphere.

Z = 2π(x\frac{∂}{∂y} − y \frac{∂}{∂x})

and X_{A,n} project on T_{A} to pairwise transverse commuting vector fields, which we also denote by Z and X_{A,n}; the orbits of both flows are periodic of period 1. Hence, for any pair (Z, X_{A,n}) there is a diffeomorphism L_{A,n}: T_{A} → T^{2} = ℝ^{2}/ℤ^{2} sending Z to ∂/∂x and X_{A,n} to ∂/∂y; this diffeomorphism is unique up to composition with a translation of T^{2}. Furthermore, the diffeomorphisms L_{A,m} ∘ L_{A,n}^{−1} are affine maps of the torus T^{2}, for all n, m ∈ ℤ, so that T_{A} is endowed with a canonical affine structure (indeed the affine map L_{A,m} ∘

L_{A,n}^{−1} on T^{2} is the map induced by the matrix ( \begin{matrix}1 & n − m \\ 0 & 1 \end{matrix} ) composed with a

translation).

Note that the orbits of Z correspond to the positive generator of the fundamental group of ℝ^{2} \ {0}; we denote by σ the corresponding element of π_{1}(T_{A}). Given any closed loop γ: [0, 1] → T_{A}, and any point x ∈ ℝ^{2} \ {0} with π_{A}(x) = γ(0), there is a lift of γ to a path in ℝ^{2} \{0} joining x to A^{k}(x), where k is the algebraic intersection number of σ with γ. Finally, observe that the homotopy classes corresponding to the orbits of X_{n,A}, when n ∈ ℤ are precisely those whose intersection number with σ is 1: in other words, there is a basis of π_{1}(T_{A}) = ℤ^{2} such that σ = (1, 0) and the orbits of X_{A,n} are homotopic to (n, 1).

2.3 A Mather invariant for diffeomorphisms of S^{2}

Denote by D_{A,B} ⊂ O the set of diffeomorphisms f ∈ O whose expression in the coordinates (x, y) coincides with some conformal linear maps A and B in neighborhoods U_{f}^{N} of N and U_{f}^{S} of S. The aim of this part is to build a Mather invariant for diffeomorphisms in D_{A,B}.

Consider f ∈ D_{A,B}. We retain the notations of the previous subsection and introduce the affine tori T_{A} and T_{B}. The orbit space (S^{2} \ {N, S}) /f is a torus T_{f} and we denote by π_{f}: S^{2} \ {N, S} → T_{f} the natural projection. Furthermore, as f agrees with A on U_{f}^{N}, the torus T_{f} may be identified with the torus T_{A} by a diffeomorphism π_{N}: T_{f} → T_{A}, and in the same way, the fact that f coincides with B in a neighborhood of S induces a diffeomorphism π_{S}: T_{f} → T_{B}.

Notice that the homomorphisms π_{N∗}: H_{1}(T_{f}, ℤ) → H_{1}(T_{A}, ℤ) and π_{S∗}: H_{1}(T_{f}) → H_{1}(T_{B}) preserve the homology class of σ (corresponding to the positive homology generator of S^{2} \ {N, S} or of ℝ^{2} \ {0}), and the homology intersection form with σ.

Consequently, for any f ∈ D_{A,B}, there is an integer n(f) such that the map Δ_{f,0,0} = L_{B,0} ∘ π_{S} ∘ π_{N}^{−1} ∘ L_{A,0}^{−1} is isotopic to the linear map of T^{2} induced by the matrix ( \begin{matrix}1 & n(f) \\ 0 & 1 \end{matrix} ).

Lemma 2.2. For any f ∈ D_{A,B} there is a C^{1}-neighborhood U of f in Diff^{1}(S^{2}) such that for any g ∈ U ∩ D_{A,B} one has n(f) = n(g).

Proof. Let us consider S^{2} contained in ℝ^{3}. We can choose a neighborhood U such that, if g ∈ U then the map

f_{t}(x) = \frac{(1 − t)f(x) + tg(x)}{∥(1 − t)f(x) + tg(x)∥}

is a smooth isotopy between f and g. Furthermore, by shrinking U if necessary, for any g ∈ U, the isotopy f_{t} belongs to O (that is Ω(g) = {N_{g}, S_{g}}).

If g ∈ U ∩ D_{A,B} then there are discs D^{N} and D^{S} centered on N and S, respectively, such that f_{t} = A on D^{N} and f_{t} = B on D^{S} so that f_{t} ∈ D_{A,B}. In particular f_{t}(D^{S}) ⊂ D^{S}, and f_{t}^{−1}(D^{N}) ⊂ D^{N}. Furthermore, there exists ℓ > 0 such that for any x ∈ S^{2} \ (D^{N} ∪ D^{S}), f_{t}^{ℓ}(x) ∈ D^{S} and f_{t}^{−ℓ}(x) ∈ D^{N}.

Let x ∈ D^{N} such that A(x) = f_{t}(x) ∈ D^{N} and A^{2}(x) ∉ D^{N}. Hence y_{t} = f_{t}^{ℓ+2}(x) ∈ D^{S} and f_{t}(y_{t}) = B(y_{t}) ∈ D^{S}. Let γ be the segment of orbit of X_{A,0} joining x to A(x) = f_{t}(x), and let γ_{t} = f_{t}^{ℓ+2}(γ). For every t, γ_{t} is homotopic (relative to {y_{t}, B(y_{t})} in S^{2} \ {N, S}) to a segment of orbit of X_{B,n(f_{t})}. As a consequence, n(f_{t}) varies continuously with t as t varies from 0 to 1. Hence n(f_{t}) is constant; that is, n(g) = n(f).

Hence there is a partition of D_{A,B} into open subsets D_{A,B,n} such that n(f) = n for f ∈ D_{A,B,n}. For f ∈ D_{A,B,n}, we define:

Δ_{f} = L_{B,n} ∘ π_{S} ∘ π_{N}^{−1} ∘ L_{A,0}^{−1}.

Then Δ_{f} is a diffeomorphism of T^{2}, isotopic to the identity.

Theorem 7 below justifies calling Δ_{f} the Mather invariant of f.

Theorem 7. Let f ∈ D_{A,B,n} be a smooth diffeomorphism such that Δ_{f} is a translation of the torus T^{2}. Then f leaves invariant two transverse commuting vector fields Z_{f} and X_{f} on S^{2} such that Z_{f} = Z in a neighborhood of {N, S}, X_{f} = X_{A,0} in a neighborhood of N and X_{f} = X_{B,n} in a neighborhood of S.

As a consequence the centralizer of f is isomorphic to S^{1} × ℝ.

Proof. Fix two discs D^{N} and D^{S} centered at N and S, respectively, in which f coincides with A and B, respectively.

