C^{1}-generic conservative diffeomorphisms have trivial centralizer

C. Bonatti, S. Crovisier and A. Wilkinson

January 20, 2008

Abstract

We prove that the spaces of C^{1} symplectomorphisms and of C^{1} volumepreserving diffeomorphisms of connected manifolds both contain residual subsets of diffeomorphisms whose centralizers are trivial. Key words: Trivial centralizer, trivial symmetries, C^{1} generic properties.

Les difféomorphismes conservatifs C^{1}-génériques ont un centralisateur trivial

Résumé : Nous montrons que l’espace des symplectomorphismes de classe C^{1} et l’espace des diffémomorphismes de classe C^{1} préservant une forme volume contiennent tous deux des sous-ensembles résiduels de difféomorphismes dont le centralisateur est trivial. Mots clé : Centralisateur trivial, symétries triviales, propriétés C^{1}-génériques.

Introduction

Let M be a connected compact manifold. The centralizer of a C^{r} diffeomorphism f ∈ Diff^{r}(M) is defined as

Z^{r}(f) := {g ∈ Diff^{r}(M) : fg = gf}.

Clearly Z^{r}(f) always contains the group < f > of all the powers of f. We say that f has trivial centralizer if Z^{r}(f) =< f >. A diffeomorphism f with trivial centralizer posesses no smooth symmetries, such as those that would arise if, for example, f embedded in a flow or were the lift of another diffeomorphism. S. Smale asked the following:

Question 1 ([Sm1, Sm2]). Consider the set of C^{r} diffeomorphisms of a compact connected manifold M with trivial centralizer.

a. Is this set dense in Diff^{r}(M)?

b. Is it residual in Diff^{r}(M)? That is, does it contain a dense G_{δ} subset?

c. Does it contain an open and dense subset of Diff^{r}(M)?

We refer to [BCW3] for a discussion on this question. To summarize, we mention some cases in which it has been answered:

• N. Kopell [Ko] solved the one-dimensional smooth case: the set of diffeomorphisms with trivial centralizer contains an open and dense subset of Diff^{r}(S^{1}) for r ≥ 2.

• For r ≥ 2 on higher dimensional manifolds, there exists some results about the set of diffeomorphisms with trivial centralizer under additional dynamical assumptions. For instance, this set contains an open and dense subset among the set of Axiom A diffeomorphisms in Diff^{∞}(M) possessing at least one periodic sink or source [PY1] or defined on a surface [F], and generic among those satisfying the no-cycles condition [F]; and it is locally residual among a class of partially hyperbolic C^{∞} diffeomorphisms with 1-dimensional center [Bu1].

• In the C^{1} setting, Y. Togawa proved [To1, To2] that generic Axiom A diffeomorphisms have trivial centralizer.

This paper deals with the first two parts of Question 1 for all compact M in the case of volume-preserving and symplectic C^{1} diffeomorphisms. If M is a symplectic manifold, then Symp^{1}(M) denotes the space of C^{1} symplectomorphisms of M. If M carries a volume μ, then we denote by Diff_{μ}^{1}(M) the space of C^{1} diffeomorphisms of M that preserve μ. The spaces Diff^{1}(M), Symp^{1}(M), Diff_{μ}^{1}(M) are Baire spaces in the C^{1} topology. Recall that a residual subset of a Baire space is one that contains a countable intersection of open-dense sets.

Theorem A. Let M be a compact, connected manifold of dimension at least 2. Then:

(a) In Symp^{1}(M) the set of diffeomorphisms f whose centralizer Z^{1}(f) ∩ Symp^{1}(M) is trivial is a residual subset.

(b) In Diff_{μ}^{1}(M) the set of diffeomorphisms f whose centralizer Z^{1}(f) ∩ Diff_{μ}^{1}(M) is trivial is a residual subset.

In a recent work [BCW2] we also solved Questions 1 a) and b) in the C^{1} dissipative case: the set of diffeomorphisms f whose centralizer Z^{1}(f) is trivial is a residual subset of Diff^{1}(M). On the other hand with G. Vago [BCVW] we answered negatively to Questions 1 c) in the dissipative and in the symplectic case: for any compact symplectic manifold M, there exists a familly of symplectomorphisms of M with large centraliser Z^{1}(f) ∩ Symp^{1}(M) and that is dense in a non-empty open subset U of Symp^{1}(M); with a different method we also constructed a family of diffeomorphisms of M with large centraliser Z^{1}(f) that is dense in a non-empty open subset U of Diff^{1}(M). The existence of such examples in the volume-preserving setting in dimension larger or equal to 3 remains open.

In [Ko, PY1, PY2, Bu1, To1, To2], the proof of the triviality of the centralizer splits in two parts: the first implies that, for a generic f and for any diffeomorphism g that commutes with f, we have g = f^{α} on a dense subset of M, where α is a function that is f-invariant; the second part considers the global dynamics and proves that α is bounded, then constant. It is for this second part that additional assumptions such as hyperbolicity are often required. Let us mention that for conservative diffeomorphisms of the 2-sphere, L. Burslem has shown [Bu2] a partial result for the topology C^{r} with r ≥ 16 (more precizely, she obtained the first step).

The proof of theorem A also decomposes in two steps. The first one is provided by the following result.

Theorem B. (a) For any diffeomorphism f in a residual set R ⊂ Diff^{1}(M), any periodic point x ∈ P er(f) is hyperbolic and for any g ∈ Z^{1}(f) there exists n ∈ ℤ such that g coincides with f^{n} on W^{s}(x).

