Vector Fields in the Interior of Kupka Smale Systems Satisfy Axiom A

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1 Journal of Differential Equations 77, (200) doi:0.006/jdeq , available online at on Vector Fields in the Interior of Kupka Smale Systems Satisfy Axiom A Hiroyoshi Toyoshiba Department of athematics, School of Education, Waseda University, Tokyo , Japan Received June 2, 999; revised October 2, 2000 In 994, S. Hu conjectured in his paper (994, Trans. Amer. ath. Soc. 342, )that Axiom A systems are dense in the interior of Kupka Smale systems. We will show that if X is in the interior of Kupka Smale systems, then X satisfies Axiom A. 200 Elsevier Science. INTRODUCTION In this paper we prove the following ain Theorem. If a vector field X is in the interior of the set of Kupka Smale systems, then X satisfies Axiom A. This problem, proposed by S. Liao, is more general than the so-called Stability Conjecture, If a vector field X is structural stable, then does X satisfy Axiom A?, which was answered by S. Hayashi [5] in 997, because, if X is structural stable, then obviously X is in the interior of the set of Kupka Smale systems. Therefore, this ain Theorem is an extension of the Stability Conjecture. Now, let be a d-dimensional compact smooth manifold without boundary and let X () be the set of C vector fields on with the C topology. We denote by KS the set of Kupka Smale systems of vector fields in X () and by int KS the interior of KS. We denote by X t,t R, the C flow on generated by X X (). We denote by Sing(X) the set of singularities of X. We denote by P(X) the set of periodic orbits of X. We denote by P(X) the closure of the periodic orbits of X and by P j (X) the closure of periodic orbits with index j (index is the dimension of the stable manifold). We denote by W(X) the nonwandering set of X X (). The set L is said to be a hyperbolic set of X X (), if it is compact, X t -invariant and there is a continuous splitting /0 $ Elsevier Science All rights reserved.

2 28 HIROYOSHI TOYOSHIBA T L=E 0 À E s À E u (E 0 (x)=r X(x), x L)invariant under D x X t such that there exists K>0,0<l <, satisfying and (D x X t ) E s x [ Kl t (D x X t ) E u x [ Kl t for all t \ 0, x L. When W(X) is hyperbolic and the periodic points are dense in W(X), we say that X satisfies Axiom A. For X X (), let B E (X, x)= {y ; d(x t (x), y) [ e for some t R}. Define S(X) as the set of points x such that for every neighborhood U of X and every e >0, there exists Y U,y P(Y), T 0 >0 and t 0,t R with t 0 <t such that Y T0 (y) =y, X=Y on B e (X, x), d(y t (y), X t (x)) [ e for all 0 [ t [ T 0,{X t (x); t 0 [ t [ t } {Y t (y); t \ 0} and (t t 0 )/T 0 > e. Note that S(X) is X t -invariant. Let G () denote the set of X X () which has a neighborhood U such that if Y U, then all the periodic orbits and singularities of Y are hyperbolic. In the proof of the ain Theorem in Sections 2 and 3, we do not caution against singularities, since singularities of X are isolated in W(X). In fact, if there is a singularity which is not isolated in W(X), by the Connecting Lemma (Lemma 2.9 in Sect. 2), we may obtain Y arbitrarily close to X which has a non transversal homoclinic point. This contradicts the assumption. 2. PROPERTY OF X IN S(X) In this section we state a property of x S(X) for X in the int KS. We note that if X is in the int KS, then X is also in G (). Lemma 2.. If X is in the int KS, then orbit(x) is a hyperbolic set for any x S(X). Here orbit(x) is the closure of orbit of x. The proof of Lemma 2. consists of several lemmas. First we have the following. Lemma 2.2. Let x be in S(X) for X in the int KS, then there exist a sequence of vector fields {Y n }, a sequence of periodic orbits {P n }, a sequence of points {a n } and a sequence of {e n } such that () Y n =X and P n is a periodic orbit of Y n, (2) a n P n and a n =x,

3 (3) T n =., where T n is the period of P n, (4) e n =0, KUPKA SALE SYSTES 29 (5) d(x t (x), Y t (a n )) [ e n for all 0 [ t [ T n. Proof. This is obvious from the definition of S(X). In Lemma 2.2 we may assume that the index of P n is l for all n. Let N g be the normal bundle to X over. For each x, the fiber N x is a subspace of T x with codimension that is perpendicular to X(x). For any u N x, let P X t (u) be the orthogonal projection of D x X t (u) onto N Xt (x). Then P X t :N g Q N g is a C 0 flow, which is linear on fibers. We use the following three lemmas in the proof of Lemma 2.6 below. Lemma 2.3 (Theorem 2. in Liao [9]). Let X be in G (). Then there exist a C neighborhood Ũ of X in G () and two numbers l=l(ũ)>0 and T=T(Ũ)>0such that for any Y Ũ and any periodic point p of Y, the following two estimates hold: (a) P Y t E s (p) P Y t E u (Y t (p)) [ e 2lt for any t \ T, (b) If y is the period of p, m is any positive integer, and 0=t 0 < t < <t k =my is any partition of the time interval [0, my] with t i+ t i \ T, then k my C log P Y t i+ t i E s (Y ti (p)) < l, k my C log P Y (t i+ t i ) E u (Y ti+ (p)) < l. Lemma 2.4. For the sequence {P n } in Lemma 2.2, we may assume the index of P n =l ] 0, d for all n. Proof. If l=0 on the contrary, we may assume that index of P n =0 for all n. Since each P n is compact, a subsequence of P n, denoted also by P n, converges to some X t -invariant closed subset F in orbit(x) by [5], with respect to the Hausdorff metric. The same as [8], we take a measure m n corresponding to a point a n such that F f(a) dm n = t Q. t F t f(y n s (a n )) ds, 0

