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Generic unfoldings with the same bifurcation diagram which are not (C0, C0)— equivalent
Nonlinear Analysis, Theory, Methods & Applications, Vol. 30, No. 3, pp. 1419--1428, 1997 Proc. 2nd World Congress of Nonlimear Analysts
Pergamon
PII: S0362-546X(97)00203-4
© 1997 ElsevierScienceLtd Printed in Great Britain. All rightsreserved 0362-546X/97 $17.00 + 0.00
G E N E R I C U N F O L D I N G S WITH THE SAME B I F U R C A T I O N D I A G R A M WHICH A R E N O T (c o, c ° ) - E Q U I V A L E N T H. ANNABI, M.L, ANNABI ~ and R. ROUSSARIE D~partement de Math~matiques, Facult~ des Sciences, 1060 Tunis, Tunisie. D~partement de Math~matiques, Universit~ de Bourgogne, Laboratoire de Topologie, UMR 5584 du CNRS, 21004 Dijon Cedex, France.
Key words and phrases
: Unfoldings, Line diffeomorphisms, Planar vector fields, Bifurcation dia-
gram, Topological equivalences.
1. INTRODUCTION
Let be (F~(x), 0) a germ at 0 of Coo diffeomorphisms on the line. Such a germ is also called a local family or unfolding at 0 C R x R k (where R k is the space of the parameter #) and the representative function is Coo in (x, #) E R x R k. We recall t h a t :
1) Two unfoldings ( F , , 0 ) and (G,,0) are (C°-fiber, Coo) conjugate if it exists a Coo diffeomorphism ~o of R k with ~o(0) = 0 and for any # E R k, some homeomorphism u~,(x) of R which conjugates F~ and G~(~,) in a neighborhood of 0 : Oe(,) o u~ = u~(~) o F~. 2) One says that these unfoldings are (C°, C°)-conjugate if ~ and the map (p, x) --+ u~(x) are continuous. T w o unfoldings are (C°-fiber, C°°)-conjugate if they have the same bifurcation diagram, up to a Coo diffeomorphism of the parameter space. In the (C°, C°)-conjugacy, one looks for a conjugacy which depends continuously on the parameter, and for this reason, one renounces to compare the bifurcation diagrams up to a Coo diffeomorphism, but simply up to some homeomorphism. It is easy to show that all the generic k-parameter unfoldings are (C°-fiber, COO)_ conjugate. We recall this result in the paragraph 3, for the generic 3-parameter unfoldings, the only case we want consider later. In [2], one showed that any such family (G,, 0), has an invariant I I ( G , ) for the (CG, C°)-conjugaey. This invariant is constructed from tile Mather's one for interval diffeomorphisms [1]. Our purpose here, is to use this invariant Pi(Gt, ) for constructing infinitely many unfoldings which are not (C°,C°)-conjugate. (This set of unfoldings has the cardinality : 2P~). In tile paragraph 2, we gives some properties of the Mather invariant for diffeomor-
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phisms. These properties are used to construct examples of non equivalent unfoldings in paragraphs 4, 5. This gives the following result, which is make more precised in theorem 1.1.. THEOREM 1.1. T h e r e exists a set parametrized by a functional space, of 2 by 2 non-(C °, C°)-conjugate generic 3-parameter local families. T h e non conjugacy of these unfoldings will be proved in paragraph 5 using the invariant I I ( G , ) . T h e construction of this invariant is recalled in the paragraph 3. One can define relations of (C°-fiber, C~°) and (C °, C°)-equivalence between vector field unfoldings, in a similar way. One has just to replace the conjugacy relation by the equivalence relation : one says that two vector fields X, Y are (topologically)-equivalent if there exists some homeomorphism sending oriented orbits of X onto oriented orbits of Y. T h e extension of the preceding result to vector fields is proved in paragraph 6 : THEOREM 1.2. T h e r e exists a set parametrized by a fimctional space, of 2 by 2 non-(C °, C°)-equivalent generic 4-parameter unfoldings of Hopf-Takens type. 2. THE MATHER INVARIANT. Let be I = [p, q] some closed interval in R . We adopt the following notations :
* S(I) is the set of Cc° diffeomorphisms f on I, with non zero jets at p and q and such that f(x) > x for any x in the interior of I. * Diff 0P is the set of C ~° diffeomorphisms a, not decreasing, which verify a(0) = 0 and a ( t + 1) = a ( t ) for V t e R . Such a diffeomorphism writes c~(t) = t + fl(t), where fl is a 1-periodic function with fi(0) = 0. * One considers the equivalence relation ~ in Diff 0P defined by : al~a2v=a3TER
such that f o r V t E R
c~(t)=al(t+T)-al(T).
We recall now the construction of the mapping :
p : S(I)
,M=DiffoR/
introduced by J. Mather. (See [1], [2] and [41). Let be f E S(I). Its restriction to the interval [p,q[ embeds in a unique C ~° flow Xd(t,x) [T]. In the same way, its restriction to ]p, q] embeds in a unique C~ flow Yy(t, x). Let be any a E]p, q[. One defines aa E Diff 0P by :
x (t, a)
=
a).
If b is another point in ]p, q[, it writes b = X I ( T , a) for some T E R . Its follows easily t h a t ab(t) = aa(t + T) - aa(T) for Vt E R. This means that a a ~ ~b : the class ~a of c~ defines a unique element p(f) in M. One calls it : the Mather invariant of f.
