DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 299–307

COMMUTATORS OF C^{r}–DIFFEOMORPHISMS

ACTING ALONG LEAVES

Tomasz Rybicki

Abstract. We study whether a C^{r}-diffeomorphism which is isotopic to the identity and preserves leaves of a foliation can be written as a product of commutators. The result and techniques depend on the class C^{r}, where 0 ≤ r ≤ ∞.

0. Introduction

Let M be an n-dimensional smooth manifold without boundary, and us denote by Diff^{r}(M)_{0} the totality of C^{r}-diffeomorphisms of M which are isotopic to id through a compactly supported isotopy. Specifically Diff^{r}(M)_{0} is the identity component in the C^{r} topology iff M is compact. A well known theorem of M.R.Herman-W.P.Thurston and J.N.Mather states that Diff^{r}(M)_{0} is a simple group, provided 0 ≤ r ≤ ∞ and r ≠ n + 1 (cf. [4], [5], [6], [7], [8]). Due to a theorem of D.B.A.Epstein [3] the Diff^{r}(M)_{0} is simple if and only if it is perfect, that is equal to its commutator subgroup.

We shall deal with nontransitive groups of diffeomorphisms in the simplest case. Given an (M, F), a k-dimensional smooth foliated structure on M, we let Diff^{r}(M, L)_{0} to be the totality of leaf preserving diffeomorphisms isotopic to id through a compactly supported isotopy of leaf preserving diffeomorphisms. The Diff^{r}(M, L)_{0} is (usually) nonsimple for obvious reasons, but the question of its perfectness is still valid and important. Our purpose is to extend the mentioned above results and methods of their proofs to the nontransitive case.

Theorem 1. Diff^{r}(M, L)_{0} is perfect provided 0 ≤ r ≤ k or r = ∞ (in case r = 0 one assumes additionally that Diff^{r}(M, L)_{0} possesses a factorization property (1.1)).

This theorem can be regarded as a starting point to more complicated nontransitive cases, e.g. the group of symplectomorphisms of a Poisson manifold. The proof of Theorem 1 is completely different in particular cases. For r = 0 the proof is a rather simple modification of an argument from [6]. For r = ∞ the proof is long and sophisticated. It depends essentially on Implicit Function Theorem in spaces of smooth mappings which enabled Herman [5] to show that Diff^{∞}(T^{n})_{0} is simple, T^{n} being the n-dimensional torus , and on the homology theory of Kan simplices by means of which the proof has been extended by Thurston from the torus to an arbitrary M (see e.g. [1]). The proof for Diff^{∞}(M, L)_{0} has been presented in [10].

1991 Mathematics Subject Classification. Primary 58D05, Secondary 22E65.

Key words and phrases. Group of diffeomorphisms, identity component, commutator, perfectness of groups, foliation.

This paper is in final form and no version of it will be submitted for publication elsewhere.

Finally in case 0 < r ≤ k our proof consists in a modification of Mather’s argument (cf.[7],[4]). The same argument is valid for the following

Theorem 2. Let (ℝ^{n}, F_{k}) be the product foliation, i.e. F_{k} = ℝ^{k} × {pt}, and let r ≠ n + 1. Then any f ∈ Diff^{r}(ℝ^{n}, L)_{0} can be written as a product of commutators [g, h] = ghg^{−1}h^{−1}, where g (resp. h) is a leaf (resp. foliation) preserving diffeomorphism of (ℝ^{n}, F_{k}).

We restrict ourselves to the product foliation for the only reason that there is no possibility of factorization for foliation preserving diffeomorphisms. To our best abilities there is no extension of Theorem 1 to r > k by means of Mather’s techniques. Theorem 2 implies only a less significant result.