4 30 HIROYOSHI TOYOSHIBA where f is an any continuous function. In the rest, this measure m n is called individual measure corresponding to a point a n (Chapter VI in [4])and we may assume that the sequence m n converges with weak-star topology to a probability measure m on. Each m n is invariant under Y n t. And m is a measure supported on F. Let {j n (a)} be a sequence of real valued continuous functions on such that Then we have j n (a)=j(a) uniformly on. () F j n (a) dm n =F j(a) dm. Because : F j n (a) dm n F j(a) dm: [: F j n (a) dm n F j(a) dm n : + : F j(a) dm n F j(a) dm: and j n (a) Q j(a) implies : F j n (a) dm n F j(a) dm n : [ F j n (a) j(a) dm n Q 0, while m n Q m implies : F j(a) dm n F j(a) dm: Q 0. oreover, since each m n is Y n t -invariant, m is X t -invariant. In fact, by () above, for any t R, F j(x t (a)) dm = F j n (X t (a)) dm n F j n (Y n t (a)) dm n + F j n (Y n t (a)) dm n

5 KUPKA SALE SYSTES 3 = F j n (X t (a)) dm n F j n (Y n t (a)) dm n 2 + F j n (Y n t (a)) dm n Remark. = F j n (Y n t (a)) dm n = F j n (a) dm n =F j(a) dm. F j n (X t (a)) dm n F j n (Y n t (a)) dm n 2 =0. In fact, since Y n =X, d(x t (a), Y n t (a)) is very small for sufficiently large n (for any a and a fixed t). On the other hand, as j n is uniformly continuous on j n (X t (a)) j n (Y n t (a)) < e for any a and sufficiently large n. As m n is an individual measure corresponding to a n (2) Now, let F j n (a) dm n = t Q. t F t j n (Y n s (a n )) ds. 0 t n T(a)= n log PY T T(a), t T (a)= T log PX T(a). Then t n T(a)=t T (a) uniformly on for the number T= T(Ũ)>0. From Lemma 2.3(b), for sufficiently large n, T T n 3 mn C k= t n T(Y n kt(a n ))+ n log PY (T T n m n T)(Y n T n (a n )) 4 < l, where m n is the greatest integer with T n m n T \ T. Since log P Y n (T T n m n T)(Y n T n (a n )) Q 0 as n for sufficiently large n, we have T m n T C mn k= t n T(Y n kt(a n )) < l 2 a n P n.

6 32 HIROYOSHI TOYOSHIBA Thus Therefore pm n C pmn k= t n T(Y n kt(a n ))2 < l 2 p=, 2,... pm n T F pmn T t n T(Y n s (a n )) ds 0 = pm n T Thus from (2) pm C n k=0 = T F T 0 pm n F (k+) T kt t n T(Y n s (a n )) ds pm n C t n T(Y n kt(y n s (a n ))) ds < l k=0 2 p=, 2,... FP n t n T(a n )dm n < l 2. Thus, from () we have F t T (a) dm= F FP n t n T(a n )dm n [ l 2 <0. Now we need a lemma from Liao [8] to proceed. Lemma 2.5 (Liao [8], Lemma 3.2). Let F be a closed subset of, invariant under X t. Assume that for a certain T (0,.), there is a probability measure m on F, invariant under X t such that: (f) F t T (a) dm <0 or F t T (a) dm <0. F F Then, F contains a periodic orbit of X attracting or repelling corresponding to the first inequality or the second of (f). Applying this lemma to our situation we obtain a repelling periodic orbit in F orbit(x). But this contradicts the assumption x S(X). Because x is a recurrent point of X, orbit(x) could not contain a repelling periodic orbit. Lemma 2.6. Let x be in S(X) for X in the int KS, then orbit(x) has a dominated splitting N g orbit(x)=g s À G u (dim G s =l) such that P X t (x) G s (x) P X t(x t (x)) G u (X t (x)) [ e 2lt, t\ T where T and l are same as in Lemma 2.3.

