Codimension-two reflection and non-hyperbolic invariant lines

Nonlinearity I (1994) 109-184. Printed in the UK

-

Codimension-two reflection and non-hyperbolicinvariant lines J Gheiner Mathematics Institute, Federal University of Rio de Janeiro, PO Box 68530, CEP 21945, Rio de Janeiro. Brazil Received 29 October 1991, in final form 16 August 1993 Recommended by R S Macby

Abstract The generic unfolding of codimension-two reflection periodic orbit (two eigenvalues ?cl and the others with norm different from 1) embedded in a Morse-Smale diffeomorphismis analysed. L a d and global bifmations are desaibed. Non-hyperbolic invariant manifolds are constructed as an essential twl to prove local and global stability of the unfolding. AMs classification scheme numbers: 58Fl4, 58F10

Contents Page 1. Introduction 2. Definitions

3. Normal form 4. ‘Blowing-up’ at /I = (0, 0) 5. Unfolding of periodic orbits 5.1. Fixed points 5.2. Periodic orbits 6. Bifurcation diagrams and phase portraits 7. Obstructions for stability 8. Reflection with U = -1 and 00 < -1 8.1. Simplification of the normal form 8.2. Definition of sub-central manifold 8.3. Existence of invariant sub-central manifolds and adapted vector fields 8.4. Inclination proposition 8.5. Cz control of transversal sections 8.6. Position of the strong stable manifold of the saddle-node 8.7. C’ and Cz behaviour of the strong stable manifold of the saddlenode 9. Global analysis-ase < -1 and v = -1 9.1. Introduction 9.2. Proof of theorem 1 9.3. Proof of theorem 2 and corollary References

0951-7715/94/010109+76$07.50

0 1994 IOP Publishing Ltd and LMS Publishing Lid

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109

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1. Introduction

One of the general aims of the theory of dynamical systems (diffeomorphisms, endomorphisms, vector fields) is to give a dynamical classification of all systems. Two dynamical systems belong to the same class if they have the same dynamical behaviour. This happens when there is a homeomorphism sending orbits of a vector field onto orbits of the other one and, in the case of transformations, there is a topological conjugacy between them. If a dynamical system belongs to the interior of its class (respect to the C‘ topology, r l), then it is called (structurally) stable. Morse-Smale’s systems are the simplest among the stable ones [P-SM] and they have been playing a crucial role in development of modern dynamics. Bifurcating MorseSmale systems yields a wide range of dynamically distinct systems. Generically, one can bifurcate from the Morse-Smale world, through a one-parameter family, either losing hyperbolicity of one of the periodic orbits or creating non-transversal intersection between stable and unstable manifolds of saddles. This approach was taken in [N-P-V, [N-PI with the hypothesis that the limit set of the first bifurcation is simple (finite number of periodic orbits). Hyperbolic theory is used to obtain invariant manifolds, as in [H-P-SI, and to control the behaviour of unstable foliations, as in the hlemma [PI. In all cases where stability of the family is achieved, hyperbolicity is used, although the bifurcating periodic orbits are not hyperbolic. Hyperbolicity is the key concept to achieve stability in one parameter families without cycles. In [N-PI and [P-TI bifurcations from cycles are analysed. In this article we investigate the unfolding of a diffeomorphism Fo on a compact manifold M, which is Morse-Smale, except for one periodic orbit being a reflection (two eigenvalues equal to i l and the others off the unit circle). This diffeomorphism is considered as the starting diffeomorphism in a generic two parameter Cm family of C m diffeomorphisms Fw, p G U where U is an open neighbourhood of (0.0) in W2. For the stable two-parameter families of diffeomorphisms, which will be analysed, invariant manifolds cannot be obtained through hyperbolic theory. We construct nonhyperbolic invariant manifolds and study their behaviour and the behaviour of the nonhyperbolic transversal foliations (that do not satisfy the h-lemma). We obtain good control of these elements, with respect to the C’ topology, inside a small neighbourhood of the bifurcating periodic orbit. This work belongs #toa broader context of trying to get a global picture of all the codimension-two bifurcations for diffeomorphisms. In this area other researches have been developed, such as deep local analysis of the codimension-two Hopf bifurcation in [C], semi-local stability analysis of a saddle-node with quadratic tangency in [B-P] and global analysis of generalized saddlenodes and flips in [GI, [G2]. The notions of stability and weak (mild) stability for families are the standard ones (see [N-P-TI or section 2). For p = 0, Fo is classified in six different cases, according to the topological swucture and orientation of the associated vector field after blowing-up. In the first three cases a codimension 1 Hopf curve shows up in the bifurcation diagram. Consequently the families are not locally weakly stable. In the fourth case there exist curves of quasi-transversal intersection between saddles that are born from the reflection. This is an obstruction to achieving stability, as moduli of stability show up in these cases. In the fifth case we conjecture local unstability of the family due to the oscillation of invariant manifolds of saddle-nodes and saddles as I*. + 0. Only local weak stability is guaranteed. We recall the notions of cycle and criticality. A cycle is a sequence po. .. .,p k of periodic orbits with pi # p j for 0 < i < j c k , PO = pk , pi+* C w”(pi) for 0 6 i < k. Let p be the reflection orbit. In the sixth case there exists a unique &invariant differentiable

>

Re@ction and non-hyperbolic invariant lines

111

foliation Fss of W'(p, Fo) containing W""(p,Fo)as a leaf. We say that p is s-critical if there exists a hyperbolic periodic orbit q such that Wu(q,Fo)intersects some leaf of F" non-transversally. Analogously, we define u-criticality. We say that p is semi-critical if it is s- or u-critical. We say that p is bi-critical if it is s- and u-critical. We say that p is non-critical if it is not semi-critical. The sixth caSe constitutes an open set in the space of two parameter families. For this case the following theorem gives conditions for global stability.

Theorem A. Suppose Fo has no cycles. (a) F is weakly stable i f p is semi-critical. (b) F is stable i f p is non-critical. (c) If dimension ( M ) = 2, F is stable ifand only i f p is non-critical. The following theorem describes the global bifurcations when the hypothesis of theorem A are not satisfied.

Theorem B . (a) r f FO has cycle then Fo has a Smale's horseshoe as invariant subset. (b) If p is bi-critical there exist, in the bifurcation diagram, Cm curves corresponding to quasi-transversal intersections between stable and unstable manifoldr of periodic orbits (that do not unfold from p). p = (0,O) is accumulation point qf these curves. F is not weakly stable. In the sixth case the union of the stable and unstable manifolds of the reflection p restricted to the central manifold of FO at p constitute a Cm one-dimensional invariant manifold W r . The local dynamics of F: at the central manifold of p is drawn in figure I.

\

e------

.

/

4

t-

. c

H

.e I

-

-,

.

4---

Let X be the unique Cm vector field adapted to Fa/ WF (this is the vector field defined at WF such that XI = Fo/ W r where XI is the time-one map of the X-flow). Let G, be another two-parameter family of diffeomorphisms such that CO has a reflection orbit j and 6F ' is the corresponding invariant manifold. Let .? be the vector field adapted to Go/WF. Suppose there exists a conjugacy H : U x M H U x M between Fw and G, ( H ( p ,x ) = ( h ( p ) ,H,(x)) and h(0,O) = (0, 0)). The following result of rigidity is parallel to the saddle-node case:

Proposition. HO : (W,"" - p ) b+

(WF- j)is a Cm diffeomorphism and Ho*(Xo) = i o .

So it is natural to ask for the existence of a C' one-dimensional manifold invariant by F,, varying continuously with p in the C' topology and coincident with W r for p = 0. We

cannot expect to construct continuous conjngacies without these manifolds. The following theorem answer this question.

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J Gheiner

Theorem C. For all p E U there exists a Cz, one-dimensional, F,-invariant submanifold inside the central manifold of F,. These submnnifolds, denominated Wr,contain thefrred points of Fp They vary continuously with p in the C' topology and are uniformly bounded in the C2 topology.

The C2 limitation allows, by [Y], the construction of a family X, of C' vector fields, defined at WF,adapted to F,, and varying continuously with p. These vector fields permit us to extend the rigidity at the p = 0 level to the others p E U. Let C be a transversal section to Wc.Although the h-lemma [PI is not valid in this situation we show that E, under iterations of F;, n E M,obeys a weaker inclination lemma. This result together with (heorem C enable us to consbuct the conjugacies needed to proof theorem A. The difficulty to prove theorem C and the inclination lemma comes from the central manifold being a neutral (non-hyperbolic) ambient. There is a big subset of U, labelled by (l),such that for p E (l),F, has no local periodic orbits. So hyperbolic invariant manifold theory [H-P-SI cannot be applied.