Namely, let Diff^{r}(ℝ^{n}, F_{k}) be the group of all foliation preserving diffeomorphisms of (ℝ^{n}, F_{k}). Then any f ∈ Diff^{r}(ℝ^{n}, F_{k}) is uniquely decomposed as f = gh, where g ∈ Diff^{r}(ℝ^{n−k}) and h ∈ Diff^{r}(ℝ^{n}, L). By Diff^{r}(ℝ^{n}, F_{k})_{0} we denote the totality of f such that g ∈ Diff^{r}(ℝ^{n−k})_{0}, h ∈ Diff^{r}(ℝ^{n}, L)_{0}.

Corollary. Diff^{r}(ℝ^{n}, F_{k})_{0} is a perfect group, provided r ≠ n + 1, r ≠ n − k + 1.

It is a straightforward consequence of the above decomposition.

Remarks. (1) The problem of the simplicity of Diff^{n+1}(M)_{0} (cf. [7,III]) is closely related to the perfectness of Diff^{r}(M, L)_{0}, for r > k + 1. If one would have that such groups are perfect then Diff^{n+1}(M)_{0} would be simple. Indeed, let F ∈ Diff^{n+1}(ℝ^{n})_{0} be sufficiently near the identity. Then denoting

F (x_{1}, x′) = (g(x_{1}, x′), h(x_{1}, x′)),

where x′ = (x_{2}, …, x_{n}), g : ℝ^{n} → ℝ, h : ℝ^{n} → ℝ^{n−1}, we may assume that G : (x_{1}, x′) ↦ (g(x_{1}, x′), x′) is a diffeomorphism. Hence if we denote H(x_{1}, x′) = (x_{1}, h(G^{−1}(x_{1}, x′)) then

F (x_{1}, x′) = (g(x_{1}, x′), h(x_{1}, x′)) = H(g(x_{1}, x′), x′) = H ∘ G(x_{1}, x′),

so that H = F ∘ G^{−1} is a diffeomorphism. Furthermore, both G and H can be written as a product of commutators, and so can be F.

(2) Theorem 1 still holds for C^{0,r}-diffeomorphisms where r > k. By a C^{0,r}- diffeomorphism we mean a leaf preserving homeomorphism which is of class C^{r} when restricted to leaves. The proof follows by a further modification of that of Theorem 1.

Throughout we shall assume the acquaintance of some definitions and constructions from [7, I and II] as we shall quote some part of material from this paper.

COMMUTATORS OF C^{r}–DIFFEOMORPHISMS ACTING ALONG LEAVES 301

1. Preliminaries

Due to the following factorization lemma all our considerations can be reduced to the case of the product foliation (ℝ^{n}, F_{k}).

Lemma 1.1. If f ∈ Diff^{r}(M, L)_{0} is sufficiently near the identity, and supp(f) ⊂ U_{1} ∪ … ∪ U_{r}, where U_{i} open balls, then there is a factorization f = f_{s} ⋯ f_{1} such that supp(f_{j}) ⊂ U_{i(j)} for j = 1, …, s.

The proof for r ≥ 1 is the same as in [9, Lemma 3.1]. In the much more difficult case of homeomorphisms it is not visible how an argument from [2] could be generalized for leaf preserving homeomorphisms.

For U being a finite interval in ℝ^{n} we denote

Diff_{U}^{r}(ℝ^{n}, L)_{0} = {f ∈ Diff^{r}(ℝ^{n}, L)_{0} : ∃f_{t} with supp(f_{t}) ⊂ U},

where f_{t}, t ∈ I, is an isotopy in Diff^{r}(ℝ^{n}, L) joining f with id.

For any map u : ℝ^{n} → ℝ^{n} of class C^{r} we have the sequence of seminorms

∥ u ∥_{s} = sup_{x} ∥ D^{s}u(x) ∥, s = 1, …, r,

where D^{s}u(x) denotes the s-th differential of u at x, and ∥ . ∥ is the usual norm on the space of s-linear mappings between normed vector spaces. These seminorms may be infinite in general. However we restrict ourselves to the space of all C^{r} maps u : ℝ^{n} → ℝ^{n} having support contained in a fixed finite interval U. Then ∥ . ∥_{s} become norms. Notice that the C^{r} topology on this space is completely defined by the norm ∥ . ∥_{r}. Of course ∥ u + id ∥_{s}=∥ u ∥_{s} for s ≥ 2.