7 KUPKA SALE SYSTES 33 Proof of Lemma 2.6. Now let x be a point of S(X). We can take a sequence of periodic orbits {P n }, a sequence of vector fields {Y n }, and a sequence of point {a n } satisfying the conditions of Lemma 2.2 for x. And from Lemma 2.4, we can assume that index of P n =l( ] 0ord ) for all n. oreover, we can assume that Y n is in Ũ (in Lemma 2.3)for all n. We have P Y n t (a n ) E s (a n ) P Y n t (Y n t (a n )) E u (Y n t (a n )) [ e 2lt (t \ T). For any t R, the subspaces E s (Y n t (a n )) and E u (Y n t (a n )) converge to subspaces of N Xt (x) which we define as G s (X t (x)) and G u (X t (x)) respectively. Now we attach for every y orbit(x) two subspaces G s (y) and G u (y) with the following properties: (a)dim G s y+dim G u y=dim, (b) (P X t (y)) G s y=g s X t (y) s=s, u for any t R, (c) P X t G s y P X t G u X t (y) [ e 2lt t \ T, (d) G s (y) À G u (y)=n y for any y orbit(x). By Proposition I.3 of ańẽ [3], we can extend this splitting to orbit(x) and from properties (a) (d) it follows that the subspaces G s (y) and G u (y) depend continuously on y orbit(x). Now we obtain a dominated splitting G s À G u over orbit(x) and by Lemma 3.7 in Wen [2], this dominated splitting is uniquely determined by the sequence of periodic orbits P n The next lemma is an essential step for the proof of Lemma 2.. Lemma 2.7. Let x be in S(X) for X in the int KS and N g orbit(x)=g s À G u (dim G s =l) be the dominated splitting obtained in Lemma 2.6. IfG s is contracting then G u is expanding. Proof of Lemma 2.7. From the construction of the dominated splitting, we obtain a sequence of arcs(x, X Tn (x)), a sequence of periodic points {a n } and a sequence of vector fields {Y n } such that d(x t (x), Y n t (a n )) [ e n for all 0 [ t [ T n as in Lemma 2.2. Since, X and Y n are in G (), by the same argument as in pp of [0], we obtain a vector field Ȳ n close to Y n such that the following condition holds. P Ȳ n t (f) N* restricted to the Ȳ n -orbit P n of a n (which is periodic)has an -invariant splitting Ḡ s À Ḡ u such that P Ȳ m(ȳ n n mj(a n )) Ḡ u (Ȳ n mj(a n )) = P X m(x mj (x)) G u (X mj (x)), P Ȳ n m (Ȳ n m(j )(a n )) Ḡ s (Ȳ n m(j )(a n )) = P X m(x m(j ) (x)) G s (X m(j ) (x))

8 34 HIROYOSHI TOYOSHIBA for all [ j [ [T n /m], P Ȳ n (T n [T n /m]+)(ȳ n T n (a n )) Ḡ u (Ȳ n T n (a n )) = P X (T n [T n /m]+)(x Tn (x)) G u (X Tn (x)), and P Ȳ n T n [T n /m]+(ȳ n [T n /m] (a n )) Ḡ s (Ȳ n [T n /m] (a n )) = P X T n [T n /m]+(x [Tn /m] (x)) G s (X [Tn /m] (x), where m is the smallest integer such that m \ T (T is given in Lemma 2.3). Since index of P n associated to Y n is l, index of P n associated to Ȳ n is also l. In particular, dim Ḡ s =l. As we assume that G s is contracting we may conclude that Ḡ s =E s where E s À E u is hyperbolic splitting of Ȳ n on P n. Therefore Ḡ u =E u. From Lemma 2.3(b)and T n =., for sufficiently large n (3) k n D P X m(x mj (x)) G u (X mj (x)) j= k n = D P Ȳ m(ȳ n n mj(a n )) Ḡ u (Ȳ n mj(a n )) [ e l 2 (k n )m j= (k n =[T n /m]). Thus, we have (4) inf n n C log (P X m) G u (X mj (x)) [ c j= for some c>0. Since S(X) is invariant under X t,ify=x t (x) for some t \ 0, then y has the same property as x. Therefore (4) holds for a dense subset in orbit(x). Before stating the next lemma, we define an admissible neighborhood of compact invariant set L. We say that a compact neighborhood V of L is an admissible neighborhood if N* 4 t R X t (V) has one and exactly one homogeneous dominated splitting N* 4 t R X t (V)= Ê À Fˆ extending the splitting N* L=E À F. We need the following lemma from añé [2].

