CHAOS

VOLUME 11, NUMBER 3

SEPTEMBER 2001

Some topological invariants for three-dimensional flows Emmanuel Dufrainea) Universite´ de Bourgogne, Laboratoire de Topologie, U.M.R. 5584 du C.N.R.S., B.P. 47870-21078 Dijon Cedex, France

共Received 19 December 2000; accepted 15 May 2001; published 12 July 2001兲 We deal here with vector fields on three manifolds. For a system with a homoclinic orbit to a saddle-focus point, we show that the imaginary part of the complex eigenvalues is a conjugacy invariant. We show also that the ratio of the real part of the complex eigenvalue over the real one is invariant under topological equivalence. For a system with two saddle-focus points and an orbit connecting the one-dimensional invariant manifold of those points, we compute a conjugacy invariant related to the eigenvalues of the vector field at the singularities. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1385918兴

induced by X ⬘ preserving the time orientation of the orbits. When the homeomorphism is time preserving, namely when for each t in R we have the relation

In the study of dynamical systems, the concept of topological invariants, that is quantities which remain unchanged under continuous changes of variables, is particularly relevant. These quantities may be interesting for at least two reasons. On one hand, in some simple cases, one can exhibit a complete set of invariants: For example, in the neighborhood of a heteroclinic orbit †as in Bonatti and Dufraine „2001…; van Strien „1982…, etc.‡ or for vector fields without periodic orbit on the two-dimensional torus, etc. On the other hand, they are the main tool to distinguish a priori similar complicated dynamical behaviors. Dimension three is the smallest dimension where we can find ‘‘chaotic’’ vector fields, in this case, a complete classification is hopeless. As the classical methods are not always relevant in those intricate situations, it is interesting to find new methods to obtain topological invariants, using linking number, Conley index, and others †see Gambaudo and Ghys „1997… for some examples‡. The main results of this paper deal with a homoclinic orbit to a saddle focus „a saddle with a complex eigenvalue… on a three-dimensional manifold. The Silnikov ratio is the absolute value of the real part of the complex eigenvalue over the real one; Silnikov „1965… proved that if this ratio is smaller than 1, there are infinitely many Smale horseshoes in the neighborhood of the homoclinic orbit, i.e., the system under consideration is ‘‘chaotic.’’ It turned out that this ratio is a topological invariant; we give here a simple proof of this fact. Moreover we compute new conjugacy invariants in this situation and in the case of vector fields with heteroclinic orbit between saddle foci.

hⴰX t ⫽ X t⬘ ⴰh, the flows are conjugate and h is a conjugacy. When the conjugacy h is a C r diffeomorphism, we say that the flows are C r conjugate. One of the most useful tools to obtain topological invariants is the rotation number of a homeomorphism of the circle. Let f be a homeomorphism of the circle, and let F be a lift of f to R and choose x a point of R. The limit of 关 F n (x)⫺x 兴 /n exists, for all x in R, and is equal to a number ␳ ( f ) which does not depend, up to an integer, on either the choice of x or the choice of F 关e.g., Godbillon 共1983兲兴. If f and f ⬘ are two homeomorphisms of the circle such that there exists h an orientation preserving homeomorphism such that hⴰ f ⫽ f ⬘ ⴰh then ␳ ( f )⫽ ␳ ( f ⬘ ). For instance, let X and X ⬘ be two C 2 vector fields on the two-dimensional torus without periodic orbit. There exist closed transversals ␥ and ␥ ⬘ on which we can define f and f ⬘ , the Poincare´ maps of X and X ⬘ on those curves. The numbers ␳ (X) and ␳ (X ⬘ ) are the rotation numbers of f and f ⬘ , respectively. Therefore, if X and X ⬘ are topologically equivalent then ␳ (X)⫽ ␳ (X ⬘ )mod Q. For three-dimensional vector fields, the natural extension of rotation number is the linking number. Let X be a volumepreserving vector field on S 3 , and ⌬(x,X t (x)) a system of paths joining x to X t (x). For x 1 and x 2 , two points of S 3 , let us define k 1 (T 1 ,x 1 )⫽ 兵 X t (x 1 ), t苸 关 0,T 1 兴 其 艛⌬(x 1 ,X T 1 (x 1 )) and k 2 (T 2 ,x 2 )⫽ 兵 X t (x 2 ), t苸 关 0,T 2 兴 其 艛⌬(x 2 ,X T 2 (x 2 )). There exists a system of paths ⌬(x,X t (x)), such that for almost every pair of points x 1 , x 2 and for almost every T 1 ⬎0 and T 2 ⬎0, the linking number between k 1 (T 1 ,x 1 ) and k 2 (T 2 ,x 2 ) exists, and the limit limT 1 ,T 2 →⫹⬁ (1/T 1 T 2 ) link(k 1 (T 1 ,x 1 ), k 2 (T 2 ,x 2 )) exists almost everywhere and is integrable with respect to (x 1 ,x 2 ). The Arnold invariant, Arnold 共1986兲, is the space average on S 3 ⫻S 3 of this limit. It is well known that it is a C 1 -conjugacy invariant. But it is still not known whether the Arnold invariant is a conjugacy invariant or not; as it bounds