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REMARKS. * All elements in S(I) are C°-conjugate. * If f - I d has an order k at p and q, the flows XI and Yf are determined by their ( 2 k - 1)jets [3]. It follows from this that the Mather invariant is, in this case, a differentiable invariant for the C 2k-t-conjugacy. * T h e mapping p : S(I) ~ M is surjeetive, once one has fixed the jets of f to p and q [4]. This is obtained in the following way : the flows X, Y are determined, up on C°%conjugacy, by the given jets ; if c~ E Diff,, one can ghle by c~ a fundamental domain IX(0, a), X(1, a)] of X with some fundamental domain of Y ; by this construction we obtain a diffeomorphism f C S(I) with p(f) = ~. * Let us notice that f embeds in a CeC-flow if and only if p(f) is trivial in the sense it is equal to the class of identity. One says in this case that f is integrable. Let be fo some integrable diffeomorphism, equal to the time 1 of a vector field Z. One considers also a C~ function B on I, with a compact support in ]a, f°(a)[, where a E]p, q[. One supposes that f = fo + B C S(I). We are going to give an explicit expression for the Mather invariant of f. For this, one parametrizes the interval IF, q[ by R, using the trajectory Z(t, a) of Z. T h e function fo writes : f°(t) = t + 1 in this variable t E R and the function f writes : f(t) = t + 1 + t3(t) for some C°° function/3 with support in ]0, 1[. Let us write /~(t), the 1-periodic function which coincides with/3 on ]0, 1[. One has the following result : LEMMA 2.1.
p(f) = ~
for
a(t) = t + ~ ( t ) .
PROOF. One has to consider the flows Xf, YI on a fundamental domain [a, f(a)] parametrized by t E [0, 1]. Because B has its support in ]a, f°(a)[, one has I(a) = f°(a) and X f = Z on [a, f0 (a)]. T h e reason of the last claim is that X f on this interval is the direct image of Z by f0 on the interval [ ( f 0 ) - i (a), a]. From this, it follows t h a t :
X/(t,a) = t pour
t _< 1.
(2.1)
Also, the vector field YI on [a, f°(a)] is the inverse image of Z by f . This gives :
f(Yf(t,a)) = t + l
(2.2)
for t > 0 .
The Mather invariant of f is associated to a function a defined by : Y/(a(t), a) = Xf(t, a) = t, From (2.2), we have that f ( Y f ( a ( t ) , a ) ) = a(t) + 1. Using now (2.3) we obtain :
for
Z [0, 1I.
(2.3)
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Second World Congress of Nonlinear Analysts /
\
f ~ Y f ( c ~ ( t ) , a ) ) = f ( t ) = t + 1 + 3(t). From this it follows that c~(t) = t + 3(t) on [0,1]. Let be /5 a C a function with support equal to
, ~ . If one writes again by 3 the
1-periodic function which coincides with 3 on [0, 1], the function c~ = Id + 3 has a unique representative equal to Id + ~ where ~/ has a support equal to functions 3 with support equal to
,~
~, ~ . Then the set of
can be identify with the subset M~ of classes in
M which write Id + 3. Up to now, we will make this identification. 3.
TOPOLOGICAL
I N V A R I A N T F O R L O C A L FAMILIES.
We consider a generic 3-parameter local family (F~, 0). It follows, from the preparation theorem, t h a t there exists C a functions Q(x, #), a(#), b(#), c(p) with a(0) = b(0) = c(0) = 0, such that ( F , , 0) is C a - c o n j u g a t e t o : x + Q(x, # ) ( x 4 + a(/z)x 2 + b(p)x + c(p)). The genericity means that the map p --* (a(#), b(#), c(p)) has a maximal rank at 0. Up to a diffeomorphism of the parameter space, one can suppose that # -- (a, b, c). The bifurcation diagram of the family F . ( x ) = x + Q(x, . ) ( x 4 + ax
+ bx + c)
is given by the following equations :
x +ax 2 + bx+c= 0 4x a + 2ax + b = O. This diagram E, the swallow tail, is the same for all generic local families. We call (g) the line of self-intersections of E : for each # = (a, O, a2/4) E (~), a < O, the diffeomorphism F , has two double semi-stable fixed points Pa = - ( - a / 2 ) 1/2 and qa = (--a/2) 1/2. The restriction F . ] I a of F . on Ia = [pa, q~] belongs to S(I~). At the family (F.) one associates the mapping I I ( F . ) : (g) ~ M, defined by : II(F~,)(A) : p(Fx [Ix). One has proved in [2] that this mapping II so defined on the set of unfoldings ( F , ) is invariant by (C°, C°)-conjugacy. This means that if (F,) and (G,) are two such unfoldings which are (C °, C°)-conjugated by ¢(z, #) = (uu(z), qo(#)), then for VA E (e) one h a s :
H(F.)(X) = n ( a . ) ( v ( a ) ) . The germ of I I ( F , ) at the origin will be called topological invariant of the local family (F.). Let us notice that if the unfolding (Fu) is integrable (equal to the time 1 of a family of flows), then the mapping II(Fu) is constant and its value is equal to the class of identity.
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In the next paragraph we will construct local families with non-trivial invariant. T h e y will be obtained by perturbation of the integrable family H , , which is associated to the following families of vector fields, whose differential equation is equal to : 2g ~- Z # ( x )
4. A
-~- x 4 -4- a x
bx
2 +
+ c.
C O N S T R U C T I O N OF N O N - I N T E G R A B L E LOCAL FAMILIES.
THEOREM 4.1. Let b e / 3 E M~. T h e n there exists a C~ 3-parameter germ (B~(x), 0), with a zero jet at 0 (C~-flat), such that the generic local family H~ = H u + B~ has a topological invariant which is given by the function as above by a _< 0, near 0. PROOF. Let t)e P = ((:,:,
1 :,:
R,
r(t, a)
=
3(t)e},
where the line (g) is parametrized
(e)}.
