Finitness theorem for Multi-K-Bi-Lipschtiz equivalence of map-germs
aa r X i v : . [ m a t h . AG ] J un FINITENESS THEOREM FOR MULTI- K -BI-LIPSCHITZEQUIVALENCE OF MAP GERMS LEV BIRBRAIR*, JO ˜AO CARLOS FERREIRA COSTA**,EDVALTER DA SILVA SENA FILHO***, AND RODRIGO MENDES***
Abstract.
Let P k ( n, p ) be the set of all real polynomial map germs f = ( f , ..., f p ) :( R n , → ( R p ,
0) with degree of f , ..., f p less than or equal to k ∈ N . The main resultof this paper shows that the set of equivalence classes of P k ( n, p ), with respect to multi- K -bi-Lipschitz equivalence, is finite. Introduction
Given an equivalence relation to classify map germs in the context of Singularity theory,one initial problem is the following:
Problem.
To decide if the classification under investigation has or not countablenumber of equivalence classes, becoming finite under some restrictions.If the answer is affirmative, we say that the classification is tame or has the finite-ness property. The finiteness property is an initial step in any possible attempt of theunderstanding the classification of singularities.Let P k ( n, p ) be the set of all real polynomial map germs f = ( f , ..., f p ) : ( R n , → ( R p ,
0) with degree of f , ..., f p less than or equal to k ∈ N . For some equivalence relationsthe finitness property does not hold. Henry and Parusinski [5] showed existence of modulifor R -bi-Lipschitz equivalence of analytic functions. By other hand, in [1] the authorsshowed that with respect to K -bi-Lipschitz equivalence, the finiteness property holds forthe set of all real polynomial map germs with bounded degree. Later on, Ruas and Valette[7] showed a more general result about the finiteness of Lipschitz types with respect to K -bi-Lipschitz equivalence of map germs.In this paper we introduce the notion of multi- K -bi-Lipschitz equivalence to investigatethe finiteness property in the set P k ( n, p ). This equivalence is closed related with thenotion of contact equivalence for q -tuple of map germs introduced by Sitta in [8]. Theapproach of [8] was motivated by Dufour’s work [4]. Mathematics Subject Classification.
Key words and phrases.
Bi-Lipschitz contact equivalence, finiteness theorem, Lipschitz classification.*Research supported under CNPq 302655/2014-0 grant and by Capes-Cofecub.**Research supported by FAPESP and CAPES.***Research supported by Capes.
A pair (or couple) of map-germs can be defined as follows:( f, g ) : ( R n , → ( R p × R q , . It can be seen also as a divergent diagram( R q , g ←− ( R n , f −→ ( R p , . Divergent diagrams appear in several geometrical problems and it have many applica-tions.Recently, in [2] the authors showed the finiteness property for the called bi- C - K -equivalence of pairs of map germs. This equivalence relation is the topological versionof topological contact equivalence adopted to a pair of map germs. An overview of thetheory involving classical equivalence relations of pairs of map germs can be found in [3].The main result of this paper shows that the set of equivalence classes of P k ( n, p ), withrespect to multi- K -bi-Lipschitz equivalence, is finite. As a consequence, we obtain thefiniteness property with respect to K -bi-Lipschitz equivalence.2. Preliminaries and notations
Definition 2.1.