For any x ≠ S there exists m(x) < 0 such that f^{m(x)}(x) ∈ D^{N}. One defines Z_{f}(x) = f_{∗}^{−m(x)}(Z(f^{m(x)})) and X_{f}(x) = f_{∗}^{−m(x)}(X_{A,0}(f^{m(x)})). As Z and X_{A,0} are invariant by A, one proves that the vectors Z_{f}(x) and X_{f}(x) are independent of the choice of m(x). As a consequence, one deduces that they depend smoothly on x ∈ S^{2} \ {S} and that they commute on S^{2} \ {S}. Furthermore the restrictions of Z_{f} and X_{f} to D^{S} are invariant by f, and hence by B, so that they induce two vector fields on T_{B} whose images by L_{B,n} are Δ_{f}(\frac{∂}{∂x}) = \frac{∂}{∂x} and Δ_{f}(\frac{∂}{∂y}) = \frac{∂}{∂y}, respectively; that is, they agree with the projections of the restrictions Z and X_{B,n} to D^{S}. Thus Z_{f} = Z and X_{f} = X_{B,n} on D^{S}, proving that the centralizer C(f) contains the subgroup of linear conformal maps {Z_{f}^{t} ∘ X_{f}^{t}, (t, s) ∈ S^{1} × ℝ}.

Conversely, the following easy lemma shows that any diffeomorphism g ∈ C(f) agrees with a linear conformal map h in a neighborhood of S. Since g and h both commute with f, one deduces that g and h coincide on the whole sphere. Hence C(f) is the group of conformal linear maps, which is isomorphic to S^{1} × ℝ.

Lemma 2.3. Let B = R_{b} ∘ h_{β}, with b ∈ [0, 2π) and β ≠ 1 be a linear conformal contraction whose angle b is different from 0 and π. Then, any local diffeomorphism g defined in a neighborhood of 0 and that commutes with B coincides with a conformal linear map.

2.4 Vanishing of the Mather invariant

This part is now very close to the 1-dimensional case: we consider f ∈ D_{A,B,n}, a disk D_{f}^{S} centered on S on which f = B and want to perturb the homeomorphism Δ_{f}.

Let h: S^{2} → S^{2} be a diffeomorphism whose support is contained in a disk D ⊂ D_{f}^{S}, disjoint from all B^{m}(D) for m > 0. The disk D projects homeomorphically onto a disk D′ ⊂ T_{B}, and finally onto a disk D̃ = L_{B,n}(D′) ⊂ T^{2}. Let ψ be the diffeomorphism of T^{2} with support in D̃ whose restriction to D̃ is the projection of h. We says that ψ is the projection of h on T^{2} and conversely, that h is the lift of ψ with support in D.

Fix k > 0 such that D is disjoint from B^{k}(D_{f}^{S}).

Lemma 2.4. Consider a disk D and a diffeomorphism h as above. Fix k > 0 such that D is disjoint from B^{k}(D_{f}^{S}). Then, f ∘h is a diffeomorphism in D_{A,B,n} with B^{k}(D_{f}^{S}) ⊂ D_{f∘h}^{S}, and whose Mather invariant is

Δ_{f∘h} = ψ ∘ Δ_{f}.

Corollary 2.5. Let D_{0}, …, D_{ℓ} ⊂ D_{f}^{S} be a finite sequence of disks such that

• for every i, the disk D_{i} is disjoint from B^{k}(D_{i}) for k > 0;

• for all i < j the disk D_{i} is disjoint from B^{k}(D_{j}), k ≥ 0

For every i, let h_{i} be a diffeomorphism of S^{2} with support in D_{i}, and let ψ_{i} be the projection of h_{i} on T^{2} (by L_{B,n} ∘ π_{S} ∘ π_{f}).

Then the Mather invariant of f ∘ h_{0} ∘ ⋯ ∘ h_{ℓ} is

Δ_{f∘h_{0}∘···∘h_{ℓ}} = ψ_{ℓ} ∘ ⋯ ∘ ψ_{0} ∘ Δ_{f}.

Reciprocally, for any disk D̃ ⊂ T^{2} with diameter strictly less than 1, each connected component of (L_{B,n} ∘ π_{S} ∘ π_{f})^{−1}(D̃) projects diffeomorphically onto D̃, and f induces a permutation of these components. For i > 0, let D_{i} denote the (unique) component of (L_{B,n} ∘ π_{S} ∘ π_{f})^{−1}(D̃) such that f^{−i}(D_{i}) ⊂ D_{f}^{S} but f^{−(i+1)}(D_{i}) is not contained in D_{f}^{S}. For any diffeomorphism ψ with support in D̃ we will denote by θ_{i}(ψ): S^{2} → S^{2} the lift of ψ with support in D_{i}.

The next lemma is the unique reason we required that the derivative of f at N, S be complex, hence conjugate to conformal linear maps:

Lemma 2.6. Let D̃ ⊂ T^{2} be a disk with diameter strictly less than 1 and let i, j ∈ ℕ. Then:

sup_{x∈S^{2}} ∥D_{x}θ_{i}(ψ) − Id∥ = sup_{x∈S^{2}} ∥D_{x}θ_{j}(ψ) − Id∥.

Proof. θ_{i}(ψ) is conjugated to θ_{j}(ψ) by B^{j−i}, which is the composition of a homothety with a rotation; the C^{1}-norm is preserved by conjugacy by isometries, and by homotheties, and hence is preserved by the conjugacy by B^{j−i}.

Corollary 2.7. For any ε > 0 there is a C^{1}-neighborhood V_{ε} ⊂ Diff(T^{2}) of the identity map such that for any diffeomorphism ψ ∈ V_{ε} with support in a disk D̃ ⊂ T^{2} with diameter strictly less than 1, and for any i ≥ 0, the lift θ_{i}(ψ) satisfies :

sup_{x∈S^{2}} ∥D_{x}θ_{i}(ψ) − Id∥ < ε.

Definition 2.8. Let ψ_{1}, …, ψ_{ℓ} be ℓ diffeomorphisms of T^{2} such that the support of every ψ_{i} is contained in a disk D̃_{i} with diameter strictly less than 1; a lift of the sequence ψ_{1}, …, ψ_{ℓ} is a sequence of lifts h_{1} = θ_{i_{1}}(ψ_{1}), …, h_{ℓ} = θ_{i_{ℓ}}(ψ_{ℓ}) such that, for every i < j the support of h_{i} is disjoint from all the iterates B^{k}(supp(h_{j})), for k ≥ 0.

It is straightforward to check that, for any sequence ψ_{1}, …, ψ_{ℓ} of diffeomorphisms of T^{2} such that the support of every ψ_{i} is contained in a disk D̃_{i} with diameter strictly less than 1, the sequence h_{i} = θ_{i}(ψ_{i}) is a lift.