2. Definitions M = compact Cm manifold Dif(M) = (Cm diffeomorphisms of M with the Cm topology] Let Fo E Dif(M) and p be a fixed point. Suppose DFo(p) has all eigenvalues with norm different from 1 except two. These eigenvalues are 1 and -1 and there exists a surface S, Fa- invariant, tangent at p to the correspondent eigenspace, such that the 2-jet of FOf S at p has non-null resonant (symmetric) terms. In this case p is called a reflection. There exist analogue definitions for a p periodic orbit. (I = open neighbourhood of 0 E Rz barameter space) &(M) = (Cmmaps F : U x M H U x M such that F(p,p ) = F(p,F,(p)) and F, E Dif(M)] We endow 3u(M)with the Cm weak topology (compact-open topology). Elements of F* are denominated two-parameter families of diffeomorphisms and are denoted by (F,] or F. A = (F E 3u(M)such that FO has finite hyperbolic h i t set except for one periodic orbit p being a reflection. Transversality between stable, unstable manifolds of the periodic orbits and strong stable, strong unstable manifolds of p and generic unfolding (see section 3) of the non-hyperbolic orbit are required]. Let F and G be elements of 3 " ( M ) . They are topologically conjugate (or, simply, conjugate) if there are open neighborhoods UF and UG of 0, contained in U, and a homeomorphism H : U.F x M H U, x M such that: and H, : M H M are (a) H ( p , p ) = ( @ ( p ) , H , ( p ) ) where @ : UF H homeomorphisms (b) H o F = G o H. We say that F and G are weakly conjugate if in the precedent definition H, does not depend continuously on p. We say that F at (pa, p ) is locally conjugate to G a t (EO,j)if there are neighbourhoods, V of ( p 0 , p ) and of (j&,j) contained in U x M , and a conjugacy H : V H between Ff V and G f P. Analogously we define local weak conjugacies. Conjugacy, weak conjugacy, local conjugacy, local weak conjugacy are equivalence relations. F is stable if

v

v

Reflection and non-hyperbolic invariant lines

113

it belongs to the interior of its equivalence class by conjugacy. Similarly we define weak stabiliry. F is locally stable at (PO,p ) if for any neighbourhood V of (PO, p ) in U x M there exists a neighbourhood W of F in 3u(M)such that for every G in W there exist VF c V , VG c V , neighbourhoods of (m,p ) , and a homeomorphism H : VF H Vo of the type H ( w , q ) = (@(!A H,(q)), such that H 0 ( F / V F ) = (G/ VG)0 H . If H, does not depend continuously on p we say that F is locally weakly stable. The term local in general will mean ‘at a neighbourhood of (0, p ) E U x M’ where p is .the reflection periodic orbit. o(il(x, y)Ilk) is a term that divided by Il(x, y)ll’ remains bounded above when (n,y) + (0,O). Analogue definitions hold for o(llplIk),o(ll(p, ( x , y))\lk)).o(1) means bounded term. Id and I are abbreviations for identity function. G means approximately equal (the difference of the terms divided by one of them is less than E ) .

3. Normal form

Let F E A, such that FO has a reflection at p . In this section we will work only on local results at the central manifold of F, at (0, p ) . So we can assume that Fp is a two-parameter family of diffeomorphisms of R2 and p is identified with (0,O). As central manifolds, in general, are not Cm, [F,] are C k for laxge k . Let Q = [ [ @ p ] p E ~ , $ p : BZ H Bz C’ local diffeomorphisms (defined at a neighbourhood of (0,O)) such that $0(0,0) = (0, 0), and 4 : U xRz H R2$(p,x) = $,(x)} Let = [@ : U I H U C‘ local diffeomorphisms defined at open sets U, such that (0,O) E U 1c U and @(O, 0) = (0,O)). We assume that F satisfies generic conditions on the dependence of the coefficients on p. (See [GIfor details).

Proposition (Normal form). There exist (4,) E @ and @ E ‘4J such that conjugating (F,) by (4,) and afterward changing the parameter coordinates by @ the following normal form is attained: FN(x, Y ) = (pi + ~ + ~ ~ ~ + ~ ( ~ ~ ) ~ ~ + ~ ~ ~ + b ( ~ ) x) p~ g+( xc 3(y)+o(lI ~ ) x (~x ~Y +) Il4h o(lI~l/I -Y +0(llPlIZ)Y+ X Y

where

U

+O(ll~LII)P21;.4(X?

Y)+o(lIPlI)P;4(x, y)+o(ll(x, Y)1I5))

= kl, a(0) # 0

Observation 1. P;;; degree i, j , k.

are resonant

’ (non-resonant

polynomials with monomials of

Definition. Let en k = 1,2 be equal to (1, 0) or (0, 1). A monomial x’yjek, with i + j is called resonant if it is invariant by

The proposition is demonstrated through the usual techniques as in

[a.

>2

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J Gheiner

4. ‘Blowing up’ at 1-1= (0,O)

Setting ,LL = (0,O) in the normal form we obtain

F O ( X , Y )= (x +%x2+

vy2+box3 + W Y ~ + O ( I I ( X ,

y)Il4), -Y

+XY

+o(ll(x, ~)ll’))

where v = &l;a0 = a(O.0); bo =b(O, 0); CO = c(0,O). By a theorem of Takens [TI there exists a vector field X in a neighbourhood of (0,O) E R2 such that if XI is the time one map of the associated flow and PO,21 are, respectively, the infinite jet of 2% and X1 at (0,O)and R = ( ’ 0 -1O )

then through a C m change of coordinates (conjugacy) we have I?o = R f , . We can choose X R-invariant. From the expression of FO we have X ( X ,Y ) = ( a d

+ vy2 +

O(II(X.

Y)II~),-XY

+ Y . O(II(X, Y)II’))

= AI

ao # o (1)

We use blowing-up far X (as in PI, [D-R-RI) to obtain a topological classification of Fo. After blowing-up the vector field will be denoted by We denote the formal infinite jet of x at (o,o)as 2 . We use the following proposition

x.

Proposition 1 [D-R-RI. Let G be a local diffeomorphism of R2 (in a neighbourhood of (0,O)) such that (0,O) is afuted point. Suppose that formally 5 = R o 21with R semisimple and R” = 1 (identity)for some n E N. Let (0,O) be a singularity of % of Lojasiewicz type with a characteristic orbit. Suppose that afier the blowing-up of 2 we obtain a vector field that has only hyperbolic singularities. Then G is CO conjugated to R o X I (where X is a R-invariant representant of 2).$ moreover, % has no elliptic sectors, then G is Cm conjugated to R o XI. The conjugacy H satisfies j,(H I)(O, 0) = 0.

-

When the diffeomorphismis C’ for r large but not Cm there is an equivalent proposition

ID-RI considering the formal equalities restricted to the r-jets and substituting the Cm conjugacy by conjugacy where p ( r ) 3 00 as r After blowing-up we have

3 W.

a - [sinar((ao+ ~ ) c o s ~ a + u s i n +o(r)sina]-. ~a) aa Setting r = 0 we have the equation of the singularities:

s i n a ( ( a o + ~ ) c o s ~ a + w s i n * a= )O In order to obtain hyperbolic singularities we make the additional generic assumption: no # -1 We examine six cases:

~

Reflecrion and non-hyperbolic invariant lines

115

(a) U = 1 and a. > 0

Singularities: a = 0 and ff = x

a-

a-

0) = uo-

-X(O,

ar

-X(O, a@

a

x ) = (ao

a-

a +a 1)-a@

-X(O, x ) = -aoar ar

ar

,

Figure 1.

(b)

Figure 2.

= 1 and -1 < a0 < 0 Singulxities: e = 0 and CY = x . The same formulae as those of (a) are valid for the partial derivatives of y.

U

Fiyre 3.

(c)

,

Figure 4.

= 1 and a0 i-1 .. Singularitles: 01 = O

U

aa@ a-X(O,

01

= x ; 011 = tan-'(-ao

-X(O,

0) = -(a0

ar

0) = uoar

a-x(o, a@

a-

-X(O,

a@

a-

a

a-x(o, ar

1);; ( ~ = 2 tan-' -(-no

a-

-X(O,

a@

a-

-X(O, ar

a a@

a') = zao(sin2q)(cosa1)-

a

2

012)

-X(O,LY,)

ar

a + 1)-a@

-

= Zuo(sin w~)(cos@d-

a@

a

a

= (uocos3(Y1)- + q ( @ l ) ar a@

a + &)- a a@

@*) = ( U ~ C O a2)S~

ar

n) = (a0 x ) = -ao-

- 1);.

a + 1)-a@ a ar

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J Gheiner

where (o is the partial derivative with respect to r of the second coordinate of 51 for r = 0.

Figure 5.

Egure 6.

a-

-X(O, aa

a-

-X(O, ar

0) = -(U0

+ 1)-aaa

a-x(o, aa a-X(O,

a

0) = &ar

ar

a-

-x(O,al) = (2ao+4)cosalsinZal-

aa

a-

-X(O, ar

a-

-x(o,~Z)

aa

a-x(o, ar

3T)

= (Qo

n)= -ao-

+ 1)-aaa a ar

a

aa

a + $(XI)- a aa

a,)= C0s3a1(--ao - 2)ar

= (2&+4)cosazsinzaz-

a

a aa

+

a

aZ)= cos3u ~ ( --2)~ ~ $@+ar aa

where $ is the partial derivative with respect to r of the second coordinate of y for r = 0.

Figure 7.

Figure 8.

(e) U = -1 and -1 c a0 < 0 Singularities: a = 0; a = n;a1 =tan-'(@ 1 ) : ; or2 = tan-1 -(ao + 1);. The same formulae as those of (d) are valid for the partial derivatives of y.

+

Reflection and non-hyperbolic invariant lines

Figure 9.

117

Figure 10.

(f) v = -1 and a. < -1 Singularities: a = 0; a = x. The same formulae as those of (a) are valid for the partial derivatives of fl.

Figure 11.

Figure 12.

When dimension(M) = 2' we can apply proposition 1 to conclude the following proposition Proposition 2. In cases (b) and (c) FOis locally CO conjugated to R o XI. In cases (a), (d), (e), (fl FO is Cm conjugated to R a XI.

When dimension(M) > 2 the remark after proposition 1 allows one to conclude the same proposition 2 changing Cm to

5. Unfolding of periodic orbits 5.1. Fixed points

We assume the same hypothesis as in the beginning of section 3. Proposition I . The set ofjixed points of F : U x PzH U x R2 is locally a parametrized two-dimensional manifold S given by the equations

Y(P2,X) = o(ll(P2,X)ll~). Proof. Let us consider G : R4 H RzG(p,x, y ) = F#(x,y ) form we have

- (x,y). From the normal

By the implicit function theorem we obtain local functions ~ ~ ( px. )2and , y ( f i z ,x) such thatP~(O,O)=y(O,O) = O and G ( ~ ~ L I ( P z , x ) , ~ z= (,Ox , O,) . ~Takingderivatives (~~,x)) on both sides of this equality we conclude the proposition. 0

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3 Gheiner

Let us consider G = (Gl,Gz) and detine g : U x W H U x 1by g ( p , x ) = G ~ ( p , xy (, p 2 , x ) ) . The level surface7defined by g ( p , x ) = O is the graph of p ~ ( p z , x ) .

c

v-’ Figure 1.

Figure 2.

Let us consider the curve c1 that is the geometric locus of the tangencies between straight lines parallel to the x-axis and the surface 7.From the geometry of I one can conclude that CI is a curve of saddlenodes.