Two continuous map germs f, g : ( R n , → ( R p , are said to be K -bi-Lipschitz equivalent if there exist germs of bi-Lipschitz homeomorphisms H : ( R n × R p , → ( R n × R p , and h : ( R n , → ( R n , such that H ( R n × { } p ) = R n × { } p and the following diagram is commutative: ( R n , ( id n ,f ) −→ ( R n × R p , π n −→ ( R n , h ↓ H ↓ h ↓ ( R n , ( id n ,g ) −→ ( R n × R p , π n −→ ( R n , where id n : ( R n , → ( R n , is the identity map germ of R n , π n : ( R n × R p , → ( R n , is the canonical projection germ and { } p = (0 , . . . , ∈ R p .When h = id n , f and g are said to be C -bi-Lipschitz equivalent . In other words, two map germs f and g are K -bi-Lipschitz equivalent if there exists agerm of bi-Lipschitz map H : ( R n × R p , → ( R n × R p ,
0) such that H ( x, y ) can be writtenin the form H ( x, y ) = ( h ( x ) , θ ( x, y )), x ∈ R n , y ∈ R p , where h is also a bi-Lipschitz mapgerm, such that θ ( x,
0) = 0 and H maps the germ of the graph( f ) onto the graph( g ).Recall that graph( f ) is the set defined as follows:graph( f ) = { ( x, y ) ∈ R n × R | y = f ( x ) } . Definition 2.2.
Two functions f, g : R n → R are called of the same contact at a point x ∈ R n if there exist a neighborhood U x of x in R n and two positive numbers c and c such that, for all x ∈ U x , we have c f ( x ) ≤ g ( x ) ≤ c f ( x ) . We use the notation: f ≈ g . The next Lemma is an adaptation of Theorem 2.4 given in [1]:
Lemma 2.3.
Let f, g : ( R n , → ( R , be two germs of Lipschitz functions. Supposethat there exists a germ of bi-Lipschitz homeomorphism h : ( R n , → ( R n , such thatone of the following two conditions is true:i) f ≈ g ◦ h orii) f ≈ − g ◦ h .Then, f and g are K -bi-Lipschitz equivalent.Proof. Suppose f ≈ g ◦ h (the case f ≈ − g ◦ h is analogous). Let H : ( R n × R , → ( R n × R ,
0) given by H ( x, y ) = ( h ( x ) ,
0) if y = 0( h ( x ) , g ◦ h ( x ) yf ( x ) ) if 0 < | y | ≤ | f ( x ) | ( h ( x ) , y − f ( x ) + g ◦ h ( x )) if 0 < | f ( x ) | ≤ | y | and sign ( y ) = sign ( f ( x ))( h ( x ) , y + f ( x ) − g ◦ h ( x )) if 0 < | f ( x ) | ≤ | y | and sign ( y ) = − sign ( f ( x ))( h ( x ) , y ) if f ( x ) = 0 . The map H ( x, y ) = ( h ( x ) , θ ( x, y )) defined above is bi-Lipschitz. In fact, H is injectivebecause, for any fixed x ∗ , we can show that θ ( x ∗ , y ) is a continuous and monotone function.Moreover, H is Lipschitz if 0 < | f ( x ) | ≤ | y | . Let us show that H is Lipschitz if 0 < | y | ≤ | f ( x ) | . Hence it is sufficient to show that all the partial derivatives of H existand are bounded in their domain up to a set of measure zero. Since h is a bi-Lipschitzhomeomorphism, follows that all its partial derivatives ∂h∂x i exist and are bounded inalmost every x near 0 ∈ R n , i = 1 , . . . , n . Thus, it is necessary to check only the partialderivatives of θ . Then, ∂θ∂x i = (cid:16)P n j =1 ∂g∂x j ( h ( x )) ∂h j ∂x i ( x ) f ( x ) − ∂f∂x i ( x ) g ◦ h ( x ) (cid:17) y ( f ( x )) = X n j =1 ∂g∂x j ( h ( x )) ∂h j ∂x i ( x ) yf ( x ) − ∂f∂x i ( x ) g ◦ h ( x ) f ( x ) yf ( x ) . Observe that,i) ∂h j ∂x i ( x ) is