9 KUPKA SALE SYSTES 35 Lemma 2.8. Let L be a compact invariant set of X X () such that W(X t L)=L, let N* E À F be a homogeneous dominated splitting such that E is contracting and suppose ĉ >0is such that the inequality (5) inf n n C log (P X m) F(X mj (x)) [ ĉ j= holds for a dense set of points x L, then either F is expanding(and therefore L is hyperbolic) or for every admissible neighborhood V of L and every 0<ĉ < there exists a periodic point p 4 t R X t (V) with arbitrarily large period P and satisfies ĉ [P/m] [ [P/m] D j= P X m Fˆ (X mj (p)) <, where Fˆ is given by the unique homogeneous dominated splitting N*: 3 t R X t (V)=Ê À Fˆ that extends N* L=E À F. If x S(X) for X in the int KS, then X and orbit(x) satisfies the conditions in Lemma 2.8. Applying the above lemma, we have two cases to consider. However, by añé [2], we know that the second case leads to a contradiction for X in G (), completing the proof of Lemma 2.7. We may apply the same argument and obtain the followng: Corollary. Under the same condition as in Lemma 2.7, if G u is expanding, then G s is contracting. Now we proceed to the main part of the proof of Lemma 2.. By Lemma 2.7 and Corollary, if dominated splitting N* orbit(x)=g s À G u is not hyperbolic, then neither is G s contracting nor is G u expanding. Since G s is not contracting, we can find y orbit(x) and a sequence j such that (ff) j n log P X mj n (y) G s (y) \ 0, where m is the same as in (f). Without loss of generality we may suppose that the sequence {j n } is such that there exists a X m -invariant probability measure m on orbit(x) such that F fdm= orbit(x) j n j n C f(x mi (y))

10 36 HIROYOSHI TOYOSHIBA for every continuous f : orbit(x) Q R. Applying this equality to f defined by f(a)=log P X m(a) G s (a) we obtain: (6) F fdm= orbit(x) \ j n j n C log P X m(x mi (y)) G s (X mi (y)) j n log P X mj n (y) G s (y) \ 0. On the other hand, by Birkhoff s Theorem: (7) F fdm=f orbit(x) orbit(x) n n C log P X m(x mi (a)) G s (X mi (a)) dm. Here, we need the following lemma. Lemma 2.9 (Ergodic Closing Lemma for time one map, Lemma VII.6 in Hayashi [5]). m(s(x) 2 Sing(X))= for every X -invariant probability measure m on the borel sets of, where Sing(X) denotes the set of singularities of X. By Lemma 2.9, orbit(x) 5 S(X) is an X -invariant total probability subset of orbit(x). Hence if we define an X -invariant probability measure n on orbit(x) by n= m m C X i (m), where m is a X m -invariant probability measure on orbit(x), we obtain 0=n(orbit(x) orbit(x) 5 S(X)) = m m C m(x i (orbit(x) orbit(x) 5 S(X))) = m m C m(orbit(x) orbit(x) 5 S(X)) =m(orbit(x) orbit(x) 5 S(X)).

11 This together with (6) and (7) implies: KUPKA SALE SYSTES 37 0 [ F orbit(x) 5 S(X) n n C log P X m(x mi (a)) G s (X mi (a)) dm(a). Hence there exists z S(X) 5 orbit(x) such that (8) n n C log P X m(x mi (z)) G s (X mi (z)) \ 0. Now we may take l < l 0 <0such that for n 0 large enough: n 0 (9) C log P X n m(x mi (z)) G s (X mi (z)) > l m. If z is a periodic point, then the index of z is smaller than j. In fact if the index of z is not smaller than j, then orbit(z)has a hyperbolic splitting N* orbit(z)=e s À E u (dim E s \ j). On the other hand as G s À G u (dim G s =j) is a dominated splitting, we have G s E s. Then (8) is contradicting (b)in Lemma 2.3. If z is not a periodic point, then we have a Y arbitrarily close to X which has a periodic point p with a large period y such that d(x t (z), Y t (p)) [ e for all (0 [ t [ y). By the same argument as in proof of Lemma 2.7, we obtain a vector field Ȳ C -close to Y satisfying (f). That is, N* restricted to the Ȳ-orbit of p has a P Ȳ t -invariant splitting Ḡ s À Ḡ u such that for p P P Ȳ m(ȳ m(j+) (p)) Ḡ u (Ȳ m(j+) (p)) = P X m(x m(j+) (z)) G u (X m(j+) (z)), P Ȳ m(ȳ mj (p)) Ḡ s (Ȳ mj (p)) = P X m(x mj (z)) G s (X mj (z)) for all 0 [ j [ [y/m] 2, and P Ȳ (y [y/m]+)(ȳ y (p)) Ḡ u (Ȳ y (p)) = P X (y [y/m]+)(x y (z)) G u (X y (z)), Then we have P Ȳ y [y/m]+(ȳ [y/m] (p)) Ḡ s (Ȳ [y/m] (p)) = P X y [y/m]+(x [y/m] (z)) G s (X [y/m] (z)). P Ȳ y(ȳ y (p)) Ḡ u (Ȳ y (p)) (k=[y/m] ) P Ȳ m(ȳ mk mi (p)) Ḡ u (Ȳ mk mi (p)) 2 [ D k P Ȳ (y mk)(ȳ y (p)) Ḡ u (Ȳ y (p))

