J Geom Anal (2009) 19: 963–993 DOI 10.1007/s12220-009-9093-8

Bi-Lipschitz Sufficiency of Jets Guillaume Valette

Received: 8 October 2007 / Published online: 1 July 2009 © Mathematica Josephina, Inc. 2009

Abstract We give some theorems of bi-Lipschitz or C 1 sufficiency of jets which are expressed by means of transversality with respect to some strata of a stratification satisfying the (L) condition of T. Mostowski. This enables us to prove that the number of metric types of intersection of smooth transversals to a stratum of an (a) regular stratification of a subanalytic set is finite. Keywords Stratified sets · Sufficiency of jets · Bi-Lipschitz equivalence Mathematics Subject Classification (2000) 32C05 · 58A20 · 58A35 1 Introduction In this paper we study the problem of sufficiency of jets. This is a classical problem of singularity theory. The question is to determine whether a function germ is characterized (up to some equivalence relation) by its Taylor expansion at the origin. Many authors have given versatile criteria during the three last decades [2, 4–10, 13, 14, 18, 19]. We give here several results about sufficiency of jets. We study the intersection of a stratified space with a transversal to a given stratum, the zero locus of maps and the maps themselves up to a homeomorphism. We focus on the bi-Lipschitz or C 1 equivalence in each case. This paper is partially supported by the Polish KBN grant 0235/P03/2004/27. G. Valette () Department of Mathematics, University of Toronto, 40 St. George street, Toronto, ON M5S 2E4, Canada e-mail: [email protected] G. Valette Instytut Matematyczny PAN, ul. Sw. Tomasza 30, 31-027 Kraków, Poland

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Our criteria for sufficiency of jets are proved by studying stratifications satisfying the (L) condition of T. Mostowski. This condition has been introduced to obtain the bi-Lipschitz triviality along the strata. Existence of stratifications satisfying this condition has been proved for complex analytic or real subanalytic sets [11, 12]. It is important to note that for determinacy of transversal (Sect. 5) we will not assume that the given stratum is a stratum of an (L) regular stratification (which would be a rather strong assumption). We will just fix an (L) regular stratification which is compatible with our given stratum. Actually such a stratification always exists for a given subanalytic set X. Then, we will put transversality conditions with respect to this stratification generalizing the results of [14]. In the two strata case we will get some results of C 1 determinacy. It is interesting to note that these theorems will apply not only on manifolds but also on the more general class of spaces having an isolated singularity at the origin. Moreover the criteria obtained are better than the ones given in [18] or [13] in the sense that the order of determinacy is lower. We end this paper by two further applications of the results of Sect. 4. The first one is a finiteness result which is also a generalization of a result of [14]. We prove that the number of Lipschitz types of intersection of smooth direct transversals at a given point is finite when the stratification satisfies the Whitney (a) condition (Theorem 7.1.1). The second one is about Kuo and Trotman’s blowing up. This is a transformation of stratified spaces introduced by T.C. Kuo and D. Trotman which is known to improve the regularity. In [14] the authors provide explicit criteria using the (t) condition about stratifications. This condition has been introduced by R. Thom and is weaker than the classical ones (Whitney and Kuo–Verdier). It does not guarantee the topological triviality along the strata. It is proved in [4, 14, 19] that the (t) condition may induce the (w) condition of Kuo–Verdier. This observation is then used by the authors to prove some determinacy theorems for transversals to a stratum of a stratification. We emphasize in this paper that Lipschitz type conditions on stratifications have similar properties. The paper is organized as follows. In the first section we introduce definitions, notations and show some basic results. In the second we study Lipschitz stratifications. In Sect. 4, after recalling the notion of pull-back through a deformation of transversals we show our pull-back theorems for Lipschitz type conditions. These theorems enable us to state all the determinacy theorems of Sects. 5 and 6. In Sect. 5 we deal with bi-Lipschitz sufficiency of jets and in Sect. 6 we focus on C 1 sufficiency of jets. Notations Throughout this paper n and m will be two fixed integers, d = m + n, Y will denote {0Rn } × Rm , N will denote Rn × {0Rm } so that Rd = N + Y . We denote by π the orthogonal projection onto Y . d dGivenl q ∈ R , lwe will write d(q; Y ) for the distance from q to Y . Let dGd = l=m Gd where Gd is the Grassmannian of vector spaces of dimension l in R .

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Given a vector subspace A of Rd we will denote by πA the orthogonal projection on A. We denote by Z+ the set of integers with the symbol + as index (i.e. a+ with a ∈ Z). We also fix a subanalytic subset X of Rd . All considered stratifications will be assumed to be subanalytic. The letters i and j will stand for two elements of Z satisfying i ≤ j . We extend to Z ∪ Z+ the order of Z as follows. Let k and k  be two integers, then  if k ≤ k  , k < k  if k < k  , and k < k . We also extend the by convention k+ ≤ k+ + + addition by setting:  k + k+ = (k + k  )+  k+ + k+ = (k + k  )+ .

Let f and g be two functions defined on a subset A of Rd . We will write f  g whenever there exist a strictly positive constant C and a neighborhood U of the origin such that f (q) ≤ Cg(q) for all q ∈ U ∩ A. We will write f (q)  g(q)k+ if there exists a function ψ defined at each point where f and g are defined, and a neighborhood U of the origin such that f (q) ≤ g(q)k ψ(q) with ψ(q) tending to 0 at the origin. Let k, k  ∈ Z ∪ Z+ . With the above notations if |f (q)|  d(q; Y )k and |g(q)|    d(q; Y )k then |f (q).g(q)|  d(q; Y )k+k .

2 Definitions and Preliminaries Definition 2.0.1 A direct transversal to Y is the germ of a mapping v : N → Y , at least C 2 satisfying v(0) = 0. We will denote by v its graph.

Definition 2.0.2 We call a horn neighborhood of order j of v a neighborhood of type: H (v; j ) = {q = (x; t) ∈ Rn × Rm : |vs (x) − ts | ≤ C|x|j , s = 1, . . . , m}, where C is a positive constant.

Definition 2.0.3 Let A be a subset of Rd containing Y and let k be an element of Z ∪ Z+ . A function w : A → R is called rugose with exponent k if for any point q of A: |w(q) − w(π(q))|  d(q; Y )k .

(2.1)

It is called Lipschitz with exponent k if it is rugose with exponent k + 1 and if for any 1 couple (q; q  ) ∈ A × A such that |q − q  | ≤ 2c d(q; Y ) (for some c > 1): |w(q) − w(q  ))|  d(q; Y )k |q − q  |.

(2.2)

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Remark 1 (1) If the function is constant on Y we may omit the hypothesis “rugose with exponent k + 1” in the above definition as long as we withdraw the restriction on the couples (q; q  ) and we change the inequality for: |w(q) − w(q  ))|  θ (q; q  )k |q − q  |,

(2.3)

where θ (q; q  ) = max(d(q; Y ); d(q  ; Y )). (2) A mapping which is Lipschitz with exponent 0+ and C 1 outside Y is C 1 in the sense of Whitney. We recall a classical result about partitions of unity (see for instance [3] (3.1.13)). Proposition 2.0.4 There exists a positive constant C such that for any family (Ul )l∈L of open subsets of Rd there exists a C ∞ partition  of unity (αl )l∈L whose supports form a locally finite covering of the covering U (of l∈L Ul ), and whose derivatives satisfy: |dx αl | ≤

C , h(x)

where h(x) = sup{min{1, d(x; Rd \ Ul )}/ l ∈ L}. Proposition 2.0.5 Let X be a closed subanalytic subset of Rd containing Y and let k ≤ i. Let w0 ∈ R and let w be a function defined on X, rugose with exponent i + 1 and Lipschitz with exponent k such that w|Y ≡ w0 . Then w may be extended to the whole of Rd into a function still rugose with exponent i + 1 and Lipschitz with exponent k. Proof First by Remark 1, (2.2) is true with no restriction on (q; q  ) as long as we replace d(q; Y ) by θ (q; q  ). We fix a constant C such that for any couple (q; q  ) ∈ X × X the function w satisfies: |w(q) − w(q  )| ≤ C|q − q  |θ (q; q  )k . Let Ap = {q ∈ Rd / 3.21 p < d(q; Y ) < Ap ∩ X × X

1 2p }.

(2.4)

By (2.4), for any couple (q; q  ) in Ap ×

|q − q  | . (2.5) 2kp Thus, using Banach’s extension theorem (cf. [1]), we can extend w|Ap ∩X to the whole of Ap , to a function 2Ckp Lipschitz. Let wˆ be this extension. The function wˆ satisfies (2.5). Hence for any couple of points (q; q  ) in Ap × Ap : |w(q) − w(q  )| ≤ C

|w(q) ˆ − w(q ˆ  )| ≤ 3C|q − q  |θ (q; q  )k . We define the following subsets of Ap :  Bp = q ∈ Ap /d(q; Ap ∩ X) ≤

2 3.2p(i−k+1)

(2.6) 

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and

967

 Cp = q ∈ Ap /d(q; Ap ∩ X) ≥

 . p(i−k+1) 1

3.2

By the preceding proposition there exists a partition of unity (α; 1 − α) adapted C to the covering of Ap given by (Bp ; Cp ) which satisfies |dx α| ≤ h(x) where h(x) 1 is the function defined in the Proposition 2.0.4. But since h(q) ≥ 3.2p(i−k+1) we get p(i−k+1) |dx α| ≤ 3C2 for all x in Ap . Then by the Mean Value Theorem there exists a positive constant C (independent of p) such that:

|α(q) − α(q  )| ≤ C2p(i−k+1) |q − q  | ≤ 3Cd(q; Y )k−i−1 |q − q  | for any couple (q; q  ) ∈ Ap × Ap . Moreover by definition if q ∈ Bp then d(q; Ap ∩ X) ≤ since q belongs to Ap :

2 , 3.2p(i−k+1)

which implies

d(q; Ap ∩ X) ≤ 2d(q; Y )i−k+1 .