(b) For any diffeomorphism f in a residual set Rsymp ⊂ Symp^{1}(M), any hyperbolic periodic point x ∈ P er(f) and any g ∈ Z^{1}(f), there exists n ∈ ℤ such that g coincides with f^{n} on W^{s}(x).

(c) For any diffeomorphism f in a residual set R_{μ} ⊂ Diff_{μ}^{1}(M), any hyperbolic periodic point x ∈ P er(f) and any g ∈ Z^{1}(f), there exists n ∈ ℤ such that g coincides with f^{n} on either W^{s}(x) or W^{u}(x).

The second step is provided by the following genericity result about conservative diffeomorphisms. This immediately concludes the proof of theorem A.

Theorem ([BC, ABC]). For any compact connected manifold M, there are residual sets R̃symp ⊂ Symp^{1}(M) and R̃_{μ} ⊂ Diff_{μ}^{1}(M) such that, every f ∈ R̃symp ∪ R̃_{μ} has a hyperbolic periodic point p with

\overline{W^{s}(p)} = \overline{W^{u}(p)} = M.

More generally, Theorem B naturally applies to the class of C^{1} diffeomorphisms satisfying a property we call periodic accessibility: this property says that any two points x, y in a dense subset E ⊂ M of non-periodic points may be joined by a finite sequence x_{0} = x, x_{1}, …, x_{n} = y, x_{i} ∈ E such that for any i ∈ {0, …, n − 1}, the points x_{i} and x_{i+1} belong to the closure \overline{W^{s}(O_{i})} of the stable manifold or to the closure \overline{W^{u}(O_{i})} of the unstable manifold of a hyperbolic periodic orbit O_{i}.

Question 2. Is periodic accessibility generic in Diff^{r}(M)?

As a weaker problem^{1}, one can also ask if, for generic diffeomorphims, the union of the stable manifolds of the periodic points are dense in M.

As an immediate corollary of Theorem B, it follows that if f ∈ R satisfies the periodic accessibility property, then Z^{1}(f) is trivial. Since the periodic accessibility property is satisfied by Axiom A diffeomorphisms, one can recover in this way Togawa’s result that the C^{1} generic Axiom A diffeomorphism has trivial centralizer. The proof in the general case [BCW3] is much more delicate and involves new ideas.

Theorem B (a) was previously proved by Togawa [To1, To2]. We present here a variation of his argument, based on an unbounded distortion property that is discussed in section 1; we note below that this proof also handles the “Lipschitz centralizer” and this will be crucial for [BCW3]. When we started with this work we were not aware of Togawa’s papers and we used a stronger unbounded distortion property which is much more difficult to obtain. We think that our former approach has independent interest and could have further application: it is written in the first version [BCW1] of this article. The conservative cases (b) and (c) of Theorem B can be derived as for case (a) with an extension result presented in section 2: any perturbation of the dynamics of a symplectomorphism inside the stable manifold of a periodic point can be realized as a perturbation of the dynamics on M.

1This problem is often proposed as a conjecture by the first author.

1 The unbounded distortion property

Kopell’s proof in [Ko] that the set of diffeomorphisms having a trivial centralizer is open and dense in Diff^{r}(S^{1}) for r ≥ 2 uses the fact that a C^{2} diffeomorphism f of [0, 1] without fixed points in (0, 1) has bounded distortion, meaning: for any x, y ∈ (0, 1), the ratio

\frac{|f^{n}′(x)|}{|f^{n}′(y)|}  (1)

is bounded, independent of n and uniformly for x, y lying in a compact set. A bounded distortion estimate lies behind many results about C^{2}, hyperbolic diffeomorphisms of the circle and codimension-1 foliations.

Suppose that r ≥ 2. Since Morse-Smale diffeomorphisms are open and dense in Diff^{r}(S^{1}), the proof that the set of diffeomorphisms having a trivial centralizer is open and dense in Diff^{r}(S^{1}) essentially reduces to showing that (C^{r} open and densely) a C^{r} diffeomorphism f : [0, 1] → [0, 1] without fixed points in (0, 1) has trivial centralizer. The bounded distortion of such an f forces its centralizer to embed simultaneously in two smooth flows containing f, one determined by the germ of f at 0, and the other by the germ at 1; for an open and dense set of f ∈ Diff_{+}^{r}[0, 1], these flows agree only at the iterates of f. The r ≥ 2 hypothesis is clearly necessary for bounded distortion.

The central observation and starting point of this paper is that the centralizer of a C^{1} diffeomorphism of [0, 1] with unbounded distortion is always trivial. We elaborate a bit on this. Notice that if x and y lie on the same f-orbit, then the ratio in (1) is bounded, independent of n. We show that, C^{1} generically among the diffeomorphisms of [0, 1] without fixed points in (0, 1), the ratio (1) is uniformly bounded in n only if x and y lie on the same orbit; that is, for a residual set of f, and for all x, y ∈ (0, 1), if x ∉ O_{f}(y) = {f^{n}(y) | n ∈ ℤ}, then

lim sup_{n→∞} \frac{|f^{n}′(x)|}{|f^{n}′(y)|} = ∞.  (2)

Assume that this unbounded distortion property holds for f. Fix x ∈ (0, 1). A simple application of the Chain Rule shows that if gf = fg, then the distortion in (1) between x and y = g(x) is bounded; hence x and g(x) must lie on the same f-orbit. From here, it is straightforward to show that g = f^{n}, for some n (see Lemma 1.2 below). As in [Ko], a small amount of additional work shows that a residual set in Diff^{1}(S^{1}) has trivial centralizer.