bounded up to a set of measure zero, for all i, j = 1 , ..., n ; L. BIRBRAIR, J.C.F. COSTA, E.S. SENA FILHO, AND R. MENDES ii) since 0 < | y | ≤ | f ( x ) | then yf ( x ) is bounded;iii) since f ≈ g ◦ h , the expression g ◦ h ( x ) f ( x ) is bounded;iv) ∂g∂x j , ∂f∂x i are bounded for all i, j = 1 , ..., n, because f and g are Lipschitz.From (i)-(iv) we can conclude the ∂θ∂x i exist and it is bounded up to a set of measurezero, for all i = 1 , . . . , n . Observe that the partial derivative of θ with respect to y ∈ R isalso bounded.Since the map H is Lipschitz outside of the set of the measure zero, the followingexpression holds: k H ( x, y ) − H ( u, v ) k≤ k k ( x, y ) − ( u, v ) k ∀ ( x, y ) , ( u, v ) ∈ V \ U, where V is an open neighborhood of the origin in ( R n × R , U is a set of measurezero and k is a real constant positive. We need to show that the last inequality remainsvalid for all ( x, y ) , ( u, v ) ∈ V . In fact, take ( x , y ) , ( u , v ) ∈ V ∩ U. Since U is a set ofmeasure zero, there exist sequences ( x n , y n ) , ( u m , v m ) ∈ V \ U such that ( x n , y n ) → ( x , y )e ( u m , v m ) → ( u , v ).Moreover, k H ( x n , y n ) − H ( u m , v m ) k≤ k k ( x n , y n ) − ( u m , v m ) k , ∀ n, m ∈ N . Since H is a continuous map, taking n → ∞ and then m → ∞ follows that k H ( x , y ) − H ( u , v ) k≤ k k ( x , y ) − ( u , v ) k . Hence, H is Lipschitz in all V .Since H − can be constructed in the same form as H , we conclude that H − is alsoLipschitz. Then, H is bi-Lipschitz. Moreover, by construction of H , follows that:i) H ( x, f ( x )) = ( h ( x ) , g ◦ h ( x )) andii) H ( R n × { } ) = R n × { } .Hence f and g are K -bi-Lipschitz equivalent. (cid:3) Definition 2.4.
Let q = ( q , q , ..., q p ) ∈ N p . A ( p, q )-multi pair of map germs is a familyof p maps { F , . . . , F p } , where F i : ( R n , → ( R q i , , i = 1 , . . . , p is a map germ. A ( p, q ) -multi pair of map germs can be considered as a map germ F = ( F , ..., F p ) : ( R n , → ( R q × R q × · · · × R q p , , where q is called multi index of a ( p, q ) -multi pair of map germs. Remark 2.5.
A map germ f : ( R n , → ( R p , can be considered as a ( p, q ) -multi pairof function germs, i.e., take q = (1 , . . . , p -times. Definition 2.6.
A family of p +1 germs of bi-Lipschitz homeomorphisms of type ( h, H , ..., H p ) , where h : ( R n , → ( R n , and H i : ( R n × R q i , → ( R n × R q i , is called a contactfamily if H i ( x, y i ) = ( h ( x ) , ˜ H i ( x, y i )) for all i = 1 , ..., p , x ∈ R n , y i ∈ R q i . Definition 2.7.
Let q = ( q , ..., q p ) be a multi index. A ( p, q )-multi pair of bi-Lipschitzhomeomorphisms is a germ of a bi-Lipschitz homeomorphism, generated by a contactfamily of p + 1 bi-Lipschitz homeomorphisms in the form H : ( R n × R q × ... × R q p , → ( R n × R q × ... × R q p , , given by H ( x, y , ..., y p ) = ( h ( x ) , ˜ H ( x, y ) , ..., ˜ H p ( x, y p )) where ˜ H i , i = 1 , . . . , p , are as in Definition 2.6.We call the bi-Lipschitz map germ h of a common factor of the ( p, q )-multi pair ofbi-Lipschitz homeomorphisms . Definition 2.8.
