Centralizers of C^{1}-generic diffeomorphisms

C. Bonatti, S. Crovisier and A. Wilkinson

September 28, 2006

Abstract

On the one hand, we prove that the spaces of C^{1} symplectomorphisms and of C^{1} volume-preserving diffeomorphisms both contain residual subsets of diffeomorphisms whose centralizers are trivial. On the other hand, we show that the space of C^{1} diffeomorphisms of the circle and a non-empty open set of C^{1} diffeomorphisms of the two-sphere contain dense subsets of diffeomorphisms whose centralizer contains a subgroup isomorphic to ℝ.

Key words: Trivial centralizer, trivial symmetries, Mather invariant, C^{1} generic properties.

Introduction

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

C(f) := {g ∈ Diff^{r}(M) : fg = gf}.

Clearly C(f) always contains the group < f > of all the powers of f. We say that f has trivial centralizer if C(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. Smale asked the following:

Question 0.1 ([Sm1, Sm2]) 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)?

3. Is T^{r}(M) open in Diff^{r}(M)?

This question has been answered in several special cases. To summarize these results in rough chronological order, we have:

• T^{r}(S^{1}) is open and dense in Diff^{r}(S^{1}) for r ≥ 2 [Ko];

• T^{1}(M) is residual among the Axiom A diffeomorphisms in Diff^{1}(M); in particular, T^{1}(S^{1}) is residual in Diff^{1}(S^{1}) [To1, To2];

• T^{∞}(M) is open and dense among the Axiom A diffeomorphisms in Diff^{∞}(M) possessing at least one periodic sink or source [PY1];

• T^{∞}(M) is open and dense among the Anosov diffeomorphisms in Diff^{∞}(𝕋^{n}), where 𝕋^{n} is the n-torus [PY2];

• T^{∞}(M) is locally residual among the partially hyperbolic diffeomorphisms with 1-dimensional center [Bu].

There are two main results in this paper. In the first (Theorem 0.2), we give a complete answer to the first two parts of Question 0.1 for all compact M in the case of volume-preserving and symplectic C^{1}-diffeomorphisms. In the second result (Theorem 0.7), we answer the third part of Question 0.1 for the circle S^{1} and the sphere S^{2}, again in the case r = 1.

A) Trivial centralizer for C^{1}-generic symplectomorphisms and volume-preserving diffeomorphisms

In order to state our first main result precisely we will need some notation. 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. If M is a symplectic manifold, then Symp^{1}(M) denotes the space of C^{1} symplectomorphisms of M. The spaces Diff^{1}(M), Diff_{μ}^{1}(M), Symp^{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 0.2 Let M be a compact, connected manifold of dimension at least 2. Then:

(a) T^{1}(M) ∩ Diff_{μ}^{1}(M) is residual in Diff_{μ}^{1}(M).

(b) T^{1}(M) ∩ Symp^{1}(M) is residual in Symp^{1}(M).

Theorem 0.2 is a corollary of parts (b) and (c) of the following result:

Theorem 0.3 (a) There is a residual set R ⊂ Diff^{1}(M) such that, for any diffeomorphism f ∈ R, for any g ∈ C(f) and any periodic point x ∈ P er(f), the point x is hyperbolic and, there exist m, n ∈ ℤ such that g coincides with f^{n} on W^{s}(x) and with f^{m} on W^{u}(x).

(b) There is a residual set Rsymp ⊂ Symp^{1}(M) such that, for any diffeomorphism f ∈ Rsymp for any g ∈ C(f) and any hyperbolic periodic point x ∈ P er(f), there exist m, n ∈ ℤ such that g coincides with f^{n} on W^{s}(x) and with f^{m} on W^{u}(x).

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

Theorem 0.3 (a) was previously proved by Togawa [To1, To2], using different methods. Togawa’s methods, combined with the results in Appendix A, can also be used to prove parts (b) and (c) of Theorem 0.3. While using Togawa’s results would shorten considerably the proof of Theorem 0.3, we believe our approach, in particular Propositions 1.4 and 1.9, has independent interest. It would be interesting to see if these results have further application. We discuss the motivation and background to this approach in Section 1.

Proof of Theorem 0.2. Theorem 0.2 follows immediately from Theorem 0.3 and:

Theorem 0.4 ([BC, ABC]) For any compact connected manifold M, there are residual sets R̃_{μ} ⊂ Diff_{μ}^{1}(M) and R̃symp ⊂ Symp^{1}(M) such that, every f ∈ R̃_{μ} ∪ R̃symp has a hyperbolic periodic point p with

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

More generally, Theorem 0.3 naturally applies to the class of C^{1} diffeomorphisms satisfying a property we call periodic accessibility. A diffeomorphism f on a compact manifold satisfies the periodic accessibility property if there is a dense subset E ⊂ M of non-periodic points such that any pair of points x, y ∈ E may be joined by a finite sequence x_{0} = x, x_{1}, …, x_{n} = y, x_{i} ∈ E and a sequence p_{i} of hyperbolic periodic orbits such that for any i ∈ {0, …, n − 1} one has:

\{x_{i}, x_{i+1}\} ⊂ \overline{W_{orb}^{s}(p_{i})} or \{x_{i}, x_{i+1}\} ⊂ \overline{W_{orb}^{s}(p_{i})}.

Question 0.5 Is periodic accessibility generic in Diff^{r}(M)?

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

Theorem 0.3 has the immediate corollary:

Corollary 0.6 Furthermore, if f ∈ R satisfies the periodic accessibility property then C(f) is trivial.

The periodic accessibility property is satisfied by Axiom A diffeomorphisms, by C^{1}-generic tame diffeomorphisms (i.e. by C^{1}-generic diffeomorphisms having finitely many homoclinic classes), and by C^{1}-generic conservative (volume preserving or symplectic) diffeomorphisms. In this way, one can recover Togawa’s result that the C^{1}-generic Axiom A diffeomorphism has trivial centralizer.

B) Large centralizer for a locally C^{1} dense set of diffeomorphisms

Our next main result addresses the third part of Question 0.1: is T^{r}(M) open in Diff^{r}(M)? In the case of the circle, recall that Kopell proved that T^{r}(S^{1}) is open-dense in Diff^{r}(S^{1}) for r ≥ 2, and Togawa proved that T^{1}(S^{1}) is residual in Diff^{1}(S^{1}). It is natural to ask whether Togawa’s result can be strengthened to show that T^{1}(S^{1}) is open-dense.

Our next main result shows that the answer is “no”: the answers to the third part of Question 0.1 are genuinely different in the C^{1} and C^{2} topologies, at least for the circle. We are also able to answer the third part of Question 0.1 for the 2-sphere. Specifically, we have:

Theorem 0.7 T^{1}(S^{1}) and T^{1}(S^{2}) are not open. Moreover:

(a) There is a dense subset D^{1} ⊂ Diff^{1}(S^{1}) such that every f ∈ D^{1} leaves invariant a C^{∞} Morse-Smale vector field. In particular, C(f) contains a subgroup isomorphic to ℝ.

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

There is a dense subset D^{2} ⊂ O such that every f ∈ D^{2} is the time 1 map of a Morse-Smale C^{∞}-vector field. In particular, C(f) contains a subgroup isomorphic to ℝ.

Structure of the paper

In order to prove Theorem 0.3, it is enough to show that along the invariant manifolds of the periodic points, f satisfies an unbounded distortion property. This is discussed in Section 1. As a simpler setting, we also deal with contractions of ℝ^{d} whose unique periodic point is 0. In Section 2, we will see that C^{1}-generic contraction of ℝ^{d} has the unbounded distortion property; this can be generalized to the dynamics inside the invariant manifolds of the periodic points, since by Appendix A, any perturbation of the dynamics inside the stable manifod of a periodic point can be realized as a perturbation of the dynamics on M. Theorem 0.7 will be proved in Section 3.

1 The unbounded distortion property

Kopell’s proof in [Ko] that T^{r}(S^{1}) is open-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 T^{r}(S^{1}) is open-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 (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 and Corollary 1.3 below). As in [Ko], a small amount of additional work shows that a residual set in Diff^{1}(S^{1}) has trivial centralizer. The details of this argument we have just described for S^{1} are contained in this section and Section 2.2.

The bulk of this paper 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 invariant manifolds

Let f : M → M be a C^{1} diffeomorphism, and let p ∈ M be a hyperbolic periodic point of f. For x ∈ W^{s}(p) we denote by Jac^{s}(f)(x) the Jacobian of the map induced by T_{x}f between T_{x}W^{s}(p) and T_{f(x)}W^{s}(f(p)).

Definition 1.1 A hyperbolic periodic point p ∈ M has the stable manifold distortion property if, for every x, y ∈ W^{s}(p) \ {p} not in the same f-orbit,

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

As mentioned in the previous subsection, unbounded distortion forces trivial centralizers:

Lemma 1.2 Let p be a hyperbolic periodic point of period k ∈ ℕ with the stable manifold distortion property, and let g ∈ C(f). If g(p) = p, then there exists an m ∈ ℤ such that g = f^{km} on W^{s}(p).

Proof of Lemma 1.2. We claim that for every x ∈ W^{s}(p), g preserves the f^{k}-orbit of x. From this claim it follows that for every x ∈ W^{s}(p), there exists an integer m(x) such that g(x) = f^{km(x)}(x), and there is a unique such m(x) if x ≠ p. Continuity of f implies that the function m is locally constant on W^{s}(p) \ {p}. If dim(W^{s}(p)) > 1, then W^{s}(p) \ {p} is connected, and m is constant. If dim(W^{s}(p)) = 1, then m is constant on each of the connected components of W^{s}(p) \ {p}; in this case, since g is differentiable at p and |f^{k}′(p)| ≠ 1, the values of m on the two components must coincide.