(2.7)

ˆ + (1 − α(q))w0 . On Bp , the function wˆ is rugose We now set wp (q) := α(q)w(q) of exponent i + 1. Given a point q of Bp , let q0 be the point which realizes the distance to Ap ∩ X. Then: d(q0 ; Y ) ≤ d(q; Y ) + |q − q0 | ≤ d(q; Y ) + 2d(q; Y )i−k+1

(by (2.7))

≤ 3d(q; Y ) for k ≤ i.

(2.8)

Then we get using (2.6): |w(q) ˆ − w0 | ≤ |w(q) ˆ − w(q0 )| + |w(q0 ) − w0 | ≤ C|q − q0 |θ (q; q0 )k + C(.q0 ; Y )i+1 ≤ C|q − q0 |d(q; Y )k + Cd(q0 ; Y )i+1

(by (2.8))

≤ d(q; Ap ∩ X)d(q; Y )k + Cd(q; Y )i+1 ≤ d(q; Y )i+1

(by (2.7)).

In consequence wp is also a function rugose with exponent i + 1. Let us show that wp is a function Lipschitz with exponent k. Let (q; q  ) ∈ Ap × Ap , we have: |wp (q) − wp (q  )| ≤ |α(q)w(q) ˆ − α(q  )w(q ˆ  )| + |(α(q) − α(q  ))w0 | ˆ − w0 )| ≤ |α(q)(w(q) ˆ − w0 ) − α(q  )(w(q) ˆ − w0 | + |α(q  )||w(q) ˆ − w(q ˆ  )| ≤ |α(q) − α(q  )||w(q) ≤ C|q − q  |d(q; Y )k−i−1 d(q; Y )i+1 + |q − q  |d(q; Y )k ≤ |q − q  |d(q; Y )k .

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This proves that w  is Lipschitz with exponent k on Ap . Again applying Proposition 2.0.4 we find a partition of unity (φp )p∈N adapted to  the covering Rd \ Y by the Ap ’s. Then as d(Ap \ q=p Aq ; Rd \ Ap ) ≥ 3.21 p for any 3C q ∈ Ap we have: |dq φp | ≤ 3C.2p . This implies that |dq φp | ≤ d(q;Y ) for q ∈ Ap . For q∈ / Ap this is still true since dφp (q) = 0. This implies by the Mean Value Theorem: |φp (q) − φp (q  )| 

|q − q  | d(q; Y )

(2.9)

1 for any couple (q; q  ) ∈ Ap × Ap satisfying |q − q  | ≤ 2c d(q; Y ). We thus set w(q) = p∈N φp (q)wp (q). As all the wp ’s are rugose with exponent i + 1, so is the function w. It remains to see that w is Lipschitz with exponent k. Let q and q  be two points of Rd . First remark that each point of Rd belongs to at most two of the Ap ’s and consequently, in the sum defining w, at most two terms are nonzero at each point q. 1 d(q; Y ). Then Let (q; q  ) ∈ R2d satisfying |q − q  | ≤ 2c

       φp (q)(wp (q) − w0 ) − φp (q  )(wp (q  ) − w0 ) |w(q) − w(q  )| =  p∈N   p∈N

      ≤ |φp (q) − φp (q  )|(wp (q) − w0 ) + φp (q  )|wp (q) − wp (q  )|   p∈N

≤ 4C

|q − q  | .d(q; Y )i+1 + 4C|q − q  |d(q; Y )k d(q; Y )

≤ 4C|q − q  |d(q; Y )k

for i ≥ k.

For i ∈ Z+ or k ∈ Z+ the proof is analogous.



Note that the Lipschitz constants of the extension may be expressed as a function of the constant initially given and the ratio may be bounded by some constants depending only on the dimension of the ambient space. Remark 2 The extension w, ˆ rugose with exponent i + 1 and Lipschitz with exponent k, may fulfill as well: (1) wˆ is C ∞ on Rn \ X. (2) Let (Wl )l∈L be a collection of open disjoint manifold of X, such that w is differentiable at any point of Wl for any l. Then wˆ may be chosen differentiable at any point of Wl in Rd for all l. To prove it, it is enough to replace in the above proof the extension provided by Banach’s Theorem, by an extension which is C ∞ in the complement of X. Actually we can make it differentiable by considering its convolution with a smooth map.

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2.1 Lipschitz Distributions Let X be a subset of Rd . A l-distribution of Rd over X will be a continuous distribution  : X → Gld . We need a definition about distributions. Given a direct transversal v, we denote by Dv the m-distribution generated by the vectors (∂x vs − ∂s ), s = 1, . . . , m, where v = (v1 ; . . . ; vm ). Definition 2.1.1 Let  : X → Gld be an l-distribution. Then  is said to be rugose with exponent i if |(Pq⊥ )|Y |  d(q; Y )i where Pq denotes the orthogonal projection on (q) and |.| is the norm of linear maps. It is said to be Lipschitz with exponent k if it is rugose with exponent k + 1 and if 1 for any couple (q; q  ) ∈ X × X such that |q − q  | ≤ 2c d(q; Y ), we have: |Pq − Pq  | ≤ C|q − q  |d(q; Y )k . It is important to note that the integer k may be negative. Note that the case k = 0+ corresponds to distributions that extend to C 1 distributions to the ambient space if the distribution is C 1 on its domain. This is a consequence of the Whitney extension theorem. This will enable us to provide theorems for C 1 sufficiency of jets. It is well known that Lipschitz vector fields are integrable and generate a biLipschitz one parameter group. Clearly a distribution is Lipschitz (with exponent 0) if and only if it admits a family of Lipschitz sections ζ1 , . . . , ζm which induce a basis of Y at each point of Y . The one parameter groups generated by this family of vector fields give rise to an isotopy which induce a trivialization of X along Y (considering the variables of Y as parameters). So we can state: Lemma 2.1.2 Let X be a closed subset stratified by a stratification S and let  : X → Gm d be a Lipschitz m-distribution tangent to S. Then X is bi-Lipschitz trivial along Y . If X \ Y is smooth and if  is actually Lipschitz with exponent 0+ and C 1 on X \ Y then X is C 1 trivial along Y .

3 Some Regularity Conditions Throughout this section X will denote a subset of Rd , and  will be a mapping from X into Gd . For instance it can be an m-distribution over X or the union of the tangent bundles of the strata of a stratification S. In the latter case we will write (S) instead of . Given such a subset we will denote by Pq the orthogonal projection onto q . The mapping v will be a fixed direct transversal. We recall that Dv denotes the distribution generated by the vectors (∂x vs − ∂s ), s = 1, . . . , m.

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3.1 The (t i ) Conditions The (t i ) conditions generalize the (t) condition introduced by R. Thom which was devoted to the case of C ∞ transversals. We recall their definitions which originate in [14] or [19]. Let U and V be two vector subspaces of Rd and let us recall the following definition [14, 19]: ⊥

|π (u)| τ (U ; V ) = sup inf tan θ (u; v) = sup V |πV (u)| u∈U −{0} v∈V −{0}

(3.1)

where θ (u; v) denotes the angle between u and v. Let (T ) := {v ∈ Lin(N; Y )/ v  T }. Define also δ(A; B) =

sup

d(u; B)

u∈A,|u|=1

and μ(A; B) =

inf

u∈A,|u|=1

d(u; B ⊥ )

for A and B vector subspaces of Rd . The following identity is very easy to check: δ(A ∩ B; C ∩ B) ≤

δ(A ∩ B; C) μ(B ⊥ ; C)

(3.2)

for any given vector subspaces A, B, C of Rd . We start by recalling the definition of the (t i ) conditions (see [14]). We will say that the sequence (ys ) is i-flat with respect to a sequence (xs ) if |ys | |xs |i . Definition 3.1.1 Let v be a direct transversal. We will say that (v; ) is (t i ) if no sequence (xs ; ys ; (xs ;ys ) ) tending to 0 satisfies: |ys − v(xs )| is i-flat with respect to (xs ) is d(dxs v; ((xs ;ys ) )) is (i − 1)-flat with respect to (xs ). In the case  = (S), the condition (t i ) can be characterized by the fact that any direct transversal to Y is transverse to the other strata. We will write (v; S) instead of (v; (S)).