I. INTRODUCTION

We recall that two smooth vector fields X and X ⬘ on a smooth compact manifold M are topologically equivalent if there exists a homeomorphism h:M →M mapping the orbits of the flow X t induced by X on the orbits of the flow X ⬘t a兲

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© 2001 American Institute of Physics

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FIG. 1. Homoclinic orbit of saddle-focus type. FIG. 2. Heteroclinic orbit.

by below the energy of the vector field, it could be interesting to give a topological meaning of this quantity. For a first attempt in this direction, see Gambaudo and Ghys 共1997兲. Another situation where there are many topological invariants is the semilocal situation: equivalence or conjugacy in the neighborhood of a particular orbit. The first example of such a phenomenon appears in Palis 共1978兲, where the author gives conjugacy invariants for vector fields and diffeomorphisms of the plane in a neighborhood of a saddle connection. We can find classical works on the existence of topological invariants in similar situations in Afrajmovich and Il’Yashenko 共1994兲; Beloqui 共1986兲; Ceballos and Labarca 共1992兲; Togawa 共1987兲; and van Strien 共1982兲, for instance. In the semilocal situation, we often give interest to either topological equivalence of vector fields or topological conjugacy of diffeomorphisms. Thus it is also interesting to find conjugacy invariants for vector fields to obtain some indications for the existence of conjugacy invariants of diffeomorphisms with the same ‘‘pattern’’ of orbits. Indeed conjugacy invariants are often simpler to find for vector fields than for diffeomorphisms. In this paper we give a systematic approach of two semilocal situations in dimension three, using knot-like arguments. As for the Arnold invariant, we close pieces of orbits with paths to obtain knots, the main difficulty is to control the contribution of those paths to the linking number. More precisely we consider the following two sets of vector fields on a manifold M 3 . 共1兲 Homoclinic orbit of saddle-focus type: We denote by S the set of C 2 vector fields X satisfying the following. 共a兲 共b兲

There exists a point p such that X(p)⫽0. The eigenvalues of the linear part of X at p are given by (␭ and ⫺ ␳ ⫾i ␻ ) or (⫺␭ and ␳ ⫾i ␻ ), with ␳ and ␭⬎0. The point p belongs to the ␻ -limit and the ␣ -limit sets of an orbit ⌫, and p is the only singularity of X in a neighborhood of ¯⌫ 共see Fig. 1兲.

共2兲 Heteroclinic connection: We denote by J the set of C 2 vector fields X, such that: 共a兲 共b兲

X has two singularities p 1 and p 2 , with ⫺ ␳ 1 ⫾i ␻ 1 , ␭ 1 and ␳ 2 ⫾i ␻ 2 , ⫺␭ 2 as eigenvalues of its linear part at p 1 and p 2 ( ␳ i and ␭ i positives兲. The invariant manifolds corresponding to ␭ i (i⫽1,2) are composed of two connected components, such a component is a separatrix. We assume that those manifolds have a common separatrix: there is a heteroclinic orbit ⌫ between p 1 and p 2 共see Fig. 2兲.

In situation 1, we first obtain Theorem 1: 兩␻兩 is a conjugacy invariant: if X and X⬘, two vector fields in S, are conjugate by h, then 兩␻兩⫽兩␻⬘兩. In fact, if h preserve the orientation, ␻⫽␻⬘ and if h reverse the orientation, ␻⫽⫺␻⬘. The second result in this situation is Theorem 2: The ratio ␳/␭ is invariant under topological equivalence. Notice that this result is already contained in Ceballos and Labarca 共1992兲 and Afrajmovich and Il’Yashenko 共1994兲. In those two papers, the proof of this theorem was based on the study of a Poincare´ map. Using knot-like argument, we prove the theorem following an idea of Togawa 共1987兲, we believe in a much simpler way. In situation 2, we define the Palis invariant p⫽ ␳ 1 / ␳ 2 , we give the proof of the invariance under conjugacy of this ratio, and we obtain a new conjugacy invariant: Theorem 3: 兩 ␻ 1 ⫹ p ␻ 2 兩 is a conjugacy invariant. In fact, if X and X⬘, two vector fields in J, are conjugate by h, then ␻ 1 ⫹ p ␻ 2 ⫽ ␻ ⬘1 ⫹ p ␻ ⬘2 if h preserve the orientation, and ␻ 1 ⫹ p ␻ 2 ⫽⫺( ␻ ⬘1 ⫹ p ␻ ⬘2 ) if h reverse the orientation.