It remains to prove the claim. We may assume that x ≠ p. The relation gf^{n} = f^{n}g implies that,

Jac^{s}(g)(f^{n}x)Jac^{s}(f^{n})(x) = Jac^{s}(f^{n})(gx)Jac^{s}(g)(x).

In particular, for all m ≥ 0, we have

|\frac{Jac^{s}(g)(f^{mk}x)}{Jac^{s}(g)(x)}| = |\frac{Jac^{s}(f^{mk})(gx)}{Jac^{s}(f^{mk})(x)}| .

Since f^{mk}(x) lies in a compact region of W^{s}(p) for all m ∈ ℕ, the left hand side of this expression is uniformly bounded in m. On the other hand, it is easy to see that if f has the stable manifold distortion property, then so does f^{k}. This implies that the right hand side of the equation above is unbounded, a contradiction. This proves the claim. □

Corollary 1.3 Let p be a hyperbolic periodic point of period k ∈ ℕ with the stable manifold distortion property, and let g ∈ C(f). If g(O(p, f)) = O(p, f), then there exists an m ∈ ℤ such that g = f^{m} on W^{s}(O(p, f)).

Proof of Corollary 1.3. Since g preserves the f-orbit of p, we have g(p) = f^{j}(p) for some integer j. Let G = f^{−j}g. Then it is easy to see that G commutes with f, and G fixes every point on the f-orbit of p. By Lemma 1.2, there is an integer m such that the restriction of G to W^{s}(p) coincides with f^{mk}. Again, since f and G commute, the restriction of G to W^{s}(f^{i}p) is conjugate by f^{i} to the restriction of G to W^{s}(p). Consequently, G coincides with f^{mk} on the stable manifold W^{s}(O(p, f)) of the orbit of p, and so g coincides with f^{mk+j} on W^{s}(O(p, f)). □

Note that for any integer k ≥ 1, a C^{1}-generic diffeomorphism f has only finitely periodic orbits of period k and, by transversality, that all these orbits have different exponents. In particular, any diffeomorphism g ∈ C(f) preserves each of these orbits and satisfies the assumption of corollary 1.3.

To prove Theorem 0.3, we are thus reduced to proving:

Proposition 1.4 There is a residual set R ⊂ Diff^{1}(M) such that, for any diffeomorphism f ∈ R, every periodic point x ∈ P er(f) is hyperbolic and has the stable manifold distortion property.

There is a residual set R_{symp} ⊂ Symp^{1}(M) such that, for any diffeomorphism f ∈ R_{symp}, every hyperbolic periodic point x ∈ P er(f) has the stable distortion property.

There is a residual set R_{μ} ⊂ Diff_{μ}^{1}(M) such that, for any diffeomorphism f ∈ R_{μ}, and any hyperbolic periodic point x ∈ P er(f), if W^{s}(x) has codimension at least dim(M)/2, then x has the stable manifold distortion property.

1.2 Contractions of ℝ^{d}

Let B^{d} denotes 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}.

Remark 1.5 Since the origin is fixed, the metric ∥ · ∥_{1} is equivalent to the metric defined by

∥f − g∥′_{1} = sup_{x∈B^{d}} ∥D_{x}f − D_{x}g∥.

In general, we will prefer to work with this second one.

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}, hence a Baire space.

Let f be a diffeomorphism of a manifold M, p be a periodic point of f and n^{s} its stable dimension. A stable chart for p is a local chart ψ: ℝ^{d} → M such that if one denotes by π the projection of ℝ^{d} onto the n^{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^{n^{s}} → ℝ^{d−n^{s}}.

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

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

then, θ belongs to C^{n^{s}}.

Proposition 1.6 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 and to the chart ψ induces a continuous map Θ: U → C^{n^{s}}. This map is open.

In the conservative setting, the same property holds.

Theorem 1.7 Let Θ: U → C^{n^{s}} be a family of contractions associated to a periodic point p and a stable chart ψ as in proposition 1.6. Then, the map Θ: U ∩ Symp^{1}(M) → C^{n^{s}} is open.

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

This will be proved in Sections A.1 and A.2.

Proposition 1.8 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 n_{i}^{s}.

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

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

Proposition 1.9 There is a residual set R_{0} ⊂ C^{d} such that, for all f ∈ R_{0}, if x, y ∈ \overline{B(0, 1)} \ \{x, y\} with x ∉ O_{f}(y), then

lim sup_{n→∞} |\frac{Jac(f^{n})(x)}{Jac(f^{n})(y)}| = ∞.  (3)

The proof that Proposition 1.9 implies Proposition 1.4 is quite immediate in the non-conservative case: the dynamics in any stable manifold is diffeomorphically conjugate to a contraction of ℝ^{n^{s}}; one concludes by noting that any perturbation of the dynamics inside the stable manifold extends to a perturbation of the dynamics on M. In the conservative case, this last property is much more delicate and its proof will be postponed until Appendix A.

1.3 K-distortion and the Baire argument

In this subsection, we explain how to reduce the unbounded distortion property (3) in Proposition 1.9 to a property satisfied in finite time, which we call the K-distortion property. Using a Baire argument, we then reformulate Proposition 1.9 in terms of this K-distortion property to obtain our main perturbation result (Theorem 1.13).

Definition 1.10 Let B ⊂ ℝ^{d} be a compact region and let f : B → B be an embedding. Given compact sets Λ, Δ ⊂ B, we say that Λ and Δ are dynamically disjoint for f if f^{n}(Λ) ∩ f^{m}(Δ) = ∅ for any n, m ∈ ℕ.

Definition 1.11 Let Λ, Δ ⊂ B^{d} \ {0} be compact sets that are dynamically disjoint for f ∈ C^{d}. We say that Λ and Δ satisfy the K-distortion property for f at time N if, for any x ∈ Λ, y ∈ Δ there exists n ∈ {0, …, N} such that :

|\frac{Jac f^{n}(x)}{Jac f^{n}(y)}| > K.

More briefly, Λ, Δ have the K-distortion property for f if there exists an N so that they satisfy the K-distortion property for f at time N

The properties of dynamical disjointness and K-distortion persist under perturbations of both the diffeomorphism and the compact sets.

Proposition 1.12 Let Λ, Δ ⊂ B^{d} \ {0} be dynamically disjoint for f ∈ C^{d}.

1. There exist neighborhoods U_{f} ⊂ C^{d} of f, U_{Λ} of Λ, and U_{Δ} of Δ such that all compact sets Λ′ ⊂ U_{Λ}, Δ′ ⊂ U_{Δ} are dynamically disjoint for all g ∈ U_{f};

2. suppose that Λ and Δ satisfy the K-distortion property for f at time N. Then we can choose U_{f}, U_{Λ}, U_{Δ} so that all compact sets Λ′ ⊂ U_{Λ}, Δ′ ⊂ U_{Δ} satisfy the K-distortion property for g ∈ U_{f} at time N.

Proof of Proposition 1.12. For the first item, it is enough to show that g^{n}(Λ) ∩ Δ = ∅ for n ∈ ℕ. One considers D, a neighborhood of 0 satisfying f(\overline{D}) ⊂ D disjoint from Δ. Since Λ is compact, for large N we have f^{N}(Λ) ⊂ D. For g close enough to f, one gets g^{n}(Λ) ⊂ D for each n ≥ N, which implies the required property.

The second item is an easy continuity argument. □

We next reformulate Proposition 1.9 in terms of K-distortion.

Theorem 1.13 (Main perturbation result) Let Λ, Δ ⊂ B^{d} \ {0} be compact sets that are dynamically disjoint for f ∈ C^{d}. Then for every neighborhood U of f in C^{d}, and for every K > 0, there exist g ∈ U such that Λ and Δ satisfy the K-distortion property for g.

To prove that Theorem 1.13 implies Proposition 1.9, we employ a standard Baire argument. Let U_{0} be a countable basis of (relatively) open balls for the topology on B^{d}, and for U_{1}, U_{2} ∈ U, let

O(U_{1}, U_{2}) = \{f ∈ D^{d} | \overline{U}_{1} and \overline{U}_{2} are dynamically disjoint for f\}.

Proposition 1.12 implies that O(U_{1}, U_{2}) is open in D^{d}, and clearly:

D^{d} = ⋃_{U_{1},U_{2}∈U} O(U_{1}, U_{2}).  (4)

Note that O(U_{1}, U_{2}) is nonempty if and only if \overline{U}_{1} and \overline{U}_{2} are disjoint, and henceforth any two such sets we discuss will be assumed to be disjoint.

Given U_{1}, U_{2} ∈ U, open sets V_{1}, V_{2} with \overline{V}_{1} ⊂ U_{1} and \overline{V}_{2} ⊂ U_{2}, and K > 0, we define O(U_{1}, U_{2}, V_{1}, V_{2}, K) be the set of all f ∈ O(U_{1}, U_{2}) such that:

\overline{V}_{1} and \overline{V}_{2} satisfy the K-distortion property at time N,

for some N > 0.

Theorem 1.13 immediately implies:

Proposition 1.14 O(U_{1}, U_{2}, V_{1}, V_{2}, K) is open and dense in O(U_{1}, U_{2}).

For each U ∈ U, let V_{U} be a countable basis of open sets in U, consisting of sets whose closures are contained in U. Proposition 1.14 implies that

R_{U_{1},U_{2}} = ⋂ O(U_{1}, U_{2}, V_{1}, V_{2}, K)

is residual in O(U_{1}, U_{2}), where the intersection is taken over all V_{1} ∈ V_{U_{1}}, V_{2} ∈ V_{U_{2}} and K ∈ ℕ. Let

R_{0} = ⋂_{U_{1},U_{2}∈U} (R_{U_{1},U_{2}} ∪ (D^{d} \ O(U_{1}, U_{2})));

clearly R_{0} is residual in D^{d}.