(a,O, 4 ) ,
The line ( g ) b e i n g parametrized by , ( a ) :
a < O,
the surface P itself is
parametrized by (z, a) E R x R - . Let be:
h a ( x ) ~- Hp.(a)(X), z ( x , a) : Za(X ) : Zl~(a)(X ) a n d Za(t , x) -~- Z•(a)(t
Let us notice that for any a, the function ha belongs to pa = -
-
, qa =
-
S(I~,)
where Ia =
, x). [Pa,qa]
with
I n / a one chooses the fundamental domain Ja = [0, ha(0)]
and we c a l l : J = U ( J a x {a}). a
Let also be r(t, a) = z~(t, 0). This function r gives us a parametrization of the interior of Ia, by t E R . T h e interval J~ corresponds to t ~ [0, 1]. Let be t(x, a) the inverse function of r, defined on J. Let be some/3 E Me. One writes r ( t , a) = ~(t)e~. One defines b~(x,a) = r(t + 1 + r(t,a),a)- r(t + 1,a) for t = t(x,a) and (x,a) e J, and b~(x, a) = 0 elsewhere. The function h~(x) = h~(x) + be(x, a) has a restriction to Ia whose Mather invariant is the germ of the mapping t , • , F(t, a) (see paragraph 2). We are going to prove that the function ~ is C~ and C~°-flat at (0, x) for any x. T h e only non trivial thing to verify is the claim that bz is C~°-flat at (0, 0). We need some preliminaries. If f ( ~ l , 42) is a 2-variable C°~ function, one will write : 01sl
° s f - oc'
f
where s = (sl, s2) E N 2 and [ s I= sl + s 2 . For a continous function f with compact support in l:t 2, one writes : [If[[= Sup
I f ( ( 1 , (2) 1-
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Second World Congress of Nonlinear Analysts
R e t u r n now to the function F. We notice t h a t for all k E N , there exists Mk > 0 such that : VsEN 2
Indeed, cg*F = Ps
M..i~l
,
, t eZ where P~
H0*FI]_
(1) ,t
(4.1)
is p o l y n o m i a l in 1 of degree < 2 I s [ whose a
coefficients are C ~ functions in t, with s u p p o r t in [0, 1]. We are going now to e s t i m a t e the vector field z~(x) :
a~ z ~ ( x ) > 1--6 if
(x,a) EJand
-~
1
(4.2)
Indeed : a2
a
2
za(x) = ~ + ax ~ + T = (*~ + 3) and :
t(x, a) We have t ( - a , a )
_> ~
4
= ri
ds
~
Jo
Za(S) "
1
> 1 if a E l - x , 0 [ a n d ha(0) <] a I.
So, for all x E Ja one has : 1 a2 '
Finally, one has the following e s t i m a t e s on the partial derivatives of t(x, a). T h e r e exist c o n s t a n t s Ark > 0 and nk E N for Vk E N such t h a t : Vs E N 2
,
II0~tll
< NI, I l a I-n*~k •
(4.3)
Indeed, one proves by r e c u r r e n c e t h a t :
Vs E N 2,
z(r,a)OSt(x,a) = Q s ( z , a )
with r ( t ( x , a ) , a ) = x
(4.4)
where Qs is a p o l y n o m i a l in the p a r t i a l derivatives :
Onz(r,a)
, Omr(t,a)
, OVt(x,a) with I v l < l s l -
To prove (4.4) one differentiates the e q u a t i o n r(t(x, a), a) : x :
z(r, a ) . ~Ot7 (x,a) = 1 , 8t Or z(r,a)-~a (x,a) + ~a (t,a) = O. F o r m u l a s (4.5) i m p l y inequality (4.3) for k = 1, s o m e N1 a n d n l = 2.
(4.5)
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More generally, for any k the inequality (4.3) follows from (4.4) by recurrence, using formula (4.2). We can r e t u r n now to the proof of the flatness of b~ at (0, 0). T h e Taylor formula with integral remaining t e r m allows to write : b~(x, a) = r(t,a)
/0
z r(t( + 1 + s r(t,a)),%as~
(4.6)
where t = t(x, a) and (x, a) E J. T h e function b~ is the p r o d u c t of two functions :
~ ( x , a ) = r ( t ( x , a ) , a ) et e(~,a) =
z
r(t(x,a) + l + s r(t(z,a),a)),a
ds.
For the proof of flatness we can restrict ourself to the compact defined by (x, a) E J with 1
lal_<-.
4 We are going to prove first t h a t there exist some constants Ck > 0 such t h a t : Vs • N 2 , 110%11
_< e
2~ o
.
(4.7)
Indeed, for any s, 0s~/ is a s u m of monomials. Each of these rnonomials is a p r o d u c t of partial derivatives, Ù ' W ( t , a ) with ] m [> 0 (for one of the t e r m s at least), and O'~t(x,a). Using estimates (4.1) and (4.3), one obtains (4.7). We now prove t h a t there exist some constants Dk > 0 and integers nk such t h a t : vs e g 2
IIO~[I < Dk l a ]-nk •
(4.8)
Indeed, one has for any s,
T h e integrant in the above formula is a sum of monomials. Each of t h e m is a p r o d u c t of partial derivatives with one of the following form :
• ) O'~z(r(t + 1 + s F, a), a), which is bounded as a restriction of a continuous function on a compact,
• ) Onr(t + 1 + s F,a), which is bounded for the same reason because t • [0, 1] and
• ) ovr(t,a)
, Oqt(x,a) ,
multiplied by some power of s. T h e n the estimation (4.8) follows again from (4.1) and (4.3). One has to notice t h a t existence of factors of forms 0 p F(t, a) implies the necessity of a negative exponent in (4.8).
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We can now conclude the proof. For each s, O~b~ is a combination of monomials. Each of them is a product of partial derivatives of "7 and ~ and must contain a factor from "7- It follows from (10) and (11) that there exist constants Ek > 0 such that :
Vs E N 2,
IlO~b~ll < e ~
(4.9)
So, the function b~ is Coo-fiat at (0,0). It follows from this, that this function is C °o and fiat all along the axis 0x in P. One can extend it in a C °O function in R 4, which will be fiat. at 0 E R 4, by the formula :
B~(x)=b~ I (x)=b~(x,-]p])
with I P I = X/a2+ b2+c2.
This concludes the proof of theorem 1. 5. INFINITELY MANY NON CONJUGATE UNFOLDINC, S.