II. HOMOCLINIC ORBIT OF SADDLE-FOCUS TYPE

Let X be a vector field of S. We assume that the eigenvalues of the linear part of X at p are ␭ and ⫺ ␳ ⫾i ␻ with ␳ and ␭⬎0. We get the other case by reversing time. From the invariant manifold theorem, Hadamard 共1901兲, s there exists a local stable surface near p, W loc (p) and a oneu dimensional local unstable manifold, W loc(p). We can extend those local manifolds to obtain W u (p) and W s (p), which are immersed in M. In fact, ⌫⫽W u (p)艚W s (p). In a neighborhood U of the point p, we take C 0 invariant coordinates; it is possible to construct C 0 invariant coordinates using the ␭ lemma. We call cylindrical coordinates such coordinates given by (r, ␪ ,z), with r苸 关 0,r 0 兴 , ␪ 苸R, z u (p) is 苸 关 ⫺z 0 ,z 0 兴 . In those coordinates, we assume that W loc s the set 兵 (0,␪ ,z),z 其 , W loc(p) is the set 兵 (r, ␪ ,0),r, ␪ 其 , and ⌫艚U belongs to U ⫹ ⫽ 兵 (r, ␪ ,z),z⭓0 其 . A. Construction of a neighborhood of the homoclinic orbit

For given cylindrical coordinates, we construct neighborhoods of ¯⌫ by taking the union of two sets: a neighborhood of the singularity p and a tubular neighborhood of the regular part of ¯⌫.

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Chaos, Vol. 11, No. 3, 2001

Let ␧⬎0, U ␧ is the set 兵 (r, ␪ ,z),r⭐r 0 , 兩 z 兩 ⭐␧ 其 . The set ˜ ⳵ U ␧ is the union of D⫽ 兵 (r, ␪ ,z),r⭐r 0 ,z⫽␧ 其 , D ⫽ 兵 (r, ␪ ,z),r⭐r 0 ,z⫽⫺␧ 其 , and C⫽ 兵 (r, ␪ ,z),r⫽r 0 , 兩 z 兩 ⭐␧ 其 . We denote by B the intersection of ⌫ with C, and B 1 the intersection of ⌫ with D. The regular part of the homoclinic orbit is outside U ␧ , between B 1 and B. We take R ␧ a tubular neighborhood of this piece of orbit: R ␧ is an embedding of D 2 ⫻ 关 0,1兴 with disks of diameter less than ␧. We assume that the regular part of ⌫ outside U ␧ is the set 兵 0 其 ⫻ 关 0,1兴 , D 2 ⫻ 兵 0 其 is a subset of D, and that D 2 ⫻ 兵 1 其 is a topological disk on C. We obtain a neighborhood V ␧ ⫽U ␧ 艛R ␧ of ⌫, and for ␧ small enough, V ␧ is a solid torus. As we restrict our study to the local behavior around ⌫, this allows us to avoid any hypothesis of the embedding of ⌫ in M 3 . In particular, the knot type of ⌫ in M 3 has no influence. B. The twisting number

We recall that in a solid torus, D is a meridinal disk if ⳵ D gives a nontrivial cycle in the first homology group of the boundary of the solid torus. The curve ⳵ D is called a meridian. ¯ , isotopic to a loop cutDefinition II.1: A loop, in V ␧ ⶿⌫ ting just once a meridinal disk is called a one-loop. s (p), between B We denote by L, the straight line, in W loc and p; and ⌳ is the loop we obtain by replacing the piece of ⌫ between B and p by L. The core of V ␧ is isotopic to ⌳. Lemma II.2: Let ␧⬎0, there exists an isotopy H ␧t :M →M such that H ␧0 is the identity map of M, and H ␧1 (⌫) ⫽⌳; moreover, H ␧t is the identity outside U ␧ (for each t in [0,1]). Proof: There exists a homeomorphism g of the disk of s (p) mapping the part of ⌫ between B and p radius r 0 in W loc onto L, such that g is the identity on the boundary circle and g(p)⫽ p. Like every homeomorphism of the disk which is the identity on the boundary, the homeomorphism g is isotopic to identity by g t (g 0 is the identity and g 1 ⫽g) with g t (p)⫽p and g t equals the identity on the unit circle for each t in 关 0,1兴 . In U ␧ , we take H ␧t (r, ␪ ,z) ⫽(g t(␧⫺ 兩 z 兩 )/␧ (r, ␪ ),z). Outside U ␧ , we extend H ␧t with the identity. This is possible because H ␧t is the identity on ⳵U␧ . 䊐 Therefore, the homoclinic orbit ¯⌫ is isotopic to the core ¯ )⯝Z2 . There is a reof V ␧ , then we can say that H 1 (V ␧ ⶿⌫ ¯ on ⳵ V , then the isotopy class of two loops traction of V ␧ ⶿⌫ ␧ ¯) on ⳵ V ␧ gives us an explicit isomorphism between H 1 (V ␧ ⶿⌫ 2 and Z . One of those loops is well defined: the meridian, ␮ , given by the isotopy class of 兵 (r 0 , ␪ ,z 0 ), ␪ 苸R其 ⫽ ⳵ D. As the neighborhood U is orientable, the orientation of this loop is also well defined, because the orientation of D is given by the vector X(B 1 ). We have to choose the other generator, ⌶. We remark here that this choice is not canonical. Remark II.3: With the isomorphism given by the choice of the two generators, the isotopy class of a one-loop is given by a couple (n,1). If ⌶ is given, we can state the following: Definition II.4: The twisting number of a one-loop L with ¯⌫ in V ␧ is given by the component of the isotopy class of