We shall follow Mather [7] but we replace the group Diff^{r}(ℝ^{n})_{0} by the leaf preserving one, Diff^{r}(ℝ^{n}, L)_{0}. For simplicity we shall denote

Diff(L)_{0} = Diff^{r}(ℝ^{n}, L)_{0}, Diff_{U}(L)_{0} = Diff_{U}^{r}(ℝ^{n}, L)_{0}.

We begin with some definitions. A modulus of continuity is a continuous strictly increasing ℝ-valued function α defined on the interval [0, ϵ), such that α(tx) ≤ tα(x) for any x ∈ [0, ϵ), t ≥ 1 with tx ∈ [0, ϵ). Now a mapping f : (X, ρ) → (Y, ρ′) between two metric spaces is called α-continuous if there is a constant C > 0 such that

ρ′(f(x), f(y)) ≤ Cα(ρ(x, y)).

DEFINITION. Let α be a modulus of continuity. A mapping u : ℝ^{n} → ℝ^{n} is of class C^{r,α} if it is of class C^{r} and its r-th derivative is locally α-continuous.

We have that sums, products, compositions and C^{1}-inverses of C^{r,α}-mappings are still C^{r,α}.

We shall denote by Diff^{r,α}(ℝ^{n})_{0} the identity component of the group of all compactly supported C^{r,α}-diffeomorphisms, and Diff^{α}(L)_{0} = Diff^{r,α}(ℝ^{n})_{0} ∩ Diff(L)_{0}, Diff_{U}^{α}(L)_{0} = Diff^{r,α}(ℝ^{n})_{0} ∩ Diff_{U}(L)_{0}.

Recall the definition of ∥ . ∥_{α}, the norm on the space of C^{r,α}-mappings

∥ f ∥_{α}= sup \{\frac{∥ D^{r}f(x) − D^{r}f(y) ∥}{α(|x − y|)} : x, y ∈ ℝ^{n}, x ≠ y\}.

The importance of the notion of C^{r,α}-smoothness follows on the one hand from the fact that it enables us to apply Schauder Fixed Point Theorem, and on the other that we have

Proposition 1.2. Diff^{r}(ℝ^{n})_{0} = ⋃_{α} Diff^{r,α}(ℝ^{n})_{0}, and, consequently, we get Diff(L)_{0} = ⋃_{α} Diff^{α}(L)_{0}, where α runs over all the moduli of continuity.

2. Proof of Theorems

Following [7,II] we give the proof of Theorem 1 and Theorem 2 for r ≤ n. The remaining part of Theorem 2 can be shown by making use of [7,I] with rolling-up operators Ψ_{i} instead of unrolling operators P_{i}.

From now on we assume that A is a constant greater than 1. For simplicity we shall omit A in the notations. Denote

D_{i} = [−2A, 2A]^{n−i} × [−2, 2]^{i} ⊂ ℝ^{n}

for i = 1, …, n. Then we have

D_{n} = [−2, 2]^{n} ⊂ D_{n−1} ⊂ ⋯ ⊂ D_{0} = [−2A, 2A]^{n}.

We shall make use of Mather’s operator P_{i} which ”unrolls” diffeomorphisms along the x_{i}-axis, i = 1, …, n. Basic features of the definition will be recalled in the next section in order to show the following useful properties of the operators.

(i) The domain of P_{i} is a C^{1}-neighbourhood of the id in the group of C^{1}- diffeomorphisms compactly supported in D_{i}; the range is in the group of C^{1}- diffeomorphisms compactly supported in D_{i−1}.

(ii) The P_{i} preserve the identity.