The bulk of this section is devoted to formulating and proving a higherdimensional version of the argument we have just described. The interval is replaced by an invariant manifold (stable or unstable) of a periodic point. The derivative f′ in (2) is replaced by the Jacobian of f along the invariant manifold.

1.1 Unbounded distortion along stable manifolds of hyperbolic periodic points

Denote by Lip(M) is the set of lipeomorphisms of M: these are the homeomorphisms g of M such that g and g^{−1} are Lipschitz. For any hyperbolic periodic orbit O ⊂ M of a diffeomorphism f ∈ Diff^{1}(M) and for x ∈ W^{s}(O) we denote by Jac^{s}(f)(x) the Jacobian of the map induced by T_{x}f between T_{x}W^{s}(O) and T_{f(x)}W^{s}(O). The topology considered on W^{s}(O) in the following is the manifold topology (and not the topology inherited from M).

Definition 1.1. A hyperbolic periodic orbit O ⊂ M of a diffeomorphism f ∈ Diff^{1}(M) has the stable manifold unbounded distortion property if there exists a dense set D ⊂ W^{s}(O) with the following property: for any points x ∈ D and y ∈ W^{s}(O) \ O not in the same f-orbit and for any N ≥ 1 there exists n ≥ 1 such that

|log(Jac^{s}(f^{n})(x)) − log(Jac^{s}(f^{n})(y))| > N.

The two next lemmas are stated for lipeomorphisms but for this paper it would be enough to work with diffeomorphisms. The reason we consider lipeomorphisms is that this slightly more general version will be needed in [BCW2]. As mentioned in the previous subsection, unbounded distortion forces trivial centralizers along the stable manifold:

Lemma 1.2. Let f ∈ Diff^{1}(M) be a diffeomorphism and O be a hyperbolic periodic orbit having the stable manifold unbounded distortion property. Then for any g ∈ Lip(M) such that g ∘ f = f ∘ g and g(O) = O, and for any connected component W of W^{s}(O) \ O, there exists an i ∈ ℤ such that g = f^{i} on W.

Remark: Since O is hyperbolic, if g is differentiable at p, then there exists i ∈ ℤ such that g = f^{i} on W^{s}(O).

Proof. Let D ⊂ W^{s}(O) be the dense set as in definition 1.1. We first note that the set W^{s}(O) is invariant by g.

We claim that: For every x ∈ D \ O, the map g preserves the f-orbit of x. In order to prove it, one considers for n ∈ ℕ large a small ball B centered at x. The relation gf^{n} = f^{n}g implies that

\frac{V ol(g(f^{n}(B)))}{V ol(f^{n}(B))} = \frac{V ol(f^{n}(g(B)))/V ol(g(B))}{V ol(f^{n}(B))/V ol(B)} \frac{V ol(g(B))}{V ol(B)}.

Since g is a lipeomorphism and f^{n}(B) lies in a compact region of W^{s}(O) for all n ∈ ℕ, the two ratios \frac{V ol(g(f^{n}(B)))}{V ol(f^{n}(B))} and \frac{V ol(g(B))}{V ol(B)} are comparable. This implies that

the quantity \frac{V ol(f^{n}(g(B)))/V ol(g(B))}{V ol(f^{n}(B))/V ol(B)} is bounded. By choosing the ball B small enough it is close to |\frac{Jac^{s}(f^{n})(gx)}{Jac^{s}(f^{n})(x)} |, which is therefore also bounded. Hence if y = g(x), then the quantity |log(Jac^{s}(f^{n})(x)) − log(Jac^{s}(f^{n})(y))| is bounded in n. Since g preserves W^{s}(O), the point y = g(x) belongs to W^{s}(O) \ O and the stable manifold unbounded distortion property implies that y belongs to the f-orbit of x, proving the claim.

For i ∈ ℤ we consider the closed subset P_{i} = {x ∈ W^{s}(O) \ O, g(x) = f^{i}(x)} of W^{s}(O) \ O. The claim above shows that their union P contains D, hence is dense in W^{s}(O) \ O. Moreover for i ≠ j the intersection P_{i} ∩ P_{j} is empty (any point in the intersection would be j − i-periodic).

We claim that: For every i ∈ ℤ, the set P_{i} is open in P. Consider a point x ∈ P_{i} and a neighborhood U ⊂ W^{s}(O) of x that is disjoint from all its iterates. Consider also a point y ∈ P ∩ U close to x. Since g(x) belongs to f^{i}(U), this is also the case for g(y). On the other hand there exists j ∈ ℤ such that g(y) = f^{j}(y). Since the iterates of U are pairwise disjoint, it follows that j = i. The map g hence agrees with f^{i} on a neighborhood of x in P ∩ W^{s}(O), proving the claim.

Let us consider a non-empty set P_{i}. Since P is dense in W^{s}(O), it follows from the second claim that the interior of P_{i} in W^{s}(O) is a non-empty f-invariant set in which g = f^{i}. Since O is a hyperbolic periodic orbit, the Lipschitz constant of f^{i} in a neighborhood of O is arbitrarily large when |i| goes to infinity. Using that g and g^{−1} are uniformly Lipschitz, one deduces that only a finite number of sets P_{i} are non-empty. Consequently W^{s}(O) \ O is the disjoint union of finitely many closed sets P_{i}. Hence any connected component W of W^{s}(O) \ O is contained in a single set P_{i}, proving that g = f^{i} on W.