Topological invariants for three-dimensional flows

445

¯ ) on the well defined meridian generator. L in H 1 (V ␧ ⶿⌫ ¯ )⫽n. If 关 L兴 H 1 ⫽(n,1), twist(L,⌫ Let C 1 be the subset of C of points A such that the Poincare´ map of A on C is well defined. We denote by ␶ the time of first return on C, ␶ depending on A. We take a path ⌬(A,X ␶ (A)) joining A to X ␶ (A) on C⶿ 兵 B 其 . We assume that this path cut at most one time the segment W s ⫽ 兵 (r 0 ,0,z), 兩 z 兩 ⭐z 0 其 and at most one time W loc (p). Then we denote by ␥ A the following one-loop:

␥ A ⫽ 兵 X t 共 A 兲 ,t苸 关 0,␶ 兴 其 艛⌬ 共 A,X ␶ 共 A 兲兲 . Lemma II.5: Let (A n ) n苸N be a sequence of points of C 1 which converges to B. If the limit lim n→⫹⬁

¯兲 twist共 ␥ A n ,⌫

␶n

exists, it does not depend on the choice of ⌶ or on the choice of coordinates. ¯ ), we Proof: First, if ⌶⬘ is another generator of H 1 (V ␧ ⶿⌫ ¯ ) the twisting number given by ⌶⬘. denote by twist⬘ ( ␥ A ,⌫ The number of intersections between ⌶ and ⌶⬘ is equal to a ¯ )⫺twist( ␥ ,⌫ ¯ ) 兩 ⭐K. As constant K, then 兩 twist⬘ ( ␥ A ,⌫ A then we get limn→⫹⬁ ␶ n ⫽⫹⬁, ¯ )⫺twist( ␥ ,⌫ ¯ ) 兩 ⫽0. limA n →B (1/ ␶ n ) 兩 twist⬘ ( ␥ A n ,⌫ An Let (r, ␪ ,z) and (r ⬘ , ␪ ⬘ ,z ⬘ ) be two systems of cylindrical coordinates, we can assume that the cylinders C and C ⬘ are equal, following the flow to modify one of the systems of coordinates. For a given point A on C 1 , we get two different one-loops, ␥ A and ␥ A⬘ . Each coordinate system gives one segment W 共for (r, ␪ ,z)) and W ⬘ 共for (r ⬘ , ␪ ⬘ ,z ⬘ )) and therefore each coordinate system gives one closing path on C, ⌬(A,X ␶ (A)) and ⌬ ⬘ (A,X ␶ (A)), respectively. There an isotopy of C, which is the identity on the boundary, such that the images of W and W ⬘ by this isotopy are transversal and have a finite number of intersections. Therefore, the number of intersections 共after the isotopy兲 between ⌬(A,X ␶ (A)) and ⌬ ⬘ (A,X ␶ (A)) is bounded above and below by constants which do not depend on A. ¯ )⫺twist( ␥ ⬘ ,⌫ ¯ )兩 We obtain the inequality: 兩 twist( ␥ A ,⌫ A ⭐K ⬘ . We end the proof of the lemma with the same argument as quoted previously. 䊐 Corollary II.6: Let X and ˜X be two vector fields in S with homoclinic orbits ⌫ and ˜⌫, conjugate by a homeomorphism h. If A n is a sequence of points on C 1 , going to B, then ˜A n ˜ 1 ⫽h(C 1 ), and ⫽h(A n ) is a sequence of points on C lim n→⫹⬁

¯兲 twist共 ␥ A n ,⌫

␶n

⫽⫾ lim n→⫹⬁

twist共 ␥ ˜A n ,⌫D 兲

␶n

according to whether h is orientation preserving or not. Proof: Indeed, if X and ˜X are conjugate, then h maps u u ˜ W loc(p) on W loc (p ) and cylindrical coordinates on cylindrical coordinates, preserving time orientation. Therefore, if h ˜ , else h( ␮ ) is preserves the orientation, h( ␮ ) is isotopic to ␮ ˜ isotopic to ⫺ ␮ . This ends the proof using Lemma II.5. 䊐