Proof of Theorem 2. Consider f ∈ D_{A,B,n} and a C^{1}-neighborhood U of f. Fix ε > 0 such that, if g_{1}, …, g_{m}, m > 0, are diffeomorphisms of S^{2} with pairwise disjoint supports in S^{2} \ {N, S}, and such that sup_{x∈S^{2}} ∥Dg_{i}(x) − Id∥ < ε, then f ∘ g_{1} ∘ ⋯ ∘ g_{m} ∈ U. Let V_{ε} be the C^{1}-neighborhood of the identity map of T^{2} given by Corollary 2.7.

Using Theorem 6, we write

Δ_{f} = ψ_{1}^{−1} ∘ ⋯ ∘ ψ_{ℓ}^{−1},

for some ℓ > 0, where ψ_{i} ∈ V_{ε}, and the support of ψ_{i} is contained in a disk D̃_{i} with diameter strictly less than 1. Let (h_{1}, …, h_{ℓ}) be a lift of the sequence (ψ_{1}, …, ψ_{ℓ}); the h_{i} satisfy

sup_{x∈S^{2}} ∥D_{x}h_{i} − Id∥ < ε,

by our choice of V_{ε}.

Our choice of ε > 0 implies that g = f ∘ h_{1} ∘ ⋯ ∘ h_{ℓ} is a diffeomorphism belonging to D_{A,B,n} ∩ U. Furthermore, its Mather invariant is Δ_{g} = ψ_{ℓ} ∘ ⋯ ∘ ψ_{1} ∘ Δ_{f} = Id.

We have just shown that any f ∈ D_{A,B} is the C^{1}-limit of a sequence g_{k} ∈ D_{A,B} whose Mather invariant is the identity map; in particular, the centralizer of g_{k} is isomorphic to ℝ × S^{1}.

Since by Proposition 2.1, O contains a dense set of diffeomorphisms smoothly conjugate to elements of D_{A,B,n}, any diffeomorphism in O is the limit of diffeomorphisms g_{k} that are the time 1 map of Morse-Smale vector fields, ending the proof of Theorem 2.

3 Local embeddability in a flow on surfaces

The aim of this section is to prove Theorem 3: we consider a closed connected surface S, and let O ⊂ Diff^{1}(S) be the set of Morse-Smale diffeomorphisms as defined in the statement of Theorem 3. Since Morse-Smale systems are structurally stable, we have that O is a non-empty open subset of Diff^{1}(S).

Let D_{1} ⊂ O be the dense subset of O such that for every f ∈ D_{1} one has:

• every fixed point x of f admits a neighborhood U_{x} and smooth local coordinates defined on U_{x} such that the expression of f in restriction to U_{x} ∩ f^{−1}(U_{x}) is linear (hence coincides with the derivative Df(x));

• given any two fixed points x, y of f, one has det Df(x) ≠ det Df(y).

The eigenvalues of every sink or source q of a diffeomorphism f ∈ D_{1} are non-real. Hence we can choose the local coordinates on U_{q} in such a way that the restriction of f to U_{q} ∩ f^{−1}(U_{q}) is a conformal linear map (i.e. the composition of a rotation with a homothety). In the same way we can choose local coordinates around any saddle p so that f |_{U_{p}∩f^{−1}(U_{p})} is described by a diagonal matrix.

Proposition 3.1. Given any f ∈ D_{1} and any C^{1}-neighborhood U of f, there is g ∈ U such that g is the time one map of a smooth vector field and agrees with f outside an arbitrarily small neighborhood of the sinks and sources of f. Furthermore, the centralizer C(g) of g is isomorphic to ℝ.

Proposition 3.1 clearly implies Theorem 3. The proof of this proposition is the aim of the rest of Section 3. In particular, up to Subsection 3.3, we prove that C(g) contains ℝ, while in Subsection 3.4 we prove the equality.

We fix now a diffeomorphism f ∈ D_{1} and and a neighborhood U of f. Let σ_{f}, α_{f} and ω_{f} denote the sets of saddles, sources and sinks of f, respectively.

3.1 Vector field in a neighborhood of any saddle

Recall that by assumption σ_{f} is non-empty. In this section, we shall build an invariant neighborhood of σ_{f} endowed with a flow which will be our local model around saddles (and their invariant manifolds).

Proposition 3.2. There exists an invariant open neighborhood V_{0} and a vector field Y_{0} on V_{0} such that

• the flow of Y_{0} is complete (i.e. defined from −∞ to +∞);

• the diffeomorphism f coincides with the time one map of Y_{0} on V_{0}.

Proof. We will use the following property satisfied by Morse-Smale diffeomorphisms in O (recall that by definition of O in Theorem 3 there are no heteroclinic orbits):

(*) For any two saddles p_{1}, p_{2} ∈ σ_{f}, there exist neighborhoods B_{p_{1}}, B_{p_{2}} of p_{1} and p_{2}, respectively, such that there is no point x ∈ S \ (U_{p_{1}} ∪ U_{p_{2}}) whose backward orbit intersects B_{p_{1}} and whose forward orbit intersects B_{p_{2}}.

Let p ∈ σ_{f} be any saddle periodic point. One may assume that in the local coordinates (x, y) of U_{p}, the expression of the map f is (x, y) ↦ (λ_{p}^{u}x, λ_{p}^{s}y). Since the eigenvalues λ_{p}^{u}, λ_{p}^{s} are positive, f agrees in U_{p} with the time one map of the vector field

Y_{p}(x, y) = x ln λ_{p}^{u} \frac{∂}{∂x} + y ln λ_{p}^{s} \frac{∂}{∂y} .

For T > 0 large, we introduce the octagon Δ_{p} defined by the equations:

|x| < (λ_{p}^{u})^{−T} , |y| < (λ_{p}^{s})^{T} , \frac{ln |x|^{−1}}{ln λ_{p}^{u}} + \frac{ln |y|^{−1}}{ln(λ_{p}^{s})^{−1}} > 3T.

By property (*), the forward orbit of f(Δ_{p})\Δ_{p} does not intersect Δ_{p}. Hence, one can extend the vector field Y_{p} to V_{p} = ∪_{k∈ℤ}f^{k}(Δ_{p}), so that it is equal to f_{∗}^{k}(Y_{p}) on f^{k}(Δ_{p}) \ f^{k−1}(Δ_{p}) and to f_{∗}^{−k}(Y_{p}) on f^{−k}(Δ_{p}) \ f^{−k+1}(Δ_{p}) for any k > 0. The open set V_{p} is invariant by the flow of Y_{p}, and by construction the restriction of f to V_{p} coincides with the time one map of Y_{p}.

Using (*) again, we deduce that if each domain Δ_{p} has been chosen small enough, then the open sets V_{p} are pairwise disjoint. Hence, one can define on the union V_{0} = ⋃_{p∈σ_{f}} V_{p} a vector field Y_{0} as required that coincides with Y_{p} on any V_{p}.

3.2 Vector field on the punctured surface S \ (ω_{f} ∪ α_{f})

In this section we prove the following proposition:

Proposition 3.3. There exists a vector field Y defined on the punctured surface S \ (ω_{f} ∪ α_{f}) such that:

• the flow of Y is complete;

• the diffeomorphism f is the time one map of Y on S \ (ω_{f} ∪ α_{f}).