Let us suppose that (H~ ~, 0) and ,(H~. ,0) are (C°, C°)-conjugate by (g~(x), ~(#)). Let be ~(a) = ~(#(a)) the restriction of { to (~) parametrized by a, as above. It was proved in [/~] that I I ( H ~ ) ( a ) = I I ( H ~ ) ( j ( a ) ) for all a E (f), sufficiently near 0. Here, this formula writes : 1
l
ca ~l(t) = e¢-rz7 ~2(t) for a near 0. This implies that there exists A > 0 , such that ~2(t) -= Afl~(t). So, it is natural to introduce the projective space PM~ = Mc/~ where ~ is the relation : ~I~-(/~2 ~
~/~ > 0 such that ~2 = A~I.
This allows us to write our result in the following form : THEOREM 5.1. If 131 and f12 are 2 different classes in PMe, then the local families (H~', 0) and ~(H~2", 0~j are not (C6, C°)-conjugate. REMARKS. *) The space PMc has the cardinality 2 R. *) The families (H~, 0) are Coo but not analytic. It would be very interesting to obtain analytic examples. 6. NON EQUIVALENT HOPF-TAKENS UNFOLDINGS. We consider now Hopf-Takens generic unfoldings (X~, 0) of order 4-focus [4]. In complex coordinate z, the differential equations of such an unfolding can be written : --
z( 0 +
Iz I
I z 14
I z 16
z Isl + o(I z 110)
with p = (#0, #1, #2, #3) E R 4.
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The vector field X . has a return map P~ on the axis 0x+ = {x >__0}, in a neighborhood of the origin, with Pu(0) = 0. The study of (X., 0) up to topological equivalence reduces to the study of ( P . , 0) up to (C°-fibers, C °~) or (C°, C°) conjugacy. In [4], it was proved that the bifurcation diagram of the generic local family ( X . , 0) is Coo-conjugate to the bifurcation diagram of the polynomial family (X N, 0) with equation :
= z(,o +;,.1 I z I
I
Iz°lilz
s I).
This diagram contains a surface (L) on which the return map p N of X;N has two semi-stable distinct fixed points (corresponding to two semi-stable orbits of x N ) . One can parametrize (L) b y :
=
=
, #l=-PS,
w i t h ( s , p ) E {(s,p) E R 2 1 s > 0 ,
#2
+2p, #3= p>0
and s 2 > p } .
Let be a = (s,p). For all a E (L), let be xl(a), x2(a) the two semi-stables fixed points of P~(a) with 0 < xl(a) < x2(a) and I~ = [xl(a), x2(a)]. One can now define the topological invariant I I ( X , ) as a germ at 0 of mapping fi'om L toM: n(x.)(a)
=
I
For the family X y , the return map P~(x) is equal to X N (277, x) (where X N (t, x) is the trajectory of X N through z = (x,0)). So this invariant is trivial: II(X;,N)(a) -- Id. For any fl E Me, we are going to construct a perturbation Y f of X N, C °° and flat at {z = 0} such that the topological invariant of X ,fl = Xff + Y f is equal to fl(t)e-~3 (where [a [= p 2 x / ~ - ~ ) . For this, one first constructs a perturbation Bfi of the return map Pu, C°° and fiat along {# = 0} U {x = 0}. As in paragraph 4, one first constructs a family bfl(x, a) on R x L, such that the topological invariant of P~(a)(X) + b~(x, a) is equal to fl(t)e '~. Because 0 ~ Ia, the function ~ ( x , a) is fiat at {x = 0} for all a. From this, it follows that one can extend it in a C °o function Bfi ( B ~.(~) = bfl(x,a)), flat on {# = 0} tJ {x = 0}. It is now easy to suspend the map P~(x) = Pu(x) + Bfi(x) in a vector field Xfi --X N + Yff where Yff isCoo and flat on {z = O} U {it = 0}. We make this construction in polar coordinates (r, 0). Let be )~N, the family X N written in these coordinates. One can consider Pff as a composed map : Pff = Pu o (Id + K~) with Kfi(x) a C ~ function, flat on {x = 0} U {it = 0}.
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Let be W = {(r,O) I 0 < 0 < ~r,r > 0}. We consider a parametrized diffeomorphism We can choose ¢ , to be fiat to identity on {r = 0} and such that ~# I 0x+ = Id. This diffeomorphism defines a new parametrization of 0x_ = {x < 0} such that the transition of )(if, from 0x+ to 0x_ is equal to identity, and then such that the transition from 0x_ to 0x+ is equal to P~. One can now define a vector fiekl family S~ on W, such that the flow O (O(t, to), r(t, ro)) of ~ + S~, from the point (r0, 0), and for t e [0, 7r] is equal t o :
o(t, to) = t r(t, r0) ro+
(t)K
(ro)
where ¢(t) is a positive Coo, function, such that ¢(t) -= 0 in a neighborhood of t = 0 and ~b(t) ~ 1 in a neighborhood of t = 7r. Clearly enough, S~ is Coo and fiat on {r = 0} U {# = 0} and has its support in {(r, 0) 1 0 < 0 < T r , r>0}. Let be now )(~, the family which is equal to ~#,
~-~ + S
on W and is equal to X#N
elsewhere. This family is a C °O flat perturbation of ) ~itN along {r = O} U {Iz = O} It defines a flat perturbation of X Nu,at {z = 0}, with return map on Ox+, equal to P~.
ACKNOWLEDGMENTS
We thank F. Dumortier for helpfull discussions we have with him, during the preparation of this paper.
R E F E R E N CES
1. J. MATHER, Commutators of diffeomorphisms, Comm. (1973).
Math. Helv., vol.48, 195-233
. R. ROUSSARIE, Weak and continuous equivalences for families of line diffeomorphisms, In M.I. Camacho, M.J. Pacifico ~ F. Takens, Dynamical systems and bifurcation theory, Pitman Research Notes in Mathematics series 160, 377-385 (1987). 3. F. TAKENS, Unfoldings of certain singularities of vector fields. Generalized Hopf bifurcations, J.D.E. n°14, 476-493 (1973). 4. J.-CH. Y o c c o z , Centralisateurs et conjugaison diff~rentiable des diff~omorphismes du cercle, Th~se, Universit~ d'Orsay (1985) et "Petits diviseurs en dimension 1", Astdrisque, vol.231 (1996).