12 38 HIROYOSHI TOYOSHIBA k =D ( P Ȳ m(ȳ mk mi (p)) Ḡ u (Ȳ mk mi (p)) P Ȳ m(ȳ m(k ) mi (p)) Ḡ s (Ȳ m(k ) mi (p)) ) D k P X m(k ) mi(z)) G s (X m(k ) mi (z)) 2 P Ȳ (y mk)(ȳ y (p)) Ḡ u (Ȳ y (p)) [ e 2lmk e l0 2 mk P Ȳ (y mk)(ȳ y (p)) Ḡ u (Ȳ y (p)). Note that we have used (9) to induce the last inequality above. If y is very large, then k can be also large so that: e 2lmk e l0 2 mk P Ȳ (y mk)(ȳ y (p)) Ḡ u (Ȳ y (p)) <, Thus, Ḡ u E u (where E s À E u is hyperbolic splitting of P for Ȳ). Therefore dim E u \ dim Ḡ u =(dim ) l. That is dim E s [ l. If dim E s =l, then Ḡ u =E u, Ḡ s =E s. By Lemma 2.3(b)and (9) above,we have k e l 2 mk > D P Ȳ m(ȳ mi (p)) E s (Ȳ mi (p)) k =D P X m(x mi (z)) G s (X mi (z)) (from (9)) >e l0 2 mk. This is a contradiction. Thus we obtain: index of P<l. By the same argument as above, we have again a sequence {YŒ n }, a sequence {P n} of periodic orbits, a sequence {a n} with a n P n, a sequence {T n} such that T, a sequence of arcs(z, X T (z)), and a sequence n e n Q 0 such that () YŒ n =X and P n is a periodic orbit of YŒ n with period T n, (2) a n=z, (3) d(x t (z), Y n t (a n)) [ e n for all 0 [ t [ T n. And, for all n, we can assume that index of P n=lœ with lœ <l. As before, we can obtain the dominated splitting G s Œ À G u Œ over orbit(z)( orbit(x)) with dim G s Œ=lŒ. And if G s Œ is contracting, then by Lemma 2.7 orbit(z) is a hyperbolic set. Otherwise we can again find w orbit(z) 5 S(X) such that

13 KUPKA SALE SYSTES 39 orbit(w) orbit(z) orbit(x) and has a dominated splitting Ĝ s À Ĝ u with dim Ĝ s <lœ <l. In case that Ĝ s is not contracting, repeating this argument on the way down we eventually obtain a sequence {Y n } of vector fields, and a sequence {P n } of periodic orbits, such that Y n =X, index of P n =0 for all n, contradicting Lemma 2.5. Therefore, we can find a point r S(X) 5 orbit(x) such that orbit(r) has a hyperbolic splitting N* orbit(r)= E s À E u (dim E s < dim G s ). On the other hand since G u is not expanding, applying the same argument, we obtain a point s S(X) 5 orbit(x) such that orbit(s) has a hyperbolic splitting N* orbit(s)=ê s À Ê u (dim Ê u < dim G u ). Obviously, r is not equal to s. So, we obtain two disjoint hyperbolic sets orbit(r) and orbit(s) in orbit(x). (In particular, r and s may be periodic points.)since r and s are recurrent, there exist sequences {p n}, {p 2 n} converging to r, s respectively and shadowing the orbit r, s respectively. So, we assume that ṅ= orbit(p n) and ṅ= orbit(p 2 n) are two hyperbolic sets. Because x is recurrent and r, s are in ṅ= orbit(p n) and ṅ= orbit(p 2 n) respectively, there exists a point which leaves arbitrarily close to ṅ= orbit(p 2 n) and arrives arbitrarily close to ṅ= orbit(p n). Now, we need the following lemma. Lemma 2.0 (Connecting Lemma). Let L and L 2 be disjoint hyperbolic sets for X in X (X). Assume that for any two neighborhoods U, V of L and L 2 respectively, there is a point x in U and t \ 0 so that X t (x) V. Then, there is YC -near X so that W u (L ) 5 W s (L 2 ) ]. Where W u (L ) is a unstable manifold of L and W s (L 2 ) is a stable manifold of L 2. Applying this Lemma 2.0, we obtain YC -close to X and. W u 0 n=. orbit(pn)2 2 5 W s 0 n= orbit(pn)2 ]. Therefore, there is that p W u (sœ) 5 W s (rœ), where rœ and sœ are points in orbit(p n), orbit(p 2 n) respectively. Then, we can obtain periodic ṅ= ṅ= points p,p 2 arbitrarily close to rœ,sœ respectively. By the continuity of stable and unstable manifolds, perturbing Y, we may assume that W u (p 2) 5 W s (p ) ]. But this intersection is not transversal, contradicting the assumption that X is in the int KS. So, we complete the proof of Lemma 2..