Suppose that f ∈ R_{0}. Let x, y ∈ B^{d} \ {0} such that y ∉ O(x) and let K > 0 be given. We show that there exists an n ∈ ℕ such that

|\frac{Jac(f^{n})(x)}{Jac(f^{n})(y)}| > K.

Since f is a contraction and y ∉ O(x), there exist U_{1}, U_{2} ∈ U_{0} such that x ∈ U_{1}, y ∈ U_{2}, and \overline{U}_{1}, \overline{U}_{2} are dynamically disjoint for f. This means that f ∈ O(U_{1}, U_{2}), and the definition of R_{0} then implies that f ∈ R_{U_{1},U_{2}}. Let V_{1} ∈ V_{U_{1}}, V_{2} ∈ V_{U_{2}} be neighborhoods of x and y, respectively. Since f ∈ O(U_{1}, U_{2}, V_{1}, V_{2}, K), we obtain that for some n ∈ ℕ

|\frac{Jac(f^{n})(x)}{Jac(f^{n})(y)}| > K,

which completes the proof of Proposition 1.9. □

2 Proof of Theorem 1.13 (Main perturbation result)

Before proving Theorem 1.13 we introduce notations and concepts that will be used in the whole section. We then isolate the proof in some particular cases. We do this for two reasons: first, it will allow us to illustrate some of the main ideas of the general case while avoiding serious technical issues, and second, these special cases will be ingredients in the proof of the general case.

2.1 Preliminaries

For our purposes, a map that is linear near the origin is especially easy to work with, because a linear map has constant Jacobian.

2.1.1 Linearization near the origin. A well-known feature of the C^{1} topology is that a diffeomorphism may be C^{1} approximated by its derivative in a neighborhood of a fixed point: we will say that f ∈ C^{d} has a linear germ if there exists a linear map A : ℝ^{d} → ℝ^{d} and a neighborhood U of 0 in B^{d} with f(U) ⊂ U such that f|_{U} = A|_{U}.

By Proposition 1.12, if the two compact sets Λ, Δ ⊂ B^{d} \ {0} are dynamically disjoint for some contraction f_{0}, one can approximate f_{0} by a contraction f that has a linear germ A = D_{0}f and such that Λ, Δ remain dynamically disjoint.

2.1.2 Bring Λ, Δ into the linearized region. Once one considers a contraction f having a linear germ, we show that one can reduce the proof of Theorem 1.13 to the case the contraction is a linear map A.

Let U be a forward invariant set where f coincides with a linear map A and choose some numbers 1 > r_{2} > r_{1} > 0 satisfying B(0, r_{2}) ⊂ U. Since f is a contraction, there exists an integer m > 0 such that f^{m}(Λ ∪ Δ) ⊂ B(0, r_{1}). We would like to work with the sets f^{m}(Λ) and f^{m}(Δ) in place of Λ and Δ, and the following simple lemma allows us to do so.

Lemma 2.1 Let f ∈ C^{d}, and let Λ, Δ be dynamically disjoint for f. For every K > 0 and integer m ≥ 0, there exists a neighborhood V of f and K′ > 0 such that, for all g ∈ V, if g^{m}(Λ) and g^{m}(Δ) have the K′-distortion property for g, then Λ and Δ have the K-distortion property for g.

Note that changing the Riemannian metric on ℝ^{d} only affects our choice of K and the C^{1}-size of the neighborhood U in C^{d}. Hence, one can assume that B^{d} is mapped into B(0, 1) by A so that A also is a contraction.

Let us assume that Theorem 1.13 has been proven for the linear map. Then any small perturbation g of A in C^{d} can be glued to f inside B(0, r_{2}) thanks to the following standard lemma:

Lemma 2.2 Given f ∈ C^{d}, numbers 1 > r_{2} > r_{1} > 0, and ε > 0, there exists ε′ > 0 such that, for every embedding g : B(0, r_{2}) → ℝ^{d} satisfying

sup_{x∈B(0,r_{2})} ∥D_{x}g − D_{x}f∥_{0} < ε′,

and g(0) = 0, there exists a diffeomorphism g′ ∈ C^{d} such that

1. ∥g′ − f∥_{1} < ε

2. g′ = f on B^{d} \ B(0, r_{2})

3. g′ = g on B(0, r_{1}).

2.1.3 Localize the perturbations. We introduce some terminology that will be used in the rest of the paper.

Definition 2.3 Let 1 > r_{2} > r_{1} > 0. The spherical shell in B^{d} of outer radius r_{2} and inner radius r_{1} is the set:

S(r_{1}, r_{2}) = B(0, r_{2}) \ B(0, r_{1}).

The modulus m(S(r_{1}, r_{2})) of the shell S(r_{1}, r_{2}) is defined by:

m(S(r_{1}, r_{2})) = log (\frac{r_{2}}{r_{1}} ) .

Let S(r_{1}, r_{2}) be a spherical shell. We say that x ∈ B^{d} is inside S(r_{1}, r_{2}) if ∥x∥ ≤ r_{1} and outside S(r_{1}, r_{2}) if ∥x∥ ≥ r_{2}. The set of points inside of a spherical shell S is denoted by I(S), and the set of points outside of S is denoted by O(S).

If S(r_{3}, r_{4}) is another spherical shell then we write S(r_{1}, r_{2}) ≺ S(r_{3}, r_{4}) if r_{2} ≤ r_{3}. We say that x ∈ ℝ^{d} is in between S(r_{1}, r_{2}) and S(r_{3}, r_{4}) if x is outside S(r_{1}, r_{2}) and inside S(r_{3}, r_{4}); that is, if r_{2} ≤ ∥x∥ ≤ r_{3}.

In this terminology, the map g′ given by Lemma 2.2 coincides with g inside the shell S(r_{1}, r_{2}) and coincides with f outside of S(r_{1}, r_{2}).

2.1.4 Tidy perturbations. In order to compose several perturbations, we will often require the following property:

Definition 2.4 Let B ⊂ R^{d}, let f : B → B be an embedding and let X ⊂ B. We say that an embedding g : B → B is tidy relative to f and X if,

1. g(x) = f(x), for all x ∈ B \ X, and

2. g^{m}(x) = f^{m}(x) for all x ∈ B \ X and m ≥ 1 such that g^{m}(x) ∈ B \ X.

Notice that if X is forward invariant by f (i.e. f(X) ⊆ X), then g is tidy relative to f and X if and only if g = f on B \ X.

2.2 The conformal case

We prove here Theorem 1.13 in the particular case when f = A is a conformal linear contraction: we have A = α.I for some constant α ∈ (0, 1) and some isometry I. The main reason why the conformal case is simpler is that conjugacy by a linear conformal map preserves the C^{1}-norm. Also note that when d = 1, this case is the general case. We will prove the following more precise statement.

Proposition 2.5 Let A ∈ C^{d} be a linear conformal contraction and (Λ, Δ, r) be a triple such that Λ, Δ ⊂ B(0, 1) are two dynamically disjoint compact sets, disjoint from the ball B(0, r). Fix constants ε, K > 0.

Then there exists β ∈ (0, 1) such that, for any 0 < s < r, there exists g ∈ C^{d} with the following properties:

1. ∥Dg − A∥_{0} < ε;

2. Λ and Δ have the K-distortion property for g;

3. g is tidy relative to A and the spherical shell S(βs, s).

2.2.1 The case Δ is a small ball. Once again, it is instructive to consider a simple case; we assume first that Δ is a round ball D_{1}, contained in a fundamental domain for the action of A, so that A^{i}(D_{1})∩D_{1} = ∅, for all i ≠ 0. Let D_{0} ⊂ G be another round ball with the same center as D_{1} and containing \overline{D_{1}} in its interior, chosen to be dynamically disjoint from Λ.

Let η be a diffeomorphism of B^{d} satisfying:

P1. η is the identity map on B^{d} \ D_{0};

P2. the restriction of η to the ball D_{1} is an affine conformal contraction whose fixed point is the center of D_{0} and D_{1}; in particular, η(D_{1}) is contained in the interior of D_{1}, and the Jacobian of η in D_{1} is a constant μ strictly less than 1.

P3. ∥D(A ∘ η) − A∥_{0} < ε, and ∥D(A ∘ η^{−1}) − A∥_{0} < ε.

It is easy to see that the distortion for one or more iterates of A ∘ η between x ∈ Λ and y ∈ Δ is equal to μ^{−1}. To get the distortion greater than K, we perform a sequence of such perturbations, each supported on a forward image A^{i}(D_{0}).

To this end, choose m > 0 such that μ^{−m} > K. For i ≥ 0, the diffeomorphism η_{i} = A^{i}ηA^{−i} is supported on A^{i}(D_{0}) and has distortion μ inside A^{i}(D_{0}). Furthermore, since conjugacy by a linear conformal map preserves the C^{1} norm, the C^{1}-distance from A ∘ η_{i} to A is the same for all i ≥ 0, and therefore less than ε. Now fix some integer n ≥ 0 and let

g_{n,m}(x) = \begin{cases}A ∘ η_{n+i}(x) & if x ∈ A^{n+i}(D_{0}), for some i ∈ \{0, … m − 1\}, \\ A ∘ η_{n+m+i}^{−1}(x) & if x ∈ A^{n+m+i}(D_{0}), for some i ∈ \{0, … m − 1\}, \\ A(x) & otherwise. \end{cases}

It is straightforward to check that the diffeomorphism g = g_{n,m} satisfies

∥Dg − A∥_{0} = sup_{i∈{0,...,m−1}} ∥D(A ∘ η_{n+i}) − A∥_{0} < ε.

Moreover, for all x ∈ Λ, y ∈ D_{1} = Δ we have

|\frac{Jac g^{n+m}(x)}{Jac g^{n+m}(y)}| > K.

Hence Λ, Δ have the K-distortion property for g.