Ci

Remark 3 It is proved in [14] that: τ (Y ; T ) = 1/d(0; (T )). This implies in particular that the condition (t 0 ), is the condition (w) of Kuo–Verdier [17]. Moreover, for i ≥ 1 the (t i ) for the couple (0; ) condition may be characterized by: 1  |x|1−i μ(Y ; q )

(3.3)

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in some horn neighborhood of the 0 map of order i. Note that in particular if i = 1 then the (t i ) condition asserts that μ(Y ; T ) is bounded below away from zero on X in a horn neighborhood of 0 of order 1. Observe that if we work up to the mapping φv (x; y) := (x; y − v(x)), we reduce the study to the zero transversal. Therefore, in the case of an arbitrary transversal we get the following characterization of the condition (t i ) for (v; ). The condition (t i ) holds for (v; ) if and only if in a horn neighborhood of v of order i we have: 1  |x|1−i . μ(Dv (q); q )

(3.4)

The (t i ) condition may provide very explicit criteria for sufficiency of jets with respect to topological equivalence. In this paper we are interested in C 1 or bi-Lipschitz equivalence. For this, we need to introduce another condition which is expressed by means of another function μ1 . Let us define for m > 1, A and B vector subspaces of Rd : μ1 (A; B) = inf{|πB (u)|.|πB (u )|/u, u ∈ A, |u| = |u | = 1, u ⊥ u }. Define also μ1 to be μ if m = 1. j In view of (3.4) it is natural to set the definition of the (t1 ) conditions as follows. j

Definition 3.1.2 We will say that (v; ) satisfies the (t1 ) condition if in some horn neighborhood of order j of v : 1  d(q; Y )1−j . μ1 (Dv (q); q ) j

(3.5) i,j

For simplicity if (v; ) is (t i ) and (t1 ) we will shorten it in (t1 ). Remark 4 Observe again that if we work up to the mapping φv (x; y) := (x; y −v(x)), we reduce the study to the zero transversal. More precisely, if we set q := dφv (q ) we get a distribution on φv−1 (X). Then (0;  ) is (t1 ) if and only if so is the couple (v; ). i,j

3.2 Lipschitz Type Conditions We are going to construct distributions which are tangent to Lipschitz stratifications in the sense of Mostowski. We will work with a stratification of Rd , S = {S0 , . . . , Sd } l with either dim Sj = j or Sj = ∅. We set Xl := j =1 Sj . We recall the definition of the condition of Mostowski [11]. For convenience, we write it for a couple (; S), where  is map  : Rd → Gd , instead of just a stratification S but it comes to the same in the case where q is the tangent space to the stratum which contains q. Let  : Rd → Gd with q ⊆ Tq Sp where Sp is the stratum containing q. We denote by Pq the projection on q . Following [11] we denote by dj (q) the distance from q to Xj .

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Definition 3.2.1 Let c > 1 be a fixed constant. A c-chain of a point q ∈ Ss relative to a stratification , is a strictly decreasing sequence of indices s = s1 > s2 > · · · > sr = l and a sequence of points {qsp } where qsp ∈ Ssp and qs1 = q, such that each sp is the greatest integer (< sp−1 ) for which: dk (q) ≥ 2c2 dsp (q)

∀k,

l ≤ k < sp

and

|q − qsp | ≤ cdsp (q). If there is no confusion, we will call the sequence of points {qst } a c-chain of q, or simply a chain of q. The length of such a chain is r, the number of its elements. We will need two identities about chains that are straightforward to check from the definition. We have: |qjs − qjs+1 | ≤ 2n+1 c2 (n + 1)djs −1 (q) djs −1 (q) ≤ 2djs −1 (qjs )

for 2 ≤ s + 1 ≤ r for 2 ≤ s.

(3.6)

Definition 3.2.2 (See [11, 12]) Let c > 1 be a fixed constant. We say that (S; ) is (L) regular if there is some constant C > 0 such that for every c-chain qs1 = q, qs2 , . . . , qsr relative to , and each p, 1 ≤ p ≤ r, |Pq⊥s Pqs2 · · · Pqsp | ≤ C 1

and, for all q  ∈ Ss1 such that |q − q  | ≤

1 2c

|q − qs2 | d(q; Xsp −1 )

(3.7)

d(q; Xs1 −1 ), then:

|(Pq − Pq  )Pqs2 · · · Pqsp | ≤ C

|q − q  | d(q; Xsp −1 )

(3.8)

where dl−1 ≡ 1. In particular, |Pq − Pq  | ≤ C

|q − q  | . d(q; Xs1 −1 )

(3.9)

We recall that a stratification is said to be compatible with a subset if this subset is a union of strata. As we have said, in the case  = (S) (the union of the tangent bundles of the strata), we have by the definitions that S is (L) regular (in the sense of Mostowski) if and only if (S; (S)) is. Assume that i ≥ 1 (and recall that j ≥ i). We define inductively the following sequences of numbers: ns := card{m < t ≤ s : Xt−1  Xt }

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and αs := ns (2 − i − j ) + j − i − 1.

(3.10)

This enables us to define the following coverings of the Xs ’s, s ≥ m. First let Us0 := {q ∈ Xs : d(q; Xs−1 ) ≥ e0 d(q; Y )j −1−αs−1 }. Let Um0 = ∅ and for 1 ≤ l ≤ s − m let l−1 ) ≤ e d(q; Y )j −1−αs−1 }. Usl := {q ∈ Xs : d(q; Us−1 l−1 if Xs = Xs−1 (where e0 , e are some strictly positive constants) and Usl := Us−1 whenever Xs = Xs−1 . Note that, given e, we can choose e0 in such a way that {Us0 , . . . , Uss−m } covers Xs . If it is needed we will write Usl (e) instead of Usl to specify e.

Lemma 3.2.3 Given c > 1, there exists ν > 0 such that, for every q ∈ Usl (e), if qs1 , . . . , qsr is a c-chain for q then qs2 , . . . , qst belong to Usl (ν.e), where t is the greatest integer for which st ≥ s − l. Proof We do the proof for qs2 , the argument may be iterated to yield the result for the other qsp ’s using (3.6). For simplicity set l2 = l − (s1 − s2 ). First, as −αs is increasing we have: −1 d(q; Usl22−1 ) ≤ (s1 − s2 ) e d(q; Y )j −1−αs2 −1 .

(3.11)

Remark also that by definition of USl : ds2 (q) ≤ d(q; Usl22 ) ≤ e(s1 − s2 ) d(q; Y )j −1−αs2 . This implies that: d(q; qs2 ) ≤ c ds2 (q) ≤ ce(s1 − s2 )d(q; Y )j −1−αs2 . Then, together with (3.11) we have: −1 −1 d(qs2 ; Usl22−1 ) ≤ d(qs2 ; q) + d(q; Usl22−1 )

≤ (s1 − s2 )(c e d(q; Y )j −1−αs2 + e d(q; Y )j −1−αs2 −1 ). Thus, it is enough to choose ν ≥ (c + 1)(s1 − s2 ).



We now prove a proposition which will be useful at the end of the next section. Proposition 3.2.4 Assume that i ≥ 1 and that (; S) is (L) regular. Then, there exists a horn neighborhood of v of order i, say H , such that for each s ≥ m and l ≤ s − m there exists an m-distribution Dsl on Usl ∩ H , Lipschitz with exponent at i,j least αs (see (3.10)), with Dsl (q) ⊂ q for all q, and such that (v; Dsl ) is (t1 ) if (v; ) is.

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Proof We are going to construct by induction on s > m the m-distributions Dsl on Xs , 0 ≤ l ≤ s − m. We start by checking that we may find a horn neighborhood H of v of order i such that: H ∩ Xm = {0}.

(3.12)

Replacing v by its Taylor expansion we may assume that it is polynomial. Thus, if (3.12) failed for any H , there would exist an analytic arc γ (t) ∈ Xm satisfying: |v(γ1 ) − γ2 | |γ1 |i ,

(3.13)

γ := (γ1 ; v(γ1 )). Then |γ (r) − γ (r)| r i (we where γ = (γ1 ; γ2 ) in N ⊕ Y . Let parameterize these paths by their distance to the origin). This means that γ ) r i−1 . δ(Tγ (r) γ ; T γ (r)

(3.14)

 γ  (r) in T Let Qr be the normal space to γ v (r) and let Pr := γ (r) ⊕ Qr . Then, by (3.14): i−1 . δ(Pr ; T γ (r) v ) r

(3.15)

Note that γ  (r) ∈ Tγ (r) Xm ∩ Pr so that dim Tγ (r) Xm ∩ Pr ≥ 1. Hence, the linear mapping λ : N → Y whose graph is Pr belongs to (Tγ (r) Xm ). By (3.15) we deduce that d(dx v; (Tγ (r) Xm )) r i−1 . As r is equivalent to d(q; Y ) along γ and q is a subset of Tq Xm for any q ∈ Xm , this contradicts the definition of the (t i ) condition. Hence we must have on some horn neighborhood H of v of order i: 1  d(q; Y )−i . d(q; Xm ) But the (L) condition implies that on H ∩ Xs0 we must have whenever (q; q  ) 1 d(q; Y ): satisfies |q − q  | ≤ 2c |Pq − Pq  |  |q − q  |d(q; Y )−i . Let s1 be the first integer greater than m such that ns1 = 1. 0 l We can define Ds01 : Us01 → Gm d by Ds1 (q) := Pq (Dv (q)) (if l > 0 then Uns1 is empty). By the above inequality and (3.3) we get for w ∈ Dv (q) and q ∈ H ∩ Xs1 : |Pq (w) − Pq  (w)|  |q − q  |d(q; Y )1−2i . |Pq (w)| This implies that the m-distribution Ds01 is Lipschitz with exponent (1 − 2i). As ns1 = 1 we see that αs1 = 1 − 2i which completes the first step of the induction. Assume the result true until s − 1 > m. If Xs = Xs−1 then ns = ns−1 and there is nothing to prove. Hence we may assume Xs−1 = Xs and ns = ns−1 + 1.