Lemma 1. The eigenvalues of D Fp u t f i e d points of FF are given by + ~ z + 2 ~ ( ~ ) ~ + ~ ( 1 1 ~ ~ ~ ~ 1 1 ~ )

A+(P.Z,X)

A-(Pz,x) = -1 + x +O(llPZ,X1I2).

Proof. Let TF(,x,y ) and Dw(x, y ) be the trace and the determinant of DF,(x, y ) . Taking partial derivatives at the normal form of F, and using the parametrization of S we obtain

+

+ +

+

T p k Y ) = wz + o ( l ~ z 1 ~ )( W p ) 1 ) ~ o ( 1 ~ 1 ~ ) ~ ( l ~ z l ) o ( l ~ l ) Dp(x. Y ) = -1

- PZ + 0(l@212) + (1 - W P . )+)o(lxl2) ~ + o(IP~I)o(IxI).

Using the formula

A$(x,y)=

TFf ( T i - 4DF)”’ ~

~~

2

we obtain the expressions for A*.

0

Proposition 2. There exists a local curve C I : iR H U x R2 of saddle-nodes given by the coordinates: Cl(P2) = (/1I(PZL2).PZ.X(LL2).Y(PZ))

Cl(0)

= (O,O,O,O)

In (U - E ) FF has two firedpoints (that are hyperbolic orpip).

Reflection and mn-hyperbolic invarimf llnes

119

Proof. The proposition follows from the geometry of S or from implicit function theorem applied to hf(,u~, x ) given by the lemma.

0

Observation. The second derivative of Fp along the central manifold of the saddle-node is approximately equal to 2a0. Proposition 3. There exists a local curve cz : W H U x RZoffips given by the coordinates: CZ(P2)

a ( 0 ) = (O>O, 0,O)

= (PI(P2), P2, X ( P Z ) , Y(ILZ))

P ~ ( M )= O ( I P ~ I ~ )

X(PZ) = 0(lP212) y(w2) = o ( 1 ~ ~ 1 ~ ) ) . In other words ( x ( P z ) . ~

( ~ 2 1is )a p i p f o r

F(p,(pI).pl) ifm # 0.

Proof. Direct application of implicit function theorem to A.-(pz,x) given by the 0 lemma. We collect in the following proposition additional information about the dynamics and the geometry of the invariant manifold of the saddle-nodes or flips corresponding to the negative eigenvalue. Proposition 4. Let ( x ( p 2 ) . y(p2)) be a saddle-node or apipfor Fp. ( a ) The eigenvectors v + ( p ~and ) u-(p2) corresponding to the positive and negative eigenvalues at this point are given by U+(PLZ) = (1, o ( l 1 ~ 2 1 ~ ) ) U + ( P ~= ) (1,

V-(ELZ)

O(IPZP))

= (~(IPZI~), 1)

U-(P~) = (~(IELZI’), 1)

saddle-node flip.

(b) The cuwature c(p2) of the invariant manifold corresponding to U-(& point has the form c(&) =

2aov fo(1) P2

4P2) =

-+ 00)

-2v lL2

at t h e f i e d

saddle-node

pip

where o(1) means bounded term. (c) The third derivative of F,’ along the central manifold of the flip has the form (12VIPZ)

+ o(1).

Proof. ( a ) Taking partial derivatives at the normal form and substituting the coordinates ( x ( p 2 ) , y ( p 2 ) ) of the saddle-node or flip we obtain D F p ( x , Y)

(

1 +o(lP2I2) -1 k2 O ( I P ~ I ~ )

0(1P2l2)

1

+ +

0(1P2l2)

2+

I’1

1

saddle-node

O(IP~I~)

DF,(x, y ) = O( I

O(IPZ12)

-1

+ O(lWZ12)

flip.

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J Gheiner

For the saddle-node, using the lemma and proposition 2, we solve the equations for OL and ,!?

For the flip, using the lemma and proposition 3, we solve the equations for a and p

0

The result follows directly. (b) Let rpm : R H U- ( ~ 2 )given

Rz be a parametrization of the invariant manifold correspondent to

by rpp2(t)

= (x(&)I Y(PZ))

+ t . u-OL3 + Y ( f ) . V + ( W Z )

where y ( 0 ) = y'(0) = 0. F+ o qp2 : R orientation. The curvature C(PZ) is given by

H

Rzis another parametrization with

Substituting the values of L*(pz) in the saddlenode and flip cases we obtain

"'Y

2aov

= -+o(l) /*2

-2v

y"0) = -+o(l), PZ

By (1) the same estimates of y"(0) hold for c(pz).

reversed

Re$ection and non-hyperbolic invariant lines

121

( c ) Let r* be the straight lines passing through the flip (x(pz).y(fi2)) having, We identify r - with W using the parametrization respectively, the directions U*(&). Il(t) = (X(PZ)* Y(P2)) + t - u-(Pz). Let tG2 : R2 H W be the projection on r - following the direction of r+.

In the flip case tGZ(X,

Y) = y(1 +O(lPZ16))

+ O(lPZ13)X

Direct calculation gives 12u

D ~ ( E ~ F ~= ~~ ~ + ~o (( ~ O) ) if PLt # 0

0

112

Obsemufioa. We observe that the estimate of the third derivative is independent of parametrizations having the same derivative at the Aip. 5.2. Periodic orbits Let G : U XR'H RZsuch that G ( ~ , x , y ) ~ F ~ ( x , y ) - I d ( x , y ) . Let ( G ~G , ~)SG. From the normal form we obtain G l ( P L , XY) , = (2PI

+ Pl"+

( W P ) P I +2P2+o(llPl12))x

+ ( M P ) +0(llPll))X2 + (2u + 0(llPlI))Y2 + ( 2 ~+ 2a2(P) ) + o ( l l ~ l ~ ) )+x (~~ c ( P-) 2v + ~ ( P ) +v O(IIPII))XY~ +O(llPI12)Y

G z ( ~ ~Y)x ,= o(llklI)P?

+O(llPIl)Pzl;(X,Y)

+0(ll(X.Y)1l4)

+ (-PI + o ( l l ~ l l ~ +~IPII)PIx ))~

+ (-2 + o(ll!-ll))xy + (1 - d P ) + o(llPll))xzY

+ (-U

+O(lIPll))Y3

+ O(llPII)Pzl;(X,

Y) +o(ll(x,

Y)1I4).

By the implicit function theorem there exist local functions xi:Ut+R

for p E U .

and

y;:UwR

i=l,2

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J Gheiner

Taking partial derivatives at (1) and (2) we get

Therefore

We define zL:U

I

H

R

ZI(P) = G ~ ( ~ L x I ( Pyi(A) ), ZZ(P) = G ~ P XZ(P), ,

After substitution by the expressions for xi(& P;

and yi(p), i = I, 2 we obtain

+

Z1(PlrP2)= 2P1 - j-& O(IIPII)Pl +O(1IP1l3) ZZ(P1. PZ)

=PI

.O(llPIl2) + O ( l l P l h .

Proposition 1. There exist U neighbourhood of (0.0) in the parameter space and V neighbourhoodof (0,O)in R2and two C ' ( r large) diffeomorphismsq;: U x V + U xRz i = 1,2 of theform W ( P ,x , Y) = ( p ,v;,(x, y)) such that

G I OVI(P,X,Y)= z I ( P ) + ~ ~ o ~ ~ + ~ ~ Y ~ Gz 0 ab, x . Y)

= ab)- b Y

v i h 0,O)= (P, Xi(P), Y ~ ( P ) ) Dzrpi(O,O, 0) =Id

(Dz means partial derivative respect to V ) .

Proof. Morse l e m a with parameters.

0

Let, for i = 1,2, G;,(x,y)%G;&.x,y); ~ j , ( x , y )d=dq i ( p , x , y ) . The preceding proposition shows that the level curves of Gi, are the image of the level CUNS of Gi,o9;, through vi,. The level curves of GI, o PI, are ellipses if no and U have the same signal and are hyperboles if a0 and v have opposite signals. The level curves of Gz, o a, are hyperboles (here we consider also the possibility of a pair of straight lines as degenerated hyperboles). Let cz be the flip curve in U , parameter space. We use = for equality of functions.

Lemma I.

ZZ(P1, PZ)

= Ofor (Pl, PZ) E cz.

Reflection and non-hyperbok invariant lines

123

Proof. Let > 0 and v = -1. If z2(po) > 0 for some E c2, then the geometry of the level curves of Gip, must be one of the following two possibilities

{&Po

= 0) = RI~OU % P o

( G P O

Figure 1.

= 01 = R 3 p o U %Po

Figure 2.

”’

Kclli

In both options when fi is slightly perturbed the number of intersection points does not vary. This number is equal to the number of local periodic orbits with period less than or equal to two. This is a contradiction with POcorresponding to a flip bifurcation. The other U cases have analogous proofs. Therefore the correct picture for p E c2 and p # (0,O) is given by figure 3.

G2 =O

G2.= O-Figure 3.

Figure 4.

Let cI and cz be the curves of saddle-nodes and flips at the U parameter space. c1 and c2 are given by propositions 2 and 3 of section 5.1 considering only the ‘U part of the image’. For a0 > 0 and v = -1 we obtain figure 4. Let xi:U H R be defined by X I (&I, ~ 2 gfii. ) We define another system of coordlnates by

l,2=p2 *I

= LLI

-XI

0 CZ(fi2).

In the new system the flip curve is equal to the bz-axis and by the preceding lemma we can write in the new coordinates a(lii1, L2) = liil

’

h(lii1,

liid

Let fi = (p,,p2) such that z l ( f i ) i:0. We define: PI,, as the point of greatest curvature of the right component (RI,, in figure 1) of the hyperbole (Cl, 0 qi,)(x, Y ) = 0. P2, as the point of greatest curvature of the other component (G, in figure 1).

I24

J Gheiner

&

as the point of greatest curvature of the right component (R3, in figure 1) of the hyperbole (6,0 vpz,)(x, Y ) = 0. P4, as the point of greatest curvature of the other component (%,, in figure 1) Pip =

Lemma 2. h(O,,&)

I

qlrr(Pi,)

if i = I, 2

(oz,,(Pj,)

if i = 3.4.