Let q ∈ ω_{f} be a sink of f. In the local coordinates we fixed on U_{q}, the expression for the diffeomorphism f is given by a conformal matrix B = R_{b}∘h_{β} with b ∈ [0, 2π) and 0 < β < 1. Hence the results of Subsection 2.2 apply: f is in the neighborhood of q the time one map of each of the vector fields X_{B,n} for any n ∈ ℤ. We also defined the vector field Z = 2π(x\frac{∂}{∂y} − y \frac{∂}{∂x}).

Denote by π_{q} the projection of W^{s}(q)\{q} on the orbit space T_{q} = (W^{s}(q)\ {q})/f, which is a torus T^{2}. The vector fields X_{B,n} and Z project to vector fields whose orbits are all periodic. We can choose a basis for the homology H_{1}(T_{q}, ℤ) such that the class of the orbits of Z is σ = (1, 0) and the class of the orbits of X_{B,n} is (n, 1).

Now consider an invariant open neighborhood V_{0} and a vector field Y_{0} on V_{0} as given by Proposition 3.2. We can also choose a smaller neighborhood V of σ_{f} that is invariant by Y_{0} (by taking small balls centered at each saddle and saturating by the flow of Y_{0}). We want to focus on the traces of V and V_{0} on the orbit spaces T_{q}. We emphasize the following facts.

- The set π_{q}(V_{0} ∩ W^{s}(q) \ {q}) is foliated by the orbits of the projection (π_{q})_{∗}Y_{0}, which are closed, have period 1 and define the same non-zero homology class.

- As f is not a north-south diffeomorphism on the sphere, there is at least one unstable separatrix of a saddle p that is contained in the basin of q.

- The set π_{q}(V ∩ W^{s}(q) \ {q}) is a neighborhood of the projection in T_{q} of the unstable separatrices that are contained in the basin of q. Hence it is compactly contained in π_{q}(V_{0} ∩ W^{s}(q) \ {q}) and invariant by the flow of (π_{q})_{∗}Y_{0}.

This implies the following:

Lemma 3.4. The orbit space T_{q} can be endowed with a vector field Q_{q} such that

• the restriction of Q_{q} to π_{q}(V ∩ W^{s}(q) \ {q}) coincides with (π_{q})_{∗}(Y_{0});

• all the orbits of Q_{q} are closed and of period 1.

Each vector field Q_{q} lifts to a vector field Y_{q} on the open set W^{s}(q) \ {q} = π_{q}^{−1}(T_{q}), and by construction coincides with Y_{0} on the intersection of W^{s}(q)\{q} with V. Hence, we have defined a vector field Y on V ∪⋃_{q∈ω_{f}} W^{s}(q) whose time one map agrees with f. Any point y of S \ (ω_{f} ∪ α_{f} ∪ V) is a wandering point and its ω-limit set is a sink (otherwise y would belong to one of the invariant manifolds of a saddle, hence to V, which is a contradiction). This shows that Y is now defined on the whole punctured surface S \(ω_{f} ∪α_{f}), proving Proposition 3.3.

3.3 Gluing the vector fields around sinks and sources

The aim of this section is to perform a small perturbation of f in a small neighborhood of the sinks and sources of f (but keeping f unchanged in a smaller neighborhood of the sinks and sources) so that the vector field Y provided by Proposition 3.3 can be extended to a smooth vector field on S.

Fix a sink q ∈ ω_{f} and keep the notations of the previous section. The dynamics in a neighborhood of q agree with those of a conformal linear map B. By projecting the vector field Y on the torus T_{q}, we obtain a vector field Ŷ_{q} = (π_{q})_{∗}(Y). Each orbit of Y is a path joining a point y to f(y). consequently, the orbits of Ŷ_{q} on T_{q} are (simple) curves and are in the same homology class as the orbits of a vector field X̂_{q} obtained by projecting the vector field X_{B,n_{q}}, for some n_{q}.

Lemma 3.5. There exists a smooth diffeomorphism ψ_{f,q} of T_{q} that is isotopic to the identity map and such that (ψ_{f,q})_{∗}(Ŷ_{q}) = X̂_{q}.

Proof. The orbits of Ŷ_{q} and X̂_{q} are all periodic of period 1 and are in the same homology class. Let σ be a cross-section of X̂_{p} obtained by projecting an orbit of the vector field Z = 2π(x\frac{∂}{∂y} − y \frac{∂}{∂x}). One chooses a complete smooth cross section σ_{q} of Ŷ_{q} that is in the same homology class as σ and cuts every orbit of Ŷ_{q} in exactly one point. We also choose an orientation-preserving diffeomorphism ψ_{f,q}: σ_{q} → σ. This diffeomorphism extends in a unique way in the announced diffeomorphism of the torus T_{q}.

Remark 3.6. • The diffeomorphism ψ_{f,q} is not unique: its depends on the choice of Y and of the cross section σ_{q} in the neighborhood of q.

• If ψ_{f,q} is the identity map, then the vector field Y agrees with the linear vector fields X_{q} in a neighborhood of q.

Given any f ∈ D_{1}, we will perform a perturbation g_{q} ∈ D_{1} of f whose associated diffeomorphism ψ_{g,q} is the identity map.

Proposition 3.7. Consider f ∈ D_{1}, a smooth complete vector field Y on S \ (α_{f} ∪ ω_{f}) such that f is the time one map of the flow of Y and a sink q ∈ ω_{f}. For every C^{1}-neighborhood U of f, and for every neighborhood O of q there is g_{q} ∈ U ∩ D_{1} with the following properties:

• there is neighborhood O′ ⊂ O of q such that g_{q} = f on O′ ∪ (S \ O);

• there is a smooth complete vector field Ỹ defined on S \ (α_{f} ∪ ω_{f}) ∪ {q} coinciding with Y outside O and such that g_{q} is the time one map of the flow of Ỹ.

One obtains the first part of Proposition 3.1 by applying Proposition 3.7 to f for each sink and to f^{−1} for each source, successively.

The argument for modifying ψ_{f,q} is almost identical to the proof of Theorem 2 at Section 2.4, and we just sketch it.

Sketch of the proof of Proposition 3.7. By reducing O, we may assume that it is a small disk centered at q and contained in U_{q}. Let D ⊂ O be a disk whose iterates f^{i}(D), i ∈ ℕ, are pairwise disjoint and all contained in O. We denote by D̂ = π_{q}(D) the projection of D on T_{q}. Notice that π_{q} induces a diffeomorphism from D to D̂. Let h be a diffeomorphism of S coinciding with the identity map on the complement of D. We denote by ĥ the diffeomorphism of T_{q} that is the identity map on the complement of D̂ and is π_{q} × h × π_{q}^{−1} on D̂. Let g_{h} = f ∘ h, and let Y_{h} be the vector field on S \ (α_{f} ∪ ω_{f}) that coincides with Y in the complement of ⋃_{i>0} f^{i}(D) and with (f^{i} ∘ h)_{∗}(Y) on f^{i}(Q) for i > 0. The following lemma is the analogue of Lemma 2.4.