14 40 HIROYOSHI TOYOSHIBA 3. HYPERBOLICITY OF P(X) In this section we prove the following Lemma 3.. If X is in the int KS, then P(X) is hyperbolic. It is well known that if X is in G () then, X has only a finite number of repelling periodic orbits and attracting periodic orbits. Therefore, P 0 (X)=P 0 (X). Obviously, P 0 (X) is hyperbolic. Now we assume that j P i (X) is hyperbolic. By induction, we have only to show that P j (X) is a hyperbolic set. Lemma 3.2 (Theorem B in [20]). P j (X)=, then P j (X) is hyperbolic. Let X be in int KS. If j P i (X) 5 Proof of Lemma 3.2. Now, we assume that P j (X) is not hyperbolic, then by the same argument of ańẽ [0], we can find a point x S(X) 5 P j (X) such that n n C log P X m(x mi (x)) G s (X mi (x)) \ 0 where G s À G u is the dominated splitting over P j (X). Because x S(X), we can take a vector field Y arbitrarily near X which has a periodic point y close to x. Then, again by the same argument of ańẽ [0], we have Index y [ j. If index y=j, we know that this is a contradiction to the condition that X is in (Int KS ) G (). Therefore, we can assume that index y<j. From the argument in Section 2, orbit(x) is a hyperbolic set whose index is less that j. But this is a contradiction to the hypothesis. L Now, we assume that j P i (X) 5 P j (X) ]. As j P i (X) is hyperbolic, j P i (X)=L 2 2 L k where each L j,[ j [ k, is a basic set. A basic set means an isolated, transitive, hyperbolic set. Suppose that L i 5 P j (X) ]. Let {P n }(P n P j (X)) be a sequence of periodic orbits such that P n converges to some closed set F with respect to the Hausdorff metric and we can assume that L i 5 F ]. Then F has a dominated splitting such that N* F=G s À G u (dim G s =j). Let m n be an individual measure corresponding to a n P n and the sequence {m n } converge with weak-star topology to m supported on F. Each m n is invariant under X t. Let j(a) be a real valued continuous function on. Then we have (0) F j(a) dm n =F j(a) dm. oreover, since each m n is X t -invariant, m is X t -invariant.

15 KUPKA SALE SYSTES 4 As m n is an individual measure corresponding to a n () F j(a) dm n = t Q. t F t j(x s (a n )) ds. 0 Now let j(a)= log T PX T (a) G s (a) where this T is given by Lemma 2.3. By Lemma 2.3(b), for sufficiently large n, m T T n 3 n C j(x kt (a n ))+ k=0 T log PX T n m n T(X mn T(a n )) G s (X mn T(a n )) 4 < l, where m n is the largest integer with T n m n T \ T. For sufficiently large n, we have Thus T m n T m C n j(x kt (a n )) < l k=0 2 lm n lm n l C j(x kt (a n ))2 < k=0 2 l=, 2... Therefore lm n T F lmn T j(x s (a n )) ds 0 = lm n T = T F T 0 lm C n F (k+) T j(x s (a n )) ds k=0 kt lm n lm n C j(x kt (X s (a n ))) ds < l k=0 2 l=, 2,... Thus from () F j(a n )dm n < l 2. Thus, from (0) we have F j(a) dm= F Now we need the following lemma to proceed. FP n j(a n )dm n [ l 2 <0.

16 42 HIROYOSHI TOYOSHIBA Lemma 3.3 (Ergodic Closing Lemma). For any X X (). m(sing(x) 2 S(X))= for every X-invariant probability measure m. Applying this lemma, we have (2) F j(a) dm <0. F 5 S(X) On the other hand, by the Birkhoff s Theorem (3) F j(a) dm=f F 5 S(X) F 5 S(X) n n C j(x it (a)) dm <0. Hence there exists a point p in F 5 S(X) such that (4) n C n j(x it (p)) < 0. Now we assume that p is a hyperbolic periodic point. Then, from (4), the index of p is not less than j. AsL i and orbit(p)are disjoint, let U(L i ) and U(p) be any two neighborhoods of L i, p respectively. Because of p F and F 5 L i ], sequence {P n } converges to not only p but also L i. So, we may find a point q, which is in a point of the periodic orbit P n,inu(p) and t \ 0 so that X t (q) in U(L i ). Applying the Connecting Lemma, we obtain a vector field YC -close to X such that W u (p) 5 W s (L i ) ] which is not transversal, contradicting the assumption that X is in the int KS. Next if p is not a periodic point, then since p S(X), we obtain a sequence of vector fields {Y n }, a sequence of periodic orbits {P n } as in Lemma 2.2. By Lemma 2. orbit(p) is a hyperbolic set. If index of P n [ j, then we have a hyperbolic splitting such that N* orbit(p)=e s À E u with dim E s [ j. On the other hand, orbit( p) has also a dominated splitting N* orbit(p)=g s À G u with dim G s =j. Apparently, G s E s. But by (2), G s is contracting. So the above hyperbolic splitting E s À E u over orbit(p) is not true. Therefore index of P n \ j for all n. By Lemma 2., orbit(p) is a hyperbolic set in F and has hyperbolic splitting N* orbit(p)=e s À E u with dim E s \ j. By the same argument as above, we obtain YC -close to X such that W u (orbit(p)) 5 W s (L i ) ]. Now, because orbit(p) is hyperbolic, there is a periodic point p shadowing orbit(p). oreover, by the same argument as in the last part in Section 2, we can take a periodic point p 2 L i with W u (p ) 5 W s (p 2 ) ]. Then this intersection is not transversal. But this is a contradiction. Therefore we can conclude that