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The next step is to compute this limit and show that it does not depend on the choice of the sequence of points. For ¯ ), this purpose, we compute the class of ␥ A in H 1 (V⶿⌫ choosing the coordinates and the generator ⌶ in a convenient way. First of all, we take C 1 coordinates in a neighborhood of the saddle p such that X is linear in those coordinates. This is possible using a theorem of Belitskij 共1979兲, which asserts that in a neighborhood of the saddle p, if the relations ” Re ␭ j⫹Re ␭k are satisfied for all the eigenvalues ␭ i of Re ␭i⫽ the linear part of X at p, the vector field is C 1 conjugate to its linear part. For a saddle focus, this condition is always satisfied since two eigenvalues have the same real part 共the complex ones兲 and the other eigenvalue has nonzero real part of opposite sign. Lemma II.7: Let A be a point of C 1 such that X ␶ (A) belongs to C, if we denote by ␶ ⫽T⫹s where T is the time spent in U, we have the following: There exists K⬍⬁ which does not depend on A, such that

冏

¯ 兲⫺ twist共 ␥ A ,⌫

冋 册冏

␻T ⬍K. 2␲

Here [.] denotes the integer part. Remark II.9: K depends on the choices we made on ⌶, on the coordinates, and on the isotopy. Proof: Let A be a point of C 1 . To compute the quantity ¯ ), we have to make an isotopy from ¯⌫ to ⌳. By twist( ␥ A ,⌫ Lemma II.2, there exists ␧⬎0 such that the isotopy H ␧t keeps ␥ A unchanged for every t in 关 0,1兴 . By definition, ¯ ) is equal to the first coordinate of the isotopy twist( ␥ A ,⌫ class of ␥ A in H 1 (V ␧ ⶿⌳). Up to a rotation of the coordinates around the z axis, we can assume that B is the point (r B , ␪ ⫽0, z⫽0). Therefore, the intersection of U and ⌳ is the set 兵 (r,0,0),r苸 关 0,r B 兴 其 艛 兵 (0,␪ ,z),z苸 关 0,z 0 兴 其 . Let ⌶ be the union of the path 兵兵 (r,0,z 0 ),r 苸 关 r 1 ,r B 兴 其 艛 兵 (r B ,0,z),z苸 关 z 1 ,z 0 兴 其其 (r 1 and z 1 are chosen such that this path is in ⳵ U ␧ 艚 ⳵ V ␧ ) and a path in ⳵ R ␧ such that ⌶ is a loop, and a generator of H 1 ( ⳵ V ␧ ). There exists a retraction of V ␧ ⶿⌳ on ⳵ V ␧ keeping the square 兵 (r,0,z),r 苸 关 0,r B 兴 ,z苸 关 0,z 0 兴 其 invariant. Then the isotopy class of ␥ A in H 1 (V ␧ ⶿⌳) is the algebraic sum of three terms: 关 ␻ T/2␲ 兴 for the part of ␥ A belonging to U, a constant k 共with respect to A) for the part in R, and ⫾1 added by the path ⌬(A,X ␶ (A)). The proof of the lemma follows. 䊐 C. The conjugacy invariant

Theorem 1: 兩 ␻ 兩 is a conjugacy invariant: if X and X ⬘ , two vector fields in S, are conjugate by h, then 兩 ␻ 兩 ⫽ 兩 ␻ ⬘ 兩 . In fact, if h preserve the orientation, ␻ ⫽ ␻ ⬘ and if h reverse the orientation, ␻ ⫽⫺ ␻ ⬘ . Proof: Let A n be a sequence of points on C 1 which tends to B, and such that there exist ␶ n ⬎0 with X ␶ n (A n )苸C. By Lemma II.7, we have lim n→⬁

¯ 兲 ⫺ 关共 ␻ /2 ␲ 兲 T 兴 兩 兩 twist共 ␥ A n ,⌫ n

␶n

⫽0.