Lemma 3.8. With the notation above, one has:

• g_{h} belongs to D_{1} and agrees with f in a neighborhood of F ix(f);

• g_{h} is the time one map of the flow of the vector field Y_{h};

• (ψ_{f,q} ∘ ĥ^{−1})_{∗}(Ŷ_{h}) = X̂_{q}, so that one can choose ψ_{g_{h},q} = ψ_{f,q} ∘ ĥ^{−1}.

Proof. The unique difficulty here is to show that g_{h} is the time one map of Y_{h}. For that, let H be the diffeomorphism of S \ (α_{f} ∪ ω_{f}) that is f^{i} ∘ h ∘ f^{−i} on f^{i}(D) for i > 0 and the identity map in the complement of ⋃_{i>0} f^{i}(D). Note that g_{h} is conjugate to f by H and Y_{h} = H_{∗}(Y).

Conversely, for any disk D̂ ⊂ T_{q} with small diameter, the connected components of π_{q}^{−1}(D̂) are diffeomorphic to D̂; we denote by D_{i} the component that is contained in B^{i}(O) but not in B^{i+1}(O). For any diffeomorphism ĥ of T_{q}, with support in the disk D̂, we denote by L_{i}h the lift of ĥ that is supported in D_{i}.

Let now consider a diffeomorphism ψ_{f,q} of T_{q} associated to f and Y. Since ψ_{f,q} is isotopic to the identity map, the fragmentation lemma (Theorem 6) allows us to write ψ_{f,q} as the composition ψ_{f,q} = ĥ_{k} ∘ ⋯ ∘ ĥ_{1} of finitely many diffeomorphisms ĥ_{i} arbitrarily C^{1}-close to the identity map and each with support in an arbitrarily small disk. We then build the lifts h_{i} = L_{2i}(ĥ_{i}) whose supports are pairwise disjoint. Let g = f ∘ h_{1} ∘ ⋯ ∘ h_{k}. Then g belongs to D_{1} and agrees with f in the complement of O and on f^{2k+3}(O). Furthermore, since the diffeomorphisms ĥ_{i} can be chosen close to the identity, g belongs to U. Finally, applying inductively Lemma 3.8, we see that g is the time one map of a vector field Ỹ that coincides with the linear vector field X_{q} in a neighborhood of q. Hence it may be extended smoothly at q, ending the proof.

3.4 End of the proof of theorem 3

Let D_{2} ⊂ D_{1} be the subset of diffeomorphisms that are the time one map of the flow of a smooth vector field. The previous sections proved that D_{2} is dense in D_{1}, hence in the open subset O of Diff^{1}(M). In order to prove Theorem 3, we will first compute C(f) for f ∈ D_{2}.

Let f ∈ D_{2} be the time one map of a smooth vector field X, and consider g ∈ C(f). By our assumptions on D_{1} (two different fixed points have different determinant), f and g have the same fixed points. Consider now any saddle p of f. The unstable separatrices are defined to be the connected components of W^{u}(p) \ {p}, and we denote them by W_{+}^{u}(p), W_{−}^{u}(p). Note that g preserves or exchanges the two separatrices of p.

Proposition 3.9. Consider f ∈ D_{2}. If g ∈ C(f) preserves an unstable separatrix of a saddle p ∈ σ_{f}, then g belongs to the flow of X.

As in the proof of Proposition 1.11 of Section 1.5, we deduce:

Corollary 3.10. For f ∈ D_{2} the centralizer is either the flow of X (hence is isomorphic to ℝ) or is isomorphic to ℝ × ℤ/2ℤ.

Every unstable separatrix of a saddle p is contained in the basin of a sink q of f. In this section, we endow each orbit space T_{q} with affine coordinates (r, s) ∈ ℝ^{2}/ℤ^{2} such that the vector field X projects to \frac{∂}{∂s}, and the vector field Z, whose expression in the local coordinates at q is Z = 2π(x\frac{∂}{∂y} − y \frac{∂}{∂x}), projects to \frac{∂}{∂r}. Note that the unstable separatrix of p is precisely one orbit of X and induces on T_{q} a circle {r} × ℝ/ℤ.

Proof of proposition 3.9. Assume that g(W_{+}^{u}(p)) = W_{+}^{u}(p), for some saddle point p and let q be the sink of f whose basin contains this separatrix. The projection of the separatrix W_{+}^{u}(p) on T_{q} will be denoted by {r_{0}} × ℝ/ℤ.

Since g commutes with f, it induces on T_{q} a diffeomorphism g/_{f}. By Lemma 2.3, g is locally the composition of an homothety and a rotation; hence the expression of g/_{f} in the (r, s) coordinates is a translation: (r, s) ↦ (r + α, s + β).

The fact that g leaves invariant W_{+}^{u}(p) implies that g/_{f}({r_{0}} × ℝ/ℤ) = {r_{0} + α} × ℝ/ℤ = {r_{0}} × ℝ/ℤ, so that α = 0. In particular, this implies that g/_{f} leaves invariant every orbit of \frac{∂}{∂s}. Consequently, g leaves invariant every X-orbit contained in the basin W^{s}(q).

Let q_{0} = q. Let q_{1} be a source such that W^{u}(q_{1}) ∩ W^{s}(q_{0}) ≠ ∅. In the same way we endow the torus T_{q_{1}} with affine coordinates, and g induces on T_{q_{1}} a translation. The set W^{u}(q_{1}) ∩ W^{s}(q_{0}) ≠ ∅ is open and invariant by X; hence it contains an X-orbit, which is invariant by g. This orbit induces on T_{q_{1}} a circle of the form {r_{1}} × ℝ/ℤ, invariant by g/_{f}. This proves as before that g leaves invariant every X-orbit contained in W^{u}(q_{1}).

Since S is connected and f is Morse-Smale, for every sink or source q there is a finite sequence q_{0}, q_{1}, …, q_{n} = q of alternating sources and sinks such that W^{u}(q_{i}) ∩ W^{s}(q_{i+1}) or W^{s}(q_{i}) ∩ W^{u}(q_{i+1}) is non empty for each i = 0, …, n − 1. The discussion above hence proves that g leaves invariant every X-orbit contained in the basin or a sink or of a source, hence leaves invariant every X-orbit.

This shows that for every point x ∈ S \ F ix(f), there exists t(x) ∈ ℝ such that g(x) = X^{t(x)}(x). Furthermore, the continuous map x ↦ t(x) is locally constant in the punctured neighborhoods of the sinks and of the sources; hence (using the fact that g and X commute with f) the map t is constant on every basin of a sink or a source. Since t takes the same value on any two intersecting basins, it follows that t(x) is constant on the complement of the fixed points. Thus g belongs to the flow of X.