Bi-Lipschitz Sufficiency of Jets

975

Fix 0 < l ≤ s − m (we will construct Ds0 later on). Using Proposition 2.0.5 we can l−1 (given by the induction hypothesis) to the ambient space to a distribution extend Ds−1 D which is still Lipschitz with exponent αs−1 . We set for q ∈ H ∩ Usl : Dsl := Pq (D ). i,j We first check that the condition (t1 ) is fulfilled by Dsl . We have to check that l−1 l ) still hold for Ds|H . Let the inequalities (3.4) and (3.5) (which hold for Ds−1 ∩U l s

q ∈ H ∩ Usl and let qj1 , . . . , qjp be a chain of q. As in Lemma 3.2.3, let t be the greatest integer for which st ≥ s − l. Then, by the definition of chains and by the definition of Usl : d(qst ; Xs−l ) ≥ ε d(q; Y )j −1−αs−1

(3.16)

for some strictly positive constant ε. By Lemma 3.2.3 we know that qs2 , . . . , qst belong to Usl (ν.e) (we may apply the l−1 (ν.e) and construct Dsl only on Usl (e)). We may also induction hypothesis to Us−1 assume that qs2 , . . . , qst belong to a given horn neighborhood of order i by taking H small enough. As D is Lipschitz with exponent at least αs−1 (see Remark 1 (1)), given a unit vector w in D (q), we may find w(qs2 ), . . . , w(qst ) in D (qs2 ), . . . , D (qst ) respectively such that for each u: |w(qsu+1 ) − w(qsu )| ≤ Cd(qsu+1 ; Xsu )d(qsu ; Y )αs−1 .

(3.17)

Now thanks to (3.7), (3.16), and (3.17) we have: |Pq (w(q)) − w(q)| ≤

t 

|Pq⊥s Pqs2 · · · Pqsu (w(qsu+1 ) − w(qsu ))| 1

u=1

+ |Pq⊥s Pqs2 · · · Pqst (w(qst ))| 1

≤ C e|q − qs2 | d(q; Y )αs−1 . Using again (3.17) and the definition of Usl we may derive that (for some constant C) we have: |Pq (w(q)) − w(qs2 )| ≤ C e|q − qs2 | d(q; Y )αs−1 ≤ c Ce d(q; Y )j −1 . i,j

i,j

(3.18)

l−1 is (t1 ) this implies that Dsl is (t1 ) as well, provided e is chosen small Since Ds−1 enough (recall that i ≤ j ). Let us now check that the constructed distribution Dsl , l = 1, . . . , s is Lipschitz with exponent αs . Fix q and q  in H ∩ Usl satisfying |q − q  | ≤ 12 d(q; Y ). Note that (3.18) implies that:

δ(Dsl (q); Dsl (qs2 )) ≤ C|q − qs2 |.d(q; Y )αs−1 . Therefore, if |q − q  | > 12 d(q; Xs2 ) we are done. So we may assume that: 1 |q − q  | ≤ d(q; Xs2 ). 2

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G. Valette

As D is Lipschitz with exponent at least αs−1 , for w unit vector of D (q), we may find a unit vector w  ∈ D (q  ) for which: |w − w  | ≤ |q − q  |d(q; Y )αs−1 .

(3.19)

But thanks to the definition of the (L) condition we may write:  t      |Pq (w) − Pq  (w)| ≤  (Pq − Pq  )Pqs2 · · · Pqsu (w(qsu+1 ) − w(qsu ))   u=1

+ |(Pq − Pq  )Pqs2 · · · Pqst (w(qst ))| ≤ C|q − q  |d(q; Y )αs−1 , using (3.8), (3.16), (3.17) and the definition of chains. Together with (3.19) this implies that |Pq (w) − Pq  (w  )| ≤ C|q − q  |d(q; Y )αs−1 . By (3.17) (for u = 0) and (3.18), as w is a unit vector, we have |Pq (w)| ≥ ε, for a strictly positive number ε independent of q and w. Together with the preceding inequality this implies that: |μq − μq  | ≤ C|q − q  |d(q; Y )αs−1 ≤ C|q − q  |d(q; Y )αs ,

(3.20)

(since ns−1 < ns ) where μq is the orthogonal projection on Dsl (q). If l = 0 it is enough to set Ds0 (q) := Pq (Dv (q)). By definition, we see that the i,j (t1 ) condition is clearly satisfied by this distribution when it is satisfied by (v; ). Furthermore, as Dv is a Lipschitz for w unit vector of Dv (q), we may find a unit vector w  ∈ Dv (q  ) for which |w − w  | ≤ C|q − q  |. But thanks to (3.9) this implies that (for any e0 ) we have on H ∩ Us0 for a constant C |Pq (w) − Pq  (w  )| ≤ C |q − q  | d(q; Y )1−j +αs−1 . Now, as (v; ) is (t i ) we have: |Pq (w)| ≥ εd(q; Y )i−1 , for some strictly positive constant ε independent of the unit vector w in Dv (q). This implies that: |μq − μq  | ≤ C|q − q  | d(q; Y )2−i−j +αs−1 , where μ denotes the orthogonal projection on Ds0 . But αs−1 − i − j + 2 = αs .



In the above proof e has to be small enough but e0 is arbitrary so that we have a family of distributions which are defined on a covering of Xs for every s.

Bi-Lipschitz Sufficiency of Jets

977

4 Pullback and Regularity Conditions 



In this section m will be an integer and Y  will be {0Rn } × Rm . We will write Rd for N ⊕ Y  . We fix a direct transversal v and a h-distribution , with h positive integer. As in the previous section Pq will be the orthogonal projection on q .   Given a mapping f : Rd → Rd (resp. f : Rd → Rd ) we will denote by fN its component along N , fY (resp. fY  ) its component along Y (resp. Y  ), and fYs (resp. fYs ) the sth component fY (resp. fY  ) with respect to the canonical basis of Y (resp. Y  ). We define as in [14] the notion of deformation of transversals.  Let ρ ∈ N and let U be a neighborhood of the origin in Rm . Let h1 , . . . , hm : N → Y be smooth functions on N satisfying |hr (x)| = o(|x|ρ ) (resp. |hr (x)| = O(|x|ρ )) and |dx hr | = o(|x|ρ−1 ) (resp. |dx hr | = O(|x|ρ−1 )) for any r. Then F : N × U → N × Y, F (x; u) = (x, f (x, u)) = (x, fu (x))

 m  = x, v(x) + ur hr (x) r=1

is called a deformation of order ρ+ (resp. of order ρ). Deformations of transversals induce a pull-back transformation over the graphs of applications from N × Y into Gd . We define it as in [14]. Let S = {S0 , . . . , Sd } be a stratification of a closed set X ⊂ Rd . If for each point p ∈ U the mapping F is transverse to the strata we may define F ∗ S as the stratification of F −1 (X) given by the manifolds F −1 (Sk ).  Similarly, if at each point p ∈ Rd , the linear mapping dp F is transverse to F (p) we may define F ∗  by setting F ∗ p := dp F −1 (F (p) ). In particular if  is an m-distribution, we thus define an m -distribution. We may also push forward a transversal u by F by setting F∗ u(x) := F (x; u(x)). Important Throughout this section ρ will be an element of Z ∪ Z+ , and F a deformation of order ρ. We will assume that ρ ≥ j and max(ρ − 2; 0) ≥ k. We recall a result proved by D. Trotman and L. Wilson in [14]. The terminology is a bit different, it is adapted to the one of the present paper which is more convenient to generalize these results for Lipschitz conditions. The reader is referred to [14] for the proof of this theorem. The idea is that the pull-back by a deformation of transversals increases the regularity. Theorem 4.0.5 [14] If (v; ) satisfies the condition (t i ) and ρ ≥ i then F ∗  is rugose with exponent ρ − i + 1. 4.1 Lipschitz Distributions We denote by r0 (v) the greatest integer satisfying |v(x)|  |x|r0 (v) . We set φv (x; u) = (x; u − v(x)).

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G. Valette

The following proposition will be useful since it will reduce the problem to the case where v = 0. Proposition 4.1.1 Assume that k ≤ r0 (v) − 2. Then  is Lipschitz with exponent k if ∗  is. and only if φ−v Proof Note that v for couples (x; x  ) ∈ N × N satisfying |x − x  | ≤ 12 |x| we have: |dx v − dx  v|  |x − x  ||x|r0 (v)−2 .