17 KUPKA SALE SYSTES 43 j P i (X) 5 P j (X)= and P j (X) is hyperbolic. By the induction step, we obtain d P i (X) is a hyperbolic set. 4. W(X)=P(X) In this section we prove the following Lemma 4.. If X is in the int KS, then P(X)=W(X). Let L + (X) be the closure of the set of w-it points that are not a singularity for X. At first we shall prove Lemma 4.2. If X is in the int KS, then L + (X) P(X). Proof. Now suppose that q L + (X) P(X). Then, we have a point x such that q w(x) and a sequence of arcs(x n,x Tn (x n )) which are in orbit(x) and may be closed by the Pugh s Closing Lemma and satisfies x n =q and X Tn (x n )=q. Hence, we have a sequence of vector fields {Y n }, a sequence of periodic orbits {P n } obtained by closing the arcs(x n,x Tn (x n )) such that () Y n =X, P n is a periodic orbit of Y n with period T n, (2)index of P n =l for all n. oreover we may assume that a sequence of arcs(x n,x Tn (x n )) and a sequence of periodic orbits {P n } converge to some closed set H( w(x)) with respect to Hausdorff metric. Same as in Section 3, H has a dominated splitting N* H=G s À G u with dim G s =l. Take a neighborhood W of H so small that there exists a continuous splitting N* W=Ĝ s À Ĝ u extending N* H=G s À G u. We may assume that P n W for all n and d(n)= sup{d(e s (p), Ĝ s (p)) p P n } converges to zero when. We take an individual measure m n corresponding to a n (a n P n ). Each m n is invariant under Y n t. The sequence {m n } converges with weak-star topology to m which is a probability measure supported on H. Setting j n (a)= n T log PY T E s (a) and j(a)= log T PX T Ĝ s (a), where this T is given by Lemma 2.3, then we have (5) F j n (a) dm n =F j(a) dm. Because :F j n (a) dm n F j(a) dm: [ : F j n (a) dm n F j(a) dm n : +: F j(a) dm n F j(a) dm:

18 44 HIROYOSHI TOYOSHIBA and j n (a) j(a) Q 0, (a P n ) implies : F while m n Q m implies j n (a) dm n F j(a) dm n : [ F j n (a) j(a) dm n Q 0, : F j(a) dm n F j(a) dm: Q 0. oreover, since each m n is Y n t -invariant, m is X t -invariant. In fact, by (5) above, for any t R, F j(x t (a)) dm Remark. = F j n (X t (a)) dm n F j n (Y n t (a)) dm n + F j n (Y n t (a)) dm n = F j n (X t (a)) dm n F j n (Y n t (a)) dm n 2 + F j n (Y n t (a)) dm n = F j n (Y n t (a)) dm n = F j n (a) dm n =F j(a) dm. F j n (X t (a)) dm n F j n (Y n t (a)) dm n 2 =0. In fact, since Y n =X, d(x t (a), Y n t (a)) is very small for sufficiently large n (for any a and a fixed t). On the other hand, j n (X t (a)) j n (Y n t (a)) < e for any a P n and sufficiently large n. As m n is an individual measure corresponding to a n (6) F j n (a) dm n = t Q. t F t j n (Y n s (a n )) ds. 0

19 By the same argument as in Section 3, we have KUPKA SALE SYSTES 45 (7) F H T log PX T (a) G s (a) dm <0. And, we have a point p S(X) 5 H such that (8) n nt C log P X T (X it (p)) G s (X it (p)) < 0. oreover we see that either p is a periodic point in H with index of p \ l or orbit(p) is a hyperbolic set in H with the index of orbit(p) \ l. Therefore, we always have a hyperbolic set in H whose index is not smaller than l. We need the following lemma from añé [2]. Lemma 4.3. Let L be a compact invariant set of X and E N* L be a continuous invariant subbundle. If there exists S>0such that F L S log PX S E dm <0 for every ergodic m which is X S -invariant, then E is contracting. Now, we assume that G s is not contracting and take T as in Lemma 2.3. Then, by the above lemma there exists an ergodic measure n that is invariant under X T such that F H 5 S(X) T log PX T G s dn \ 0 Then we obtain a point b H 5 S(X) such that n nt C log P X T (X it (b)) G s (X it (b)) \ 0 Then, if b is a periodic point then the index of b<l. And if b is not a periodic point, then we obtain a sequence of vector fields {Y n }, a sequence of periodic orbits {P n }, a sequence of T n, and a sequence e n as in Lemma 2.2. And, at this time, the index of P n <l. Because if the index of P n \ l, then orbit(b) is a hyperbolic set and has a hyperbolic splitting N* orbit(b)= E s À E u with dim E s \ l. Therefore >orbit(b) has two dominated splitting G s À G u, E s À E u. Then G s E s. Hence n n C log P X T (X it (b)) G s (X it (b)) < 0