In order to conclude the proof of Theorem 3, it remains to show that there is a dense subset D ⊂ D_{2} of diffeomorphisms such that, for each f ∈ D, C(f) is precisely the flow of the corresponding vector field. We first note that for the existence of an extra symmetry, it is necessary that for any saddle p ∈ σ_{f}, the two unstable separatrices belong to the basin of a same sink q. Assuming that this topological condition is satisfied, the proof is very similar to the argument on the circle: we will exhibit a new invariant of differentiable conjugacy which vanishes if the diffeomorphism has an extra symmetry. We will then show that arbitrarily small perturbations allow us to modify this invariant.

We first define precisely this invariant. For every sink q ∈ ω_{f}, we consider the orbit space T_{q} of W^{s}(q) \ {q}, with its structure of an affine torus. For every saddle p whose unstable manifold has a unique sink q in the ω-limit set, let {r_{+}(p)}×S^{1} and {r_{−}(p)}×S^{1} be the projections of the separatrices W_{+}^{u}(p) and W_{−}^{u}(p) on the affine torus T_{q}. We consider the distance |r_{+} − r_{−}| ∈ [0, \frac{1}{2}] between the two points r_{+}, r_{−} ∈ S^{1} = ℝ/ℤ.

With these notations, for every saddle p in σ_{f} we define

δ^{u}(p) = \{ \begin{matrix}\frac{1}{2} − |r_{+}(p) − r_{−}(p)| if W_{+}^{u}(p), W_{−}^{u}(p) have the same ω-limit set; \\ +∞ otherwise. \end{matrix}

As we’ll explain in the proof of Lemma 3.11, the number δ^{u}(p) measures whether the projections of the two separatrices on an affine torus can be exchanged by a rotation.

The invariant of the dynamics we will work with is defined by

δ^{u}(f) = sup_{p∈σ_{f}} δ^{u}(p).

We conclude the proof of Theorem 3 with the following two lemmas.

Lemma 3.11. Let f be in D_{2}. If its centralizer C(f) is isomorphic to ℝ × ℤ/2ℤ, then δ^{u}(f) = 0

Proof. Let g be the element of order 2; it exchanges the unstable separatrices of every saddle p of f. In particular, W_{+}^{u}(p) and W_{−}^{u}(p) are contained in the basin of the same sink q. Moreover, g projects on T_{q} to the translation by (1/2, 0). This implies that we must have δ^{u}(p) = 0 and we are done.

Lemma 3.12. For f in a dense subset D ⊂ D_{2} we have δ^{u}(f) ≠ 0.

Proof. Let f ∈ D_{2} be such that δ^{u}(f) = 0 and let X be the flow associated with f. The unstable separatrices of any saddle p are contained in the basin of a sink q. By an arbitrarily small perturbation of X with support in the complement of a neighborhood of the fixed points, one can change the projection in T_{q} of any of the two separatrices. Then the time one map of the perturbed flow is a diffeomorphism g ∈ D_{2} arbitrarily close to f and such that δ^{u}(g) ≠ 0.

4 Huge centralizers in dimension larger than 3

4.1 Reduction to the existence of periodic islands

Theorem 4 is a consequence of the next result:

Theorem 8. Let M be a compact manifold of dimension d ≥ 3. Then there is a non-empty open subset O ⊂ Diff^{1}(M) and a dense subset D_{0} ⊂ O such that any diffeomorphism f ∈ D_{0} has a periodic point x such that Df^{n}(x) = Id ∈ GL(T_{x}M) where n is the period of x.

Before explaining the proof of Theorem 8 we explain here why it implies Theorem 4.

Corollary 4.1. Let M be a compact manifold of dimension d ≥ 3. Then there is a non-empty open subset O ⊂ Diff^{1}(M) and a dense subset D ⊂ O such that any diffeomorphism f ∈ D has the following property:

there is an embedded ball D_{f} ⊂ M of dimension d and an integer n > 0 such that f^{i}(D_{f}) ∩ D_{f} = ∅ for i ∈ {1, …, n − 1} and the restriction of f^{n} to D_{f} is the identity map.

Proof. Given any f_{0} in the set D_{0} given by Theorem 8, given any C^{1}- neighborhood U of f_{0} and given any neighborhood V of the orbit Orb(x, f_{0}), there exists f ∈ U that agrees with f in the complement of V, and such that f^{n} is the identity map in a neighborhood of x.

End of the proof of Theorem 4. Consider f in the set D constructed in Corollary 4.1, and let D be a periodic ball of period n such that f^{n} coincides with the identity map on D (hence on the f-orbit of D). Let φ: ℝ^{d} → M be a smooth embedding such that φ(𝔻^{d}) = D. To any diffeomorphism h ∈ Diff^{1}(ℝ^{d}, ℝ^{d}\𝔻^{d}) we associate h_{0} : M → M, the diffeomorphism equal to the identity map in the complement of D and equal to φhφ^{−1} on D. For each i ∈ ℤ we set h_{i} = f^{i} ∘ h_{0} ∘ f^{−i}; notice that h_{i} is a diffeomorphism of M with support contained in f^{i}(D), and h_{i+n} = h_{i}. We denote by h_{φ}: M → M the diffeomorphism of M that coincides with h_{i} on f^{i}(D), for every i ∈ ℤ, and with the identity map in the complement of ⋃_{i} f^{i}(D).

By construction, h_{φ} commutes with f. Then h ↦ h_{φ} is an injective homomorphism from Diff^{1}(ℝ^{d}, ℝ^{d} \ 𝔻^{d}) to C(f).

4.2 Existence of periodic orbits tangent to the identity map

The open set O produced in Theorem 8 and Corollary 4.1 is analoguous to those built in [BD]. We start by recalling some notions.

Let f be a diffeomorphism, and let x ∈ P er_{hyp}(f) a hyperbolic periodic point. Given another hyperbolic periodic point y of f, we say that x and y are homoclinically related and we write x ∼ y if the stable and the unstable manifolds of the orbit of x transversely intersect the unstable and the stable manifolds of the orbit of y, respectively. Let Σ(x, f) = {y ∈ P er_{hyp}(f), y ∼ x}. The homoclinic class H(x, f) is the closure H(x, f) = \overline{Σ(x, f)}.

A point x ∈ M is chain recurrent if for every δ > 0 there exists a δ-pseudo-orbit x = x_{0}, x_{1}, …, x_{k} = x. The chain recurrence class C(x, f) of a chain recurrent point x is the set of points y such that, for every δ > 0, there is a δ-pseudo orbit starting at x and ending at y and a δ-pseudo orbit starting at y and ending at x.