Let us consider a point x on the outside of the shell S = S(α^{n+2m}r, α^{n}r) and i > 0 such that h^{i}(x) is inside the shell. If x does not belong to the orbit of D_{0} for A, then g_{n,m} coincides with A on the orbit of x. Let us assume now that x belongs to A^{k}(D_{0}) for some k < n: a straightforward computation shows that g_{n,m}^{(n−k)+2m}(x) = A^{(n−k)+2m}(x) which is the first point of the orbit of x that lies inside the shell S. Now, inside the shell g_{n,m} = A and since the inside of S is forward-invariant under A, it follows that g_{n,m}^{i}(x) = A^{i}(x). Hence, g_{n,m} is tidy with respect to the shell S.

If β = α^{2m+1} and if n is the smallest integer such that α^{n} < s, the map g = g_{n,m} is tidy relative to A and the shell S(βs, s). This ends the proof of Proposition 1.13 in this case.

2.2.2 Cover Δ with small balls. For arbitrary Λ, Δ, the strategy (to which we will return in later arguments) is to create distortion between Λ and Δ in small increments. Each increment will consist of a perturbation supported on a spherical shell and will produce distortion between Λ and a small piece of Δ.

Notice that for every r ∈ (0, 1], the spherical shell S(αr, r) is a fundamental domain for the action of A; to simplify notations, in this section it will be denoted by G_{r}. The following construction is an easy consequence of the fact that Λ and Δ are dynamically disjoint for A.

There is a family {(D_{0}^{j}, D_{1}^{j}, r_{j})}_{j∈{1,...,k}} with the following properties:

• for every j, r_{j} is a number in (0, 1), D_{0}^{j} is a round disk contained in the fundamental domain G_{r_{j}}, and D_{1}^{j} is a round disk, centered at the same point as D_{0}^{j} and contained in the interior of D_{0}^{j};

• Δ is contained in ⋃_{j=1}^{k} D_{1}^{j};

• for every j, Λ is disjoint from the orbit ⋃_{n∈ℤ} h^{n}(D_{0}^{j}).

2.2.3 Create distortion between Λ and balls in the cover of Δ. We have seen that for any ball D_{0}^{j} in the cover of Δ, there is a perturbation g_{j} of A producing K-distortion between points in Λ and points in D_{1}^{j}. These perturbations can be chosen to be supported in disjoint spherical shells S_{j}. We will use the fact that g_{j} is tidy relative to A and S_{j} in order to ensure that these perturbations can be considered independently.

Consider an integer ℓ > 0 such that the shell S(α^{ℓ}, 1) (composed of ℓ successive fundamental domains of A) contains Λ ∪ Δ. Let β_{j} be the constants associated to the triples (D_{0}^{j}, D_{1}^{j}, r^{j}) by the proof of Proposition 2.5 given at Section 2.2.1. Fix now a sequence s_{1}, …, s_{k} such that s_{1} = min(s, α^{ℓ}), and s_{j+1} = α^{2ℓ+1}β_{j}s_{j}, for every j ∈ {1, …, k − 1}. Corresponding to this sequence of numbers is a sequence of shells S_{j} = S(β_{j}s_{j}, s_{j}), nested as

S_{k} ≺ S_{k−1} ≺ ⋯ ≺ S_{1} ≺ S(α^{ℓ}, 1).

By our choice of s_{j}, in between any two successive shells S_{j+1} and S_{j}, there are 2ℓ + 1 successive fundamental domains

G_{α^{2ℓ}β_{j}s_{j}} ≺ ⋯ ≺ G_{αβ_{j}s_{j}} ≺ G_{β_{j}s_{j}}.

For each j ∈ {1, …, k} we denote by g_{j} the diffeomorphism constructed at Section 2.2.1 for (D_{0}^{j}, D_{1}^{j}) and the shell S_{j}. Now let g be the map defined as follows:

g(x) = \begin{cases}g_{j}(x) & if x ∈ S_{j}, for some j ∈ \{0, … k\}, \\ A(x) & otherwise. \end{cases}

Then ∥Dg − A∥_{0} < ε. Moreover, one easily proves by induction on j the fact that if x ∈ B^{d} \B(0, s_{1}) and A^{n}(x) is outside the shell S_{1}, or between S_{j+1} and S_{j}, or inside the shell S_{k}, then g^{n}(x) = A^{n}(x). In particular, g is tidy with respect to A and the shell S(β_{k}s, s) where β = α^{(2ℓ+1).k} ∏_{j=1}^{k} β_{j}.

It remains to prove that Λ and Δ satisfy the K-distortion property for g. Recall that Λ ∪ Δ ⊂ S(α^{ℓ}, 1) and that there are 2ℓ + 1 fundamental domains between any two shells S_{j} and S_{j−1}. Hence, for every j ∈ {2, …, k} there exists n_{j} > 0 such that A^{n_{j}}(Λ ∪ Δ) is between the shells S_{j} and S_{j−1}. In particular g^{n_{j}} = A^{n_{j}} on Λ ∪ Δ. We also define n_{1} = 0.

Consider x ∈ Λ and y ∈ Δ . Then there exists j such that y ∈ D_{1}^{j}. By assumption on g_{j} there exists N_{j} ∈ {n_{j}, …, n_{j+1}} such that

|\frac{Jac A^{N_{j}}}{Jac g_{j}^{N_{j}}(y)}| > K.

Claim 1 For every m ∈ {n_{j}, …, N_{j}} one has g^{m} = A^{m} in a neighborhood of x.

Proof. The map g_{j} coincides with A on the complement of the orbit of D_{0}^{j}, which is disjoint from Λ. Hence, as long as A^{m}(x) does not belong to S_{j+1}, one has g^{m} = A^{m} in a neighborhood of x. It remains to show that A^{N_{j}}(x) does not belong to the ball B(0, s_{j+1}). As there are 2ℓ + 1 fundamental domains between the shells S_{j+1} and S_{j}, if A^{N_{j}}(x) ∈ B(0, s_{j+1}), then the point A^{N_{j}}(y) cannot belong to the shell S_{j}. As g_{j} is tidy with respect to S_{j}, this contradicts the fact that Jac g_{j}^{N_{j}}(y) ≠ Jac A^{N_{j}}. This contradiction concludes the proof of the claim. □

As a direct consequence of this claim, we obtain:

|\frac{Jac g^{N_{j}}(x)}{Jac g^{N_{j}}(y)}| > K.

Hence Λ and Δ satisfy the K-distortion property for g.

2.3 The generic linear case, avoiding a codimension 1 or 2 submanifold

The arguments in the previous section do not generalize immediately to the case where the linear contraction A is not conformal, since conjugation by a nonconformal linear map does not preserve the C^{1} norm. Nonetheless, by a small perturbation in C^{d}, one can assume that the linear contraction A is reduced: it has simple spectrum, implying that ℝ^{d} splits into a direct sum of invariant subspaces on which A is conformal. We will denote by F the 1− or 2−dimensional invariant space of A corresponding to the eigenvalues of A of smallest modulus, and we will denote by E the sum of the other eigenspaces.

For such a reduced linear contraction, we can use an inductive argument and the result of Section 2.2. Difficulties arise because dynamical disjointness is not preserved by projectiononto invariant subspaces. In this subsection, we treat another special case of Theorem 1.13: the linear contraction A is a reduced and the sets Λ, Δ avoid the codimension 1 or 2 submanifold E determined by the weakly contracting eigenspaces of A.

We write A = (A_{E}, A_{F}), as a product of linear contractions of E and F: since A has simple spectrum, A_{F} is conformal. We denote by U_{E} and U_{F} the unit balls of the spaces E and F and by π_{F} : ℝ^{d} → F the linear projection of the product E ×F on its second factor. Note that we can change the Euclidean norm on ℝ^{d} and assume that E and F are orthogonal.

2.3.1 The case π_{F} (Λ), π_{F} (Δ) are dynamically disjoint. In this case, since A_{F} is conformal, Proposition 2.5 implies that there is a perturbation g_{F} ∈ C^{dim(F)} of A_{F}, tidy with respect to A_{F} and a shell S_{F} ⊂ U_{F}, so that π_{F} (Λ) and π_{F} (Δ) have the K-distortion property for g_{F}. Then, since the linear map A_{E} has no distortion, the sets Λ and Δ also have the K distortion property for the product map g = (A_{E}, g_{F}). The embedding g is tidy with respect to A and U_{E} × S_{F} and satisfies the conclusions of Theorem 1.13.

2.3.2 Decomposition of Λ and Δ. In general, even though Λ and Δ are dynamically disjoint for A, their projections π_{F} (Λ) and π_{F} (Δ) might not be dynamically disjoint for A_{F}. A naïve way to try to fix the proof is to first perturb A|_{U_{E}×U_{F}} so that the projections π_{F} (Λ) and π_{F} (Δ) become dynamically disjoint. There are two problems with this approach:

1. the projections π_{F} (Λ) and π_{F} (Δ) might in fact coincide, so that a very large perturbation of A would be required to disjoint them;

2. any perturbations that change the relative position of π_{F} (Λ) and π_{F} (Δ) will destroy the invariance of the splitting ℝ^{d} = E ⊕ F in U, not to mention the linear conformality of the projection A_{F}.

To cope with these difficulties, we will cover Λ and Δ with finitely many compact, dynamically-defined pieces Λ^{1}, …, Λ^{k_{1}} and Δ^{1}, …, Δ^{k_{2}}. We perform a sequence of perturbations; at each step we arrange for one of the pairs (Λ^{i}, Δ^{j}) to haven the K-distortion property. Recall that in the conformal case, we chose pieces in our cover of Δ to be round balls; in this case, pieces in our covers of Λ and Δ will be of the form \overline{D × S}, where D is ball in U_{E} and S is a spherical shell in U_{F}.

We introduce more notation. Let α be the norm of the linear conformal contraction A_{F} of F. For r ∈ (0, 1], we denote by G_{r} the shell S(αr, r) ⊂ U_{F}, which is a fundamental domain for the action of A_{F} in U_{F}; its modulus is μ = − log α.