Let q = ( q , ..., q p ) be a multi index. Two ( p, q ) -multi pairs of map germs F, G : ( R n , → ( R q × ... × R q p , are said to be m ulti- K -bi-Lipschitz equivalent if thereexists a ( p, q ) -multi pair of bi-Lipschitz homeomorphism H : ( R n × R q × ... × R q p , → ( R n × R q × ... × R q p , as in Definition 2.7, such that the following diagram is commutative: ( R n , h (cid:15) (cid:15) ( id n ,F ) / / ( R n × R q × ... × R q p , H (cid:15) (cid:15) π n / / ( R n , h (cid:15) (cid:15) ( R n , ( id n ,G ) / / ( R n × R q × ... × R q p , π n / / ( R n , where id n is the identity map germ of R n ; π n is the usual projection in R n ; h is thecommon factor of ( p, q ) -multi pair of bi-Lipschitz homeomorphisms and for all i = 1 , ..., p we have ˜ H i ( R n × { } q i ) = R n × { } q i , Remark 2.9.
Let q = ( q , ..., q p ) be the multi index and let F = ( F , ..., F p ) , G =( G , ..., G p ) : ( R n , → ( R q × ... × R q p , be two ( p, q ) -multi pairs of map germs. Then:i) When q = q ∈ N , the definition of multi- K -bi-Lipschitz equivalence coincides withthe definition of K -bi-Lipschitz equivalence.ii) If F and G are multi- K -bi-Lipschitz equivalent (as multi pair of map germs) then F and G are K -bi-Lipschitz equivalent (as map germs).2. If F and G are multi- K -bi-Lipschitz equivalent, then the map germs F i and G i are K -bi-Lipschitz equivalent, for all i = 1 , ..., p. Main results and finiteness property
Theorem 3.1.
Let f = ( f , ..., f p ) , g = ( g , ..., g p ) : ( R n , → ( R p , be two germs ofLipschitz maps. Suppose there exists a bi-Lipschitz homeomorphism H : ( R n × R p , → ( R n × R p , such that:i) The sets R n × { } p , R n × R p − × { } , R n × R p − × { } × R , ..., R n × { } × R p − areinvariant under H . In other words, H satisfies L. BIRBRAIR, J.C.F. COSTA, E.S. SENA FILHO, AND R. MENDES H ( R n × { } p ) = R n × { } p , H ( R n × R p − × { } ) = R n × R p − × { } , . . . ,H ( R n × { } × R p − ) = R n × { } × R p − . ii) H ( graph ( f )) = graph ( g ) .Then, the germs f and g are multi- K -bi-Lipschitz-equivalent. To prove the Theorem 3.1 we need some preliminary results.Let f = ( f , ..., f p ) , g = ( g , ..., g p ) : ( R n , → ( R p ,
0) be two germs of Lipschitz mapsand suppose there exists a germ of bi-Lipschitz homeomorphism H : ( R n × R p , → ( R n × R p , R n × R p : V + k = { ( x, y , ..., y p ) ∈ R n × R p | y k > } and V − k = { ( x, y , ..., y p ) ∈ R n × R p | y k < } , k = 1 , . . . , p . Assertion 1.
For each k = 1 , . . . , p , one of the following conditions holds:i) H ( V + k ) = V + k and H ( V − k ) = V − k orii) H ( V + k ) = V − k and H ( V − k ) = V + k , Proof.
If the condition does not hold, there are points a, b, c, d ∈ R n × R p , such that a ∈ V + k , b, c, d ∈ V − k with H ( a ) = b, H ( c ) = d . Consider a path λ : [0 , → V − k connecting the points b and d . Therefore, H − ◦ λ is a path in R n × R p , connecting thepoints a and c passing by R n × R k − × { } × R p − k . However, this is not possible because H ( R n × R k − × { } × R p − k ) = R n × R k − × { } × R p − k and H ( H − ◦ λ ) = λ ⊂ V − k . (cid:3) Assertion 2.
Let h : ( R n , → ( R n ,
0) be defined by h ( x ) = π n ( H ( x, f ( x ))). Then h is a bi-Lipschitz map germ. Proof.