For any periodic point y ∈ P er(f), let π(y) be its period, and let

J_{f}(y) = \frac{1}{π(y)} log |DetDf^{π(y)}(y)|,

be the sum of the Lyapunov exponents of y.

Recall that an f-invariant set Λ admits a dominated splitting if there is an Df-invariant decomposition T M|_{Λ} = E ⊕ F of the tangent bundle T M over Λ as a direct sum of two invariant subbundles E and F such that:

• the dimension dim(E(x)) is independent on x ∈ Λ;

• the vectors in E are uniformly less expanded than the vectors in F; that is, there exists N ∈ ℕ such that for any x ∈ Λ and any non-zero vectors u ∈ E(x) and v ∈ F (x):

\frac{∥Df^{N}(u)∥}{∥u∥} < \frac{1}{2} \frac{∥Df^{N}(v)∥}{∥v∥}.

The bundles E and F of a dominated splitting are always continuous and extend continuously to a dominated splitting over the closure of Λ (elementary properties of dominated splitting are described in [BDV, Appendix B.1]). As a direct consequence, if a set does not admit a dominated splitting, then the same holds for any dense subset of it.

Now Theorem 8 follows from

Proposition 4.2. Let M be a compact manifold of dimension dim(M) ≥ 3. There is a non empty open subset O ⊂ Diff^{1}(M) and a continuous function f ∈ O ↦ x_{f} ∈ M such that, for every f ∈ O:

• x_{f} is a hyperbolic periodic saddle point of f with J_{f}(x) > 0;

• there exists y_{f} ∈ Σ(x_{f}, f) such that J_{f}(y_{f}) < 0;

• the chain recurrent class C(x_{f}, f) does not admit a dominated splitting.

We now deduce Theorem 8 from Proposition 4.2

Proof of Theorem 8. We just repeat briefly here the proof given in [BD]. Fix a diffeomorphism f ∈ O, and a neighborhood U ⊂ O of f. We will prove that U contains a diffeomorphism g having a periodic orbit whose derivative at the period is the identity.

According to [BC], for C^{1}-generic diffeomorphisms, the chain recurrent class of every periodic orbit is equal to its homoclinic class. Hence there exists f_{0} ∈ U such that C(x_{f_{0}}, f_{0}) = H(x_{f_{0}}, f_{0}).

For every ε > 0 we consider the set

Σ_{ε}(f_{0}) = {y ∈ Σ(x_{f_{0}}, f_{0}), |J_{f_{0}}(y)| < ε}.

Any two points in Σ(x_{f_{0}}, f_{0}) are homoclinically related. As a consequence, given a finite set X ⊂ Σ(x_{f_{0}}, f), there is a hyperbolic basic set of f_{0} containing X. From this fact and from the hypotheses J_{f_{0}}(x_{f_{0}}) > 0 and J_{f_{0}}(y_{f_{0}}) > 0, we deduce that, for every ε > 0, the set Σ_{ε}(f_{0}) satifies the two following properties:

1. the set Σ_{ε}(f_{0}) is dense in Σ(x_{f_{0}}, f_{0}), and hence in H(x_{f_{0}}, f_{0}) = C(x_{f_{0}}, f_{0}); it follows that Σ_{ε}(f_{0}) does not admit a dominated splitting;

2. the set Σ_{ε}(f_{0}) admits transitions as defined in [BDP]. This is implied by the fact that, given any finite subset X ⊂ Σ_{ε}(f_{0}), there is a hyperbolic basic set K_{X} containing X whose periodic orbits are contained in Σ_{ε}(f_{0}): K_{X} ∩ P er(f_{0}) ⊂ Σ_{ε}(f_{0}).

Since Σ_{ε}(f_{0}) admits transitions and does not admit a dominated splitting,

[BDP] implies that, for every δ > 0, there is a periodic point x ∈ Σ_{ε}(f_{0}) and a δ-small perturbation g_{0} of f_{0} agreeing with f_{0} on the orbit of x (and in the complement of an arbitrarily small neighborhood of the orbit of x) such that Dg_{0}^{π(x)}(x) is an homothety, where π(x) is the period of x. Notice that, for δ > 0 small enough, we have |J_{g_{0}}(x)| < 2ε.

As ε can be chosen arbitrarily small, for ε << δ there is a δ-small perturbation g of g_{0} coinciding with g_{0} and f on the orbit of x (and in the complement of an arbitrarily small neighborhood of the orbit of x) such that Dg^{π(x)}(x) is the identity map. For δ small enough, g belongs to U concluding the proof.

It remains to explain how one can build the open set O announced in Proposition 4.2. Our construction (as in [BD]) is based on the coexistence in a single homoclinic class of periodic orbits having complex eigenvalues of any rank. Let us explain this notion.

Let f be a diffeomorphism and x a periodic point of f of period π(x). An eigenvalue of x is a (real or complex) eignevalue of the derivative Df^{π(x)}(x). We say that x has a complex eigenvalue of rank (i, i+1), for i ∈ {1, …, dim(M)− 1}, if there is a Df^{π(x)}-invariant splitting T_{x}M = E ⊕ F ⊕ G such that:

• dim(E) = i − 1, dim(F) = 2, dim(G) = dim(M) − i − 1;

• the restriction of Df^{π(x)}(x) to F has a pair of complex (non-real) conjugated eigenvalues; let λ be the modulus of these eigenvalues;

• the eigenvalues of the restrictions of Df^{π(x)}(x) to E (resp. G) have modulus strictly less (resp. larger) than λ.

Lemma 4.3. Let K be a f invariant set. Assume that K contains a periodic point x having a complex eigenvalue of rank (i, i+1). Assume that T M = E⊕F is a dominated splitting on K. Then dimE ≠ i.

Proof. Just notice that E(x) and F (x) are Df^{π(x)}(x)-invariant, and that the eigenvalues corresponding to E(x) are strictly less that those corresponding to F (x).

Corollary 4.4. Let K be an f-invariant set containing periodic points x_{i} having a complex eigenvalue of rank (i, i + 1) for every i ∈ {1, …, dimM − 1}. Then K does not admit a dominated splitting.

Hence Proposition 4.2 is a direct consequence of:

Proposition 4.5. For any compact manifold M with dimM ≥ 3, there is a non empty open subset O ⊂ Diff^{1}(M) and a continous map f ∈ O ↦ x_{f} ∈ M such that, for every f ∈ O:

1. the point x_{f} is a hyperbolic periodic point of stable index (dimension of the stable manifold) equal to 1;

2. J_{f}(x_{f}) > 0 and there is a hyperbolic periodic point y_{f}, homoclinically related to x_{f}, such that J_{f}(y_{f}) < 0

3. for every i ∈ {2, …, dimM − 1}, there is a hyperbolic periodic point x_{i,f} homoclinically related to x_{f}, and having a complex eigenvalue of rank (i, i + 1).

4. the chain recurrence class C(x_{f}, f) contains a hyperbolic periodic point z_{f} of stable index equal to 2, and having a complex eigenvalue of index (1, 2).