Pergamon
PII: S0362-546X(97)00203-4
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G E N E R I C U N F O L D I N G S WITH THE SAME B I F U R C A T I O N D I A G R A M WHICH A R E N O T (c o, c ° ) - E Q U I V A L E N T H. ANNABI, M.L, ANNABI ~ and R. ROUSSARIE D~partement de Math~matiques, Facult~ des Sciences, 1060 Tunis, Tunisie. D~partement de Math~matiques, Universit~ de Bourgogne, Laboratoire de Topologie, UMR 5584 du CNRS, 21004 Dijon Cedex, France.
Key words and phrases
: Unfoldings, Line diffeomorphisms, Planar vector fields, Bifurcation dia-
gram, Topological equivalences.
1. INTRODUCTION
Let be (F~(x), 0) a germ at 0 of Coo diffeomorphisms on the line. Such a germ is also called a local family or unfolding at 0 C R x R k (where R k is the space of the parameter #) and the representative function is Coo in (x, #) E R x R k. We recall t h a t :
1) Two unfoldings ( F , , 0 ) and (G,,0) are (C°-fiber, Coo) conjugate if it exists a Coo diffeomorphism ~o of R k with ~o(0) = 0 and for any # E R k, some homeomorphism u~,(x) of R which conjugates F~ and G~(~,) in a neighborhood of 0 : Oe(,) o u~ = u~(~) o F~. 2) One says that these unfoldings are (C°, C°)-conjugate if ~ and the map (p, x) --+ u~(x) are continuous. T w o unfoldings are (C°-fiber, C°°)-conjugate if they have the same bifurcation diagram, up to a Coo diffeomorphism of the parameter space. In the (C°, C°)-conjugacy, one looks for a conjugacy which depends continuously on the parameter, and for this reason, one renounces to compare the bifurcation diagrams up to a Coo diffeomorphism, but simply up to some homeomorphism. It is easy to show that all the generic k-parameter unfoldings are (C°-fiber, COO)_ conjugate. We recall this result in the paragraph 3, for the generic 3-parameter unfoldings, the only case we want consider later. In [2], one showed that any such family (G,, 0), has an invariant I I ( G , ) for the (CG, C°)-conjugaey. This invariant is constructed from tile Mather's one for interval diffeomorphisms [1]. Our purpose here, is to use this invariant Pi(Gt, ) for constructing infinitely many unfoldings which are not (C°,C°)-conjugate. (This set of unfoldings has the cardinality : 2P~). In tile paragraph 2, we gives some properties of the Mather invariant for diffeomor-
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phisms. These properties are used to construct examples of non equivalent unfoldings in paragraphs 4, 5. This gives the following result, which is make more precised in theorem 1.1.. THEOREM 1.1. T h e r e exists a set parametrized by a functional space, of 2 by 2 non-(C °, C°)-conjugate generic 3-parameter local families. T h e non conjugacy of these unfoldings will be proved in paragraph 5 using the invariant I I ( G , ) . T h e construction of this invariant is recalled in the paragraph 3. One can define relations of (C°-fiber, C~°) and (C °, C°)-equivalence between vector field unfoldings, in a similar way. One has just to replace the conjugacy relation by the equivalence relation : one says that two vector fields X, Y are (topologically)-equivalent if there exists some homeomorphism sending oriented orbits of X onto oriented orbits of Y. T h e extension of the preceding result to vector fields is proved in paragraph 6 : THEOREM 1.2. T h e r e exists a set parametrized by a fimctional space, of 2 by 2 non-(C °, C°)-equivalent generic 4-parameter unfoldings of Hopf-Takens type. 2. THE MATHER INVARIANT. Let be I = [p, q] some closed interval in R . We adopt the following notations :
* S(I) is the set of Cc° diffeomorphisms f on I, with non zero jets at p and q and such that f(x) > x for any x in the interior of I. * Diff 0P is the set of C ~° diffeomorphisms a, not decreasing, which verify a(0) = 0 and a ( t + 1) = a ( t ) for V t e R . Such a diffeomorphism writes c~(t) = t + fl(t), where fl is a 1-periodic function with fi(0) = 0. * One considers the equivalence relation ~ in Diff 0P defined by : al~a2v=a3TER
such that f o r V t E R
c~(t)=al(t+T)-al(T).
We recall now the construction of the mapping :
p : S(I)
,M=DiffoR/
introduced by J. Mather. (See [1], [2] and [41). Let be f E S(I). Its restriction to the interval [p,q[ embeds in a unique C ~° flow Xd(t,x) [T]. In the same way, its restriction to ]p, q] embeds in a unique C~ flow Yy(t, x). Let be any a E]p, q[. One defines aa E Diff 0P by :
x (t, a)
=
a).
If b is another point in ]p, q[, it writes b = X I ( T , a) for some T E R . Its follows easily t h a t ab(t) = aa(t + T) - aa(T) for Vt E R. This means that a a ~ ~b : the class ~a of c~ defines a unique element p(f) in M. One calls it : the Mather invariant of f.