Since g is a Lipschitz map, the projection π n | graf ( g ) is a bi-Lipschitz map. Bythe same argument, a map x ( x, f ( x )) is bi-Lipschiz. By hypothesis, the map H isbi-Lipschitz. Hence h is bi-Lipschitz. (cid:3) Assertion 3.
One of the following assertions is true:i) f i ( x ) ≈ g i ◦ h ( x ) orii) f i ( x ) ≈ − g i ◦ h ( x ), for all i = 1 , ..., p. Proof.
Since H is bi-Lipschitz, there exists two positives numbers c and c , such that c | f ( x ) | ≤k H ( x, f ( x ) , ..., f p ( x )) − H ( x, , f ( x ) , ..., f p ( x )) k≤ c | f ( x ) | . By above construction, k H ( x, f ( x ) , f ( x ) , ..., f p ( x )) − H ( x, , f ( x ) , ..., f p ( x )) k == k ( h ( x ) , g ◦ h ( x ) , ..., g p ◦ h ( x )) − H ( x, , f ( x ) , ..., f p ( x )) k≥ | g ◦ h ( x ) | . Therefore, | g ◦ h ( x ) | ≤ c | f ( x ) | . Using the same procedure for the map H − , we obtainthat | g ◦ h ( x ) | ≥ ˜ c | f ( x ) | , where ˜ c is a a real positive number.Then, ˜ c | f ( x ) | ≤ | g ◦ h ( x ) | ≤ c | f ( x ) | . By Assertion 3 we have that for all x ∈ R n , sign ( f ( x )) = sign ( g ◦ h ( x )) or sign ( f ( x )) = − sinal ( g ◦ h ( x )) . Therefore, f ( x ) ≈ g ◦ h ( x ) or f ( x ) ≈ − g ◦ h ( x ).Repeating the same process for all i = 1 , . . . , p , we obtain f i ( x ) ≈ g i ◦ h ( x ) or f i ( x ) ≈ − g i ◦ h ( x ) . (cid:3) Proof of Theorem 3.1.
By Assertion 3 and Lemma 2.3, follows that f i and g i are K -bi-Lipschitz equivalent, for all i = 1 , ..., p. Then, for each i = 1 , . . . , p , there exist a germ ofbi-Lipschitz homeomorphisms H i : ( R n × R , → ( R n × R , i ) H i ( x, y i ) = ( h ( x ) , ˜ H i ( x, y i )) , with ˜ H i : ( R n × R , → ( R , i = 1 , ..., p.ii ) H i ( x, f i ( x )) = ( h ( x ) , g i ◦ h ( x )), i = 1 , ..., p.iii ) H i ( R n × { } ) = R n × { } , i = 1 , ..., p. Define the map H : ( R n × R p , → ( R n × R p ,
0) given by H ( x, y , ..., y p ) = ( h ( x ) , ˜ H ( x, y ) , ..., ˜ H p ( x, y p )) . Considering q = (1 , . . . , p -times, we obtain that H is a ( p, q )-multi pair of bi-Lipschitz homomorphism, generated by the contact family of bi-Lipschitz homeomor-phisms, { h, H , ..., H p } . Moreover, i ) H ( x, f ( x ) , ..., f p ( x )) = ( h ( x ) , g ◦ h ( x ) , ..., g p ◦ h ( x )) and ii ) H ( R n × { } p ) = R n × { } p .Hence, f and g are multi- K -bi-Lipschitz equivalent. (cid:3) L. BIRBRAIR, J.C.F. COSTA, E.S. SENA FILHO, AND R. MENDES
Theorem 3.2. (Finiteness theorem) Let P k ( n, p ) be the set of all real polynomial mapgerms f = ( f , ..., f p ) : ( R n , → ( R p , , with degree of f , ..., f p less than or equal to k ∈ N . Then the set of the equivalence classes of P k ( n, p ) , with respect to multi- K -bi-Lipschitz-equivalence is finite.Proof. Let F : ( R n , → ( R p , ∈ P k ( n, p ). We associate to F the following family ofalgebraic subsets: X F = { R n ×{ } p , R n × R p − ×{ } , R n × R p − ×{ }× R , . . . , R n ×{ }× R p − , graf ( F ) } . Consider the set of all the families of algebraic subsets defined as above, that is, F = { X F , X F , . . . , X F i , . . . } . We define the following equivalence relation in F : X F i and X F j are called multi- V -bi-Lipschitz equivalent if there exists a bi-Lipschitzhomeomorphism H : ( R n × R p , → ( R n × R p ,
0) satisfying the conditions i) and ii) ofTheorem 3.1.By Vallete Lipschitz Triviality Theorem (cf. [6]), the number of equivalence classeswith respect to multi- V -bi-Lipschitz equivalence, is finite. Applying the Theorem 3.1follows that F i and F j are multi- K -bi-Lipschitz equivalent and so we conclude the proofof theorem. (cid:3) Corollary 3.3.