For k ≥ 1, we define a family of spherical shells S_{i,k} ⊂ U_{F}, indexed by integers i ≥ 0, by:

S_{i,k} = S(α^{(i+1)/k}, α^{i/k}).

Notice that S_{i,k} has modulus μ/k, for all i, k, and that G_{1} = ⋃_{i=0}^{k−1} S_{i,k}. Notice also that A_{F}^{j} (S_{i,k}) = S_{i+kj,k}, and so the partition

U_{F} \ {0} = ⋃_{i=0}^{∞} S_{i,k}

is forward invariant under A_{F}.

The proof of Theorem 1.13 in the case we are considering here can be reduced to the following result.

Proposition 2.6 Let A be a reduced linear contraction, ℝ^{d} = E ⊕ F its associated decomposition and a constant ε > 0. Then, there exists k_{0} ∈ ℕ such that the following holds.

For some u, w ∈ U_{E}, δ > 0, k ≥ k_{0} and i, j ∈ ℕ, we consider the sets

Λ = \overline{B_{E}(u, δ) × S_{i,k}} and Δ = \overline{B_{E}(w, δ) × S_{j,k}},

and assume that Λ̂ = \overline{B_{E}(u, 2δ) × S_{i,k}} and Δ̂ = \overline{B_{E}(w, 2δ) × S_{j,k}} are contained in B^{d} \ {0} and are dynamically disjoint.

Then, for any K > 0, and any neighborhood V_{F} of 0 in F, there exists a shell S ⊂ V_{F} and an embedding g: U_{E} × U_{F} → U_{E} × U_{F} such that:

1. ∥Dg − A∥_{0} < ε;

2. g is tidy with respect to A and U_{E} × S;

3. Λ, Δ have the K-distortion property for A.

Let us show how this proposition implies Theorem 1.13. First, Proposition 2.6 associates to ε > 0 an integer k_{0}. Using that Λ, Δ are dynamically disjoint and avoid the space E, one then obtains easily the following construction:

There exist δ > 0 and two families {Λ^{1}, …, Λ^{k_{1}}} and {Δ^{1}, …, Δ^{k_{2}}} with the following properties:

• for every i, j, there exist u_{i}, w_{j} ∈ E and m_{i}, ℓ_{j} ∈ ℕ such that

Λ^{i} = \overline{B_{E}(u_{i}, δ) × S_{m_{i},k_{0}}} and Δ^{j} = \overline{B_{E}(w_{j}, δ) × S_{ℓ_{j},k_{0}}}.

• Λ is contained in ∪_{i=1}^{k_{1}}Λ^{i} and Δ in ∪_{j=1}^{k_{2}}Δ^{j};

• for every i, j, Λ̂^{i} = \overline{B_{E}(u_{i}, 2δ) × S_{m_{i},k_{0}}} and Δ̂^{j} = \overline{B_{E}(w_{j}, 2δ) × S_{ℓ_{j},k_{0}}} are contained in B^{d} \ {0} and are dynamicaly disjoint.

Fix an integer ℓ > 0 such that the shell S(α^{ℓ}, 1) contains π_{F} (Λ ∪ Δ). We order all the possible pairs (Λ^{i}, Δ^{j}) as a list (P_{1}, …, P_{k_{1}k_{2}}) and apply Proposition 2.6 inductively for each pair P_{m}. We obtain a sequence of embeddings g_{m} which are tidy with respect to A and domains U_{E} × S_{m}. Since the shells S_{m} can be chosen in arbitrarily small neighborhoods of 0, one can assume that they are nested as

S_{k_{1}k_{2}} ≺ S_{(k_{1}k_{2})−1} ≺ ⋯ ≺ S_{1} ≺ S(α^{ℓ}, 1),

and that between any two successive shells S_{m+1} and S_{m}, there are 2ℓ + 1 successive fundamental domains.

One can define the contraction g and end the proof of Theorem 1.13 by gluying the g_{m} according to the domains U_{E} × S_{m} as in Section 2.2.3.

2.3.3 Move thin shells. Before proving Proposition 2.6, we need to prove the following lemma about perturbations inside the space F.

Lemma 2.7 Given ε_{0} > 0, there exists k_{0} ∈ ℕ such that, for all k ≥ k_{0}, and for every i ≥ 2k, there is a diffeomorphism ψ : U_{F} → U_{F} such that:

1. ψ(\overline{S}_{i,k}) ∩ (\overline{S}_{i+1,k} ∪ \overline{S}_{i,k} ∪ \overline{S}_{i−1,k}) = ∅;

2. ψ = Id in the complement of a fundamental domain G_{r}, which is a shell containing S_{i+1,k} ∪ \overline{S}_{i,k} ∪ \overline{S}_{i−1,k}; 3. ∥Dψ − Id_{F} ∥_{0} < ε_{0}.

To construct ψ, we will use the following lemma, which will be used again in the next subsection.

Lemma 2.8 Let μ_{0} > 0, s_{0} ≥ 2 and ε_{0} > 0 be given. Then there exists ξ > 0 such that for any collection of conformal linear maps f_{0}, f_{2}, … f_{s_{0}} ∈ D^{d} satisfying

sup_{i=1,...,s_{0}} ∥f_{i} − f_{0}∥ < ξ,

and any collection of spherical shells S^{1} ≺ S^{2} ≺ ⋯ ≺ S^{s_{0}} ⊂ B^{d} of modulus at least μ_{0}, there exists an embedding ψ ∈ D^{d} such that:

1. ∥Dψ − f_{0}∥_{0} < ε_{0},

2. ψ(x) = \begin{cases}f_{0}(x) & if x is inside S_{1}, \\ f_{i}(x) & if x is between S_{i} and S_{i+1}, for i = 1, …, s_{0} − 1, \\ f_{s_{0}}(x) & if x is outside S_{s_{0}}. \end{cases}

Proof. We prove it in the case s_{0} = 1; the general case is obtained similarly. Let λ = e^{μ_{0}} > 1 and consider a smooth function ρ: [0, +∞) → [0, 1] which is 0 on [0, 1] and 1 on [λ, +∞) and whose derivative is bounded by 2/(λ − 1).

Let f_{0}, f_{1} ∈ D^{d} be two conformal linear maps. For r ∈ (0, λ^{−1}), consider ψ ∈ D^{d} defined by x ↦ (1 − ρ(∥x∥/r))f_{0}(x) + ρ(∥x∥/r)f_{1}(x); that is,

ψ(x) = f_{0}(x) + ρ (\frac{∥x∥}{r}) (f_{1}(x) − f_{0}(x)) .

This map coincides with f_{0} on the ball B(0, r) and with f_{1} on the complement of the ball B(0, λr). Since the modulus of S_{1} is equal to μ_{0} = log(λ), one can choose r such that S_{1} = S(r, λr). Moreover notice that

∥Dψ − f_{0}∥_{0} ≤ (1 + \frac{2λ}{λ − 1}) ∥f_{1} − f_{0}∥.

If one sets

ξ = ε_{0} (1 + \frac{2λ}{λ − 1})^{−1}

one thus gets ∥Dψ − f_{0}∥_{0} < ε_{0} and this completes the proof. □

We now give the proof of the first lemma.

Proof of Lemma 2.7. Observe that for any k, the conformal dilation d_{k} : F → F defined by d_{k}(v) = α^{−3/k}v moves the spherical shell S_{i,k} to a shell that is disjoint from the union \overline{S}_{i+1,k} ∪\overline{S}_{i,k} ∪\overline{S}_{i−1,k}. For a given i, k, the map ψ we construct will coincide with d_{k} on the set \overline{S_{i,k}} and with the identity outside of a fundamental domain G_{r} containing S_{i,k} in its interior. We will choose k_{0} so that for k ≥ k_{0}, the distance from d_{k} to the identity is small. To insure that the distance from ψ to the identity does not depend on i, we choose the fundamental domain G_{r} according to the following simple lemma.

Claim 2 For k ≥ 5 and i ≥ 2k, there is a fundamental domain G_{r} ⊂ B(0, α) ⊂ U_{F} such that \overline{S}_{i,k} ⊂ int(G_{r}), and the complement of S_{i,k} in G_{r} is the union of 2 disjoint shells:

G_{r} \ S_{i,k} = S^{a} ∪ S^{b},

where S^{a} ≺ S^{b}, and setting μ = − log α we have

min{m(S^{a}), m(S^{b})} ≥ μ/3.

Proof. Fix k ≥ 5 and let i_{0} be an integer satisfying 4k/3 ≤ i_{0} ≤ 5k/3 − 1. The fundamental domain G_{α} ⊂ U_{F} contains S_{i_{0},k}, and it is easily checked that the two shells S^{a,k} ≺ S^{b,k} defined by

G_{α} \ S_{i_{0},k} = S^{a,k} ∪ S^{b,k}

have modulus at least μ/3. For any i ≥ 2k we have i ≥ i_{0} and the shell S_{i,k} is the image of S_{i_{0},k} under a linear conformal contraction; the images of G_{α}, S^{a,k} and S^{b,k} under this conformal map satisfy the conclusions of the lemma for S_{i,k}. □

Let ξ > 0 be the constant specified by Lemma 2.8, for μ_{0} = μ/3, s_{0} = 2 and ε_{0}. Choose k_{0} ≥ 5 so that

∥d_{k_{0}} − Id∥ = |α^{−3/k_{0}} − 1| < ξ.