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REMARKS. * All elements in S(I) are C°-conjugate. * If f - I d has an order k at p and q, the flows XI and Yf are determined by their ( 2 k - 1)jets [3]. It follows from this that the Mather invariant is, in this case, a differentiable invariant for the C 2k-t-conjugacy. * T h e mapping p : S(I) ~ M is surjeetive, once one has fixed the jets of f to p and q [4]. This is obtained in the following way : the flows X, Y are determined, up on C°%conjugacy, by the given jets ; if c~ E Diff,, one can ghle by c~ a fundamental domain IX(0, a), X(1, a)] of X with some fundamental domain of Y ; by this construction we obtain a diffeomorphism f C S(I) with p(f) = ~. * Let us notice that f embeds in a CeC-flow if and only if p(f) is trivial in the sense it is equal to the class of identity. One says in this case that f is integrable. Let be fo some integrable diffeomorphism, equal to the time 1 of a vector field Z. One considers also a C~ function B on I, with a compact support in ]a, f°(a)[, where a E]p, q[. One supposes that f = fo + B C S(I). We are going to give an explicit expression for the Mather invariant of f. For this, one parametrizes the interval IF, q[ by R, using the trajectory Z(t, a) of Z. T h e function fo writes : f°(t) = t + 1 in this variable t E R and the function f writes : f(t) = t + 1 + t3(t) for some C°° function/3 with support in ]0, 1[. Let us write /~(t), the 1-periodic function which coincides with/3 on ]0, 1[. One has the following result : LEMMA 2.1.
p(f) = ~
for
a(t) = t + ~ ( t ) .
PROOF. One has to consider the flows Xf, YI on a fundamental domain [a, f(a)] parametrized by t E [0, 1]. Because B has its support in ]a, f°(a)[, one has I(a) = f°(a) and X f = Z on [a, f0 (a)]. T h e reason of the last claim is that X f on this interval is the direct image of Z by f0 on the interval [ ( f 0 ) - i (a), a]. From this, it follows t h a t :
X/(t,a) = t pour
t _< 1.
(2.1)
Also, the vector field YI on [a, f°(a)] is the inverse image of Z by f . This gives :
f(Yf(t,a)) = t + l
(2.2)
for t > 0 .
The Mather invariant of f is associated to a function a defined by : Y/(a(t), a) = Xf(t, a) = t, From (2.2), we have that f ( Y f ( a ( t ) , a ) ) = a(t) + 1. Using now (2.3) we obtain :
for
Z [0, 1I.
(2.3)
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\
f ~ Y f ( c ~ ( t ) , a ) ) = f ( t ) = t + 1 + 3(t). From this it follows that c~(t) = t + 3(t) on [0,1]. Let be /5 a C a function with support equal to
, ~ . If one writes again by 3 the
1-periodic function which coincides with 3 on [0, 1], the function c~ = Id + 3 has a unique representative equal to Id + ~ where ~/ has a support equal to functions 3 with support equal to
,~
~, ~ . Then the set of
can be identify with the subset M~ of classes in
M which write Id + 3. Up to now, we will make this identification. 3.
TOPOLOGICAL
I N V A R I A N T F O R L O C A L FAMILIES.
We consider a generic 3-parameter local family (F~, 0). It follows, from the preparation theorem, t h a t there exists C a functions Q(x, #), a(#), b(#), c(p) with a(0) = b(0) = c(0) = 0, such that ( F , , 0) is C a - c o n j u g a t e t o : x + Q(x, # ) ( x 4 + a(/z)x 2 + b(p)x + c(p)). The genericity means that the map p --* (a(#), b(#), c(p)) has a maximal rank at 0. Up to a diffeomorphism of the parameter space, one can suppose that # -- (a, b, c). The bifurcation diagram of the family F . ( x ) = x + Q(x, . ) ( x 4 + ax
+ bx + c)
is given by the following equations :
x +ax 2 + bx+c= 0 4x a + 2ax + b = O. This diagram E, the swallow tail, is the same for all generic local families. We call (g) the line of self-intersections of E : for each # = (a, O, a2/4) E (~), a < O, the diffeomorphism F , has two double semi-stable fixed points Pa = - ( - a / 2 ) 1/2 and qa = (--a/2) 1/2. The restriction F . ] I a of F . on Ia = [pa, q~] belongs to S(I~). At the family (F.) one associates the mapping I I ( F . ) : (g) ~ M, defined by : II(F~,)(A) : p(Fx [Ix). One has proved in [2] that this mapping II so defined on the set of unfoldings ( F , ) is invariant by (C°, C°)-conjugacy. This means that if (F,) and (G,) are two such unfoldings which are (C °, C°)-conjugated by ¢(z, #) = (uu(z), qo(#)), then for VA E (e) one h a s :
H(F.)(X) = n ( a . ) ( v ( a ) ) . The germ of I I ( F , ) at the origin will be called topological invariant of the local family (F.). Let us notice that if the unfolding (Fu) is integrable (equal to the time 1 of a family of flows), then the mapping II(Fu) is constant and its value is equal to the class of identity.
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In the next paragraph we will construct local families with non-trivial invariant. T h e y will be obtained by perturbation of the integrable family H , , which is associated to the following families of vector fields, whose differential equation is equal to : 2g ~- Z # ( x )
4. A
-~- x 4 -4- a x
bx
2 +
+ c.
C O N S T R U C T I O N OF N O N - I N T E G R A B L E LOCAL FAMILIES.
THEOREM 4.1. Let b e / 3 E M~. T h e n there exists a C~ 3-parameter germ (B~(x), 0), with a zero jet at 0 (C~-flat), such that the generic local family H~ = H u + B~ has a topological invariant which is given by the function as above by a _< 0, near 0. PROOF. Let t)e P = ((:,:,
1 :,:
R,
r(t, a)
=
3(t)e},
where the line (g) is parametrized
(e)}.