(4.1)

This implies that dx φv also satisfies this property. Now a straightforward computation ∗  is still Lipschitz with exponent k if k ≤ r (v) − 2. shows that φ−v  0 4.2 Pullback of Distributions We recall that F is a deformation of order ρ with ρ ≥ max(j ; k + 2). Theorem 4.2.1 Assume that  is Lipschitz with exponent k ≤ r0 (v) − 2. If (v; ) is i,j (t1 ) then there exists an m-sub-distribution of F ∗  which is Lipschitz with exponent k + ρ + min(2 − (i + j ); 0). We postpone the proof of this theorem and shall prove some preliminary results. Notations We denote by D the distribution generated by the vectors

m   Pq tl ∂x hl,s − ∂s , s = 1, . . . , m, l=1

by Dr (p) the subspace generated by the vectors





m 

tl ∂x hl,s − ∂s ,

with s = r,

l=1

where q = F (x; t), and by Dr (p) the vector subspace Pq (Dr (p)). Let πr,p be the orthogonal projection onto the orthogonal complement of Dr (p) in D(q). Proposition 4.2.2 We assume the same hypotheses as in Theorem 4.2.1 for the transversal v = 0. We also assume i ≥ 1. Then there exists a neighborhood U of Y  such that for any r = 1, . . . , m:   (i) μ(Y ; q )  |πr,p ( m l=1 tl ∂x hl,r − ∂r )|. 1  (ii) For any couple (p; p ) ∈ U ∩ X × U ∩ X satisfying |p − p  | ≤ 2c d(p; Y  ): 



m 2     tl ∂x hl,r − ∂r  , πr,p  

k−i−j +2 

|πr,p − πr,p |  |p − p |d(q; Y )

l=1

where q = F (p) = (x; t1 ; . . . ; tm ).

Bi-Lipschitz Sufficiency of Jets

979

Proof It is clear from the definition of the pull-back that, as ρ ≥ j , the inverse image of a horn neighborhood of v of order j is a neighborhood of Y  . Hence, if U is a i,j sufficiently small neighborhood of Y  the inequalities of the condition (t1 ) will hold on its image by F . We show (i) by way of contradiction. Suppose: 

m       tl ∂x hl,r − ∂r  μ(Y ; q ) (4.2) πr,p   l=1

for a sequence in U ∩ X. Let p ∈ U ∩ X, q = F (p) and let  πr,p be the orthogonal projection onto Dr (p). We have  πr,p + πr,p = Pq . Let w(p) be the vector of Dr (p) which projects onto    πr,p (∂r − m l=1 tl ∂x hl,r ) via Pq . Then

Pq





m 

tl ∂x hl,r − ∂r

= Pq (w) + πr,p

l=1





m 

tl ∂x hl,r − ∂r

l=1

and so by (4.2):  m  

m            tl ∂x hl,r − ∂r − w(p)  = πr,p tl ∂x hl,r − ∂r  μ(Y ; q ). Pq     l=1

By (3.3) we have

l=1

1 μ(Y ;q )

 d(q; Y )1−i . Moreover since p ∈ H , we have |∂x hl,r |  |x|ρ−1 ,

for each l between 1  and  m . As ρ ≥ i we get | m l=1 tl ∂x hl,r | μ(Y ; q ). Thus: |Pq (∂r − w(p))| μ(Y ; q ). As w ∈ Dr (p) the norm of (∂r − w(p)) is bounded below away from zero. This contradicts the definition of μ(Y ; q ) and proves the point (i). Now let us show (ii). Note that, as  is Lipschitz with exponent k we have, thanks to (i) (and the condition (t i )): 

m 2       k−2i+2 tl ∂x hl,r − ∂r  . |Pq − Pq  |  |q − q |d(q; Y ) πr,p   l=1

This already means that (ii) holds if, in the left-hand side, we replace and πr,p by Pq . Since  πr,q + πr,q = Pq , it is enough to show (ii) with the projection  πr,p instead of πr,p in the left-hand side. 1 Let (p; p  ) ∈ U ∩ F ∗ X × U ∩ F ∗ X such that |p − p  | ≤ 2c d(p; Y  ). We set    q = F (p), q = F (p ) so that, as F is Lipschitz |q − q | ≤ C|p − p  | and there1 d(p; Y  ), (q; q  ) fulfills an inequality of the fore since (p; p  ) satisfies |p − p  | ≤ 2c

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G. Valette

same type (with respect to Y ) provided c is chosen sufficiently big. Now, as  is Lipschitz with exponent k, we have: 

 m      tl ∂x hl,r   |q − q  |d(q; Y )k . (Pq − Pq  ) ∂r −  

(4.3)

l=1

Moreover for all l = 1, . . . , m , we have: |tl ∂x hl,r − tl ∂x  hl,r | ≤ |tl − tl ||∂x hl,r | + |∂x  hl,r − ∂x hl,r ||t  r|  |p − p  |(d(q; Y )ρ−1 + d(q; Y )max(ρ−2;0) )  |p − p  |d(q; Y )max(ρ−2;0) .

(4.4)

As max(ρ − 2; 0) ≥ k, putting together (4.3) and (4.4) we get the following inequality:  m 

m         tl ∂x hl,r − ∂r − Pq  tl ∂x  hl,r − ∂r   |p − p  |d(q; Y )k . Pq   l=1

l=1

Moreover it is easy to derive from the definition of μ1 that: μ1 (Y ; q ) ≤ d(Pq (∂r ); Vr ). min{|Pq (w)|/w ⊥ ∂r , |w| = 1, w ∈ Y }, where Vr denotes the subspace of q generated by the Pq (∂s ), s = r. And as for   j any integer s ≤ m, | m l=1 tl ∂x hl,s | μ1 (Y ; q ) (due to the condition (t1 ) and ρ − j ≥ 0), in particular we have  m      tl ∂x hl,s  min (|Pq (w)|)    |w|=1 l=1

  m     and  tl ∂x hl,r  d(Pq (∂r ); Vr ),   l=1

w⊥∂r

in a sufficiently small neighborhood of the origin. Hence:







d(Pq (∂r ); Vr ) ∼ d Pq ∂r −

m 



tl ∂x hl,r ; Vr .

(4.5)

l=1

(∼ means that the ratio is bounded above and below.) Moreover if w ∈ Y is a unit vector such that w ⊥ ∂r , we have: 

 m      d(w, Dr (q)) ≤ max  tl ∂x hl,s  , s   l=1

so that: min (|Pq (w)|) ∼

|w|=1 w∈Dr (q)

min

|w|=1 w∈Y,w⊥∂r

(|Pq (w)|).

(4.6)

Bi-Lipschitz Sufficiency of Jets

981

By (4.5) et (4.6) we deduce that:







μ1 (Y ; q )  d Pq ∂r −

m 



tl ∂x hl,r ; Vr

l=1

min |Pq (w)|.

|w|=1 w∈Dr (q)

(4.7)

But by definition of Dr any element w of Dr (p) satisfies Pq (w) =  πr,p (w). For  = D (p)⊥ ∩ P . We have: simplicity set Vr,p r q







m 

d Pq ∂r −



tl ∂x hl,r ; Vr

l=1





≤ d Pq ∂r −





m 

tl ∂x hl,r

l=1







≤ d Pq ∂r −

m 



; Vr

tl ∂x hl,r



; Vr

l=1

+ δ(Vr ; Vr )  m      tl ∂x hl,s  + max s   l=1

  m 

 m          = πr,p ∂r − tl ∂x hl,r  + max tl ∂x hl,s . s     l=1

l=1

Moreover, by (i):   m      μ(Y ; q )  πr,p (∂r − tl ∂x hl,r )   l=1

and as

 m      max tl ∂x hl,s   d(q; Y )ρ μ(Y ; q ) s   l=1

we may write:



d Pq ∂r −



m 

 tl ∂x hl,r ; Vr

l=1





 m      tl ∂x hl,r .  πr,p ∂r −   l=1

And we deduce from (4.7) that: 

m       μ1 (Y ; q ) ≤ πr,p tl ∂x hl,r − ∂r . min |Pq (w)|.  |w|=1  l=1

w∈Dr (q)

Thus for any w ∈ Dr (p): 

m     |Pq (w) − Pq  (w)|    |p − p  |d(q; Y )k+1−j πr,p tl ∂x hl,r − ∂r ,   |Pq (w)| l=1

(4.8)

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G. Valette

therefore: 





m      tl ∂x hl,r − ∂r . πr,p  

k+1−j 

δ(Dr (p); Dr (p ))  |p − p |d(q; Y )

l=1

This implies: 

m        k+1−j | πr,p −  πr,p |  |p − p ||x| tl ∂x hl,r − ∂r . πr,p   l=1

And hence, via (4.4), we get: 

m 

m        πr,p tl ∂x hl,r − ∂r −  tl ∂x  hl,r − ∂r  πr,p    l=1

l=1



m       k+1−j   |p − p ||x| tl ∂x hl,r − ∂r , πr,p   l=1



which, together with (i), proves (ii).

We shall need a similar result in the case where i ≥ 1. This case is actually easier and is the purpose of the following lemma. Lemma 4.2.3 We assume the same hypotheses as in Proposition 4.2.2. We assume  as well that i ≤ 1. Then there exists a neighborhood U of Y  in Rd such that at any ∗ point p ∈ F X ∩ U : |πr,p (∂r ) − ∂r |  d(p; Y  )1−i for any integer 1 ≤ r ≤ m. 