Let i ≥ 2k, and let G_{r} be the fundamental domain given by the claim above. The complement of S_{i,k} in G_{r} is the union of two spherical shells S^{a} ≺ S^{b} of modulus at least μ/3. Applying Lemma 2.8 to the shells S^{a} and S^{b} and the maps f_{0} = f_{2} = Id and f_{1} = d_{k_{0}}, we obtain a map ψ that satisfies the conclusions of Lemma 2.7. □

2.3.4 Separate Λ and Δ. We introduce a perturbation g_{1} of A that will make some forward iterates of Λ and Δ have dynamically disjoint π_{F} - projections; this perturbation will take a special form, which will allow us to make the final perturbation of Proposition 2.6 tidy.

Lemma 2.9 Let A be a reduced linear contraction, ε > 0 a constant and Λ, Δ ⊂ B^{d} \ {0} two sets satisfying the assumptions of Proposition 2.6.

Then, for any neighborhood V_{F} of 0 in F, there exists a diffeomorphism g_{1} = A ∘ φ of ℝ^{d}, a shell S_{1} ⊂ V_{F} of F which is a fundamental domain of A_{F}, and an integer ℓ_{1} ≥ 1 such that:

1. the map φ takes the form φ(u, v) = (u, θ(u, v)), for (u, v) ∈ E × F, and is supported in U_{E} × S_{1};

2. max (∥Dφ − Id∥_{0}, ∥Dφ^{−1} − Id∥_{0}) < \frac{ε}{∥A∥}, and thus ∥Dg_{1} − A∥_{0} < ε;

3. g_{1}^{ℓ_{1}}(Λ̂ ∪ Δ̂) is contained in U_{E} × I(S_{1}), and π_{F} (g_{1}^{ℓ_{1}}(Λ̂)) and π_{F} (g_{1}^{ℓ_{1}}(Δ̂)) are dynamically disjoint for the restriction of A_{F} to I(S_{1}).

Proof. Let ε_{0} = \frac{ε}{4∥A∥}; Lemma 2.7 associates to this constant an integer k_{0}. For some integer k ≥ k_{0}, we consider two compact sets

Λ = \overline{B_{E}(u, δ) × S_{i,k}} and Δ = \overline{B_{E}(w, δ) × S_{j,k}}.

The same formula with balls in E of radius 2δ defines the compact sets Λ̂, Δ̂. Recall that these two sets are assumed to be contained in B^{d} \ {0} and to be dynamically disjoint.

Let ν = ∥A_{E}^{−1}∥∥A_{F} ∥; because F is the maximally contracted eigenspace of A, we have ν < 1. Fix a large integer ℓ such that

i + kℓ ≥ 2k and ν^{ℓ} < \frac{δε_{0}}{4} .

If π_{F} (A^{ℓ}(Λ)) and π_{F} (A^{ℓ}(Δ)) are dynamically disjoint for A_{F}, then there is nothing to prove (we choose S_{1} to be any spherical shell, set g_{1} = A and choose ℓ_{1} ≥ ℓ such that g_{1}^{ℓ_{1}}(Λ̂ ∪ Δ̂) is contained in U_{E} × I(S_{1})).

If, on the other hand, π_{F} (A^{ℓ}(Λ)) = \overline{S_{i+kℓ,k}} and π_{F} (A^{ℓ}(Δ)) = \overline{S_{j+kℓ,k}} are not dynamically disjoint for A_{F}, then there exists q_{0} ∈ ℤ such that

A_{F}^{ℓ} (\overline{S_{i,k}}) ∩ A_{F}^{q_{0}} (\overline{S_{j,k}}) ≠ ∅.

This implies that for some κ ∈ {−1, 0, 1} we have

A_{F}^{q_{0}}(\overline{S_{j,k}}) = \overline{S_{i+kℓ+κ,k}}.

Since i + kℓ ≥ 2k, Lemma 2.7 for ε_{0} and the shell S_{i+kℓ,k} provides us with a fundamental domain S_{1} = G_{r} and a diffeomorphism ψ ∈ D^{dim(F)} such that

1. S_{1} ∩ ⋃_{q∈ℕ} A_{F}^{q} (\overline{S_{j,k}}) = A_{F}^{q_{0}}(\overline{S_{j,k}}) = \overline{S_{i+kℓ+κ,k}},

2. ψ(\overline{S}_{i+kℓ,k}) ∩ (\overline{S}_{i+kℓ+1,k} ∪ \overline{S}_{i+kℓ,k} ∪ \overline{S}_{i+kℓ−1,k}) = ∅, 3. ψ = Id_{F} in the complement of S_{1}, and

4. ∥Dψ − Id_{F} ∥_{0} < ε_{0}.

Fix a smooth bump function ρ_{0} : U_{E} → [0, 1] which is 1 on B(u, δ) and 0 outside B(u, 2δ), and whose derivative has a norm bounded by 2δ^{−1}. One then defines ρ : A_{E}^{ℓ}(U_{E}) → [0, 1] and a map θ : A^{ℓ}(Λ̂) → S_{1} by:

ρ(u) = ρ_{0}(A_{E}^{−ℓ}(u)),
θ(u, v) = ρ(u)ψ(v) + (1 − ρ(u))v.

Finally, for (u, v) ∈ A^{ℓ}(Λ̂), let

φ(u, v) = (u, θ(u, v));

we extend this definition by the identity to obtain a diffeomorphism φ ∈ D^{d}. It is supported in A^{ℓ}(Λ̂) ⊂ U_{E} × S_{1} and coincides with Id × ψ on A^{ℓ}(Λ).

Note that the support of ψ is contained in the ball A_{F}^{ℓ} (U_{F}) with radius ∥A_{F} ∥^{ℓ}. Hence, we have

∥Dφ − Id∥_{0} ≤ ∥Dρ∥_{0} ∥ψ − Id_{F} ∥_{0} + ∥ρ∥_{0} ∥Dψ − Id_{F} ∥_{0}
< 2δ^{−1} ∥A_{E}^{−1}∥^{ℓ} · 2∥A_{F} ∥^{ℓ} + ε_{0}
< 4δ^{−1}ν^{ℓ} + ε_{0}
< 2ε_{0} = \frac{ε}{2∥A∥}.

An elementary calculation shows that, for any diffeomorphism h of B^{d}, if ∥Dh − Id∥_{0} ≤ \frac{1}{2}, then ∥Dh^{−1} − Id∥_{0} ≤ 2∥Dh − Id∥_{0}. Hence, we get

∥Dφ^{−1} − Id∥_{0} ≤ 2∥Dφ − Id∥_{0} ≤ \frac{ε}{∥A∥}.  (5)

Let g_{1} = A ∘ φ and choose ℓ_{1} > ℓ so that g_{1}^{ℓ_{1}}(Λ̂ ∪ Δ̂) is contained in U_{E} ×I(S_{1}). Since φ is supported in A^{ℓ}(Λ̂), and since Λ̂ and Δ̂ are dynamically disjoint for A, we have for each k ≥ 0, g_{1}^{k}(Δ) = A^{k}(Δ). Since

S_{1} ∩ ⋃_{q∈ℤ} π_{F} (A^{q}(Δ̂)) = π_{F} (A^{q_{0}}(Δ̂)) ⊂ \overline{S}_{i+kℓ+1,k} ∪ \overline{S}_{i+kℓ,k} ∪ \overline{S}_{i+kℓ−1,k},

and

π_{F} (A^{ℓ}(Λ̂)) = \overline{S}_{i+kℓ,k},

it follows from the properties of ψ that ψ(π_{F} (A^{ℓ}(Λ))) and π_{F} (A^{q_{0}}(Δ)) are dynamically disjoint for A_{F}. Since g_{1} = A on U_{E} × S_{1} and g_{1} = A ∘ φ on A^{ℓ}(Λ̂), this immediately implies that π_{F} (g_{1}^{ℓ_{1}}(Λ) and π_{F} (g_{1}^{ℓ_{1}}(Δ)) are dynamically disjoint for the restriction of A_{F} to I(S_{1}). This completes the proof of Lemma 2.9. □ 2.3.5 Proof of Proposition 2.6. The desired perturbation g of A will be obtained in three steps g_{1}, g_{2}, g_{3}. We first choose an integer k_{0} ∈ ℕ according to Lemma 2.7. We apply Lemma 2.9 and obtain the first perturbation g_{1} supported on a set U_{E} × S_{1} where S_{1} is a shell in F contained in an arbitrarily small neighborhood of 0. This also provides us with two dynamically disjoint sets π_{F} (g_{1}^{ℓ_{1}}(Λ)) and π_{F} (g_{1}^{ℓ_{1}}(Δ)) for the dynamics of A_{F}.

We then apply the argument of Section 2.3.1: fixing a constant K′ > 0, we obtain a perturbation h = (A_{E}, g_{F}) of A and a shell S_{2} ≺ S_{1} of F such that g_{1}^{ℓ_{1}}(Λ) and g_{1}^{ℓ_{1}}(Δ) have the K′-distortion property for h. Moreover, h is tidy with respect to A and U_{E} × S_{2}. We define g_{2} as the map which coincides with g_{1} on B^{1} \ (U_{E} × S_{2}) and with h on U_{E} × S_{2}. If K′ has been chosen large enough, then (Λ, Δ) will have the K-distortion property at some time N > ℓ_{1} for g_{2}.

In our final perturbation, we go from the untidy map g_{2} to a map g_{3} that is tidy with respect to a larger spherical shell containing S_{1} and S_{2}, while at the same time keeping the desired distortion properties of g_{2}. We choose ℓ_{2} > N so that A_{F}^{ℓ_{2}}(\overline{S}_{1}) ⊂ int I(S_{2}). Recall that there exists a diffeomorphism φ supported on U_{E} × S_{1} such that g_{1} = A ∘ φ. We define g_{3} = g_{2} ∘ \overline{φ} where,

\overline{φ} = A^{ℓ_{2}} ∘ φ^{−1} ∘ A^{−ℓ_{2}}.