= 0.

Proof: We consider a0 > 0 and v = -1. The other cases have analogous proofs. Suppose h(0, jZ.02) # 0 for some ,i$ # 0. Then

-(O, az2

= h(0, &) # 0.

abI

Let po = (&:, ) : p E cz be the point that corresponds to (0, % , )! in the &I. p ~ coordinates. ) Let ( x i @ ) , y i ( ~ ) be ) the critical point of Gip. Then from the definition of Ptp we obtain

P I , =PI&(&) = ( x I ( K ) , Y I ( K ) )

+ p3w +

9, = (xz(K), YZ(P))

+ PI, + J,I ( D ~ p ( t & ) -Id)

i1(Drp2,(t&J

Consider the straight line rpl($l)%cpz@z) move when p varies along rrr;.

Id) . p3, dt.

Fl,ddt

(5) (6)

+ (~1.0).Let us analyse how PI, and P3,

Figure 5.

The following estimates are obtained from (5), (6):

As Dzpi (0, 0,O)= Id we can think about Pi, as the points of greatest curvature of the ‘hyp~boles’Gip=O. For PI # 0 sufficiently small we have

Reflection and non-hyperbolic invariant lines

125

and figures 6, 7 and 8.

P1 > 0

P =Po

+ (P1,O)

Figure 6.

p1=0

p=po

Figure 7.

Pl < 0

P = Po

+ (Pl,O)

Figure 8.

Figures 6, 7 and 8 constitute a contradiction with the fact that po E c2 because oneparameter bifurcation of flip type implies variation of the number of periodic orbits. U For po E cz from lemma 2 it is possible to estimate ~~P~(LLo+ -(PwII P ~ . o ) 0 Figure 9.

P = Po

+ (P1,O)

p1=0 Figure 10.

p=pO

J Gheiner

126

PI < 0

= Po

+ (Pl,O)

Figure 11.

It follows that the description of the unfolding of periodic orbits is well defined in a neighbourhood R of cz n U. We can suppose diminishing U that U c R. The following lemma summarizes OUT discussion. LR"a 3. f l p = po

+ (pl,O), with po E cz and p 6 c2 then

IIPI, - P1O ,I

>> IP3, - P3o,lI

For ao and U with the same signal we define Fj, i = 1,2, as the points with greatest and least x-coordinate of the ellipse (GI, o q l p ) ( x , y) = 0. Lemma 3 remains valid in this case.

Gq~2po;;

Let us consider a0 > 0 and U = 1. We will show that the two-periodic orbit that unfolds from the flip eventually becomes a Hopf bifurcation. If p is to the right of CI then there are no local peridic orbits. If p is between c1 and c~ there are only two fixed points. Let us suppose p is of the form p = po - ( ~ 1 ~ with 0 ) po E cz and p~ > 0. For p = po the curves Glpo and G2,0 have the aspect drawn in figure 12.

,,

G2"=o,~

~,~~~

GZp= 0 p=po

ao>O

Figure 12.

v=l

p=pO-(p1,0)

ao>O

v=l

Figure 13.

By (3) and (4) when we pass from p = po to p = po - (PI, 0) the critical points of GI, and Gz, are approximately equally translated. Using this and the estimates preceding lemma 3 we obtain Plp

- p3p

(9, - PI,$) - (p3w - p3po) = (A,,,U s ( l W 1 I ) )

where a, =. pi/&,odIp1I) < &PI,E is the diameter of U. Let Q, be the periodic orbit with period two. From the geometry of the ellipse it follows that Q, - P3, Z (0, A,!) where A,, > (&/&)PI

Refiection and non-hyperbolic invariant lines

Let po = (py, p i ) . From lemma 2 and (4) we obtain

127

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J Gheiner

Proposition 2 . For v = I there exists, in U , a curve c3 constituted of Hopf bifircationr c3(pI) = (/-!’I,

pZ(p1))

pl

0 and v = 1. Let A > 0. Let p~ e 0. Let p,, g(p.1,Ipll/h) and r,, p l / A ) . Let ,C, be the union of the horizontal segments connecting p,, and r,, to cz and the vertical segment joining pp, and r,,. There exists a correspondent curve for Q, (periodic orbit with period two) named ?, in the ‘IR x DT diagram.

cpK,pl

-1

c2

Bifurcation diagram Figure 14.

TR x DT diagram Pigum 15.

By lemma 4 we know that the subset 3,, of ?,L corresponding to the vertical segment S, of ,C, must remain inside the parabola A = 0. There exists a constant KZ > 0 such that A,, e K a m . Using this inequality, and taking A small, direct substitution gives

Taking the partial derivative with respect to pz of the formula of DT(p, x, y ) and using the estimates we have about XQ, and YQ, we obtain

a

2 , 5 > -D T ( p , XQ,., YQ,,) > 1,5

for P

E S,,.

(13)

From (12) and (13) it follows that there exists a unique Hopf point h,, E S,,.

Figure 16.

We define c3 as the union of h,, for p1 c 0 plus (0,O). From the implicit function theorem follows the differentiability of c3. For Q c 0 the proof of the existence of c3 i s similar.

Reflection and non-hyperbolic invariant lines

Corollary 1. r f

U

129

= 1 the family ( F,,] is not locally weakly stable.

Proof. It is a direct consequence of the existence of the invariant circle in the Hopf U bifurcation, and rotation number theory. From the formulae for DT and TR we obtain (TR- PT+ ~ ) ) ( K , x , Y=)

- SUY~+.O(I~I~)+O(I~I)O(I(X,Y)I) + O ( I ( X , Y ) I ~ ) . (14)

Lemma 5. Let /L = (@I, p2) E U such rhat with period two.

I > /p21and Q,,

be the periodic orbit o f F ,

V V =1 thePl(TR-(DT+l))(p,xp,,yp,) < O rfv=-I Proof.

then(TR-(DT+I))(p,xp,,y~,) > O . U

Direct substitution as in lemma 4.

Corollary 2. r f v = -1 then Q,, is a hyperbolic saddle.

Proof. Suppose a,, > 0 and U = -1. Let C be a curve in the bifurcation diagram whose exwemities are located on c2 as in figure 17. We can associate a corresponding curve 2 for in the TR x DT diagram.

e,,,

Bifurcation diagram Figure 17.

T R x DT diagram Figure 18.

+

The extremities of 2 must be located on the straight line r defined by TR = DT 1. When C is in the region ( P I (> Ip2( the curve 2 must be to the right of r . Moreover C? (excluding its extremities) cannot intersect r as there are no other bifurcations for the fixed points of F i , One can conclude this assertion observing the intersection pattern of the ‘hyperboles’ GI,= 0 and G2,, = 0 as p varies along C. These last two facts immediately imply the corollary. 0 Using all the results we have obtained in sections 5.1 and 5.2 we can characterize the bifurcation diagrams and the local dynamics of Fw for p E U .

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J Gheiner

6. Bifurcation diagrams of F,,and dynamical behaviour of F :

v=l Graph of

a0

GI,

>0 Graph of Gzr

Figure l(a).

Observation. The geometrical aspect of the graph of Gzlr does not depend on This observation is valid for the other values of U and (10.

U

and ao.

131

Reflection and non-hyperbolic invarianl lines

v=1

-32

Figure Ub).

a0

>O

132

J Gheiner

u=l

a0 < O

Graph of GI,

Figure 2.

133

Reflection and non-hyperbolic invariant lines U

= -1

a0

>0 Graph of GI,,

Figure 3.

134

3 Gheiner

Graph of GI,

25 Y ---------

135

Reflection and non--hyperbolic invanant lines

In the preceding figures we have made the simplification that Gzlr = 0 coincides with the x-axis and the y-axis. This simplification is justified by lemma 3 of section 5.2. It does not alter the relative position of the periodic orbits and the local dynamics. The arrows indicate the iteration under F,’. In the case U = -1 and 0 < a0 < -1 the figures are essentially the same as in the case U = -1 and a. < -1 except for: (a) In the regions c:, (Z), c l the unstable manifold of, resp., the saddle-node, the saddle, the non-hyperbolic saddle, has an oscillating behaviour in the region x < 0. For example, for & E c: we obtain figure 5.

v

v

Figure 5.

Figurc 6.

@) For ,u E c; U (4) U c; the stable manifold of,resp., the saddle-node, the saddle, the non-hyperbolic saddle, has an oscillating behaviour in the region x > 0. For example, for p E (4) we obtain figure 6. The smallness of la01 is the cause of this behaviour. This fact is a consequence of the geometrical lemmas that will be proved in the case U = -1 and a0 < -1.

7. Obstructions for stability (a) v = 1 We h o w , from corollary 1 of section 5.2 that for U = 1 the family [Fp}pGuis not locally weakly stable. In the case v = 1 and a0 > 0 a second obsbvction exists. It is the existence of intersections between stable and unstable manifolds of saddles for p E (3) U (4). The number of intersections in a fundamental domain of one of the invariant manifolds is an 0 in (3) U (4). invariant by conjugacy. We do not have control over this number when p i (b) U = -1 and no > 0 In this case, generically, when p E (2) and

Figure 1.

is near c: we obtain figure 1.

Figure 2.

The stable manifold of p1 oscillates around the strong unstable manifold of pz when we get close to p 2 . The situation described by figure 2 does not occur because the eigenvalue

J Gheiner

136

- 1 of DFo(0.0) forces the stable manifold to reflect around the strong unstable manifold of P l When E ( 2 ) tends to c z through a straight line parallel to the PI-axis the unstable manifold of the periodic orbit p 3 U p4 tends to the strong unstable manifold of p2 and, therefore, will eventually be tangent to the stable manifold of p , and afterward will cut it transversally. We assume that for p E (2) close to c; the stable manifold of p1 in a neighbourhood of a fundamental domain of the strong unstable manifold of pz (taken close to p2) is the graph of a Morse function (see figure 3). This is a generic condition.

Figure 3.

Figure 4.