(a,O, 4 ) ,
The line ( g ) b e i n g parametrized by , ( a ) :
a < O,
the surface P itself is
parametrized by (z, a) E R x R - . Let be:
h a ( x ) ~- Hp.(a)(X), z ( x , a) : Za(X ) : Zl~(a)(X ) a n d Za(t , x) -~- Z•(a)(t
Let us notice that for any a, the function ha belongs to pa = -
-
, qa =
-
S(I~,)
where Ia =
, x). [Pa,qa]
with
I n / a one chooses the fundamental domain Ja = [0, ha(0)]
and we c a l l : J = U ( J a x {a}). a
Let also be r(t, a) = z~(t, 0). This function r gives us a parametrization of the interior of Ia, by t E R . T h e interval J~ corresponds to t ~ [0, 1]. Let be t(x, a) the inverse function of r, defined on J. Let be some/3 E Me. One writes r ( t , a) = ~(t)e~. One defines b~(x,a) = r(t + 1 + r(t,a),a)- r(t + 1,a) for t = t(x,a) and (x,a) e J, and b~(x, a) = 0 elsewhere. The function h~(x) = h~(x) + be(x, a) has a restriction to Ia whose Mather invariant is the germ of the mapping t , • , F(t, a) (see paragraph 2). We are going to prove that the function ~ is C~ and C~°-flat at (0, x) for any x. T h e only non trivial thing to verify is the claim that bz is C~°-flat at (0, 0). We need some preliminaries. If f ( ~ l , 42) is a 2-variable C°~ function, one will write : 01sl
° s f - oc'
f
where s = (sl, s2) E N 2 and [ s I= sl + s 2 . For a continous function f with compact support in l:t 2, one writes : [If[[= Sup
I f ( ( 1 , (2) 1-
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R e t u r n now to the function F. We notice t h a t for all k E N , there exists Mk > 0 such that : VsEN 2
Indeed, cg*F = Ps
M..i~l
,
, t eZ where P~
H0*FI]_
(1) ,t
(4.1)
is p o l y n o m i a l in 1 of degree < 2 I s [ whose a
coefficients are C ~ functions in t, with s u p p o r t in [0, 1]. We are going now to e s t i m a t e the vector field z~(x) :
a~ z ~ ( x ) > 1--6 if
(x,a) EJand
-~
1
(4.2)
Indeed : a2
a
2
za(x) = ~ + ax ~ + T = (*~ + 3) and :
t(x, a) We have t ( - a , a )
_> ~
4
= ri
ds
~
Jo
Za(S) "
1
> 1 if a E l - x , 0 [ a n d ha(0) <] a I.
So, for all x E Ja one has : 1 a2 '
Finally, one has the following e s t i m a t e s on the partial derivatives of t(x, a). T h e r e exist c o n s t a n t s Ark > 0 and nk E N for Vk E N such t h a t : Vs E N 2
,
II0~tll
< NI, I l a I-n*~k •
(4.3)
Indeed, one proves by r e c u r r e n c e t h a t :
Vs E N 2,
z(r,a)OSt(x,a) = Q s ( z , a )
with r ( t ( x , a ) , a ) = x
(4.4)
where Qs is a p o l y n o m i a l in the p a r t i a l derivatives :
Onz(r,a)
, Omr(t,a)
, OVt(x,a) with I v l < l s l -
To prove (4.4) one differentiates the e q u a t i o n r(t(x, a), a) : x :
z(r, a ) . ~Ot7 (x,a) = 1 , 8t Or z(r,a)-~a (x,a) + ~a (t,a) = O. F o r m u l a s (4.5) i m p l y inequality (4.3) for k = 1, s o m e N1 a n d n l = 2.
(4.5)
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More generally, for any k the inequality (4.3) follows from (4.4) by recurrence, using formula (4.2). We can r e t u r n now to the proof of the flatness of b~ at (0, 0). T h e Taylor formula with integral remaining t e r m allows to write : b~(x, a) = r(t,a)
/0
z r(t( + 1 + s r(t,a)),%as~
(4.6)
where t = t(x, a) and (x, a) E J. T h e function b~ is the p r o d u c t of two functions :
~ ( x , a ) = r ( t ( x , a ) , a ) et e(~,a) =
z
r(t(x,a) + l + s r(t(z,a),a)),a
ds.
For the proof of flatness we can restrict ourself to the compact defined by (x, a) E J with 1
lal_<-.
4 We are going to prove first t h a t there exist some constants Ck > 0 such t h a t : Vs • N 2 , 110%11
_< e
2~ o
.
(4.7)
Indeed, for any s, 0s~/ is a s u m of monomials. Each of these rnonomials is a p r o d u c t of partial derivatives, Ù ' W ( t , a ) with ] m [> 0 (for one of the t e r m s at least), and O'~t(x,a). Using estimates (4.1) and (4.3), one obtains (4.7). We now prove t h a t there exist some constants Dk > 0 and integers nk such t h a t : vs e g 2
IIO~[I < Dk l a ]-nk •
(4.8)
Indeed, one has for any s,
T h e integrant in the above formula is a sum of monomials. Each of t h e m is a p r o d u c t of partial derivatives with one of the following form :
• ) O'~z(r(t + 1 + s F, a), a), which is bounded as a restriction of a continuous function on a compact,
• ) Onr(t + 1 + s F,a), which is bounded for the same reason because t • [0, 1] and
• ) ovr(t,a)
, Oqt(x,a) ,
multiplied by some power of s. T h e n the estimation (4.8) follows again from (4.1) and (4.3). One has to notice t h a t existence of factors of forms 0 p F(t, a) implies the necessity of a negative exponent in (4.8).
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We can now conclude the proof. For each s, O~b~ is a combination of monomials. Each of them is a product of partial derivatives of "7 and ~ and must contain a factor from "7- It follows from (10) and (11) that there exist constants Ek > 0 such that :
Vs E N 2,
IlO~b~ll < e ~
(4.9)
So, the function b~ is Coo-fiat at (0,0). It follows from this, that this function is C °o and fiat all along the axis 0x in P. One can extend it in a C °O function in R 4, which will be fiat. at 0 E R 4, by the formula :
B~(x)=b~ I (x)=b~(x,-]p])
with I P I = X/a2+ b2+c2.
This concludes the proof of theorem 1. 5. INFINITELY MANY NON CONJUGATE UNFOLDINC, S.
Let us suppose that (H~ ~, 0) and ,(H~. ,0) are (C°, C°)-conjugate by (g~(x), ~(#)). Let be ~(a) = ~(#(a)) the restriction of { to (~) parametrized by a, as above. It was proved in [/~] that I I ( H ~ ) ( a ) = I I ( H ~ ) ( j ( a ) ) for all a E (f), sufficiently near 0. Here, this formula writes : 1
l
ca ~l(t) = e¢-rz7 ~2(t) for a near 0. This implies that there exists A > 0 , such that ~2(t) -= Afl~(t). So, it is natural to introduce the projective space PM~ = Mc/~ where ~ is the relation : ~I~-(/~2 ~
~/~ > 0 such that ~2 = A~I.