(iii) If u ∈ dom(P_{i}) ∩ Diff_{D_{i}}^{α}(L)_{0} then P_{i}(u) ∈ Diff_{D_{i−1}}^{α}(L)_{0}, for i = 1, …, n. (iv) P_{i}|dom(P_{i}) ∩ Diff_{D_{i}}^{α}(L)_{0} is continuous in the C^{r} topology.

(v) For any u ∈ dom(P_{i}) ∩ Diff_{D_{i}}^{α}(L)_{0} we have

[P_{i}(u)] = [u] in the abelianization of Diff^{α}(L)_{0}

for i = n − k + 1, …, n.

(vi) For any u ∈ dom(P_{i}) ∩ Diff_{D_{i}}^{α}(L)_{0} we have

[P_{i}(u)] = [u] in the abelianization of Diff^{α}(ℝ^{n}, F_{k})_{0}

for i = 1, …, n − k.

(vii) There are δ > 0 and independent of A a large constant C > 0 such that

∥ P_{i}(u)) ∥_{α}≤ CA^{−1} ∥ u ∥_{α}

whenever u ∈ dom(P_{i}) ∩ Diff^{α}(L)_{0} and ∥ u ∥_{α}< δ for i = 1, …, n.

Choose arbitrarily f ∈ Diff(L)_{0}. Without loss of generality we may assume that f ∈ Diff_{D_{n−k}}^{α}(L)_{0}, where α is some modulus of continuity, and ∥ f ∥_{α}< ϵ, where ϵ > 0 is small. Now we may fix the constants A and ϵ such that

3C^{k} < A^{1+k−r}, 3ϵA^{r−1} < δ,

where C, δ are as in (vii).

We set

C = {u − id ∈ Diff_{D_{n−k}}^{α}(L)_{0} − id : ∥ u ∥_{α}≤ ϵ}.

Then C is convex and, due to Theorem of Ascoli, compact in the normed space of all leaf preserving C^{r}-mappings on ℝ^{n} compactly supported in D_{n−k}. Let us define a C^{r}-continuous operator Φ : C → C in the following way.

Let Ã_{1} ∈ Diff^{α}(L)_{0} be a diffeomorphism which satisfies Ã_{1}(x′, x′′) = (Ax′, x′′) on D_{0}, where x′ = (x_{1}, …, x_{k}), x′′ = (x_{k+1}, …, x_{n}). Let Ã_{2} ∈ Diff^{r,α}(ℝ^{n}, F_{k})_{0} verify Ã_{2}(x′, x′′) = (x′, Ax′′) on D_{0}. We set for u − id ∈ C

u_{0} = Ã_{1}^{−1}fuÃ_{1}

and

u_{1} = P_{n}(u_{0}), u_{2} = P_{n−1}(u_{1}), …, u_{k} = P_{n−k+1}(u_{k−1}).

Define

Φ(u − id) = u_{k} − id.

By (iii) we have u_{k} ∈ Diff_{D_{n−k}}^{α}(L)_{0}. We wish to show Φ(C) ⊂ C.

It follows from the definition of ∥ . ∥_{α} that

∥ u_{0} ∥_{α}≤ A^{r−1} ∥ fu ∥_{α}≤ 3ϵA^{r−1}

as ∥ fu ∥_{α}< 3ϵ (cf. [7]). By making use of (vii) we get

∥ u_{1} ∥_{α} ≤ CA^{−1} ∥ u_{0} ∥_{α},
…
∥ u_{k} ∥_{α} ≤ CA^{−1} ∥ u_{k−1} ∥_{α} .

Therefore

∥ u_{k} ∥_{α}≤ C^{k}A^{−k} ∥ u_{0} ∥_{α}≤ 3ϵC^{k}A^{r−k−1} < ϵ,

as required.

Now Theorem of Schauder states that there exists a fixed ”point” of Φ, that is u_{k} = u for some u − id ∈ C. Hence the definition of u_{0} and (v) imply that [fu] = [u], and consequently [f] = e, in the abelianization of Diff^{α}(L)_{0}. This completes the proof of Theorem 1.