Proof Let p ∈ Rd and q = F (p). Recall that πr,p is the projection onto Dr (p)⊥ ∩ q . Moreover since ∂r ∈ Y ∩ Dr⊥ (p), it is enough to see that δ(Y ∩ Dr⊥ (p); q ∩ Dr⊥ (p))  d(p; Y  )1−i . As i ≤ 1 and thanks to Remark 3 we have μ(Y ; q ) ≥ ε > 0. Since m | l=1 tl ∂x hl,s |  |x|ρ−1 and ρ ≥ i we may deduce that: μ(D(p); q ) ≥

ε 2

for q close the origin. But as Dr (p) ⊆ D(p) we get: ε μ(Dr (p); q ) ≥ . 2

(4.9)

Bi-Lipschitz Sufficiency of Jets

983

But: δ(Y ∩ Dr⊥ (p); q ∩ Dr⊥ (p)) ≤

δ(Y ∩ Dr⊥ (p); q ) μ(Dr (p); q )

 δ(Y ∩ A(p); q )

(by (4.9))

 d(q; Y )1−i , 

as required.

Proof of Theorem 4.2.1 Remark first that we may assume, without loss of generality,  = φv ◦ F so that F ∗ 0 = 0. Remark that: that v is the zero map. Actually we may set F ∗ )∗  = F ∗ (φ−v F ∗  = (φ−v ◦ F ), ∗  is Lipschitz with exponent k if and only if  is. and by Proposition 4.1.1 φ−v i,j Moreover (see Remark 4) the condition (t1 ) is preserved by the pull-back by φ−v . ∗ ).  with (0; φ−v Hence it is enough to show the result for F The theorem will be proved if we can find a neighborhood of Y  such that  admits a family of sections ζ1 , . . . , ζm which are Lipschitz with exponent k + ρ + i + j − 2 satisfying π(ζl ) = ∂l , l ∈ {1, . . . , m }. Given p ∈ U and r ∈ {1, . . . , m} let:

wr (p) =

∂r −

m

πr,p (∂r )

l=1 tl ∂x hl,r ; πr,p (∂r )

.

It is easy to derive from Proposition 4.2.2(ii): |wr (p) − wr (p  )|  |p − p  |d(q; Y )k−i−j +2 .

(4.10)

On the other hand from the construction of wr we know that, for r = s,



wr (p);

m 

tl ∂x hl,s − ∂s = 0

l=1

and hence for s ∈ {1, . . . , m} different from r:



wr (p);

m 

tl ∂x hl,s = wr (p); ∂s  = wr,s (p).

(4.11)

l=1

And as wr (p); ∂r − wr,N (p);

m

l=1 tl ∂x hl,r  = 1,



m  l=1

we get:

tl ∂x hl,r = wr (p); ∂r  − 1 = wr,Yr (p) − 1.

(4.12)

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G. Valette

Then we set for l ∈ 1, . . . , m :

m   hl,r (x)wr,N (p); 0; . . . ; 1; . . . ; 0 ζl (p) =

(4.13)

r=1

where the number 1 lies in lth position so that: π(ζl (p)) = ∂l . Moreover as:

 m m   ∂F (p) = Id; tl dx hl,1 ; . . . ; tl dx hl,m ∂x l=1

l=1

∂F (p) = (0; . . . ; 0; hs,1 (x); . . . ; hs,m (x)). ∂ts We may deduce using (4.11) et (4.12):

m  m   dF (p)(ζl (p)) = hl,r (x) wr,N (p); hl,r (x)wr,Y (p) r=1

=

m 

r=1

hl,r (x)wr (p).

r=1 ∗ In consequence ζl is a section mof F . Moreover (ζl (p) − ∂l ) = ( r=1 hl,r (x)wr,N (q); 0). By Proposition 4.2.2(i), we have: 1 1 .  |wr,N (p)| = m |πr,p ( l=1 tl ∂x hl,r − ∂r )| μ(Y ; q )

Now, if i ≥ 1, the inequality (3.3) shows that: 1  d(q; Y )1−i . μ(Y ; q ) If i ≤ 1 we apply Lemma 4.2.3. This proves that |wr,N (p)|  d(p; Y )1−i . In consequence, in any case we have |wr,N (p)|  d(q; Y  )1−i and so since |hl (x)|  d(q; Y )ρ we get for l = 1, . . . , m : |ζl (p) − ∂l |  d(p; Y )ρ−i+1 .

(4.14)

Hence ζl is a section rugose with exponent ρ − i + 1 which lifts ∂r . As j ≥ k, ζl is a fortiori a section rugose with exponent k + ρ − i − j + 2. Thus, it remains to prove 1 that for (p; p  ) ∈ U satisfying |p − p  | ≤ 2c d(p; Y  ) we have: |ζl (p) − ζl (p  )|  d(p; Y  )k+ρ−i−j +2 |p − p  |.

Bi-Lipschitz Sufficiency of Jets

But:

985

 m  m        hl,r (x)wl,N (p) − hl,r (x )wr,N (p ) |ζl (p) − ζl (p )| =    r=1 r=1  m       ≤  (hl,r (x) − hl,r (x ))|wr,N (p)|   r=1  m       + hl,r (x )|wr,N (p) − wr,N (p  )|   

r=1

and as the ρ-jet of hl vanishes at the origin, we may apply (4.14) and (4.10) to complete the proof.  Remark 5 Distributions that are Lipschitz with exponent 0+ and which are C 1 outside Y are actually C 1 distributions. This will enable us to provide determinacy theorems for the C 1 equivalence in the case where the stratification is composed by only two strata (see Sect. 6.1). 4.3 Pull-Back and (L) Stratification The proof of the following theorem is actually a variation of that of Theorem 4.2.1 The setting is actually the one of Sect. 3.2. Let S = {S0 , . . . , Sd } be a stratification compatible with Y (with dim Sl = l or Sl = ∅) and let  : Rd → Gd be such that q is a subspace of tangent space to the stratum containing q. Assume j ≥ i ≥ 1. Theorem 4.3.1 Let F be a deformation of order ρ. Assume that (; S) is (L) regular i,j and that (v; ) is (t1 ). Then every smooth vector field on Y  may be extended to a  vector field in a neighborhood of Y  in Rd tangent to the strata of F ∗ S and Lipschitz with exponent ρ + αd + 2 − i − j . Proof By Proposition 3.2.4 we may find m-distributions Dd0 , . . . , Ddd−m , respectively defined on the intersection of Ud0 , . . . , Udd−m with some horn neighborhood of order i, i,j Lipschitz with exponent at least αd and such that (v; Ddl ) is (t1 ) for l = 0, . . . , d −m. l Thanks to Proposition 4.2.2 (with Dd instead of ) we know that the vector field on H ∩ Udl defined by wr (p) =

∂r −

m

πr,p (∂r )

l=1 tl ∂x hl,r ; πr,p (∂r )

is Lipschitz with exponent αd − (i + j ) + 2 (see (4.10)). By Proposition 2.0.4 and the definition of the Uls ’s we may find a partition of unity adapted to the covering Ud0 , . . . , Udd−m Lipschitz with exponent at least αd . Thus a straightforward computation shows that we may paste these vector fields into one still Lipschitz with the same

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exponent. Now we can end the proof by defining the sections ζl as in the proof of Theorem 4.2.1 (see (4.13)). The same computation clearly yields that they are Lipschitz  with exponent ρ + αd + 2 − i − j .

5 Bi-Lipschitz Sufficiency of Jets Given r ∈ N, the r-jet of a mapping u : Rn → Rm , is given by (u(0); d0 u; . . . ; d0r u) where d0 u, . . . , d0r u are the successive derivatives of the mapping u at the origin. All the results of this section are consequences of those of the previous one. We are going to give some explicit criteria for sufficiency of jets. For simplicity we set for this section: ρ := i + j − 2 − αd . We assume j ≥ i ≥ 1. 5.1 Determinacy of Transversals For this section we fix a stratification of Rd , S = {S0 , . . . , Sd } compatible with Y and satisfying the (L) condition of Mostowski. Let  = (S) the union of the tangent bundles of the strata of S. Definition 5.1.1 We will say that two transversals v and u are bi-Lipschitz Sequivalent if there exists a bi-Lipschitz map H : v × [0; 1] → Rd such that for any t, Ht is a bi-Lipschitz map preserving the strata of S, H0 = Id and H1 sends u onto v . A jet is said to be bi-Lipschitz S-sufficient in C ρ whenever all the C ρ mappings u having this jet at the origin are bi-Lipschitz S-equivalent. We will write j α v(0) for the α-jet of v at 0. Theorem 5.1.2 i,j