The periodic orbits of a generic diffeomorphism f are fixed by Z^{Lip}(f).

Lemma 1.3. For any diffeomorphism f in a residual set R_{per} of Diff^{1}(M) (resp. of Symp^{1}(M), of Diff_{μ}^{1}(M)), for any periodic orbit O of f and for any g ∈ Lip(M) such that f ∘ g = g ∘ f, we have g(O) = O.

Proof. If f and g commute, then any periodic orbit O of f is sent by g on a periodic orbit O′ of f having the same period. Since g is a lipeomorphism, the eigenvalues of O and O′ for f must coincide. The set R_{per} of diffeomorphisms whose periodic orbits have different eigenvalues hence satisfies the conclusion of the lemma.

To prove Theorem B, it thus remains to show:

Proposition 1.4. For any diffeomorphism f in a residual set R ⊂ Diff^{1}(M), any periodic orbit of f is hyperbolic and has the stable manifold unbounded distortion property.

For any diffeomorphism f in a residual set R_{symp} ⊂ Symp^{1}(M), any hyperbolic periodic orbit of f has the stable manifold unbounded distortion property.

For any diffeomorphism f in a residual set R_{μ} ⊂ Diff_{μ}^{1}(M) any hyperbolic periodic O of f whose stable manifold W^{s}(O) has codimension at least dim(M)/2 has the stable manifold unbounded distortion property.

1.2 Reduction to contractions of ℝ^{d}

Let B^{d} denote the unit closed ball \overline{B(0, 1)} of ℝ^{d} and consider the Banach space of C^{1} maps B^{d} → ℝ^{d} that send 0 to 0, endowed with the C^{1} topology given by the C^{1} norm: ∥f − g∥_{1} = sup_{x∈B^{d}} ∥f(x) − g(x)∥ + ∥D_{x}f − D_{x}g∥. The set of embeddings B^{d} → ℝ^{d} fixing 0 defines an open subset that will be denoted by D^{d}.

A contraction of ℝ^{d} is an element of D^{d} that sends B^{d} into B(0, 1), so that 0 is a (hyperbolic) sink that attracts all the points in B^{d}. The set of contractions of ℝ^{d} is an open subset C^{d} ⊂ D^{d}, and hence a Baire space.

Let f be a diffeomorphism of a manifold M, p be a periodic point of f, d^{s} be its stable dimension and τ be its period. A stable chart for p is a local chart ψ: ℝ^{d} → M such that, denoting by π^{s} the projection of ℝ^{d} onto the d^{s} first coordinates, we have the following properties.

• The domain ψ(ℝ^{d}) contains p.

• In the chart ψ, the local stable manifold of p contains the graph of a C^{1} map g: ℝ^{d^{s}} → ℝ^{d−d^{s}}.

• Let v be equal to π^{s}(ψ^{−1}(p)) and let θ be the C^{1} map defined on a neighborhood of 0 by projecting on the space ℝ^{d^{s}} the dynamics of f^{τ} in the local stable manifold of p:

θ: x ↦ π^{s} ∘ ψ^{−1} ∘ f^{τ} ∘ ψ(x + v, g(x + v)) − v.

Then, θ belongs to C^{d^{s}}.

The following property is easy to check.

Lemma 1.5. Any hyperbolic periodic point p of a diffeomorphism f has a stable chart ψ. Moreover, for any diffeomorphism g in a C^{1} neighborhood U of f, the continuation p_{g} of p also admits the chart ψ as a stable chart.

The family of contractions θ_{g} associated to the periodic point p_{g} and to the chart ψ induces a continuous map Θ: U → C^{d^{s}}. This map is open.

In the conservative setting, the same property holds, but the proof is more delicate and is postponed until section 2.

Theorem 1.6. Let Θ: U → C^{d^{s}} be a family of contractions associated to a periodic point p and a stable chart ψ as in lemma 1.5. Then the map Θ: U ∩Symp^{1}(M) → C^{d^{s}} is open.

If the dimension d^{s} of the stable space of p is greater than or equal to dim(M)/2, then the map Θ: U ∩ Diff_{μ}^{1}(M) → C^{d^{s}} is open.

The major ingredient in the proof of Proposition 1.4 is the following.

Proposition 1.7. There is a residual set S^{d} ⊂ C^{d} and a dense set D ⊂ B^{d} \ {0} such that any f ∈ S^{d} has the following property: for any points x ∈ D, y ∈ B^{d} \ {0} not in the same orbit and for any N ≥ 1 there exists n ≥ 1 such that

|log(Jac(f^{n})(x)) − log(Jac(f^{n})(y))| > N.  (3)

All these results together allow us to prove that C^{1} generically, the stable manifold unbounded distortion property holds.

Proof of Proposition 1.4. For any integer n ≥ 0, there exists

• a family P_{n} of pairwise disjoint open subsets whose union is dense in Diff^{1}(M),

• for each U ∈ P_{n}, finitely many charts ψ_{1}, …, ψ_{s}: ℝ^{d} → M,

such that any diffeomorphism f ∈ U has the following properties:

• f has s periodic points of period less than n, all are hyperbolic. Each domain ψ_{i}(ℝ^{d}) contains exactly one of them, it is called p_{i,f} and its stable dimension is denoted by d_{i}^{s} and its period by τ_{i}.