In order to have Theorem 2 for r ≤ n one changes some details. The interval D_{n−k} is replaced by D_{0}, one puts u_{0} = Ã_{1}^{−1}Ã_{2}^{−1}fuÃ_{2}Ã_{1}, and u_{i+1} = P_{n−i}(u_{i}), i = 1, …, n, that is diffeomorphisms are unrolled n times. Finally we use both (v) and (vi).

3. Properties of the operators P_{i}

First we give an outline of the construction of P_{i}. We write C_{i} for ℝ^{i−1} × S^{1} × ℝ^{n−i}, i.e. C_{i} is the set of n-tuples (x_{1}, …, ϑ_{i}, …, x_{n}) where ϑ_{i} is a real defined mod 1. Let π : ℝ^{n} → C_{i} be the canonical projection. Notice that there is the obvious action of S^{1} on C_{i}, and denote by G_{i} the group of equivariant diffeomorphisms with respect to this action.

Let u ∈ Diff_{D_{i}}^{r,α}(n + 1)_{0} and ∥ u − id ∥_{1}< \frac{1}{2}. We define a diffeomorphism Γ_{i}(u) : C_{i} → C_{i} in the following way. For ϑ ∈ C_{i} we choose x ∈ ℝ^{n} such that π(x) = ϑ and x_{i} < −2. Next we choose an integer N > 0 so large that

(T_{i}u)^{N}(x)_{i} > 2.

Here T_{i} : ℝ^{n} → ℝ^{n} denotes the one time transformation of the vector field ∂_{i} = ∂/∂x_{i}, that is the unit translation along the x_{i}-axis. It is visible that N = 10 is sufficient. Then one defines

Γ_{i}(u)(ϑ) = π(T_{i}u)^{N}(x).

In particular, if |x_{j}| > 2A for some j ≠ i then Γ_{i}(u)(ϑ) = ϑ. One can also check that the operation u ↦ Γ_{i}(u) is continuous.

For Γ_{i}(u) sufficiently near id there is a unique S^{1}-equivariant diffeomorphism g : C_{i} → C_{i} such that

g | {ϑ_{i} = 0} = Γ_{i}(u) | {ϑ_{i} = 0}.

Let h = g^{−1}Γ_{i}(u). Next let h̃ be the unique diffeomorphism of ℝ^{n} verifying πh̃ = hπ and h̃ = id on {x_{i} = 0}.

Denoting by B a sufficiently large positive integer, and assuming that h̃ (and u) is C^{1}-close to id one defines diffeomorphisms h̃_{1}, …, h̃_{B} of ℝ^{n} as follows

h̃_{1} − id = \frac{1}{B} (h̃ − id),
h̃_{2} − id = \frac{1}{B − 1}(h̃h̃_{1}^{−1} − id),
⋯
h̃_{B−1} − id = \frac{1}{2}(h̃h̃_{1}^{−1} … h̃_{B−2}^{−1} − id),
h̃_{B} − id = h̃h̃_{1}^{−1} … h̃_{B−1}^{−1} − id,

so that h̃ = h̃_{B}h̃_{B−1} … h̃_{1}. (This decomposition enables us to get (vii).)

Now one chooses a function ζ ∈ C^{∞}(ℝ) which takes values in [0,1], is periodic of period 1, equal to 1 in a neighbourhood of the integers, and equal to 0 in a neighbourhood of the half-integers. We define h̃_{jk} where j = 1, …, B, k = 0, 1, by the equalities

h̃_{j0} − id = ζ(x_{i})(h̃_{j} − id) and h̃_{j1} = h̃_{j}h̃_{j0}^{−1}.

Thus we get a decomposition h̃ = h̃_{B1}h̃_{B0} … h̃_{1,1}h̃_{1,0}.