We denote by T n the time spent in U by A n , we get ␶ n ⫽T n ⫹s n with limn→⬁ T n ⫽⫹⬁ and limn→⬁ s n ⫽s. Therefore, lim

关 ␻ /2 ␲ T n 兴

␶n

n→⬁

␻ 关 ␻ /2 ␲ T n 兴 ⫽ . T ⫹s 2 ␲ n n n→⬁

⫽ lim

Thus lim n→⬁

¯兲 twist共 ␥ A n ,⌫

␶n

⫽

␻ . 2␲

This limit does not depend on the choices we made and, by Corollary II.6, we get the result. 䊐

D. The invariant of topological equivalence

This notion of twisting and the trick consisting to close an orbit to obtain a loop allows us to give an alternative proof of a theorem of Afrajmovich–Il’yashenko, and Ceballos–Labarca. Theorem 2: The ratio ␳ /␭ is invariant under topological equivalence. Let us give the idea of the proof. We consider points on C in ⳵ U returning twice on C. Let U(2,␧) denotes the set of such points in U ␧ . Let A be a point in U(2,␧), we denote by T 1 the first time of exit from U and T 2 the second one. A calculation gives lim ␧→0

T2 ␳ ⫽ ␭ A苸U(2,␧) T 1 inf

共see Lemma II.10兲. Then ␳ /␭ is a conjugacy invariant. Following Togawa 共1987兲, we replace the time length by twisting number, which is invariant by topological equivalence. Previously, we proved that asymptotically we have ¯ )⫽( ␻ /2␲ )T, then it shows that up to a multiplitwist( ␥ A n ,⌫ cation by a constant, the time can be replaced by the twisting number. Then we compute the ratio of twists around ⌫ between the second and the first turn of a two-loop, and obtain the theorem. Remarks II.9: (i) As in the previous theorem, the result is true in arbitrary manifolds of dimension three and the knot type of ⌫ has no influence. (ii) Notice that we do not need periodic orbits to obtain knots, we can avoid the Silnikov assumption (see Silnikov (1965)): ␳ ⬍␭ and extend Togawa’s theorem. Moreover we will not make use of the Alexander polynomial and then we obtain a simpler proof of this result. (iii) It is sufficient to prove Theorem 2 for ␳ /␭ ⫽1, this will be convenient for Lemma II.10. Proof: 共Theorem 2兲 As in the previous part, we will use C 1 coordinates near p, where the vector field is given by its linear part. Let A be such a point of U(2,␧), and define ␶ 1 at the time of first return on C and ␶ 2 the same time for X ␶ 1 (A). We will work with the following one-loops:

␥ A1 ⫽ 兵 X t 共 A 兲 ,t苸 关 0,␶ 1 兴 其 艛⌬ 共 A,X ␶ 1 共 A 兲兲 and

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Chaos, Vol. 11, No. 3, 2001

Topological invariants for three-dimensional flows

␥ A2 ⫽ 兵 X t 共 A 兲 ,t苸 关 ␶ 1 , ␶ 2 兴 其 艛⌬ 共 X ␶ 1 共 A 兲 ,X ␶ 1 ⫹ ␶ 2 共 A 兲兲 . We define

␩共 ␧ 兲⫽

inf A苸U(2,␧)

¯兲 twist共 ␥ A2 ,⌫ ¯兲 twist共 ␥ 1 ,⌫

cuts L in a sequence of points A n , with limn→⬁ z A n ⫽0. If ␳ /␭ ⫽1, for n sufficiently large, each point A n is in U(2) and we have log M 共 ␥ A n 兲

lim

log m 共 ␥ A n 兲

n→⬁

A

and

447

⫽

␳ . ␭ 䊐

This proves Theorem 2.

␩ ⫽ lim ␩ 共 ␧ 兲 .

III. HETEROCLINIC CONNECTION

␧→0

As in the previous section, the quantity ␩ will not depend on the choices we made 共particularly the choice of ⌬ or of the second generator for the twisting number兲, then, as twisting number is a topological quantity, it is invariant under topological equivalence. We take the notations: A⫽(r 0 , ␪ 0 , m( ␥ A )) and X ␶ 1 (A)⫽(r ␶ 1 , ␪ ␶ 1 , M ( ␥ A )). As the point A is in U(2), m( ␥ A ) and M ( ␥ A ) are positives. Let T 1 be ⫺log(m(␥A))/␭, we have ␶ 1 ⫽T 1 ⫹s 1 with lim␧→0 s 1 ⫽s⬍⬁ if A is in U共2,␧). Likewise T 2 ⫽ ⫺log(M(␥A))/␭ and we have ␶ 2 ⫽T 2 ⫹s 2 such that lim␧→0 s 2 ⫽s ⬘ ⬍⬁. By Lemma II.7, we have the two inequalities

冏 冏

冏 冏

¯ 兲 ⫺ ␻ T ⬍K , twist共 ␥ A1 ,⌫ 1 2␲ 1 ¯ 兲⫺ twist共 ␥ A2 ,⌫

␻ T ⬍K 2 . 2␲ 2

As T 1 and T 2 go to infinity, we have

␩ ⫽ lim

inf

␧→0 A苸U(2,␧)

log M 共 ␥ A 兲 . log m 共 ␥ A 兲

But in the same time, we claim Lemma II.10: lim

inf

␧→0 A苸U(2,␧)

log M 共 ␥ A 兲 ␳ ⫽ . log m 共 ␥ A 兲 ␭

Proof: Applying the mean value theorem to the Poincare´ map f, from D to C in R, we have M 共 ␥ A 兲 ⭐ 共 sup兩兩 D f 兩兩 兲 m 共 ␥ A 兲

␳ /␭

.