(1) If (v; ) satisfies the condition (t1 ) then z = j ρ v(0) is bi-Lipschitz S-sufficient in C ρ . i,j (2) If (u; ) satisfies the condition (t1 ), for any polynomial representative u of j ρ v(0) jet then z = j ρ−1 v(0) is bi-Lipschitz S-sufficient in C ρ . Proof Let h be a C r mapping with a ρ-jet vanishing at the origin and let F (x; t) = (x; v(x) + th(x)). Then F is a deformation of order ρ+ . By Theorem 4.3.1, as ρ = i + j − 2 − αd , F ∗  is a Lipschitz m-distribution which implies, by Lemma 2.1.2, that F −1 (X) is bi-Lipschitz trivial along Y  . In other words there exists a bi-Lipschitz isotopy carrying the graph of v onto the graph of v + th, for t ∈ [0; 1]. For (2) we apply the same argument, the only difference is that the deformation is of order ρ.  In [15] some theorems of determinacy with respect to other relations are also provided. For instance it is given some explicit criteria for a transversal to be finitely

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determined up to a homeomorphism which is bi-Lipschitz with exponent k, for k ≥ 0 (with an explicit order of determinacy) or which is a quasi-isometry. Some criteria using the condition (t i− ) are also provided. Remark 6 In order to make more explicit our criterion, let us recall that in our setting the (t i ) condition is fulfilled by the couple (v; ) if and only if for any q in a horn neighborhood of v of order i we have for l > m: μ(Dv (q); Tq Sl ) ≥ ε|x|i−1

(5.1)

where q = (x; t). j Similarly, the (t1 ) condition is fulfilled by the couple (v; S) if and only if for any q in a horn neighborhood of v of order j we have for l > m: μ1 (Dv (q); Tq Sl ) ≥ ε|x|j −1

(5.2)

where q = (x; t). 5.2 SV-Determinacy We are going to give a determinacy theorem for smooth mappings with respect to the SV -equivalence. This is roughly speaking the determinacy of the zero locus (see below for an exact definition). We fix a subset X of N and a stratification S := {{0}, S1 , . . . , Sd } of X. In [14] are given some analogous theorems but in the topological point of view. Here we will focus on such an equivalence involving biLipschitz maps. This equivalence preserve the strata the stratification S. Two given germs of C ρ mappings, f, g : N → Y are said bi-Lipschitz SV equivalent if there exists a bi-Lipschitz homeomorphism of N , preserving the strata, bi-Lipschitz and sending f −1 (0) onto g −1 (0). We set S0 = {Y \ {0}, S0 × {0}, . . . , Sd × {0}} so that f and g are bi-Lipschitz SV equivalent if and only if they are bi-Lipschitz S0 -equivalent. We recall that we have set ρ = i + j − 2 − αd . i,j

Theorem 5.2.1 If (v; S0 ) is (t1 ) then z = j ρ v(0) is bi-Lipschitz SV-sufficient in C ρ . Proof This theorem is a direct consequence of Theorem 5.1.2 and the above observation that f and g are bi-Lipschitz SV -equivalent if and only if f and g are bi-Lipschitz S0 -equivalent.  Remark 7 Therefore, (5.1) and (5.2) provide explicit criteria for SV -bi-Lipschitz determinacy. It suffices to replace S by S0 . Note that in this case the functions μ(Dv (x); Tq Sl ) and μ1 (Dv (x); Tq Sl ) may be explicitly expressed as follows: μ(Dv (x); Tq Sl ) = inf{|dx v(w)| : |w| = 1, w ∈ Tq Sl } and μ1 (Dv (x); Tq Sl ) = inf{|dx v(w)|.|dx v(w  )| : |w  | = |w| = 1, w ∈ Tq Sl , w  ∈ Tq Sl }.

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Also, in this case, the horn neighborhood on which (5.1) (resp. (5.2)) has to hold is of type: {x ∈ Rn : |v(x)| ≤ C|x|i }, for some constant C (resp. {x

∈ Rn

(5.3)

: |v(x)| ≤ C|x|j }).

5.3 R-Sufficiency of Jets We say that two mappings f and g from Rn to Rm are bi-Lipschitz R-equivalent whenever there exists a bi-Lipschitz mapping ϕ from N to N , preserving the strata and such that f = g ◦ ϕ. We say that a jet is bi-Lipschitz R-sufficient if all the mappings having this jet at the origin are bi-Lipschitz R-equivalent. In this section X will be a subanalytic subset of N , stratified by a subanalytic stratification of Rd , S = {S1 , . . . , Sd } compatible with X and Y , and satisfying the condition (L). Let q := Tx Spx × {0Rm } where Spx is the stratum containing x if q = (x; t). Recall that ρ = i + j − 2 − αd . i,j

Theorem 5.3.1 If (v; ) is (t1 ) then z = j ρ v(0) is bi-Lipschitz R-sufficient in C ρ . Proof We use a similar argument as in the proof of Theorem 5.1.2. Consider a mapping h having a zero ρ-jet at the origin and let F (x; t) := (x; v(x) + th(x)). By Theorem 4.3.1 then F ∗  admits a section which is bi-Lipschitz. By Lemma 2.1.2 this implies that F ∗ S is bi-Lipschitz trivial along Y .  i,j

Remark 8 To get an explicit criterion for (v; ) to be (t1 ) regular just take the one given in Remark 7 but omitting the restriction to the horn neighborhood given by (5.3).

6 The Two Strata Case We are going to give some determinacy theorems in the case where the stratification is composed by only two strata. In this case, vector fields smooth on X which are Lipschitz with exponent 0+ actually extend to C 1 vector field to a neighborhood of Y . Hence we get the C 1 -S-determinacy. Again we will study successively two types of determinacy: The S-determinacy and the R-determinacy. For this section we set: ρ := max(i + j − 2; 2). 6.1 S-Determinacy in the Two Strata Case In this section we fix a stratification of X composed by two strata S := {Y, X \ Y }. We assume that the distribution q → Tq X, is Lipschitz with exponent k ∈ Z (with k possibly positive). In other words we assume that we have for any q and q  in X 1 satisfying |q − q  | ≤ 2c d(q; Y ): δ(Tq X; Tq  X)  |q − q  |.d(q; Y )k .

(6.1)

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with k integer. Note that, thanks to the Łojasiewicz inequality, such an inequality may always be obtained by taking k sufficiently large. Now the C 1 -S-equivalence is defined in the same way as the bi-Lipschitz equivalence, just replacing bi-Lipschitz by C 1 . The notion of C 1 -S-equivalence immediately gives rise to the notion of C 1 -S-sufficiency, derived in the same way. i,j

Theorem 6.1.1 Assume max(ρ − 2; 0) ≥ k. If (v; S) satisfies the condition (t1 ) then z = j ρ−k v(0) is C 1 -S-sufficient in C ρ−k . Proof The proof is readily the same as that of Theorem 5.1.2. Just use Theorem 4.2.1 instead of Theorem 4.3.1. The C 1 determinacy is a consequence of the above observation that, in the two strata case, vector fields smooth on X which are Lipschitz with exponent 0+ actually extend to C 1 vector field to a neighborhood of Y .  When i = 1 the transversality is good and the required conditions for determinacy are simpler. In particular: Corollary 6.1.2 The C 1 type of the germ of the intersection of a (L) stratified set with a C 2 transversal to a given stratum at a given point does not depend on the considered transversal. Proof Note that the (L) condition of Mostowski implies the (w) condition of Kuo– Verdier which is the condition (t 0 ). If (t 0 ) holds then a fortiori (t 1 ) holds then by definitions (t11,1 ) also holds; so, by the preceding theorem (with ρ = k = 0), we get that the 0-jet of v is C 1 -S-sufficient.  Of course in the above theorem the hypothesis max(ρ − 2; 0) ≥ k may always be i,j obtained when we have (t1 ) since in this case we may choose i, j as large as we please. 6.2 Germs of Sets with an Isolated Singularity In this section we will be interested in the case of subsets X such that X \ {0} is a smooth manifolds. We assume as well that the tangent distribution to X is Lipschitz with exponent k. We assume now i ≥ 2 (for simplicity, some results with i < 2 could be stated but have no interest) so that now ρ = i + j − 2. We denote by S the stratification constituted by the pair (X \ {0}; Y ). We define C 1 -R-equivalence exactly like bi-Lipschitz R equivalence, just replacing biLipschitz by C 1 (see Sect. 5.3). Immediately, we also define in an analogous way C 1 sufficiency for jets. i,j

Theorem 6.2.1 Assume ρ − 2 ≥ k and k ≤ r0 (v) − 2. If (v; S) satisfies (t1 ) then z = j ρ−k v(0) is C 1 -R-sufficient in C ρ−k .