To each maximum or minimum of W s ( p I )corresponds a curve of tangency between W'(p1) and W u ( p 3Up4) i n the bifuracation diagram as in figure 4. The possible extension of these curves to (0,O)and the passibility of intersections between two such curves depend on the variation of the number of maxima and minima and on the order (and change of order) of maxima and minima by height when fi -+ (0.0). For example, the diagrams in figures 5 and 6 are not homeomorphic.

\ F i 5.

Figure 6.

The number of maxima and minima of W r ( p l )and the order by height are two invariants

for local weak conjugacies. We do not know how to control these invariants as we perturb the family. Also quasi-transversal intersections between W'(pl) and Wu(p3 U p 4 ) imply the existence of modulus of stability [PZ]. Proposition 1. Let F E sl. In the case ao > 0 and v = -1 there exist curves in the bifurcation diagram corresponding to quasi-transversal intersections between stable and unstable manifolds of saddles that unfoldfrom p . Conjecture 1. In the case a0 > 0 and v = -1, F is not locally weakly stable.

Reflecrion and non-hyperbolic invarinm lines

137

-1 and -1 2 by the remark following proposition 2 of section 4 we get an H as above only U('). As r increases we may lose the size of the neighbourhood where H is defined. Even in this situation we can recuperate the original size of the invariant x-axis iterating forward and backward by Fo. This argument shows that the x-axis is locally a Cm invariant curve by Fo.

8.2. Definition of sub-central manifold Let us fix the following notation Ds = disk with centre (0,O) and radius 8 > 0. ]I[/i = ci norm W u ( p i )= unstable manifold of pi WUi(pi)= left piece of the unstable manifold of pi W"'(pi) = right piece of the unstable manifold of p;.

For the stable manifold analogously we define: Wy(pi);W"(pi); Wr'(p1) W'"'(pi) = weak stable manifold of p ; attractor Wrwf( p i ) = left piece of W""(p;) WSWr(pi) = right piece of Wcw(p;) Ws"(pi)= strong stable manifold of pi attractor. Analogously we have: W"."(pi);Wr.'r(p;); W c ( p i ) = central manifold of p i . Analogously we have: w"(pi); W"(p;). Analogously to Wssand W f w : W u u ( p i )= strong unstable manifold of pi repeller WU" ( p ; ); WUU' ( p ;) W"'"(pi) = weak unstable manifold of pi repeller WUWI ( p i ); Wu'"r(pJ. Let p1 be the fixed point with greatest x-coordinate. We define W F by: W;' = x-axis if p = (0,O) W F = w'(p1) if I*. E c: w;' = W"'(p*) U WS"'(p,) or w: = WU'(p2) U W""(pl)

if p E (2)

Reflection and non-hyperbolic invariant Zims

w:

139

= w”‘(p2) U W”‘(p1) if p E c:

wp = W””(pz) U w”’(p1) if p E (3) wr = w””’(p2) U wsr((pl)if p E c; wc”= W’”’(p2) U W“‘(p1) or if wc”= WUW’(p2)U W”(p1)

1

’

E (4)

,

W: = W c ( p l )if p E c;. The options in the regions (2) and (4) will be determined by proposition 4 of section 8.3. The selection of W””’ and WSw‘ in ‘ (4) and (2) will be made in the same proposition. We must define W r in region (1).

8.3. Existence of invariant sub-central mangolds and adapted vectorJields Proposition I . For every p E (1) there exists an invariant C2 mangold for F,, denominated WF. The map 11 H W p is continuous in the C’ topology in the closure of (1). Proof. Let obtain

FF = (FI,, Fz,). From the simplified normal form ((1) of section 8.1) we aFl, -(x,

ax

a

F2, -(XI

ax

@ ?- -

ay

a Fza -(x, ay

0) = t1+ fi2

+ za(w)x]+O(IXI’)

+O(IIPII)O(IXI)

0) = ~ ~ l l 1 1 l l +0(llPll)X4 ~~~l~l) 0) = o(ll11ll)o(lxi) 0) = [-I +XI

+

O(lX13)

+ o(ll11l12) + O(Il11ll)O(lXl) + o(lx14).

We divide (1) into three regions: (1) 112 > IF1I

(2) P2 a(~)lq)(nk-XcY

We define

Let S,, be the horizontal segment [F,(xk, yk), (XK,yk)]. By (l), (2) and (3) it is possible to construct a C2curve C, in RZ satisfying: (a) C,, is the graph of a real function whith domain equal to the segment [F~,,(xk, y k ) , xk] (b) C, has boundary equal to (xk, yk) U F,h, yd (c) C, has tangent vectors at (xk,yk) and at F,,(xw, yk) equal to, resp., (1,O) and

DF,,(% Y

~ I. (1,O)

(d) Ilc, - Spll, 0 be fixed. We extend C, to an invariant curve W;” by the equation


0).

In the first region both eigenvalues of DF;' contribute to contract the inclination of the tangent vectors to W s r ( p l ) .In the second region the positive eigenvalue of DF;' is more expansive than the negative eigenvalue. Therefore, the C' or C2 viewpoint, the situation is 0 favourable. The proof follows the line of the proofs of propositions 1 and 2. Proposition 8. WSr(p1),for p

E

c; satisfies

llW"(pi)lli < k h z l

lI~"(pi)1lz< kz

for positive constants kl and kz. Proof

Analogous to the proof for the second region of proposition 7.

Proposition 9. WSsr(pi), for p

E

(3), satisfies

l l W " y r ( ~ ~ 0) contained in the x-axis W u ( Q o )= unstable manifold of QO in the disk DS D ' = fundamental domain of WU(Qo) V = small neighbourhood of 2) (fundamental neighbourhood) EO= transversal section to W S ( Q o )at po = [ X O , 0) xo > 0 E, = connected component of Fi(Eo)n Dscontaining Fi(xo, 0) n,:R2 --f R projection on the first coordinate pn = ( x n , 0)= F: (PO) uo = tangent vector to EOat ( X O , 0) = po un = D F , ( P o )' VO = (an, b n ) L=

121

R=(' 0 -1 O)

& = R o Fo

I[ 11' = C' norm

Proposition 1. Given E > 0 there exists N such that g n

2 N then llEnnVll, < e.

In order to prove this proposition we demonstrate two lemmas.

Resection and non-hyperbolic invariant lines

Lemma 1. Given E

=. 0 there exist N

E

M and 6 > 0 such that ifn > N and xo < 6 then

An < E.

Proof. From the simplified normal form we obtain

Then 1

+ 2aoxi + 1 + -xi

i=O

O(lXi 12)

O(lXil4)

Let

Consider no

E

N such that 1

-< x a < --aono

As

(XO,0)

C-

no

Therefore

Thus

1 -ao(no - 1) '

belongs to the stable manifold of a unidimensional saddlenode i t that

-< X i
N implies

For

If no is sufficiently large, which is equivalent to xo be sufficiently small, then A,, < E. 0 Observation. The same result is valid for a family of sections transversal to a fundamental domain of W'(Q0) provided that there exists a positive lower bound for the various h. We consider X ( x , y ) the vector field whose time one map is equal to

&. Let

0 ( x , y ) =the angle between ( x , y) and the x-axis X(X,

Y ) = ( X l ( X , Y). XZ(X, y ) )

AI@) is a multivalued function. Our aim is to draw the 'graph' of A1(0). Let gl[O, a]:H R be defined by

Rejlection and non-hyperbolic invariant lines

149

Proof. From (2) of section 8.1 we obtain

Let u ( x , y) = tan@@,y) and w ( x , y) = cot@(x,y). Let c =tan

(5- 82) z 1. Then

dipkT>G 0

The lemma follows directly from (3).

We draw the graph of gI , and the 'graph' of A I , by lemma 2, is a thin strip around that graph. See figures 1 and 2.

1 2Figure 1.

Figure 2.

From the definition of gl and a0

I(I +61)gll < Itan81

i-1

it follows that if 61 is sufficiently small then

for8 #

I e.

;

Therefore the vector field X is always transversal to the straight lines y = cx with c # 0. Thus the orbits of X intersect each straight line y = cx at a unique point and the minimal and maximal angles of the vector field X with minus the x-axis occur, approximately, at the straight lines y = &x and y = . - f i x . These angles are, approximately, *tan-'((l/2-). Therefore it is not possible for a transversal section to the x-axis to get C' close to the x-axis, in any neighbourhood of the origin, when iterated hy Ft, n E N.

Figure 3.

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J Gheiner

We will analyse the angle that the eigenvectors of D&(x, y ) make with the x-axis when ( x , y ) varies along an orbit of X. From the simplified normal form we get

Let (A&, y ) . 1) ( A ~ ( x , y ) . 1) be the eigenvalues of D&x, From the eigenvector equation

y)

where ci is the corresponding eigenvalue, we can deduce

where the - signal is chosen if i = 2 and the We consider only the region y > 0. Let

+ signal is chosen if i = 3.

Let (x, y) varies along an orbit of X. From (4) we can conclude

IAz(O)l >

1

tan0 e 6

1

2s

if

(the 6 of the disk Ds)

and

O X ;

12

is strictly increasing

--1

. a

if 1

- E e h2(8) e - - + E 1/z

1

[tanel e 6-I and (tan0 8 or 0 >

I)

if [tans] > 6-’.

By symmetry in the formulae of Az(x, y ) and h3(x, y ) we can infer that A3(0) is essentially equal to (-Az(n - 0)). From (5) we draw in figure 4 the graphs of At@) along an orbit of X.

151

Reflection and non-hyperbolic invuriant lines

011

= arctan(6)

ad = arctan-6)

a2 = arctan(6-1) 013

= arctan( -6-1)

Figure 4.

Let 00 =tan-' A.From (4) we have AS(&) (1/2&) h1(6'0). For B close to z we have, from (4), Az(0) 2 (tanB)/-(l +%). As Ai(@ "= (tanQ)/-ao we conclude Al(0) < Az(0) < 0. Analogously, for 0 close to 0 we have A,(@) > h3(0) > 0. Therefore we can draw, in figure 5, the eigenvectors w d x , y) = (A&. y). 1) and w3(x,y ) = ( h 3 ( x ,y), 1) along an orbit of X. These eigenvectors rotate anti-clockwise as we slip along an orbit of X.