• The chart ψ_{i} is a stable chart for p_{i,f}.

By Lemma 1.5, the chart ψ_{i} induces an open map Θ_{i}: U → C^{d_{i}^{s}}, so that Θ_{i}^{−1}(S^{d_{i}^{s}}) is residual in U where S^{d_{i}^{s}} is the residual subset of C^{d_{i}^{s}} provided by proposition 1.7. Suppose that f belongs to this residual set. The dynamics of f^{τ_{i}} on the local stable manifold of p_{i} are differentiably conjugate to the dynamics of a map Θ_{i}(f) ∈ S^{d_{i}^{s}}, proving that p_{i} has the stable unbounded distortion property.

The union R_{n} over U ∈ P_{n} of the sets ⋂_{i=1}^{s} Θ_{i}^{−1}(S^{d_{i}^{s}}) is residual in Diff^{1}(M) and Proposition 1.4 holds with the residual set R = ∩_{n}R_{n}. The proof in the conservative cases is similar and uses Theorem 1.6.

1.3 Huge distortion: proof of Proposition 1.7

We fix a countable dense subset D of Int(B^{d})\{0} and a countable basis of compact neighborhoods B for B^{d} \ {0}. For any x ∈ D and any compact set Λ ∈ B, the set of contractions f ∈ C^{d} such that the orbits of x and of Λ are disjoint is an open set O(x, Λ). For any N ≥ 1, the subset O(x, Λ, N) ⊂ O(x, Λ) of contractions f such that for any y ∈ Λ and some n ≥ 1 we have (3) is open. Thus the set

S^{d} = ⋂_{x,Λ,N} (O(x, Λ, N) ∪ (C^{d} \ \overline{O(x, Λ)}))

is a G_{δ} and any diffeomorphism f ∈ S^{d} satisfies the conclusion of Proposition 1.7. In order to prove Proposition 1.7 we must show that S^{d} is dense (hence residual) in C^{d}. This is a consequence of the following perturbation result.

Lemma 1.8. The set O(x, Λ, N) is dense in O(x, Λ)

Proof. Consider a contraction f ∈ O(x, Λ). By performing C^{1} small perturbation, we can assume that f is linear in a neighborhood V of 0. Now consider i_{0} ≥ 0 such that all the iterates f^{i}(x) and f^{i}(Λ) with i ≥ i_{0} are contained in V. There exists a C^{1} small perturbation with compact support g_{1} of the linear map D_{0}f such that Jacg_{1}(0) ≠ Jacf(0).

For some n ≥ i_{0} and any i ∈ {i_{0}, … n}, the map f agrees with D_{0}f in a neighborhood of f^{i}(x) and can be replaced by the map

g_{i,ε}: z ↦ f^{i}(x) + ε.g_{1} (\frac{z − f^{i}(x)}{ε}) .

By choosing ε small enough, all these perturbations have disjoint support. We thus obtain a map g that is still C^{1} close to f. Note that f and g agree on the positive orbits of x and of a neighborhood of Λ.

We have for any point y ∈ Λ,

|log(Jac(g^{n})(x)) − log(Jac(g^{n})(y))| ≥

(n − i_{0}) |log Jacg_{1}(0) − log Jacf(0)| − |log(Jac(f^{i_{0}})(x)) − log(Jac(f^{i_{0}})(y))| ,

which is larger than N, if n has been chosen large enough. Hence, g belongs to O(x, Λ, N).

2 Conservative extension results

We explain in this part how a perturbation of a conservative diffeomorphism along a submanifold W can be extended to a conservative perturbation on the whole manifold M.

This implies Theorem 1.6: the results proven in this section will be applied to the case where W is an invariant manifold of a hyperbolic periodic point p. In the volume-preserving case, we assume that dim(W) ≤ \frac{1}{2}dim(M) (note that this hypothesis is always satisfied either by the stable or by the unstable manifold of p). In the symplectic case, there is no additional hypothesis, but we use the following well-known fact.

Lemma 2.1. Let f ∈ Symp^{1}(M) and let p be a hyperbolic periodic point for f. Then W^{s}(p) and W^{u}(p) are Lagrangian submanifolds of M.

Proof. Let x ∈ W^{s}(p), and let v, w ∈ T_{x}W^{s}(p) be tangent vectors to W^{s}(p). On the one hand, since f is a symplectomorphism, we have

ω(D_{x}f^{k}(v), D_{x}f^{k}(w)) = ω(v, w),

for all k ∈ ℤ. On the other hand, as k → +∞, we have

ω(D_{x}f^{k}(v), D_{x}f^{k}(w)) → 0.

Hence ω vanishes identically on W^{s}(p). The same is true for W^{u}(p). Since W^{s}(p) and W^{u}(p) have complementary dimension and ω is nondegenerate, they must have the same dimension. Hence, both are Lagrangian submanifolds of M.

2.1 The symplectic case

Proposition 2.2. Let M be a symplectic manifold and z a point contained in a C^{1} Lagrangian submanifold W ⊂ M. Then there exists in W a disk D = \overline{B_{W} (z, r_{0})} centered at z such that, for every neighborhood U ⊂ M of D and every ε > 0, there exists δ > 0 with the following property.