Next one defines a sequence of strips ”regularly” located in the wide strip |x_{i}| < 2A as follows. Letting a be the least half-integer ≥ −2A one denotes

\begin{aligned}E_{−}^{j} & = \{x ∈ ℝ^{n} : a + 3j − 3 ≤ x_{i} ≤ a + 3j − 2\}, \\ E_{+}^{j} & = \{x ∈ ℝ^{n} : a + 3j − 3/2 ≤ x_{i} ≤ a + 3j − 1/2\}. \end{aligned}

Then B is defined as the greatest integer such that a + 3B − 1/2 ≤ 2A. Finally one defines v = P_{i}(u) as a unique diffeomorphism of ℝ^{n} satisfying

v|E_{−}^{j} = h̃_{j0}|E_{−}^{j}, v|E_{+}^{j} = h̃_{j1}|E_{+}^{j}

and

v|(ℝ^{n} − ⋃_{j} E_{−}^{j} ∪ E_{+}^{j}) = id.

This terminates the construction of P_{i}.

Remark. The clue property of Mather’s construction is that

Γ_{i}(P_{i}(u))Γ_{i}(u)^{−1} ∈ G_{i}.

This fact enables to obtain suitable conjugate relations (Proposition 3.2).

Proposition 3.1. If u ∈ dom(P_{i}), i = 1, …, n, and u preserves the leaves of F then so does P_{i}(u).

Proof consists in looking closely at the definition of P_{i}. In both cases: i ≤ k, and i > k, successive steps are ”leaf preserving” (one considers also the induced foliation on C_{i}). In case i ≤ k the (T_{i}u)^{N} is leaf preserving as T_{i} and u are so, and so is Γ_{i}(u). In case i > k the Γ_{i}(u) takes the form Γ_{i}(u)(x′, θ′) = (u_{1}(x′), θ′) for any θ = (x′, θ′) ∈ C_{n}, where θ′ = (x_{k+1}, …, θ_{i}, …, x_{n}). This follows from the fact that T_{i} is the unit translation and θ′ is mod 1. Hence Γ_{i}(u) is leaf preserving. It is even more clear that the remaining steps are also ”leaf preserving” (cf.[7,II]).

Choose a function ρ ∈ C^{∞}(ℝ) such that ρ = 1 on [−2A, 2A] and supp(ρ) ⊂ [−2A − 1, 2A + 1]. Next, abusing the notation, we define ρ ∈ C^{∞}(ℝ^{n}) by ρ(x) = ρ(x_{1}) … ρ(x_{n}), and ρ ∈ C^{∞}(ℝ^{n−k}) by ρ(x′′) = ρ(x_{k+1}) … ρ(x_{n}) where x′′ = (x_{k+1}, …, x_{n}). We set

\begin{aligned}τ_{i}(x) & = exp(ρ(x)∂_{i}), i = 1, …, k, \\ τ_{i}(x) & = exp(ρ(x′′)∂_{i}), i = k + 1, …, n. \end{aligned}

Then τ_{i} ∈ Diff^{α}(L)_{0}, i ≤ k and τ_{i} ∈ Diff^{r,α}(ℝ^{n}, F_{k})_{0}, i > k.

Next let ϕ_{i}, i = 1, …, n, be diffeomorphisms characterized by the following properties: (a) dom(ϕ_{i}) = {x ∈ ℝ^{n} : |x_{j}| ≤ 2A, i ≠ j}, (b) ϕ_{i∗}(∂_{i}) = ρ∂_{i}, (c) ϕ|D_{0} = id, (d) in case i ≤ k the ϕ_{i} is leaf preserving, (e) in case i > k the ϕ_{i} is foliation preserving.

Note that

(1)  τ_{i} = ϕ_{i}T_{i}ϕ_{i}^{−1}, i = 1, …, n,

since T_{i} = exp(∂_{i}), τ_{i} = exp(ρ∂_{i}).

The following is a modification of [7,&4], and we reproduce only a sketch of the proof.