Figure 5.

The - signal means the eigenvector is contracted by LIFO, the - signal means the signal, analogously, mean eigenvector is strongly contracted. The signal and the expansion and strong expansion. The lack of signal means neutral behaviour under iteraction by LIFO. From (4) we can deduce for Bo = tan-'(-lO[aol)

+

Proof of proposifion 1. Let

with yo z 0 p = (XO,yo) E CO -. pi = (xi.Yi) = F ~ ( P ) ug

= tangent vector to

vi = D f i ( p ) . UO.

& at p

++

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J Gheiner

By lenima 1 and continuity we can suppose that yo is sufficiently small and the angle between ug and the x-axis is approximately n/2. Let

FL, = [ ( x , y ) l y > O A O e Y < 1o1aO~} n q X

lOlaOl} n a

RI = ((X.Y)IY > O A

{

I '

% = (X,y)Iy > O A 0 > - > loa0 n 4 X

While FA(p) remains in FL, the angle between the vector ui and the x-axis stays between ((rr/2) - E ) and the angle between the eigenvector w&i. yi) and the x-axis. We claim that in the region RI the vi, as i grows, rotate anti-clockwisea definite amount bigger than a constant that depends only on 4. We could use the rotation of X to prove this fact but the proof would not be transportable for p # (0,O).We assume the iterates pi are in RI. Let

6 ( u , w)

= ,angle between v and

-

[I + 2a0xi + O ( I Y ~ I ' ) I ~t-2yi ~ 2coxiyi o ( I Y ~ I ~ ) I F ~ [-Yi + Yi4YiP)lii + 11 - x i O(lYi14)lYi

Ai+l =

UI

+

+

) I [-zY, ~.~ - [I + b o x i + O ( I Y ~ I *+

+

+

+

+ ic0xiyi + o(ihi3)1

(7)

[ - ~ i + Y ~ o ( I Y # ) I ~[I~ -xi + O ( I Y ~ I ~ I

As ui remains between the eigenvectors wz(xi, yi) and w3(xi, yi) we have from (6) E

> ii> -

2 6

while pi remains in RI Let nl = smallest integer i such that pi E R 1

nz = biggest integer i such that pi E RI We assume that for nl then in,< ii < Thus

-4;

< i < ni we have

ii

z

-$.

If this hypothesis fails for some i i

In this case we have proved our claim. So we assume the hypothesis is valid. Then from (7) we obtain

Reflection and non-hyperbolic invariant lines

153

Therefore "2-1

X",

< X", - 1.8

(9)

yi. i=nj

From the expression for Fo(x,y ) we obtain, for pi E RI

4'= (&+I, Yi+l) -(xi, Yi)

(-Y?, -xiyi).

Let A' = (A:, Ai). It follows that

In order to traverse RIwe must have

To obtain (10) we use the symmetry of Therefore

and

Therefore

From (9) we have

Fo with respect to the y-axis.

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J Gheiner

Therefore in RIthe rotation of vi as i increases from nl to n2 is greater than 0.3/l&l, We can suppose that E > 0 is much smaller than 0.3/l&l. As O(v,,, el) > (n/Z)- E we have

From (8) and (12) we get, in any case,

From the position of the eigenvectors of D & along an orbit of X (see figure 5 ) it follows that z

8 (v,,+i, el) > 2

0.2 +la01

i

> 0.

Let

Let n 3 be the smallest integer i such that pi E R 3 . For i > n3 we have pi E 77.3. We will estimate the angle 8(un,+i,el) for i > 0. Let xi = = Fi/.Ei. Using the formula for D&xi, yi) we obtain, for i 2 n,

(xi)-'

[1 -xi + NIX~I')IX + I-Y~ + Y ~ O ( I X ~ I ~3) I ii.' + 2a0xi + 0(IxiI2)I + [ - 2 ~ i+ ~ C O X ~+Yo(IxiI ~ )I From (13) we have Ii< 0 for i > 123. Let tgi = tan 8 ( x i , y i ) = yi/xi. For i > n3 we have tgi+[ tgi(l - (& -t 1)xi + tgiyi) and 0 > tgi 2 + 1.

-

hi+t =

,,,,,,,,,

[I

Q

" '

(14)

(15)

We assume that

hi

< tgi

for i 2 n3.

Then, from (14) and (15)

hj,l

> [I - 2(a0

[

+ I)x&

:; + 1

tgi+, < 1 - -(a0

1)Xi

tg,.

(18)

From (17) and (IS) it follows that if (16) is violated and we pick io the smallest i that violates (161, then xi 2 tgi, for i-> io. In this case (IS) plus the fact that the xi behave as a unidimensional saddle-node imply that given E 0 there exists il > io such that tgi < E, for i > il, so the proof of the proposition is finished.

Reflection and non-hyperbolic invariant lines

155

Consequently let us assume (16). Then from (17)

However

From the simplified normal form for FO we have, for pi

E 'R3

Let n4 be smallest positive integer such that pi E V (neighbourhood of 2, fundamental domain of W'(Q0)). 2) = [U,PI with (Y < 9 , < 0. Let px4 = (xnG,y n G ) .Let 0 < j , < j , be real numbers such that

From (20) we get

-

1 (a0

+ &I

( j +~ (n4

-4 ) h-1

From the definition of jii = 1.2 we have

J Gheiner

156

Inequalities (21), (22) and (23) do not depend on the value of yo, the y coordinate of P = ( X O , YO). From (19) and (13) we obtain

If yo $ 0 then (n4 - n3) from (22) and (23)

+CO

and n4 .f +W. As j1 and j z vary very little, it follows

Therefore In4 + 0 if yo $0. Given E z 0, if we take V sufficiently small yo will be so small that the orbit of p = ( X O , yo) will intersect V satisfying 0 > h,,,> -E. One can easily check that taking V sufficiently small corresponds to take n sufficiently big. 0 Observation 1.. Let r be a family of transversal sections parametrized by a parameter y in a compact interval of R. Proposition 1 is valid for r if r varies continuously, respect to the parameter, in the C" topology. In this case the N of the proposition will not depend on the section. Observation 2. For p E (1) it is possible to prove an analogous proposition. The differences are that instead of 'given E > 0' one must write 'there exists a continuous positive function ~ ( psuch ) that lim,+(o,o) E ( # ) = 0' and instead of 'there exists N such that if n N' one must write 'there are N 2 ( p ) > N I (p) 0 satisfying lim,+(,,o) N z ( p ) = +w such that if N z ( p ) n N,(p)'. The techniques to prove this new proposition are the same techniques.

>

Observation 3. The same techniques used to prove proposition 1 (geometrical analysis of the eigenvectors and arithmetical displacement) can be used to prove the following proposition: For each p E (3), there exists a neighbourhood V , of (0, 0), satisfying lim

P+(O,O)

(diameter of V J = 0

such that in the region D&\ V, the long semi-stable and semi-unstable manifolds of saddles remain llpl]-C' close to the x-axis (in other words the tangent vectors to these long semimanifolds, inside the specified region, have a y-component less than a constant (independent of f i ) times Il~ll). 8.5. C2 control of transversal sections Let:

Ds = disk centred at (0,O) with radius 6 z 0 in R2 where 6 is chosen small so that the quadratic terms of FO prevail upon higher-order terms Qo = (0,O) reflection fixed point of FO Po = ( X o , O ) E Ws(Qo) f7 4 Pn = Fl(P0) = G m 0)

R ' = { ( x , Y) E Dsb

> 0 A IYI < $1

Reflection and non-hyperbolic invarianr lines

157

-

FO = R o FO where R = ( ' 0 -1O ) EO= ((xo,y ) E R]transversal section to WS(&o) Ea = @(EO) nR (only the connected component that contains p a ) Ex = x-axis; E, = y-axis R2 H E, canonical projection on the x-axis nY:R2H E, canonical projection on the y-axis rp,: I, H Ex the function whose graph is Ea * the condition

ir,:

(, ) = scalar product in c = 1

I,, = nY(zl,,) Ei = ~ ~ ( 0 )

R2.

-

( (10)X 0

Idi = identity function in R or R2 as i = 1 or i = 2. Proposition 1. There exists ajirst integer io > 0 that violates the condition *.

The violation of * is unfonn in y E

Iio, in other

words,

Observation 1. For i > io the condition * is still violated V y E 4 . This follows from the proof of the proposition. Observation 2. Fori 2 io, (FO(Co)\R) has tangent vectorS making angles with E, smaller than ((x/2) -(1/25 lql)).This fact follows from proposition 1 and the geometrical position of the eigenvectors of POanalysed in section 8.4. This observation can substitute the analysis done in regions RIand 7& in the proof of proposition 1 of section 8.4. This observation permits one to combine the C2 smcture obtained in proposition 1 with the C1 strncture obtained in proposition 1 of section 8.4.

+

Proofofproposition 1. Let iiogao (-no - 1)/10. Let (xo,yo) E EO with yo > 0. Let (xn,y,J = F,"(XO,yo) E R.From the simplified normal form we obtain "-1

xo

+ (a0 - 0,02)Xi)

n-I i=O

i=O

n-1

t &Xi)

i x, i ' X 0 n(l

+

"-1

- (1 - c1X:)xi)

y o n , , - (1 c,xi',xi, < yn c yo i=O

i d

158

J Gheiner

where c1 > 0 is a constant that does not depend on (XO, yo). We can take S (of the disk Da) sufficiently small to get Vi Applying this to (1) and (2) we conclude 1Y"I IYOl > -. lxnl

As Igi(y)[


0; (1 + clxi") < Irlol.

From (4) we have 0 2 2ao. d ( g i ( Y ) ) >

From

1

* we obtain 0

Using

zooao.

< 2 a o ~ i ( g i ( y ) ) ~ ~ " ( g i i 0 sufficiently small

v L( ~ Y ) = ( g l ( r ) ) 2 d ' ( g i ( y ) )+ si(Y) where -2,02 c si(y)

,-0,98. Let mi ~ ~ u p , ~ , ~ { ~ : ' ( From y ) ] . (9) we obtain

i

m i e i . (-0.98)

Vt

From (1) we have X, > no

n; >

(1 1

(-00

=- 0.