Let X be a vector field of J. As in Sec. II, there exist C 1 coordinates in U 1 and U 2 , neigborhoods of p 1 and p 2 , respectively, where the vector field is given by its linear part. In U 1 , we write those coordinates (r, ␪ ,z), r and ␪ denote s ¯ ,¯␪ ,z ¯ ) in U 2 . U ⫹ (p 1 ) and (r the polar coordinates in W loc 1 and ⫺ U 2 are the parts of U 1 and U 2 which contain ⌫艚U 1 and ⌫艚U 2 , respectively. For a point A in one of those neighborhoods, the first coordinate r or ¯r denotes the distance d(A,⌫) between A and the heteroclinic orbit ⌫. We first recall a theorem of Palis and we give the proof, adapted from the two-dimensional case studied in Palis 共1978兲, which will be useful in the proof of the next result. Theorem III.1: (Palis) p⫽ ␳ 1 / ␳ 2 is a conjugacy invariant. Proof: Let X and X ⬘ be two vector fields of J, conjugate by h. Take (x n ) n苸N a sequence of points in U 1 converging to s (p 1 )⶿ 兵 p 1 其 , and ␶ n a sequence of times such x, a point of W loc that y n ⫽X ␶ n (x n ) is in U 2 and converge to y in u W loc (p 2 )⶿ 兵 p 2 其 . Let t n and T n be two sequences of time such that t n →⬁, T n →⬁, ␶ n ⫽T n ⫹s n ⫹t n with s n →s⬍⬁. And such that, X t n (x n )→A in U 1 with A on ⌫ and X ⫺T n (y n )→B in U 2 with B on ⌫ too. The transition between X t n (x n ) and X ⫺T n (y n ) is given by a C 1 diffeomorphism. Then, using the mean value theorem, there exists k 1 and k 2 , positive numbers such that k 1 d 共 x n ,⌫ 兲 e ⫺ ␳ 1 t n ⭐e ␳ 2 (⫺T n ) d 共 y n ,⌫ 兲 ⭐k 2 d 共 x n ,⌫ 兲 e ⫺ ␳ 1 t n . We have lim ln n→⬁

Then

Then we obtain lim

inf

␧→0 A苸U(2,␧)

log M 共 ␥ A 兲 ␳ ⭓ . log m 共 ␥ A 兲 ␭

Now, we have just to find a sequence ␥ A n of orbits in U共2兲, such that for every ␧⬎0 there is at least one orbit in U(2,␧) and log M 共 ␥ A n 兲

␳ ⫽ . lim ␭ n→⬁ log m 共 ␥ A n 兲 For this purpose, take L⫽ 兵 (r 0 , ␪ B ,z),z苸 关 0,z 0 /4兴 其 , if we denote by ⌬ 1 the map from C to D given by the vector field, ⌬ 1 (L) is a spiral of center B 1 . The image of this spiral by f

ln lim n→⬁

冉

冉

冊 冉

冊

d 共 y n ,⌫ 兲 d 共 y,⌫ 兲 ⫽ln ⬍⫹⬁. d 共 x n ,⌫ 兲 d 共 x,⌫ 兲

冊

d 共 y n ,⌫ 兲 d 共 x n ,⌫ 兲 ⫽0, tn

lim ⫺ ␳ 1 ⫹ ␳ 2 n→⬁

Tn ⫽0. tn

Therefore lim n→⬁

Tn ␳1 ⫽ . tn ␳2

For X ⬘ , we take x n⬘ ⫽h(x n ) and y ⬘n ⫽h(y n ), we get e ␳ 2⬘ T n k ⬘1 d ⬘ 共 x ⬘n ,⌫ ⬘ 兲 e ⫺ ␳ 1⬘ t n ⭐d ⬘ 共 y n⬘ ,⌫ ⬘ 兲 ⭐e ␳ ⬘2 T n k ⬘2 d ⬘ 共 x ⬘n ,⌫ ⬘ 兲 e ⫺ ␳ 1⬘ t n .