If one sets S_{3} = A_{F}^{ℓ_{2}}(S_{1}), one sees that the diffeomorphisms g_{3} and g_{2} coincide outside the set U_{E} × S_{3}, which is contained in a small neighborhood of E, if ℓ_{2} has been chosen large enough. In particular, the sets (Λ, Δ) have the K-distortion property for g_{3}.

The maps g_{3} and A coincide outside three domains. On U_{E} × S_{1} and U_{E} × S_{2}, we have ∥Dg_{3} − A∥_{0} < ε. On U_{E} × S_{3}, the claim below gives

∥Dg_{3} − A∥_{0} < ∥A∥.∥D\overline{φ} − Id∥_{0} < ∥A∥.∥Dφ^{−1} − Id∥_{0},

which is less than ε by Lemma 2.9.

Claim 3 We have ∥D\overline{φ} − Id∥_{0} < ∥Dφ^{−1} − Id∥_{0}.

Proof. Since φ^{−1} has the form φ^{−1}(u, v) = (u, \overline{θ}(u, v)), we have,

Dφ^{−1} = ( \begin{matrix}Id_{E} & 0 \\ \overline{θ}_{u} & \overline{θ}_{v} \end{matrix} ) .

This gives

Dφ^{−1} = ( \begin{matrix}Id_{E} & 0 \\ A_{F}^{ℓ_{2}} \overline{θ}_{u}A_{E}^{−ℓ_{2}} & A_{F}^{ℓ_{2}} \overline{θ}_{v}A_{F}^{−ℓ_{2}} \end{matrix} ) .  (6)

Since ∥A_{F} ∥∥A_{E}^{−1}∥ < 1, we have ∥A_{F}^{ℓ_{2}} \overline{θ}_{u}A_{E}^{−ℓ_{2}}∥ ≤ ∥\overline{θ}_{u}∥. Using that A_{F} is conformal and since E, F were assumed to be orthogonal, one gets the announced inequality. □

Let S be the smallest shell in F that contains S_{1}, S_{2} and S_{3}. Note that all these shells can be constructed in any small neighborhood V_{F} of 0 in F. It remains to prove that g = g_{3} is tidy with respect to A and U_{E} × S. Consider a point x ∈ U_{E} × U_{F} and an integer m such that π_{F} (x) is outside S and π_{F} (g_{3}^{m}(x)) is inside S. Since S_{1} and S_{3} are fundamental domains of A_{F}, there exist unique integers i_{1} and i_{3} such that g_{3}^{i_{1}}(x) ∈ U_{E} ×S_{1} and g_{3}^{i_{3}}(x) ∈ U_{E} ×S_{3}; moreover i_{3} = i_{1} + ℓ_{2}. Thus,

g_{3}^{m}(x) = A^{m−i_{1}−ℓ_{2}} \overline{φ}h^{ℓ_{2}}φA^{i_{1}}(x).

Since h is tidy with respect to A and U_{E} × S_{2}, and by definition of \overline{φ}, we have

g_{3}^{m}(x) = A^{m−i_{1}−ℓ_{2}}A^{ℓ_{2}}φ^{−1}A^{−ℓ_{2}}A^{ℓ_{2}}φA^{i_{1}}(x) = A^{m}(x).

This ends the proof of Proposition 2.6.

2.4 The general case

We now prove Theorem 1.13; as we saw in Sections 2.1 and 2.3, we may assume without loss of generality that f is a reduced linear contraction A. The proof will be by induction on the dimension d. Note that the case d = 1 is a direct consequence of Proposition 2.5,

Let us suppose that Theorem 1.13 has been proved in any dimension d′ < d and let A be a reduced linear contraction of ℝ^{d}. As in Section 2.3, one introduces the associated decomposition ℝ^{d} = E ⊕ F. Let Λ, Δ ⊂ B^{d} \ {0} be two compact sets that are dynamically disjoint for A, and fix a constant K > 0. The desired perturbation will be obtained in three steps g_{1}, g_{2}, g_{3}: using the induction hypothesis, we will first obtain distortion between Λ ∩ E and Δ ∩ E; next, we will create the distortion property between Λ ∩ E and Δ \ W and between Λ \ W and Δ ∩ E, where W is a small neighborhood of E in ℝ^{d}; in the last step, we will use the results of Section 2.3 to complete the proof of the theorem, obtaining the distortion between Δ \ W and Λ \ W.

2.4.1 Distortion property on the weak-stable space: use of the induction hypothesis. We first consider the induced dynamics A_{E} of A on E: the compact sets Δ_{E} = Δ ∩ U_{E} and Λ_{E} = Λ ∩ U_{E} are dynamically disjoint for f_{E}.

By the induction hypothesis, there exists a contraction h_{E} of E, arbitrarily close to A_{E} such that Δ_{E} and Λ_{E} have the K-distortion property for h_{E}. Note that one can again perturb h_{E} in a small neighborhood of 0 and assume furthermore that h_{E} coincides with A_{E} near 0.

We define g_{1} = (h_{E}, A_{F}); by continuity, there exist compact neighborhoods Δ̂_{E} and Λ̂_{E} of Δ_{E} and Λ_{E}, respectively, that satisfy the K-distortion property for any embedding close to g_{1}.

2.4.2 Distortion between points in the weak stable space and points not in the weak stable space. Let Λ′ = Λ \ int (Λ̂_{E}) and Δ′ = Δ \ int (Δ̂_{E}). We will obtain the K-distortion property between Λ′ ∪ Δ′ and Δ_{E} ∪ Λ_{E}. Since these sets belongs to disjoint forward invariant regions, the construction will be much easier than in Section 2.3.

Fix a neighborhood V_{1} of 0 that is forward invariant and where g_{1} coincides with A. We choose an integer ℓ_{1} such that g_{1}^{ℓ_{1}}(Λ′ ∪ Δ′) is contained in V_{1}. Working in F, we note that the set Γ = π_{F} (g_{1}^{ℓ_{1}}(Λ′ ∪ Δ′)) is disjoint from a ball B(0, r_{0}). We consider any triple of positive numbers (r_{1}, r_{2}, r_{3}) such that the shells S_{r_{i}} = S(\frac{1}{2}r_{i}, r_{i}) are contained in B(0, r_{0}) and satisfy S_{r_{3}} ≺ S_{r_{2}} ≺ S_{r_{1}}. Their modulus is equal to μ_{0} = log 2. By Lemma 2.8, we associate to μ_{0}, s_{0} = 4 and a small ε_{0}, a constant ξ > 0.

Choose a number β > 1 and conformal dilations f_{1}, f_{2} of F such that

∥f_{1} − A_{F} ∥ < ξ, ∥f_{2} − A_{F} ∥ < ξ, |\frac{det f_{1}}{det A_{F}} | > β, and |\frac{det A_{F}}{det f_{2}} | > β.

Let ψ be the map associated by Lemma 2.7 to the S_{r_{i}}, and the maps f_{0} = f_{3} = A_{F}, f_{1}, f_{2}. Then ψ is a contraction which is close to A_{F} if the constant ε_{0} has been chosen small enough, that coincides with A_{F} outside B(0, r_{1}) and in a neighborhood of 0, with f_{1} between the shells S_{r_{1}} and S_{r_{2}}, and with f_{2} between the shells S_{r_{2}} and S_{r_{3}}. The constants r_{1}, r_{2}, r_{3} are now chosen according to the following property.

Claim 4 For any constant K_{0} > 0, and given any r_{1}, there exists C_{1} > 0 such that, for any r_{2} in (0, r_{1}/C_{1}), and for any r_{3}, the map ψ satisfies

∀z ∈ Γ, ∃n > 0, |\frac{Jac ψ^{n}(z)}{Jac ψ^{n}(0)}| > K_{0}.

Furthermore, given any r_{1}, r_{2}, there exists C_{2} > 0 such that, for any r_{3} in (0, r_{2}/C_{2}), the map ψ satisfies

∀z ∈ Γ, ∃m > 0, |\frac{Jac ψ^{m}(0)}{Jac ψ^{m}(z)}| > K_{0}.

Proof. When \frac{r_{1}}{r_{2}} goes to infinity, the orbits of the points z ∈ Γ spend an interval of times {n_{1}(z), …, n_{2}(z)} between the shells S_{r_{1}} and S_{r_{2}} whose length n_{1}(z) − n_{2}(z) goes to infinity; furthermore, n_{1}(z) does not depend of r_{2}, r_{3} and is uniformly bounded on Γ by some integer n_{1}. At each iteration between S_{r_{1}}

and S_{r_{2}} the distortion |\frac{Jac ψ^{n}(z)}{Jac ψ^{n}(0)} | = |\frac{Jac ψ^{n}(z)}{Jac A_{F}^{n}} | increases by the factor β. Hence

it is enough to choose r_{2} such that n = inf_{z∈Λ} n_{2}(z) − n_{1}(z) satisfies

β^{n} |\frac{Jac ψ^{n_{1}}(z)}{Jac A_{F}^{n_{1}}} | > K_{0}.

The proof of the second part of the claim is analogous. □

We now define g_{2} = (h_{E}, ψ). Then, for any x ∈ g_{2}^{ℓ_{1}}(Λ′ ∪ Δ′) and any y ∈ g_{2}^{ℓ_{1}}(Δ_{E} ∪ Λ_{E}), there exists an integer n > 0 such that

\frac{Jac g_{2}^{n}(x)}{Jac g_{2}^{n}(y)} = \frac{Jac ψ^{n}(π_{F} (x))}{Jac A_{F}^{n} (0)} > K_{0},

and similarly there exists an integer m > 0 such that \frac{Jac g_{2}^{m}(y)}{Jac g_{2}^{m}(x)} > K_{0}. If K_{0} has been chosen large enough (with respect to ℓ_{1}), one deduces that Λ′ ∪ Δ′ and Δ_{E} ∪ Λ_{E} have the K-distortion property for g_{2}.