+ (a0 - 0.02)ri). Therefore Vi > 0

+ 0 . 0 2 ~ ~ ~ ~ ~ 0+;00 . ~ ~ x 0

* implies

(remember c = l/(-q)xo). From (IO) and (11) we have (-0.51)(~

+ i) < (-0.98)i + i < 1.1~.

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J Gheiner

This proves the first assertion of the proposition. Let io be the first integer that makes * false. 1.1

io < l . l c + 1 = -i- + 1 (-4J)xo

(3) can be written as

There are two possibilities

In this case there is nothing to prove.

Let il be the greatest i smaller than io such that

It follows that

Reflection and non-hyperbolic invariant lines

'From (4), (13), (la), (15) and (16) and the definition of

161 il

we have

But

From (1) we have

From (12) we obtain

As 1 z liiol/lal > 0.9 it follows that

Returning to (17)

(2,3)*

c1

0.02 0.02 + 1,03 .ln(2,3). cl+--. 0.9 la01 la01

Therefore

Observation 3. The same proof give us

aservmion 4. There is a similar proposition for E (1). The difference is that instead of 0 < xo < 6/2 we must consider x p < 6/2, where xw E W;, x p is a continuous function of P, xu

=nom

> 20

.if p2

> 0.

Let 0 < Eo < ((-00 - 1)/100 laol) Let 0 < E I < 0.1 Let 0 < e2 G 0.05 If 6 of the disk DS is sufficiently small, we can compare the various terms in the expressions for and ?i+1

and conclude

or (lb)

or (2c) p;-'

< < [Fi[. i t

We denote the strong stable manifold of pil? by Wss(pp,).

and

Reflection and non-hyperbolic invariant lines

163

?)[I < p p } . There exists a Lemma 1. Let 0 c &g < 0.1 Let R, = [(X, 7)/ [W, small neighbourhood U of (0,O) in the parameter space such that if p E (U n c:) then r l R,) c {(.e 9: If1 < 2P:+E'l 1j11 (WSS(Pp2) Corollary 1. ( w " ( p p , )n xE3) 2 {(x, 7):: 2 2p:L+L3)171)n R~, Proof of lemma 1. 1+2aoX+rl(p,i,7)

DFw(X, 7) =

i

j;

+ r3(prX,j )

where r i ( p . X,7) = o(lpzI2) +o(lpzl)o(lX, 71) and U sufficiently small we obtain 1 OFp(., 7)

+ 2aoX 7

-1

-

-1

+ o([X, 71')

E

(

-73

--27 ur +f 20"

+ rz(pL.X,7) +X

+r&

~)

2 , j;)

For X < p p and

7 < I.:*"

1

and the term -pz/2ao is much stronger than 2 ~ 0 2-27, , j , X. Therefore, for (.?, 7) E Re,, we can estimate eigenvalues and eigenvectors of DFw and of (OFp,)-'. The more contractive eigenvalue for Dl?,(X, 7) is approximately equal to (-1 (p2/2ao)). Thus the more expansive eigenvalue for (DFp(,f,?))-I is approximately (-I (p2/2ao)). The corresponding eigenvector is of the form ( ~ ( l p ~ l ' +1)~where ~), lo(lp~1'+'~)1< 2 Ip211+E3. These estimatives &e independent of the point (X, j;) E 'RE>. Let C = ( ( 2 ,y)/ lj;l 2 IXI} n REZ. Iterating this cone by ( k i t )with i + +co and n R,,we obtain W r s ( p M 2n)R, = considering C; = FLi(C) Ci. The estimative of the more expansive eigenvectors of OF;' imply

+

Proposition 1. Let

kl = 3(-ao

xo

= -x o

+ 1) 1-01

= p2

A2 such that A1 > A2 =-

yo =

AI -

ko

A3 such that 0

-yo = &p;-A3.

AI < A3 < ki

There exists a U neighbourhood of (0,O) in parameter space such that i f p E (c: and ppl is the corresponding saddle-node, 0) F O ,E ~ WLC(pir2) ~) (Wfocmeans local stable manifold).

(ii) GO, 30)$ wLC(ppLI)

nU)

J Gheiner

164

+

Proof. In condition (2b) we take EZ > A I . In lemma 1 we take c ( A l / 6 ( - ~ 1)). First we prove (i). Let Ti)= ~ L G O , We ~ ~just ) . consider the is that satisfy STj 7- P* 2-Ei for 0 < j < i. We assume the additional hypothesis: F;> IFi]. There are two cases to consider (I -22)xi. In this case, as AI EZ and the sequence F;is a decreasing (a) Vi lyi[ p2 sequence, (2b) and (la) are valid. Therefore, by (1) we obtain

e;,




thus

From (3) and (4) we have

from lemma 1. Let N be the first positive integer such that TN E From (5)

Let us consider

EN < PF

E?

e.

=P For

-+a.

ITN1 < TN to occur it is sufficient that

(we identify T N with pi+% without any loss in the estimates). Therefore it is sufficient that &3 < (A1/2(-ao+ 1)) - A3 (here'we used ((-0 - 1)/100)) This last condition is true due to the hypothesis

As W((p,)

EO


ko > -a0

- 0.03(-ao - 1) > 1.

Suppose X,q = pL2 !-A1 y~~~ - = p i - A s . Substituting i n (7), pL-AsfAzl'/*z

( - A ~ + A , ) ~=$

- A 5 + A 2 < ( - A 4 + A i ) % + A 4 < k o ( A s - A z ) + A l . InorderforG, i have Ai+.! < A;. Therefore hi

N3 > i

< P2

(I-%)

> N2 we

> N2.

(8)

From (1) and (2)we obtain

> &".

The last inequality is due to (8) and ljil

The same inequalities imply

Therefore,

(

AN3

-
0 small satisfying E > ZA,. Proposition I . (a) h,(y) is strictly increasing. (6)1.8% < h;(y) < 2.1E i f Ih,(y)[ i I 4 X,(Y) > 1 i f IMY)~ 1 (ci) < thvl < i f I 11,(y)l

%




>

> z.

if (x - x p r ) >

1

Proof. From the normal form for F i (see section 5.2) we get

+ PIO(IPI)) + (1 - 2 4 4 4 - ZPZ + O(lPI2))X + O(IP1)PL:Y+ ( - 2 d P ) + + (-zU + o ( ~ P ~ ) )+y 2O ( I , ~ +I o) (~~~xyi3) ,

F,-2(x,Y) = ((-2PI

O(IPl))X2

+ o(lWi2))Y - o(lPl)Pix + (Z+ o ( ~ P ~ ) )+x oy ( ~ P ~ )+x 0(iPi)Y2 2 + Y14). - o(lPl)P?

(1 - 3P1

O(ix,

Let ppi

= (xp*.

YpA

=

rpo: E~ n D~ --f E,

1

PZ (-% + 0(lPZl2), o(1~221~)

p0(y) = xpZ

EO= {(xpz, Y) E DsI.

Let pi: Zi -f R be the function whose graph is F;”(Co) res’uicted to 1, = [y: Ipi(y)l 0.01/ Iaol]. Let F i 2 = (gl, gz), Id = identity function and hz = (gz o (q;.Id))-1. Then





j o / p ; ~ : -it~ follows that A,, > j ; l / ~ i - ~ , which is absurd. Therefore (15) is true. Inequalities (13) and (15) imply item ( d ) of the proposition.

Reflection and non-hyperbolic invariant lines

173


1. This A does not depend on t and on the curve. (6) Let Vz be asmall neighbourhood of WS,,(p). By (2) and (3) the length of any curve contained in Ti \ VZ when iterated by F;, n large, gets multiplied by ?j >> 1. (7) From (5) and (6) and @e fact that F;(Tz) and F;(y:) hsve a minimum of transversality independent of n (by this we mean that if r, = F,"(ri)n F{(y:) then given U E Tra(F;(Tr)) and w E T,*(y:) then (u/llull, w/llwll) < E for some E > 0 independent of U , , w, n) it follows that the length of any curve contained in r when iterated by F:+k with n big gets multiplied at least by kl min(A", ,8) for some fixed constant 0 < kl 1. Therefore the (U 1) volume opf any (U 1)-dimensional region contained in z when iterated by F;+k is multiplied at least by (kzA"j?) for some fixed constant 0 < kz 1. We can take n and B such that ,8 > l j k z and 1" > l j k z . (8), From ( 4 ) and (7) the fibres r when iterated by Fgn+k,n large, intersect A almost parallel to their initial position. We have also a minimum uniform expansion of these fibres. Let R S ( F , ( A )n W ' ( p ) ) ,R c C. FCk(R) is the fibre space in A. As R C W s ( p ) .it follows lim,,,(diameter of F;(R))+ 0. Let 1(C) be the Iength of the curve C. (9) From (8) it follows that the= exists 0 < q 0 be sufficiently large in order that T - ' ( k ) fl U # 0. We modify U so that the boundary of U satisfies

180

J Gheiner

aU 3 (T-'(k) n U ) . See figure 5.

Figure 5.