For every C^{1} diffeomorphism ψ : D → D satisfying:

a. ψ = Id on a neighborhood of ∂D, and

b. d_{C^{1}}(ψ, Id) < δ,

there exists φ ∈ Symp^{1}(M) such that:

1. φ = Id on M \ U,

2. φ = ψ on D, and

3. d_{C^{1}}(φ, Id) < ε.

Proof. The basic strategy is first to symplectically embed the disk D as the 0-section of its cotangent bundle T^{∗}D. On T^{∗}D, the symplectic form is ω = dα, where α is the canonical one-form on T^{∗}D. Any diffeomorphism ψ : D → D lifts to a canonical symplectomorphism ψ^{∗} : T^{∗}D → T^{∗}D; namely the pull-back map (ψ, Dψ^{−1}). The natural thing to try to do is to set φ = ψ^{∗} in a neighborhood of the 0-section, symplectically interpolating between ψ^{∗} and Id using a generating function. This simple approach fails, however, because ψ is only C^{1}, and so ψ^{∗} is merely continuous. (Even assuming that ψ is C^{2} does not help: in order to control the C^{1} size of such a map, it is necessary to have some control on the C^{2} size of ψ, and we cannot assume any such control). Using a convolution, it is possible to overcome this problem. This approach mirrors that in [BGV], but in the symplectic setting.

The problem is local and one can work in ℝ^{2n} endowed with the standard symplectic form ω = ∑_{i} du_{i} ∧ dv_{i} where u = (u_{1}, …, u_{n}), v = (v_{1}, …, v_{n}). By a symplectic change of coordinates, we may assume that the disk D lies inside a disk {(u, v), ∥u∥ ≤ R, v = 0}. We define ψ using a generating function S.

We first recall the definition and properties of generating functions. Suppose that h : ℝ^{2n} → ℝ^{2n} is a C^{r} symplectomorphism, taking the form:

h(u, v) = (ξ(u, v), η(u, v)),

with ξ, η : ℝ^{2n} → ℝ^{n} and h(0, 0) = (0, 0). Assume that the partial derivative matrix \frac{∂}{∂v} η(u, v) is invertible (this is the case for instance if h preserves ℝ^{n} × \{0\}). We can solve for η = η(u, v) to obtain new coordinates (u, η) on a small neighborhood of (0, 0) in ℝ^{2n}. Since h is symplectic, the 1-form α = ∑_{i} v_{i}du_{i} + ξ_{i}dη_{i} is closed, and hence, exact. Thus there exists a C^{r+1} function S = S(u, η), unique up to adding a constant, defined in a neighborhood of (0, 0), such that dS = α. The function S is called a generating function for h.

On the other hand, any C^{r+1} function S = S(u, η) satisfying the nondegeneracy condition that \frac{∂^{2}}{∂u∂η} S is everywhere nonsingular is the generating function of a C^{r} symplectic diffeomorphism. Solving for α in the equation

dS = \frac{∂S}{∂u} du + \frac{∂S}{∂η} dη = α = vdu + ξdη,

we obtain the system:

\frac{∂S}{∂u} = v; \frac{∂S}{∂η} = ξ.

The nondegeneracy condition implies that this system can be solved implicitly for a C^{r} function η = η(u, v). We then obtain a C^{r} symplectomorphism:

h(u, v) = (\frac{∂S}{∂η} (u, η(u, v)), η(u, v)) ,

and S is a generating function for h.

It is easy to see that the generating function for the identity map is

S_{0}(u, η) = u · η = ∑_{i=1}^{n} u_{i}η_{i}.

Claim 1. For every ε > 0, there exists δ > 0 such that, if d_{C^{2}}(S, S_{0}) < δ then d_{C^{1}}(h, Id) < ε.

Proof. This follows from the implicit function theorem, and the details are omitted.

Returning to the proof of Proposition 2.2, assume that ψ : D → D is written in u-coordinates as

ψ(u_{1}, …, u_{n}) = (ψ_{1}(u_{1}, …, u_{n}), …, ψ_{n}(u_{1}, …, u_{n})).

We may assume that the domain of ψ has been extended to ℝ^{n}. To prove Proposition 2.2, it suffices to prove the following lemma.

Lemma 2.3. Given a disk D ⊂ ℝ^{n}, and a neighborhood U of D × {0} in ℝ^{2n}, there exists C > 0 with the following property.

For every C^{1} diffeomorphism ψ : ℝ^{n} → ℝ^{n}, equal to the identity on a neighborhood of ∂D, there is a C^{2} function S : ℝ^{2n} → ℝ such that:

1. d_{C^{2}}(S_{0}, S) ≤ Cd_{C^{1}}(ψ, Id),

2. S = S_{0} outside of U,

3. \frac{∂S}{∂u} (u, 0) = 0 for all u ∈ ℝ and

4. \frac{∂S}{∂η} (u, 0) = ψ(u) for all u ∈ D.

Note that condition 1. implies that S is nondegenerate, provided that d_{C^{1}}(ψ, Id) is sufficiently small. Proof of Lemma 2.3. To illustrate the argument in a simple case, we first prove the lemma for n = 1. The proof of the general case is very similar. Let

a(u) = ψ′(u) − 1.