20 46 HIROYOSHI TOYOSHIBA This contradicts the above inequality. Therefore orbit(b) is a hyperbolic set in H and has a hyperbolic splitting N* orbit(b)=e s À E u with dim E s <l. Then we have two hyperbolic sets orbit(p), orbit(b) in H. Since H is in w(x), for any two neighborhoods U, V of orbit( p), orbit(b) respectively, we have a point q in U and tœ \ 0 such that X tœ (q) V. By the same argument as in the last part of Section 2 and the Connecting Lemma, we obtain YC -close to X that has a non transversal intersection z W u (a) 5 W s (c). Where a and c are periodic points of Y. This contradicts the assumption. So, we obtain G s is contracting. Applying the same argument to G u,we obtain that G u is expanding. Therefore H is a hyperbolic set. Now q H and a sequence of arcs(x n,x Tn (x n )) converges to H with respect to Hausdorff metric. Applying the Shadowing Lemma we may obtain a periodic orbit P shadowing the arc. That is, q P(X). Now, we have concluded that L + (X) P(X), completing the proof. From this we have L + (X) P(X) and L + (X)=P(X). Since P(X) is a hyperbolic set, we obtain L + (X) 2 Sing(X)=P(X) 2 Sing(X)=L 2 2 L s(x), where each L i,[ i [ s(x), is a basic set. Next, we shall show that Lemma 4.4. If X is in the int KS, then X has no cycles. Proof. On the contrary, suppose that there is a cycle L i,...,l is of basic sets with L ij ] L ik (0 [ j<k[ s). Let b j (j=,..., s) be points of such that and b j W u (L ij ) 5 W s (L ij+ ), j=,..., s b s W u (L is ) 5 W s (L j ). Then, L ij is not a singularity for all {j,...,s}. In fact, if there is a singularity among {L ij j=,..., s}, we have a non transversal point among {b,...,b s }. Suppose that b is not transversal point, then we may assume that b W u (p ) 5 W s (p 2 ), where each p,p 2 is a periodic point or a singularity of X. This contradicts the assumption that X is in the int KS. Therefore, we may assume that all L ij,[ j [ s are not singularities. Then, all the indexes of L ij are the same. If not, we have a point in {b,..., b s } which is not transversal intersection. Using the same argument as above, this contradicts the assumption that X is in the int KS. Therefore, all indexes are the same. At least one point in {b,...,b s } is not a transversal intersection. Because if all points in {b,...,b s } are transversal intersection, then all L ij ( [ j [ s) is one basic set. Now, we assume that b is not

21 KUPKA SALE SYSTES 47 transversal intersection and the other points are transversal intersection. (If necessary, we can perturb X.)We obtain the next lemma which is the flow version of Lemma II.9 in [0]. For the next lemma we shall need the concept of an angle between subspaces of the Euclidean space. If E,E 2 are subspaces of R n such that E À E 2 =R n we define the angle a(e,e 2 ) as a(e,e 2 )= L, where L: E + Q E is the linear map such that E 2 ={v+lv v E + }; in particular, a(e,e + )=.. The next lemma is a flow version of Lemma II.9 in [0]. Lemma 4.5. If X is in G () then there exist aœ >0, neighborhood UŒ of X and TŒ >0 such that if Y UŒ and P is a periodic orbit of Y with period T Y (>TŒ) then a(e s (p), E u (p)) > aœ(p P, E s (p), E u (p) N*). If b is not transversal intersection, then we perturb X at b and obtain Y C -close to X such that b is transversal intersection and angle(e s,e u ) (E u and E s are tangent to W u (L i ), W s (L i2 ) respectively at b )is arbitrarily small. We may obtain a periodic point p arbitrarily close to b. From the continuity of stable and unstable manifolds, we have a(e s (p), E u (p)) < aœ (in Lemma 4.4). oreover, if we take p close enough to b, we can assume that the period of p is very large. This contradicts Lemma 4.5. Therefore X has no cycles. Lemma 4.6 (Theorem 4. in [5]). If L + (X) 2 Sing(X) is hyperbolic and X has no cycles, then P(X) 2 Sing(X)=L + (X) 2 Sing(X)=W(X). By this lemma, we conclude W(X)=P(X) 2 Sing(X). By Lemma 3. and Lemma 4., we have proved ain Theorem. ACKNOWLEDGENT The author is grateful to Professor R. Ito for his helpful comments and continuous encouragement. REFERENCES. C. Doering, Persistently transitive vector fields on three-dimensional manifolds, in Dynamical Systems and Bifurcation Theory (. I. Camacho,. J. Pacifico, and F. Takens, Eds.), Pitman Res. Notes ath., Vol. 60, pp , Longman, New York J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. ath. Soc. 58, J. Guckenheimer, A strange, strange attractor, in The Hopf Bifurcation and Its Application, Applied athmatical Series, Vol. 9, pp , Springer-Verlag, Berlin/New York, 976.

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