Fiyre 6.

using the same k . We define h : U 13 U Analogously we proceed with F, f,, preserving the level curves of T and f and mapping c: onto $ and c; onto Z; for i = 1, 2 , 3, 4 and mapping (l), ( 2 ) , (3). (4) onto the corresponding regions for p. To get a possible construction of h we can construct two foliations in U . The first one containing the level curves of T (f)and the curves ci (&) and the second one transversal to the first one, as in figure 6. The map h is defined preserving both foliations. Let D, be a fundamental domain of the sink q that attracts WF-.Let = (WF-nD,) (we consider' : W iterated by F;, n 2 0). II, is compact. For p E (c; U (1) U c: U (2)U c: U ((0, O))), Fl covers a neighbourhood V, of n, in W;(p). For p E ((3) Uc; U (4)), Fl covers a neighbourhood R, of the long branch of the unstable manifold of the saddle intersected with D, inside W;(p). Inside W;(p) and oh U, we raise a germ of foliation F : transversal to n, and transversal to F : except when p E ((1) Uc;). For p E ((1) Uc;) we require that, outside n,, F : intersects 3: transversally and that, along n,, F : intersects F : quasitransversally. This condition is possible due to the results of section8.7. We construct Ff invariant by F,. We define H,: W:(F) H W&,l(F) as follows, H,(E, 0) = ( E , O)H@(-E,0) = ( - E , 0). Let X ; be the flow of X,. We require H,(X;(&,O)) = RZb)(e,O),t E R and H,(X;(-&, 0)) = ki(,l(-~,o), E R. We define H,:J, H &),( to conjugate F,/JM and F,,(,J/&(,J.We extend

n,

t

H,:

U F,"(J,)

H

U k(,)(&d)

"€Z

"€2

using the conjugacy equation. We have defined, therefore, H, in the space of leaves of Fl and of 3:. We define invariant by F,. Preserving both foliations we define H, analogous foliations F: and rt mapping n, + f i h b ) , V, + v,,,], if p E (c; U (1) U c: U (2) U c: U ((0,o)]),R, + RA&), if p E ((3) U c; U (4)). For p E ((3) U c; U (4)) we define V, as the connected neighbourhood of n, in W ; ( p ) that contains R, and whose boundary coincides with the (external) components of the boundary of R, that are farther from n., We have already defined H, on the boundary of (V, \ R,). Using an extension theorem for isotopies [PA], [CE],[N-P-TI, we extend H, for (V, \ R,). We get, thus, H, defined at V,, for p E U .

FH

Reflection and non-hyperbolic invariant lines

181

Figure 7.

It is necessary to glue the method of constuction of H, used for p E (4) with the method used for p E c;. Let R; c (4) be the closed region bounded by c; U c; U aU and R; the region bounded by c; U ;c U aU.Analogously we define R: and &+.In W ; ( p ) , for p E (R;U R;), there exists a unique strong unstable foliation defined in the unstable manifold of the source that unfolds from p. For p E 72.7 we aggregate this foliation to F; in place of the previous foliation raised on J,. Analogously we define 7?; and and we make the same substitution for 3;. For p E R; we redefine H,: Vfi H &!) preserving :. For w E R;,as p slides from c; to c; along any leaf of the both foliations 3,"and F foliation that has leaves parallel to the pl-axis we gradually remove the leaves of the strong unstable foliation until we anive at c; when we are left onIy with the leaves that intersect the short branch of the stable manifold of the saddle at

G-

UF,"(J,).

*€N

This (partial) foliation is part of the strong unstable foliation and substitutes the original one as part of 3;. For p E R; we complete the definition of H, at the part of V, not covered by 3; using the isotopy extension theorem. Using the conjugacy equation we extend the definition of HF to W ; ( p ) for p E ((1) U c; U (4) U c; U ( ( O , O ) ] ) . For p E ((3) U cz U (2)) we need to define H, at a fundamental domain A,, in W;(p), of the sink that unfolds from p . For p E (c:UR:URz) we modify 3; a little bit in order that its leaves have quadratic tangencies with the strong stable foliation in the stable manifold of the saddle-node or sink. We require these tangencies to be placed along W:'. For p E (c: U R:) we define H, at the stable manifold of the saddle-node or sink preserving 3; and the strong stable foliation. Let p E ((3)Ucz U ((2)\(R:U%:))). Let ps,, be the saddle that unfolds from p (in W ; ( p ) ) . We define H,:(W"(pz,,) r l A,) H (W'(j%,,(,)) n &,)) as a diffeomorphism that satisfies the conjugacy equation. By the A-lemma [PI the leaves of 3; intersected with A , remain C' close to (W'(p2,,) r l A,) in a neighbourhood S, of (W"(p2,,) fl A,) in A,. Using a foliation transversal to FF we can extend H, to the leaves of 3; in S,. Hfi is also already defined at (WF+ n A,). We extend HF to the remainder of A , using the isotopy extension theorem. For fi E &+,as p slides along leaves of a foliation transversal to the curves ci from c: to ,c: we gradually remove leaves of 3; in a continuous way, and when we arrive at c,' we are left only with the leaves of F ,that intersect a neighbourhood S, of (W'(p2, p ) n A,) in A,. For p E &+we define Hfi at the region of A , covered by 3; preserving F; and the strong stable foliation. At the region not covered by F; we define H, using the isotopy extension theorem. We use the conjugacy equation to extend the definition of H, at A,.

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J Gheiner

We have ended, therefore, the definition of H,: Wc((O,p ) , F ) H Wc((O,6). p ) . In a neighbourhood of (0, p ) in U x M we get a C’ (big k) strong unstable foliation .1”,” whose leaves have the strong unstable manifold W’”(p, PO)as model. These leaves have dimension nu. The conjugacy H is already defined at the space of leaves Wc((O,p ) , F ) . We define the unstable manifold W ’ ( p , P,) as the unstable manifold of the saddle-node if p E c1, as the unstable manifold of the sink if p E (4) U (3). as the unstable manifold of the saddle if p E (2). as the union of leaves of i’$” that intersect W;C- if p E (1). The construction of H, at the stable manifolds (of periodic orbits) that intersect W ’ ( p , FJ follows the same approach of [N-P-TI. 0 Observation I. We observe that the structure of the proof of theorem 2(c) has two parts. The first one (the global one) is the inductive construction of the conjugacy at the stable manifold of periodic orbits using the order given by filtration and a compatible system of unstable foliations. This part is explained in m-P-r] and we did not repeat here. The second one is the construction of the homeomorphism in the parameter space, the construction of the local conjugacy at the central manifold of the reflection orbit, and the construction of local invariant foliations for the reflection. This part was done here and the compatibility of the strong unstable foliation at the reflection with the other unstable foliations does not present any difficulty. Observation2. The case n, p 0 demands a more elaborate construction of the foliation 3,” (inside the centre-stable manifold of p ) and a more elaborate construction of the conjugacy at a neighbourhood of n, - Wiz- n D, where D, is a fundamental domain of the sink that attracts WP-.One can follow the same general approach of case n, = 0. Proofoftheorem 2(b). Here it is not necessary to preserve neither the vector field X, nor the strong stable and unstable foliations of saddle-nodes. So we have much more freedom to construct the conjugacy. The techniques used to prove theorem 2(c) can be easily adapted to this case. We must take care to consider Fl compatible with the unstable foliations of hyperbolic saddles that precede p in the filtration order. 0 Proofoftheorem 2(d). The ‘if part of the statement has already been proved. Suppose F is stable and p is s-critical. Let X, be the adapted vector fields given by corollary 1 of section 8.3. By theorem 3 the conjugacy HOmust preserve XO.Let x = (W’(q, Fo) n W c ) where q is an hyperbolic saddle. Consider a sequence x, + x such that x, E W c and -P y E (Ws(q, PO)- q ) , m, t +CO. Let d( , ) be a C O metric in M. Let A be the contracting eigenvalue of DFo(q). Then

for some positive contant c. Let p E &(M) be close to F. As F is stable there exists a conjugacy H between F and p . We can choose p so that all the eigenvalues of saddles of I‘ have modulus different from pl. Let Q = Ho(q), 2 = Ho(x), 2” = Ho(xn).Then

for some positive constant E with I ~#I [AI. Therefore

This limit is not compatible with DHo mapping Xo differentiably to the same proof works for F;’.

20.If p

is U-critical I3

ReJlectionand non-hyperbolic invariant lines

183

Observation. The proof of theorem 2(d) works in the case nu = 0 and p is u-critical and in the case n.y = 0 and p is s-critical. Therefore, if n, = 0 or n, = 0 stability implies non-existence of one type of criticality.

Proof of theorem 3. Let H be a conjugacy between F and F . Suppose h(0,O) = (0,O). Let p~ E (CI \ [(O,O)]). Let c : ( - S , S ) H U be a small curve in U transversal to CI del dd at t = 0. We define Ft = Fc(t)and 6 = Fh(c(r)) (we recall H(p, q ) = (h(p), H,(q))). Let G : (-S,6) x M H (-6,s) x M be defined by G ( t , q ) = ( t , H,&)). Then G is a conjugacy between the one-parameter families IFt}and (jt]. Let X, and 2, be the adapted vector fields given by corollary 1 of section 8.3. It follows from [N-P-TI that Go, and therefore H,,, maps x,, differentiably onto %h@o). BY the continuity ofx, respect to p, if po + (0,O) along ci we get that HOmaps XO to 20. 0

Figure 8.

Acknowledgments This article is derived from our doctoral thesis at IMPA [GI;the results were announced in [GZ]. We thank our supervisor Jacob Palis for consistent support and also for the suggestion of the theme. We acknowledge all our friends at IMPA for providing a serious and relaxed environment for scientific research and teaching.

References Beloqui I and Pacific0 M J 1993 Quasi-transversal saddle-node bifurcation on surfaces Ergod. Theor. Dyn. Syst. to appear Chenciner A 1985 Bifurcations de points fixes elliptiques. ICourbes Invatiantes Publication Math. de I’IHES 61 67-127 Cerf I 1961 Bull. Soc. Math. France 89 227-380 Dumoitietier F 1978 Sinplarities of W m r Field, MonogralFos de Matemdtica 32 (Rio de laneim: IMPA) Dumortier F and Roussatie R 1981 Germs de Diffeomorphismes et de Champs de Vecteurs en Classe de Differentiabilitb Fide Ann. Inst. Fourier Crenoble 33(1) 195-267 Dumortier F, Rodrigues P R and Roussatie R 1981 Germ of Diffeomorphisms in the Plane (Springer Leetwre Notes in Mathemtics 902) (Berlin: Springer) pp 1 4 7 Gheiner J 1989 Sobre BifuEapTes de Codimensrio Do& (Informes de Motmd!Ico, drie F-033) (Ria de Janeiro: IMPA) Gheiner I 1989 Bifurcations and stability of reflections, generalized saddle-nodes and flips AR Acad Bras. Ci.61 Hisrsch M W, Pugh C C and Shub M 1977 lnvmiant Manifulds (Springer Lecture Notes in Mnthemntics 583) (Berlin: Springer)

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J Gheiner

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