Note that a is a continuous map, ∥a∥_{∞} ≤ d_{C^{1}}(ψ, Id), and a(u) = 0 if u ∉ int(D). Let Φ : ℝ → [0, 1] be a C^{∞} function satisfying:

• Φ(0) = 1 and Φ = 0 outside of (−1, 1),

• Φ^{(k)}(0) = 0, for all k ≥ 1,

• ∫_{ℝ} Φ(w) dw = 1. Fix a point u_{∗} ∈ ∂D, so that ψ(u_{∗}) = u_{∗}. For (u, η) ∈ ℝ^{2} let:

Q(u, η) = η ∫_{u_{∗}}^{u} ∫_{ℝ} Φ(w) a(x − wη) dw dx.

For η ≠ 0, one can make the change of variables w′ = x − wη and get

Q(u, η) = Sign(η) ∫_{u_{∗}}^{u} ∫_{ℝ} Φ (\frac{x − w′}{η} ) a(w′) dw′ dx.

Let ρ : ℝ^{2} → ℝ be a C^{∞} bump function identically equal to 1 on a neighborhood of D × {0} and vanishing outside of U. Consider

S = S_{0} + ρ Q.

Lemma 2.3 in the case n = 1 is a direct consequence of: Claim 2. The map Q: ℝ^{2} → ℝ is C^{2} and there is C = C(U) > 0 such that:

1. ∥Q |\overline{U}∥_{C^{2}} ≤ C∥a∥_{∞},

2. \frac{∂Q}{∂u}(u, 0) = 0, for all u ∈ ℝ, and

3. \frac{∂Q}{∂η}(u, 0) = ∫_{u_{∗}}^{u} a(x) dx = ψ(u) − u, for all u ∈ ℝ. Proof. We derive explicitly the formulas:

\frac{∂Q}{∂u} = η ∫_{ℝ} Φ(w) a(u − wη) dw
= Sign(η) ∫_{ℝ} Φ (\frac{u − w′}{η} ) a(w′) dw′,
\frac{∂Q}{∂η} = \frac{−Sign(η)}{η^{2}} ∫_{u_{∗}}^{u} ∫_{ℝ} Φ′ (\frac{x − w′}{η} ) (x − w′) a(w′) dw′ dx
= − ∫_{u_{∗}}^{u} ∫_{ℝ} Φ′(w) w a(x − wη) dw dx
= − ∫_{ℝ} Φ′(w) w ∫_{u_{∗}−wη}^{u−wη} a(x′) dx′ dw,

\frac{∂^{2}Q}{∂η∂u} = − ∫_{ℝ} Φ′(w) w a(u − wη) dw,

\frac{∂^{2}Q}{∂u^{2}} = \frac{1}{|η|} ∫_{ℝ} Φ′ (\frac{u − w′}{η} ) a(w′) dw′
= ∫_{ℝ} Φ′(w) a(u − wη) dw,

and finally:

\frac{∂^{2}Q}{∂η^{2}} = ∫_{ℝ} Φ′(w) w^{2} (a(u − wη) − a(u_{∗} − wη)) dw.

Properties 1. and 2. follow immediately from these formulas. To see 3., note that

\frac{∂Q}{∂η}|_{η=0} = − (∫_{u_{∗}}^{u} a(x) dx) (∫_{ℝ} Φ′(w)w dw)
= − (∫_{u_{∗}}^{u} a(x) dx) (− ∫_{ℝ} Φ(w)dw)
= ∫_{u_{∗}}^{u} a(x) dx.

We now turn to the case n ≥ 1 in Lemma 2.3. For i = 1, … n, let α_{i} be the continuous 1-form defined by

α_{i} = d(ψ_{i} − π_{i}),

where π_{i} : ℝ^{n} → ℝ is the projection onto the ith coordinate. As above, fix a point u_{∗} ∈ ∂D, so that ψ(u_{∗}) = u_{∗}. Then we have the formula:

ψ_{i}(u_{1}, …, u_{n}) − u_{i} = ∫_{u_{∗}}^{u} α_{i},

where the right-hand side is a path integral evaluated on any path from u_{∗} to u = (u_{1}, …, u_{n}). Furthermore, we have ∥α_{i}∥_{∞} ≤ d_{C^{1}}(ψ, Id), for all i. When n = 1, the 1-form α_{1} is just α_{1} = a(u) du, where a(u) = ψ′(u) − 1, as above.

Let Φ_{n} : ℝ^{n} → ℝ be an n-dimensional bell function:

Φ_{n}(x_{1}, …, x_{n}) = Φ(x_{1}) ⋯ Φ(x_{n}).

For each 1-form α, and t ∈ ℝ, we define a new 1-form α_{i}^{⋆t} on ℝ^{n} by taking the convolution:

α^{⋆t}(u) = t ∫_{ℝ^{n}} Φ_{n}(w) α(u − tw) dw.

We integrate along any path from u_{∗} to u and set

Q(u, η) = ∫_{u_{∗}}^{u} ∑_{i=1}^{n} α_{i}^{⋆η_{i}} = ∑_{i=1}^{n} η_{i} ∫_{ℝ^{n}} Φ_{n}(w) (∫_{u_{∗}}^{u} α_{i}(u − tw)) dw.

This is well-defined since ∫_{u_{∗}}^{u} α_{i}(u − tw) is independent of choice of path.