This allows us to write our result in the following form : THEOREM 5.1. If 131 and f12 are 2 different classes in PMe, then the local families (H~', 0) and ~(H~2", 0~j are not (C6, C°)-conjugate. REMARKS. *) The space PMc has the cardinality 2 R. *) The families (H~, 0) are Coo but not analytic. It would be very interesting to obtain analytic examples. 6. NON EQUIVALENT HOPF-TAKENS UNFOLDINGS. We consider now Hopf-Takens generic unfoldings (X~, 0) of order 4-focus [4]. In complex coordinate z, the differential equations of such an unfolding can be written : --
z( 0 +
Iz I
I z 14
I z 16
z Isl + o(I z 110)
with p = (#0, #1, #2, #3) E R 4.
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The vector field X . has a return map P~ on the axis 0x+ = {x >__0}, in a neighborhood of the origin, with Pu(0) = 0. The study of (X., 0) up to topological equivalence reduces to the study of ( P . , 0) up to (C°-fibers, C °~) or (C°, C°) conjugacy. In [4], it was proved that the bifurcation diagram of the generic local family ( X . , 0) is Coo-conjugate to the bifurcation diagram of the polynomial family (X N, 0) with equation :
= z(,o +;,.1 I z I
I
Iz°lilz
s I).
This diagram contains a surface (L) on which the return map p N of X;N has two semi-stable distinct fixed points (corresponding to two semi-stable orbits of x N ) . One can parametrize (L) b y :
=
=
, #l=-PS,
w i t h ( s , p ) E {(s,p) E R 2 1 s > 0 ,
#2
+2p, #3= p>0
and s 2 > p } .
Let be a = (s,p). For all a E (L), let be xl(a), x2(a) the two semi-stables fixed points of P~(a) with 0 < xl(a) < x2(a) and I~ = [xl(a), x2(a)]. One can now define the topological invariant I I ( X , ) as a germ at 0 of mapping fi'om L toM: n(x.)(a)
=
I
For the family X y , the return map P~(x) is equal to X N (277, x) (where X N (t, x) is the trajectory of X N through z = (x,0)). So this invariant is trivial: II(X;,N)(a) -- Id. For any fl E Me, we are going to construct a perturbation Y f of X N, C °° and flat at {z = 0} such that the topological invariant of X ,fl = Xff + Y f is equal to fl(t)e-~3 (where [a [= p 2 x / ~ - ~ ) . For this, one first constructs a perturbation Bfi of the return map Pu, C°° and fiat along {# = 0} U {x = 0}. As in paragraph 4, one first constructs a family bfl(x, a) on R x L, such that the topological invariant of P~(a)(X) + b~(x, a) is equal to fl(t)e '~. Because 0 ~ Ia, the function ~ ( x , a) is fiat at {x = 0} for all a. From this, it follows that one can extend it in a C °o function Bfi ( B ~.(~) = bfl(x,a)), flat on {# = 0} tJ {x = 0}. It is now easy to suspend the map P~(x) = Pu(x) + Bfi(x) in a vector field Xfi --X N + Yff where Yff isCoo and flat on {z = O} U {it = 0}. We make this construction in polar coordinates (r, 0). Let be )~N, the family X N written in these coordinates. One can consider Pff as a composed map : Pff = Pu o (Id + K~) with Kfi(x) a C ~ function, flat on {x = 0} U {it = 0}.
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Let be W = {(r,O) I 0 < 0 < ~r,r > 0}. We consider a parametrized diffeomorphism We can choose ¢ , to be fiat to identity on {r = 0} and such that ~# I 0x+ = Id. This diffeomorphism defines a new parametrization of 0x_ = {x < 0} such that the transition of )(if, from 0x+ to 0x_ is equal to identity, and then such that the transition from 0x_ to 0x+ is equal to P~. One can now define a vector fiekl family S~ on W, such that the flow O (O(t, to), r(t, ro)) of ~ + S~, from the point (r0, 0), and for t e [0, 7r] is equal t o :
o(t, to) = t r(t, r0) ro+
(t)K
(ro)
where ¢(t) is a positive Coo, function, such that ¢(t) -= 0 in a neighborhood of t = 0 and ~b(t) ~ 1 in a neighborhood of t = 7r. Clearly enough, S~ is Coo and fiat on {r = 0} U {# = 0} and has its support in {(r, 0) 1 0 < 0 < T r , r>0}. Let be now )(~, the family which is equal to ~#,
~-~ + S
on W and is equal to X#N
elsewhere. This family is a C °O flat perturbation of ) ~itN along {r = O} U {Iz = O} It defines a flat perturbation of X Nu,at {z = 0}, with return map on Ox+, equal to P~.
ACKNOWLEDGMENTS
We thank F. Dumortier for helpfull discussions we have with him, during the preparation of this paper.
R E F E R E N CES
1. J. MATHER, Commutators of diffeomorphisms, Comm. (1973).
Math. Helv., vol.48, 195-233
. R. ROUSSARIE, Weak and continuous equivalences for families of line diffeomorphisms, In M.I. Camacho, M.J. Pacifico ~ F. Takens, Dynamical systems and bifurcation theory, Pitman Research Notes in Mathematics series 160, 377-385 (1987). 3. F. TAKENS, Unfoldings of certain singularities of vector fields. Generalized Hopf bifurcations, J.D.E. n°14, 476-493 (1973). 4. J.-CH. Y o c c o z , Centralisateurs et conjugaison diff~rentiable des diff~omorphismes du cercle, Th~se, Universit~ d'Orsay (1985) et "Petits diviseurs en dimension 1", Astdrisque, vol.231 (1996).