2.4.3 Distortion between points of Λ, Δ in the complement of the weak stable space. From the two previous steps, we have shown that the pairs (Λ_{E}, Δ) and (Λ, Δ_{E}) satisfy the K-distortion property for any embedding close to g_{2}. Hence there are compact neighborhoods O_{Λ} and O_{Δ} of Λ_{E} and Δ_{E} respectively, such that the pairs (O_{Λ}, Δ) and (Λ, O_{Δ}) satisfy the K-distortion property for g_{2}. Let Λ′′ = Λ \ int (O_{Λ}), and let Δ′′ = Δ \ int (O_{Δ}). All that remains is to create distortion between Λ′′ and Δ′′ for an embedding g_{3} close to g_{2}.

By construction, there exists a neighborhood V_{2} of 0 that is forward invariant and where g_{2} coincides with A. We choose some integer ℓ_{2} such that g_{2}^{ℓ_{2}}(Λ′′ ∪ Δ′′) is contained in V_{2}. It remains to apply Theorem 1.13 as it was proved at Section 2.3, to the map A, the sets g_{2}^{ℓ_{2}}(Λ′′), g_{2}^{ℓ_{2}}(Δ′′) and a constant K_{0} large. We obtain an embedding g = g_{3} such that Λ, Δ have the K-distortion property. The set where g and A differ has been constructed in an arbitrarily small neighborhood of the origin.

3 Large centralizer for a locally C^{1}-dense set of diffeomorphisms

3.1 The case of the circle

Our aim in this subsection is to prove the first part of Theorem 0.7. The following lemma summarizes some very classical properties of diffeomorphisms of the circle.

Lemma 3.1 Let D_{0}^{1} ⊂ Diff ^{1}(S^{1}) be the set of f ∈ D_{0}^{1} 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); and • for every periodic point x ∈ P er(f), there is a neighborhood U_{x} of x such

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).

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

For α > 1 and β ∈ (0, 1), one introduces the set D_{α,β} of orientation preserving C^{∞} diffeomorphism 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.

Now item 1 of Theorem 0.7 is a consequence of the following proposition (see Section 3.1.3).

Proposition 3.2 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].

3.1.1 Mather invariant

We recall here a construction introduced by J. Mather [Ma] which associates to any diffeomorphism f ∈ D_{α,β} a class of diffeomorphism 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) = \frac{ln x}{ln α} for x small and φ(x) = \frac{ln(1−x)}{ln β} for x close to 1: there exists K_{0} > 0 such that φ^{−1}(x) = e^{ln α.x} for x < −K_{0} and φ^{−1}(x) = 1 − e^{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) coincides 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}.

Lemma 3.3 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} coincides 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 1 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 by 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} coincides 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} coincides with the translation t ↦ t+1 on (−∞, −K_{f}], the projection π_{−}∘π_{f} coincides 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 by 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), coincides 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 coincides with ∂/∂x in the neighborhood of ±∞ (because X coincides 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 3.4 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 by rotation”:

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 multiplication, at the right and at the left, by rotations. This class of maps is called the Mather invariant of f.

3.1.2 Vanishing of the Mather invariant: proof of Proposition 3.2

Fix f ∈ D_{α,β} and K_{f} > 0 such that θ_{f} = φ ∘ f ∘ φ^{−1} coincides 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 3.5 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 coincides with f in neighborhoods of 0 and 1 proving that g ∈ D_{α,β}. Furthermore, by construction, one may choose K_{g} = a + 1.

If 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 3.5 one gets:

Corollary 3.6 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 3.7 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 3.8 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 3.9 below. □

Lemma 3.9 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 the 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 one gets that sup_{x∈[0,1]} |D_{x}Θ_{a+n}(ψ) − 1| = sup_{x∈[0,1]} |D_{x}Θ_{a}(ψ) − 1|.

Hence one just has 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 which is the key point of our proof.

Theorem 3.10 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 3.10 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 3.2, we will also use Theorem 3.10 on the torus T^{2}.

Proof of Proposition 3.2. 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 3.8. Using Theorem 3.10, 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 3.8, we obtain that the diffeomorphism

g = f ∘ h_{1} ∘ h_{2} ∘ ⋯ ∘ h_{ℓ}

belongs to U; applying Corollary 3.6, we get that

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

3.1.3 Proof of Theorem 0.7 on the circle

By Lemma 3.1, it is enough to consider f ∈ D_{0}^{1}. 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 k > 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).

Consider a segment I ∈ I. The endpoints of I are the fixed points of the restriction f^{k}|_{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^{k}|_{I} ∘ h_{I}^{−1}.

According to Proposition 3.2, 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} coincides with φ_{I} in a small neighborhood of {0, 1} and is time one map of Y_{I,n}. One denotes g_{I,n} = h_{I}^{−1} ∘ ψ_{I,n} ∘ h_{I}. Notice that the diffeomorphism g_{I,n} coincides with f^{k} in neighborhoods of the endpoints of I and converges to f^{k}|_{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^{k−1}(I) \\ g_{I,n} ∘ f^{−k+1} & on f^{k−1}(I). \end{cases}

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

One denotes by X_{I,n} the vector field, defined on the orbit ⋃_{0}^{k−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, …, k − 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} coincide 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 coincides with X_{I_{i},n} on the orbit of I_{i}, for all i ∈ {1, …, ℓ}.

One easily verifies that X_{n} is a smooth vector field on S^{1}, invariant by f_{n}, and such that f_{n}^{k} is the time one map of X_{n}: the unique difficulty consists in checking the continuity and smoothness of the vector field X at the periodic points. As f_{n} is affine in the neighborhood of the periodic orbits one verifies that, at both sides of a periodic point x, the vector field X is the affine vector field vanishing at x and whose eigenvalue at x is ln D_{x}f.

Finally f_{n} converges to f in the C^{1} topology, completing the proof of Theorem 0.7 on the circle.

3.2 The case of the sphere 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 ^{1} 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.

3.2.1 Preparation of diffeomorphisms in O

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 neigborhoods 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 confomal maps.

1In [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 is wrong). For this reason, we choose here to build in detail the construction of this invariant on the sphere.

Lemma 3.11 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 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}.

3.2.2 Space of orbits of a conformal linear map

Let A, B ∈ GL(ℝ, 2) be two conformal matrices of norm α > 1 and β < 1, respectively. There exist a, b ∈ [0, 2π) such that A = R_{a} ∘ h_{α} and B = R_{b} ∘ h_{β}, where R_{a} and R_{b} are the rotation of angles a and b, respectively, and h_{α} and h_{β} are the homotheties of ratio α and β, respectively. 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}),

and B is the time one map of

X_{B,n} = ln β.(x\frac{∂}{∂x} + y \frac{∂}{∂y}) + (b + 2πn).(x\frac{∂}{∂y} − y \frac{∂}{∂x}).

Notice that 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

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

and X_{A,n} project on T_{A} in 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} which sends Z to ∂/∂x and X_{A,n} to ∂/∂y; this diffeomorphism is unique up to composition by 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}

is isotopic to the map induced by the matrix ( \begin{matrix}1 & n − m \\ 0 & 1 \end{matrix} )).

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).

In the same way T_{B} = ℝ^{2} \ {0}/B is a torus endowed with the vector fields obtained by projection of Z and X_{B,n} and we denote by L_{B,n}: T_{B} → T^{2} a diffeomorphism that sends Z to ∂/∂x and X_{B,n} to ∂/∂y.

3.2.3 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 A in a neighborhood U_{f}^{N} of N and with B in a neighborhood U_{f}^{S} of S. The aim of this part is to build a Mather invariant for diffeomorphisms in D_{A,B}.

We retain the notation of the previous subsection. Consider f ∈ D_{A,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 coincides 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 morphisms 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 3.12 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. 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}.

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

Theorem 3.13 below justifies calling Δ_{f} the Mather invariant of f.

Theorem 3.13 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,S} in a neighborhood of N and X_{f} = X_{B,S} 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)} ∈ D^{N}. One defines Z_{f}(x) = f_{∗}^{−m(x)}(Z(f^{m(x)})) and X_{f}(x) = f_{∗}^{−m(x)}(X_{A,N}(f^{m(x)})). As Z and X_{A,N} 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 coincide with the projections of the restrictions Z and X_{B,S} to D^{S}. Thus Z_{f} = Z and X_{f} = X_{B,S} on D^{S}, ending the proof. □

3.3 Vanishing of the Mather invariant

This part is now very close to the 1-dimensional case.

For any f ∈ D_{A,B,n} we denote by D_{f}^{S} a disk centered on S on which f = B. 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 3.14 With the notation above, the composition 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 3.15 Let D_{0}, …, D_{ℓ} ⊂ D_{f}^{S} be a finite sequence of disks such that

• for every i, 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}.

Observe that 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 3.16 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 by a rotation; the C^{1} norm is preserved by conjugacy by an isometry, and also by conjugacy by a homothety, hence is preserved by the conjugacy by B^{j−i}. □

Corollary 3.17 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 3.18 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 easy 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 0.7 on the sphere S^{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 3.17.

Using Theorem 3.10, 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 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 Lemma 3.11, O contains a dense set of diffeomorphisms smoothy 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 0.7. □

A Conservative extension results

We explain in this appendix how a perturbation of a conservative diffeomorphism along a submanifold W can be extend as a conservative perturbation on the whole manifold M.

This allows to obtain Proposition 1.4 from Proposition 1.9: the results proven in this section will be applied to the case W is an invariant manifold of a hyperbolic periodic point p. In the volume-preserving case, one will 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 A.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. □

A.1 The symplectic case

Proposition A.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 x 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 product, 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). Let us 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 5 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 A.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 A.2, it suffices to prove the following lemma. □

Lemma A.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 A.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}, η ≠ 0, 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 A.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 A.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 A.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 A.3 is now complete. □

A.2 The volume-preserving case

Proposition A.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 A.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 volume-preserving 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, 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