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Proof Let q = Tx X where q = (x; t) ∈ N ⊕ Y. If we take a deformation of transversals F (x; t) = (x; v(x) + th(x)) where h has a vanishing ρ-jet we get by Theorem 4.2.1 F ∗  is a C 1 distribution. The theorem follows from Lemma 2.1.2.  By Remarks 7 and 8 we get an explicit criterion. Note that a consequence of this theorem is that (t i ) is enough to have the sufficiency of the 3i-jet in C 3i . Note also that, as in the case of bi-Lipschitz determinacy we could also provide some criteria for C 1 -equivalence devoted to the SV -equivalence. Again, in the preceding theorem, the hypothesis ρ − 2 ≥ k and k ≤ r0 (v) − 2 may i,j always be obtained when we have (t1 ) since we may choose i, j, and k as large as we want. In [15], we also give some criteria for which k may be chosen in Z− . In the case where X = N the inequality (6.1) is true for any k ∈ Z. Thus the i,j conditions (t1 ) give the determinacy and we get the immediate following corollary. i,j

Corollary 6.2.2 If (v; N ) satisfies (t1 ) then z = j ρ−r0 (v) v(0) is R-sufficient in C ρ−r0 (v) . Proof As we have said the inequality (6.1) is true for any k ∈ Z. Hence, just put k = r0 (v) − 2. Note that by the definition of (t i ), we have inf |dx v(u)| ≥ ε|x|i−1 ,

|u|=1

for some ε > 0, on some horn neighborhood of v of order i. By the definition of r0 (v) we necessarily have i ≥ r0 (v) and so ρ = i + j − 2 ≥ r0 (v) + j − 2 ≥ k + 2 (since j ≥ 2).  It can be seen that this criterion always improves the one given by C.T.C. Wall in [18]. Let us illustrate it by an example. Example 1 Let v : R4 → R2 be defined by v(x; y; z; t) = (2x 2 − y 2 + 6zt 2 − z3 ; xy − 3z2 t + 2t 3 ). Denote by N (dx v) the sum of the squares of all minors of order 2. The criterion provided in [18] requires N (dx v) ≥ ε|x|i−2 for the i − r0 (v) + 1 determinacy. If x = y = 0 there is only one minor of order 2 which does not vanish. This minor is (z; t) = (6t 2 − 3z2 )2 + 72z2 t 2 . The best lower bound that we can get is |(z; t)|4  (z; t). The best exponent to bound N below in not less that 8. Moreover 1,1 (x; y) = 4x 2 + 2y 2 is greater or equal to |(x; y)|2 . As the norm of |q| as always equivalent either to |(x; y)| or to |(z; t)| we deduce that |q|8  N (dq v). Then, according to the theorem given in [18] the 9-jet is thus sufficient. In dimension 2, the functions μ21 and N coincide. The condition (t15 ) thus holds for the couple (v; GR (N )). Moreover by the same arguments as for the lower bound of N , we see that (t 3 ) holds for this couple. As r0 (v) = 2 we get by the above corollary that actually the 6-jet is sufficient.

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7 Two Other Consequences In this section we give further results which are also consequences of the results of Sect. 4. We start by proving a finiteness result. Finiteness of topological types of intersection of C ∞ transversals is known since the work by D. Trotman and L. Wilson. We first give a result of finiteness of metric types. We then prove give another theorem about the so-called Kuo’s blowing up. 7.1 A Finiteness Result Theorem 7.1.1 Let S be a subanalytic stratification of X of Rd satisfying the (a) condition of Whitney and let S be the a stratum. Then, for x0 ∈ S, there exists an integer r such that the number of Lipschitz types of germs at x0 of intersection of X with a C r direct transversal is finite. Proof As it is local we may assume that S is Y . As the stratification is (a) regular we may find a C 0 m-distribution  which is a subanalytic sub-distribution of the tangent to the tangent bundle of the strata and satisfying |Y ≡ Y . By the Łojasiewicz inequality we may find an integer k such that this distribution is Lipschitz with exponent k. Moreover (t11,1 ) is fulfilled since  is continuous. By, Theorem 4.2.1, this implies that the pull-back F ∗  by a deformation of transversals F of contact at least |k| is Lipschitz with exponent 0. Therefore by Lemma 2.1.2 we then may trivialize F ∗ X and thus the |k|-jet of v is bi-Lipschitz S-sufficient in C |k| for any direct transversal v. On the other hand, it is well known that the number of Lipschitz types of a global subanalytic family is finite [12, 16]. Therefore, the number of Lipschitz types of germs of intersections of a polynomial direct transversals to Y of bounded degree with a X is finite.  Remark 9 Actually we do not need the (a) regularity with respect to Y . The (t i ) regularity (for some i) for the couples (Y ; Ss ) would be enough. 7.2 Kuo and Trotman’s Blowing Up and (L) Conditions The blowing-up of T.C. Kuo and D. Trotman [7] is a transformation of stratified spaces. Actually, up to a chart of the Grassmannian it is a particular case of pull-back by a deformation of transversals. We fix a stratification {S, Y } of X composed by two strata, that is to say, we assume Y ⊆ S \ S and S is smooth. Let us define: = {(0; P ) ∈ {0R d } × Gnd /P  Y } Y }. S = {(q; P ) ∈ Rd × Gnd /q ∈ P ∩ S, P ∈ Y } is a stratification S, Y In [7] it is proved that if S satisfies the (t i ) condition then { regular. In particular, if a couple of strata S satisfies the (t) condition then ) is a (w)-regular stratification. Here we prove that: ( S; Y

(t i−1 )

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Theorem 7.2.1 If S satisfies the (L) condition of Mostowski then the tangent distrib may be extended to a distribution tangent to ution to Y S, C 1 in the sense of Whitney. 1 . is C trivial along Y In particular X as follows: Proof Up to a chart of the Grassmannian manifold we may write S and Y (1) (2)

= {0Rd } × U where U is an open subset of Rnm . Y  S = {(x; t; u) ∈ Si × U/ ni=1 xi uj i = tj ∀j ∈ {1, . . . , m}} where x = (x1 ; . . . ; xn ), t = (t1 ; . . . ; tm ), u = (uj i )1≤j ≤m .

Let

p(x; t; u) = (x; u) ∈ Rn

1≤i≤n

and  S = p( S). Now let us set

 n n   F (x; u) = x; u1i xi ; . . . ; umi xi . × Rnm

i=1

i=1

Then F is a deformation of order 1. Let  be the m-distribution on S defined by q = Pq (Y ) where Pq denotes the orthogonal projection onto Tq S. As S satisfies the condition (L), the couple (; S) fulfills it as well and by Theorem 4.2.1 we get that F ∗  is Lipschitz with exponent 1 which means that it is C 1 distribution. By S and in consequence F ∗  induces over  S a subdefinition of the pull-back F ∗ S =  distribution of the tangent distribution  S. As p restricted to S is a diffeomorphism we are done.  Example 2 Let us define the set X by: X = {(x; y; z; )/y 3 = z4 x 5 + x 7 }. Then the singular locus of X is the axis Oz . It is possible to see [2] that the stratification (X \ Oz ; Oz ) is a Lipschitz stratification in the sense of Mostowski. Then by the is C 1 trivial along the set of vector spaces above theorem the Grassmann blow-up X directly transverse to Oz . Acknowledgements

The author is grateful to his wife for her very careful reading of the manuscript.

References 1. Banach, S.: Wst¸ep do Teorii Funkcji Rzeczystych. Monografie Matematyczne. Wrocław, Warszawa (1951) 2. Juniati, D.: De la régularité des espaces stratifiés. Thèse de doctorat 3. Federer, H.: Geometric Measure Theory. Grundlehren Math. Wiss., vol. 153. Springer, Berlin (1969) 4. Kuo, T.C.: Characterizations of V -sufficiency of jets. Topology 11, 115–131 (1972) 5. Kuo, T.C.: On C 0 sufficiency of jets and potential functions. Topology 8, 167–171 (1969) 6. Kuo, T.C.: Sufficiency of jets via stratification theory. Invent. Math. 57, 219–226 (1980) 7. Kuo, T.C., Trotman, D.: On w and t s stratifications. Invent. Math. 92, 633–643 (1988) 8. Koike, S.: C 0 -suffciency of jets via blowing-up. J. Math. Kyoto Univ. 28(4), 605–614 (1988) 9. Koike, S.: On v-sufficiency and (h)-regularity. Publ. Res. Inst. Math. Sci. 17(2), 565–575 (1981) 10. Mather, J.: How to stratify mappings and jets spaces. In: Singularités d’Applications Différentiables (Sém., Plans-sur-Bex, 1975). Lecture Notes in Math., vol. 535, pp. 128–176. Springer, Berlin (1976) 11. Mostowski, T.: Lipschitz equisingularity. Diss. Math. (Rozpr. Mat.) 243, 46 (1985) 12. Parusi´nski, A.: Lipschitz stratification of subanalytic sets. Ann. Sci. Ec. Norm. Super. (4) 27(6), 661– 696 (1994) 13. Takens, F.: A note on sufficiency of jets. Invent. Math. 13, 225–231 (1971)

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14. Trotman, D.J.A., Wilson, L.C.: Stratifications and finite determinacy. Proc. Lond. Math. Soc. (3) 78(2), 334–368 (1999) 15. Valette, G.: Détermination et stabilité du type métrique des singularités. Thèse de doctorat 16. Valette, G.: Lipschitz triangulations. Ill. J. Math. 49(3), 953–979 (2005) 17. Verdier, J.-L.: Stratifications de Whitney et théorème de Bertini–Sard. Invent. Math. 36, 295–312 (1976) 18. Wall, C.T.C.: Finite determinacy of smooth map-germs. Bull. Lond. Math. Soc. 13(6), 481–539 (1981) 19. Wilson, L.: Stratifications and sufficiency of jets. In: Singularity theory (Trieste, 1991), pp. 953–973. World Scientific, River Edge (1995)