The set of the equivalence classes of P k ( n, p ) , with respect to K -bi-Lipschitz equivalence, is finite.Proof. Let
F, G ∈ P k ( n, p ). By Remark 2.5, F and G can be considered a ( p, q )-multipair of function germs (through its coordinate functions). Since multi- K -bi-Lipschitzequivalence admits the finiteness property, we can suppose that the corresponding familiesof coordinate functions are multi- K -bi-Lipschitz equivalent. Then one can consider a map H : ( R n × R p , → ( R n × R p ,
0) defined as follows: H = ( h, ˜ H ) , where h and ˜ H = ( ˜ H , . . . , ˜ H p ) such that h and ˜ H i are the maps obtained from thecorresponding multi- K -bi-Lipschitz equivalence of the coordinate functions. Then h and H are bi-Lipschitz homeomorphisms and satisfy the conditions of Definition 2.1 for F and G . Hence, F and G are K -bi-Lipschitz equivalent.Since in the proof we use the finiteness property of the multi- K -bi-Lipschitz equivalencewe have that the number of K -bi-Lipschitz classes is also finite. (cid:3) References [1] L. Birbrair, J.C.F. Costa, A. Fernandes, M. A. S. Ruas, K -bi-lipschitz equivalence of real function-germs . Proc. Amer. Math. Soc. 135 (2007), n. 4, 1089–1095. [2] L. Birbrair, J.C.F. Costa, E.S. Sena Filho, Finiteness theorem foi bi- C - K -equivalence of pairs ofmap germs . Preprint (2016).[3] J.C.F. Costa, H.A. Pedroso, M.J. Saia, A note on equivalence relations of pair of germs . RIMSKˆokyˆuroku Bessatsu B55 (2016), 17-39.[4] J.P. Dufour,
Sur la stabilit´e des diagrammes d’applications differentiables . Ann. Sci. Ecole Norm.Sup. 4a serie, 10 (1977).[5] J.-P. Henry, A. Parusinski,
Existence of moduli for bi-Lipschitz equivalence of analytic functions .Compositio Math. 136 (2003), n. 2, 217–235.[6] G. Vallete,
Hardt’s theorem: a bi-Lipschitz version . C. R. Acad. Sci. Paris, Ser. I 340 (2005), v. 12,895-900.[7] M.A.S. Ruas, G. Vallete, C and bi-Lipschitz K -equivalence of mappings . Math. Z. 269 (2011), no.1-2, 293-308.[8] A.M. Sitta, Boardman’s symbols for q -uples of differentiable applications . Rev. Mat. Estatstica 3(1985), 5-12. Departamento de Matem´atica, Universidade Federal do Cear´a (UFC), Campus do Pi-cici, Bloco 914, Cep. 60455-760. Fortaleza-Ce, Brasil
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