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448

Chaos, Vol. 11, No. 3, 2001

Emmanuel Dufraine

Then T n ␳ ⬘1 ⫽ . t n ␳ ⬘2

lim n→⬁

The proof of the following result is close to the proof of Theorem 1. 䊐 Theorem 3: 兩 ␻ 1 ⫹p ␻ 2 兩 is a conjugacy invariant. In fact, if X and X ⬘ , two vector fields in J, are conjugate by h, then ␻ 1 ⫹p ␻ 2 ⫽ ␻ ⬘1 ⫹p ␻ 2⬘ if h preserve the orientation, and ␻ 1 ⫹p ␻ 2 ⫽⫺( ␻ ⬘1 ⫹p ␻ ⬘2 ) if h reverse the orientation. Proof: Let X be a vector field in J. Let V be a solid ⫺ cylinder in M 3 , such that U ⫹ 1 and U 2 are in V, and V is a ¯ ¯ neighborhood of ⌫⶿⌫ . H 1 (V⶿⌫ )⯝Z, the generator is well defined 共even the orientation is well defined because h maps ⌫ on ⌫ ⬘ , preserving the time orientation兲, but we have to ¯ to ⳵ V as in the first part. choose a retraction from V⶿⌫ For A in ⳵ V, close enough to W s (p 1 ) we can define

␥ A ⫽ 兵 X t 共 A 兲 ,t苸 关 0,␶ 兴 其 艛⌬ 共 A,X ␶ 共 A 兲兲 , with ␶ the time of escape from V, and ⌬(A,X ␶ (A)) a path on ⳵ V which close the piece of orbit. We ask ⌬ to cut at most one time a fixed line L, on ⳵ V. According to Lemma II.6, let A n be a sequence of points which converges to a point of W s (p 1 )⶿ 兵 p 1 其 , we have lim

冏

n→⬁

冏 冏 冏

关 ␥ A n 兴 H 1 (V⶿⌫¯ )

␶n

⫽ lim

n→⬁

⫽ lim

冏

¯兲 twist共 ␥ A n ,⌫

␶n

ACKNOWLEDGMENTS

冏

twist共 ␥ h(A n ) ,⌫ ⬘ 兲

n→⬁

␶n

It is a pleasure to thank Elisabeth Pe´cou-Gambaudo, Jean-Marc Gambaudo, and Robert Roussarie for their help in each step of the preparation of this paper. .

By Lemma II.7 there exists K⬍⬁ such that ¯ ) ⫺ 关 ␻ 1 t⫹ ␻ 2 T 兴 兩 ⬍K, 兩 关 ␥ A 兴 H 1 (V⶿⌫

for all A close to W s (p 1 ). Thus, taking a sequence of points A n , going to B, a point of W s (p 1 ), and computing the speed of twisting, we obtain lim

¯ 兲⫺关 ␻ t ⫹␻ T 兴兩 兩 twist共 ␥ A n ,⌫ 1 n 2 n

n→⬁

␶n

⫽0.

We recall that in the proof of Theorem III.1 we have shown that Tn ␳1 ⫽ ⫽p, ␳2 n→⬁ t n lim

tn 关 ␻ 1t n⫹ ␻ 2T n兴 关 ␻ 1t n⫹ ␻ 2T n兴 ⫽ lim T ⫹s ⫹t t t ⫹s n n n n n n ⫹T n n→⬁ n→⬁ lim

⫽ 共 ␻ 1 ⫹p ␻ 2 兲

冉 冊

1 . p⫹1

With T n ⫹s n ⫹t n ⫽ ␶ n , limn→⬁ t n ⫽limn→⬁ T n ⫽⬁, and limn→⬁ s n ⫽s. Up to its sign, this limit is invariant under conjugacy, and we can do the same calculation for a vector field X⬘ conjugate to X to obtain that 兩 ␻ 1 ⫹ p ␻ 2 兩 ⫽ 兩 ␻ 1⬘ ⫹p ␻ 2⬘ 兩 . The theorem follows from the discussion about orientation in the proof of Corollary II.6. 䊐 There is a natural question: Is the set 兵 p⫽ ␳ 1 / ␳ 2 , 兩 ␻ 1 ⫹ p ␻ 2 兩 其 a complete set of invariants? We denote by ¯ ,¯␪ ,z ¯ ) in (r, ␪ ,z) the linearizing coordinates in p 1 , and by (r p 2 . Let ⌺ 1 ⫽ 兵 z⫽1 其 and ⌺ 2 ⫽ 兵¯z ⫽1 其 . We denote by f the transition map from ⌺ 1 to ⌺ 2 ( f is a C 1 diffeomorphism兲. The following claim is easy to prove: Claim III.1. Let X and X ⬘ be two vector fields of J, if p ⫽ p ⬘ and ␻ 1 ⫹ p ␻ 2 ⫽ ␻ ⬘1 ⫹ p ␻ ⬘2 and if the both transitions are homothety-rotations then X and X ⬘ are conjugate. Two antinomic problems arise there: either the two invariants form a complete set of invariants, or the derivative of the transition map is essential for the conjugacy, leading to a new invariant. In this way, it could be interesting to see if claim III.1 works with another linear application than a homothety rotation. A partial answer will be given in Bonatti and Dufraine 共2001兲, where a complete set of invariants for topological equivalence is exhibited.

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