Let ρ_{n} : R^{2n} → [0, 1] be a C^{∞} bump function vanishing identically outside of U and equal to 1 on a neighborhood of D. As before, the map S = S_{0} + ρ_{n}Q satisfies the conclusions of Lemma 2.3 provided the following claim holds. Claim 3. The map Q: ℝ^{2n} → ℝ is C^{2} and there is C = C(U) > 0 such that:

1. ∥Q |\overline{U}∥_{C^{2}} ≤ C max_{i} ∥α_{i}∥_{∞},

2. \frac{∂Q}{∂u}(u, 0) = 0, for all u ∈ ℝ^{n}, and

3. \frac{∂Q}{∂η_{i}} (u, 0) = ∫_{u_{∗}}^{u} α_{i} = ψ_{i}(u) − u_{i}, for all 1 ≤ i ≤ n and u ∈ ℝ. Proof. We repeat the calculations from the proof of Lemma 2 in the general setting. When t ≠ 0, the change of variable w′ = u − tw gives

α^{⋆t}(u) = Sign(t) ∫_{ℝ^{n}} Φ_{n} (u − tw′) α(w′) dw′,
\frac{d}{dt}α^{⋆t}(u) = − ∫_{ℝ^{n}} (dΦ_{n}(w).w + (n − 1)Φ_{n}(w)) α(u − tw) dw.

One deduces:

\frac{∂Q}{∂u} = ∑_{i=1}^{n} η_{i} ∫_{ℝ^{n}} Φ_{n}(w) α_{i}(u − tw) dw
= ∑_{i=1}^{n} \frac{η_{i}}{|η_{i}|^{n}} ∫_{ℝ^{n}} Φ_{n} (\frac{u − w′}{η_{i}} ) α_{i}(w′) dw′,
\frac{∂Q}{∂η_{i}} = ∫_{u_{∗}}^{u} \frac{d}{dη_{i}} α_{i}^{⋆η_{i}}
= − ∫_{ℝ^{n}} (dΦ_{n}(w).w + (n − 1)Φ_{n}(w)) (∫_{x=u_{∗}−η_{i}w}^{x=u−η_{i}w} α_{i}(w)) dw,
\frac{∂^{2}Q}{∂u ∂η_{i}} = − ∫_{ℝ^{n}} (dΦ_{n}(w).w + (n − 1)Φ_{n}(w)) α_{i}(u − η_{i}w) dw,
\frac{∂^{2}Q}{∂u^{2}} = ∑_{i=1}^{n} ∫_{ℝ^{n}} dΦ_{n}(w) α_{i}(u − η_{i}w) dw,

and finally:

\frac{∂^{2}Q}{∂η_{i}∂η_{j}} = δ_{i,j} ∫_{ℝ^{n}} (dΦ_{n}(w).w + (n − 1)Φ_{n}(w)) [α_{i}(x − η_{i}w).w]_{x=u_{∗}}^{x=u} dw.

It is not difficult to verify that 1.–3. hold.

The proof of Lemma 2.3 is now complete.

2.2 The volume-preserving case

Proposition 2.4. Let M be a Riemannian manifold endowed with a volume form μ and W be a C^{1} submanifold satisfying

dim(W) ≤ codim(W).

Centered at any point z ∈ W, there exists a disk D = \overline{B_{W} (z, r_{0})} of W such that, for every neighborhood U ⊂ M containing D and every ε > 0, there exists δ > 0 with the following property.

For every C^{1} diffeomorphism ψ : D → D satisfying:

a. ψ = Id on a neighborhood of ∂D, and

b. d_{C^{1}}(ψ, Id) < δ,

there exists φ ∈ Diff_{μ}^{1}(M) such that:

1. φ = Id on M \ U,

2. φ = ψ on D, and

3. d_{C^{1}}(φ, Id) < ε.

Proof. Let n = dim(M). By a local change of coordinates, we may assume that μ is the standard volume form dx_{1} ∧ ⋯ ∧ dx_{n} on a neighborhood of the origin in ℝ^{n}. By composing these coordinates with an isometry of ℝ^{n}, we may further assume that D is the graph of a C^{1} function h : ℝ^{k} → ℝ^{n−k}, where k ≤ n/2. The final change of coordinates

(x_{1}, …, x_{n}) ↦ ((x_{1}, …, x_{k}), (x_{k+1}, …, x_{n}) − h(x_{1}, …, x_{k}))

preserves volume. Applying this change of coordinates, we may assume that D lies in the coordinate plane {x_{k+1} = x_{k+2} = ⋯ = x_{n} = 0} ≃ ℝ^{k}. Now we apply the symplectic pertubation result (Proposition 2.2) inside the space {x_{2k+1} = ⋯ = x_{n} = 0} ≃ ℝ^{2k} to obtain a local C^{1} symplectomorphism φ_{0} of {x_{2k+1} = ⋯ = x_{n} = 0} that agrees with ψ on D. This symplectomorphism is C^{1} isotopic to the identity through symplectomorphisms {φ_{t}}_{t∈[0,1]}, where φ_{1} = Id (to obtain this isotopy, just choose a smooth isotopy of the generating function for ψ to the generating function for the identity).

Now we extend φ_{0} to ℝ^{n} using this isotopy to obtain a locally-supported volumepreserving diffeomorphism that agrees with ψ on D. More precisely, choose an appropriate C^{∞} bump function ρ : ℝ^{n−2k} → [0, 1], and set

φ(x_{1}, …, x_{n}) = (φ_{ρ(∥(x_{2k+1},...,x_{n})∥)}(x_{1}, …, x_{2k}), x_{2k+1}, …, x_{n}).

This is the desired map φ.

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Christian Bonatti (bonatti@u-bourgogne.fr) CNRS - Institut de Mathématiques de Bourgogne, UMR 5584 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, 99 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