Proof. The properties described in items (1)(2)(3) are open properties, and are easy to get by a local argument: we just describe how to obtain a finite number of periodic saddles of index 1 that are homoclinically related.

The property of item (4) comes from the notion of blender defined in [BD]. A blender Λ is:

• a (uniformly) hyperbolic compact set Λ that is also partially hyperbolic: there is an invariant dominated splitting E^{s} ⊕E^{u} ⊕E^{uu} over Λ such that the dimension of E^{u} is equal to 1; here we assume that the dimension of the stable bundle E^{s} is 1, so that dim(E^{uu}) = dimM − 2. The partially hyperbolic structure extends to a neighborhood U of Λ; and

• an open region V ⊂ U (called the caracteristic region of the blender), endowed with a cone field C^{uu} around a the bundle E^{uu};

with the following property. There exists a C^{1}-neighborhood U of f such that, for every g ∈ U and any ball D^{uu} ⊂ U of dimension dim(D^{uu}) = dim(E^{uu}), tangent to the conefield C^{uu}, and crossing the region V, then D^{uu} meets the stable manifold of the continuation Λ_{g} of Λ for g. (See [BDV, Section6.2] for a more detailed discussion of the notion of blender, and references.)

Let f be a diffeomorphism having • a blender Λ containing a periodic point x of index 1 and such that J_{f}(x) >

0,

• a hyperbolic periodic point z of index 2 such that:

– the unstable manifold W^{u}(z) crosses the caracteristic region of the blender, remaining tangent to the strong unstable cone field C^{uu},

– the stable manifold W^{s}(z) intersects transversely W^{u}(x),

– the stable eigenvalue of z is not real (hence z has a complex eigenvalue of rank (1, 2)),

• for any i ∈ {2, ..., dimM − 1}, a hyperbolic periodic point x_{i} of index 1 homoclinically related to x and having a complex eigenvalue of rank (i, i + 1),

• a hyperbolic periodic point y homoclinically related to x and such that J_{f}(y) < 0.

All these properties are robust. Hence there is a small neighborhood O of f such that the continuations of the periodic points x, y, z, x_{i} and of the blender Λ are well-defined for every g ∈ O and satisfy all the properties above. We conclude by noting that, for every g ∈ O, the continuation z_{g} of z belongs to the chain recurrence class of the continuation x_{g}.

5 Huge centralizers of symplectomorphisms

The aim of this section is to prove that the set of symplectomorphisms having a large centralizer is dense in O ⊂ Symp_{ω}^{1}(M), the open subset consisting of symplectomorphisms having a robustly totally elliptic periodic point. The argument is analogous to the idea of Theorem 4.

Proposition 5.1. There is a dense subset D ⊂ O such that, for f ∈ D there is a ball D ⊂ M and a integer n > 0 such that D ∩ f^{i}(D) = ∅ for i ∈ {1, …, n − 1}, and the restriction of f^{n} to D is the identity map.

This proposition is a consequence of the two following classical lemmas, that can be easily obtained by considering generating functions.

Lemma 5.2 (Symplectic Franks Lemma). Given a symplectomorphism f and a neighborhood U ⊂ Symp_{ω}^{1}(M) of f, there exists ε > 0 with the following property. For every point x ∈ M, every neighborhood V of x, every symplectic linear isomorphism A: T_{x}M → T_{f(x)}M with ∥A − Df(x)∥ < ε, there exists g ∈ U such that:

• g(x) = f(x) and g coincides with f in the complement of V;

• Dg(x) = A.

Lemma 5.3 (Linearizing perturbation). Given a symplectomorphism f, a neighborhood U ⊂ Symp_{ω}^{1}(M) of f, a periodic point x of f of period n and a neighborhood V of the orbit of x, there exists g ∈ U such that:

• g coincides with f on the orbit of x and in the complement of V; in particular x is periodic for g;

• there is a neighborhood V_{0} of x and a chart ψ: V_{0} → ℝ^{2d}, such that

– ψ(x) = 0 ∈ ℝ^{2d},

– ψ_{∗}(ω) is the canonical symplectic form on ℝ^{2d},

– the expression of g^{n} in this chart (i.e. the local symplectomorphism ψ∘g^{n}∘ψ^{−1}) is the symplectic linear map Dψ(x)∘Df^{n}(x)∘Dψ(x)^{−1}.

Proof of Proposition 5.1. According to Lemmas 5.3 and 5.2 for every f ∈ O, any neighborhood U of f and any robustly totally elliptic point x of f, there exists g in U agreeing with f on the orbit of x, and such that:

• Dg^{n}(x) (where n is the period of x) is a totally elliptic matrix whose eigenvalues λ_{i} have the form e^{α_{i}2iπ} with α_{i} = \frac{p_{i}}{q_{i}}, p_{i}, q_{i} ∈ ℤ, p_{i} ∧ q_{i} = 1;

• there is a neighborhood V of x and a symplectic chart ψ: V → ℝ^{2d} such that ψ(x) = 0 ∈ ℝ^{2d} and the expression of g^{n} in this chart is Dg^{n}(x).

Let m be the smallest common multiple of the integers q_{i}. Then g^{nm} is the identity map in a neighborhood of x. It follows that g admits a periodic ball such that the return map is the identity map, concluding the proof.

End of the proof of Theorem 5. Now the proof of Theorem 5 is identical to the proof of Theorem 4: consider a diffeomorphism f ∈ D and a periodic ball D on which the first return map is the identity map. Shrinking D if necessary, we can assume that there is a symplectic chart ψ: D → ℝ^{2d} inducing a symplectomorphism from D to the standard ball 𝔻^{2d} ⊂ ℝ^{2d}.

Let φ be a symplectic diffeomorphism of ℝ^{2d} that is the identity map on the complement of 𝔻^{2d}. Let g_{0} be the diffeomorphism of M that is the identity map on the complement of D and that is ψ^{−1}φψ on D, and let g_{n} = f^{n}g_{0}f^{−n}. Then the diffeomophism g_{φ} coinciding with the identity map in the complement of the f-orbit of D and with g_{i} on f^{i}(D) is a symplectomorphism of M commuting with f.

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Christian Bonatti (bonatti@u-bourgogne.fr) Gioia M. Vago (vago@u-bourgogne.fr) CNRS - Institut de Mathématiques de Bourgogne, UMR 5584 Université de Bourgogne - BP 47 870 21078 Dijon Cedex, France

Sylvain Crovisier (crovisie@math.univ-paris13.fr) CNRS - Laboratoire Analyse, Géométrie et Applications, UMR 7539, Institut Galilée, Université Paris 13, Avenue J.-B. Clément, 93430 Villetaneuse, France

Amie Wilkinson (wilkinso@math.northwestern.edu) Department of Mathematics, Northwestern University 2033 Sheridan Road Evanston, IL 60208-2730, USA