Proposition 3.2. Let i = 1, …, n and let u, v ∈ Diff_{D_{0}}^{α}(L)_{0}. If u and v are sufficiently C^{1}-close to the identity and Γ_{i}(v)Γ_{i}(u)^{−1} ∈ G_{i} then λτ_{i}uλ^{−1} = τ_{i}v where λ ∈ Diff^{α}(L)_{0}.

Proof. Following [7] we put

Λ(x) = (T_{i}v)^{N}(T_{i}u)^{−N}(x),

where N is a positive integer so large that (T_{i}u)^{−N}(x)_{i} < 2A. For any x ∈ ℝ^{n} such an N exists, and the definition is independent of N (large). Observe that suppΛ ⊂ {x ∈ ℝ^{n} : x_{i} > −2A, |x_{j}| < 2A, j ≠ i}. It follows from the definition

(2)  ΛT_{i}uΛ^{−1} = T_{i}v

and

(3)  Γ_{i}(v)Γ_{i}(u)^{−1}π = πΛ.

Furthermore Λ is a leaf preserving diffeomorphism, that is

(4)  Λ(x)_{i} = x_{i} , i > k.

In fact, in case i ≤ k, T_{i}, u, v are leaf preserving, and so is Λ. In case i > k the factor (T_{i}u)^{−N} sends the leaf x_{j} = a_{j} to the leaf x_{j} = a_{j} − N_{j}, where N_{j} = N for j = i and N_{j} = 0 for j ≠ i, which is sent back by (T_{i}v)^{N} to x_{j} = a_{j}. It is then clear that Λ changes nothing in the transversal directions.

The assumption Γ_{i}(v)Γ_{i}(u)^{−1} ∈ G_{i} and (3) yield the existence of Λ_{j} : ℝ^{n} → ℝ, j ≠ i, satisfying

Λ_{j}(x′_{i}) = Λ(x)_{j}

for x_{i} > 2A, where x′_{i} = (x_{1}, …, x̂_{i}, …, x_{n}). Next Λ′(x′_{i}) = (Λ_{1}(x′_{i}), …, Λ_{n}(x′_{i})), and let Λ′_{t} be a leaf preserving isotopy from Λ′ to id supported in the set {|x_{j}| < 2A, j ≠ i}.

Now the definition of λ is the following (in the second and third line the i-th coordinate is written last)

λ(x) = ϕ_{i}Λϕ_{i}^{−1}, x ∈ im(ϕ_{i})

= (Λ′(x′_{i}), exp(ρ̃(Λ(x)_{i} − x_{i})∂_{i})(x)_{i}), 2A ≤ x_{i} ≤ 2A + 2

= (Λ′_{t}(x′_{i}), x_{i}), 2A + 2 ≤ x_{i} ≤ 2A + 3

= x otherwise.

It is easy to see that λ is leaf preserving. Indeed, the r.h.s. of the first line is so as Λ and ϕ_{i}, i ≤ k, is leaf preserving, and ϕ_{i}, , i > k is foliation preserving. The r.h.s. of the second line is leaf preserving because of (4). Finally the r.h.s. of the third line is leaf preserving because of the assumption on Λ′_{t}.

The proof is concluded by the observation that conjugating by ϕ_{i} the equality (2) and making use of (1) imply the desired equality λτ_{i}uλ^{−1} = τ_{i}v.

Now it is visible that the properties (i-vii) hold. In fact, (i),(ii),(iv) follows by definition. Proposition 3.1 coincides with (iii), and Proposition 3.2 implies (v),(vi) as τ_{i} ∈ Diff^{α}(L)_{0} for i ≤ k, and τ_{i} ∈ Diff^{r,α}(ℝ^{n}, F_{k})_{0} for i > k. For the proof of (vii) see [7,&4].

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Institute of Mathematics, Pedagogical University ul.Rejtana 16 A, 35-310 Rzeszów, POLAND

E-mail : rybicki@im.uj.edu.pl