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Attractors of generic diffeomorphisms are persistent
View the table of contents for this issue, or go to the journal homepage for more
2003 Nonlinearity 16 301
(http://iopscience.iop.org/0951-7715/16/1/318)
Home Search Collections Journals About Contact us My IOPscience
Received 30 April 2002, in final form 29 October 2002Published 2 December 2002Online at stacks.iop.org/Non/16/301
Recommended by L Bunimovich
AbstractWe prove that given a compact n-dimensional boundary-less manifold M ,n � 2, there exists a residual subset R of Diff1(M) such that if � is an�-isolated and transitive set of f ∈ R, then � admits a continuation in ageneric neighbourhood of f ; such sets are called almost robustly transitive orgenerically transitive sets. Furthermore, if � is a transitive attractor of f , thenthe continuation of � is also an attractor.
This implies that �-isolated transitive sets of generic diffeomorphismsalways admit weakly hyperbolic dominated splittings; in particular, givenany surface diffeomorphism f in a residual subset of Diff1(M2), then every�-isolated transitive set of f (such as a transitive attractor) is hyperbolic. Wealso show that, generically in any dimension, �-isolated transitive sets are eitherhyperbolic or approached by a heterodimensional cycle, a type of homoclinicbifurcation.
Mathematics Subject Classification: 37C20, 37C70
1. Statement of the results
Throughout this paper, M denotes a compact boundary-less manifold of dimension n � 2 andDiff1(M) is the space of C1-diffeomorphisms on M with the usual C1 topology.
The spectral decomposition theorem states that the nonwandering set of any hyperbolic(i.e. axiom A) diffeomorphism f admits a partition into a finite number of transitive pairwisedisjoint sets called basic sets. If f also has no cycles, then it is �-stable, and each of its basicsets persists in a C1-neighbourhood of f .
There are several ways of defining nonhyperbolic generalizations of the concept of basicset, such as robustly transitive sets, as in [BDP] or [BD1]. The weakest such generalization isto simply require that a set be �-isolated and transitive.
Definition 1. A compact invariant set � of f ∈ Diff1(M) is transitive if there is a point q ∈ �
such that the future f -orbit of q is dense in �. This is equivalent to the following condition:given any two nonempty open subsets U and V of �, there is a natural number k ∈ N suchthat f k(U) ∩ V �= ∅.
We say that � is �-isolated if there exists an open neighbourhood V of � such that
�(f ) ∩ V = �.
Moreover, � is isolated or locally maximal if there exists an open neighbourhood V of �
such that
� =⋂
k∈Z
f k(V ).
Our first result deals with the persistence of �-isolated transitive sets of ‘typical’(i.e. generic) diffeomorphisms.
Theorem A. There exists a residual subset R of Diff1(M) such that if � is an �-isolatedtransitive set of f ∈ R then there exist a neighbourhood V of � and a neighbourhood W off in Diff1(M) such that if g ∈ W ∩ R then
�(g) ∩ V ≡ �g
is a compact transitive set of g.Moreover, if {gk} ⊂ (W ∩ R) converges to f in the C1 topology, then �gk
converges to �
in the Hausdorff topology.
The proof of theorem A relies on generic properties of homoclinic classes, defined insection 2.
Transitive sets that persist in a generic neighbourhood as in theorem A above wereintroduced in [A] in the study of ‘basic sets’ of nonhyperbolic diffeomorphisms that admitspectral decompositions (i.e. their nonwandering sets admit partitions into finitely manypairwise disjoint transitive sets). To be precise, a transitive compact set � of a diffeomorphismf ∈ Diff1(M) is almost robustly transitive or generically transitive if there are a neighbourhoodV of �, a neighbourhood W of f in Diff1(M), and some residual subset R of W , with f ∈ R,such that if g ∈ R then �g ≡ �(g) ∩ V is a compact transitive set of g. Hence theorem Aessentially says that generically any �-isolated transitive compact set is generically transitive.
Generic transitivity is a weak form of robust transitivity. In fact, there are no knownexamples of generically transitive sets that are not robustly transitive. This naturally raises thefollowing question.
Question. Does generic transitivity imply robust transitivity?
We now define a weak type of hyperbolicity, first introduced by [BDP].
Definition 2. Let � be a compact invariant set of f ∈ Diff1(M). Let E, C and F beDf -invariant sub-bundles of T�M such that T�M = E ⊕ C ⊕ F .
Then, E⊕C⊕F is a dominated splitting over � if (C⊕F) dominates E and F dominates(E ⊕ C) in the usual sense. The sub-bundle E is uniformly volume contracting if there existssome k ∈ N such that | det(Df k|E)| < 1 at all points of �; the sub-bundle F is uniformlyvolume expanding if it is uniformly volume contracting for f −1.
We say that � is volume partially hyperbolic if it admits a dominated splitting T�M =E ⊕ C ⊕ F , with E and F nonempty, such that E is uniformly volume contracting and F isuniformly volume expanding. Note in particular that if E or F is one-dimensional, then it ishyperbolic (i.e. uniformly contracting/expanding).
Attractors of generic diffeomorphisms 303
Finally, � is partially hyperbolic if it admits a dominated splitting T�M = E ⊕ F , whereE is uniformly contracting or F is uniformly expanding.
Theorem 4 of [BDP] states that robustly transitive sets are volume partially hyperbolic.The next result is a generic version of that theorem; it says that, generically, any genericallytransitive set is volume partially hyperbolic.
Proposition A. There exists a residual subset R of Diff1(M) such that if � is a genericallytransitive set of f ∈ R, then � is volume partially hyperbolic (or else the orbit of a periodicsink or source).
The proof of proposition A is an adaptation of the proof of theorem 4 of [BDP]; it is setout in the last section of this paper.
Combining theorem A with proposition A we immediately obtain the following result.
Corollary A. There exists a residual subset R of Diff1(Mn) such that if � is an �-isolatedtransitive set of f ∈ R then � is volume partially hyperbolic (or else the orbit of a periodicsink or source).
In particular, if n = 3 then every �-isolated transitive set of f is partially hyperbolic,and if n = 2 then every �-isolated transitive set of f is a hyperbolic set2.
Thus, generically, any nonhyperbolic basic set (here meaning any �-isolated transitiveset) admits a volume hyperbolic dominated splitting. In the case of surface diffeomorphisms,they are hyperbolic, and therefore robustly transitive. This (essentially) means that genericallytransitive sets of surface diffeomorphisms are robustly transitive, and thus the question abovehas an affirmative answer in the context of two-dimensional diffeomorphisms.
An important class of transitive �-isolated sets is topological attractors, defined as follows.
Definition 3. An attractor of f ∈ Diff1(M) is an invariant compact transitive set � ⊂ M suchthat there is a neighbourhood U of � (which we call an attracting neighbourhood of �) with
cl(f (U)) ⊂ U and⋂
n∈N
f n(U) = �.
It is easily seen that attractors are always �-isolated. The simplest attractors are periodicsinks. Given any axiom A diffeomorphism with no cycles, at least one of its basic sets is anattractor in the sense above; there are also many examples of nonhyperbolic attractors (thebest-known of which are the Henon and the Lorenz attractors). Note that corollary A impliesthat C1-generically in dimension 2 there exist no ‘Henon-type’ attractors. (Henon attractorsare two-dimensional attractors in our sense, but are nonhyperbolic.)
We also obtain a stronger version (for attractors) of theorem A; it says that generic attractorsare not only persistent, but also persistent relative to some uniform attracting neighbourhood.
Theorem B. There exists a residual subset R of Diff1(M) such that if � is an attractor off ∈ R then there exist an attracting neighbourhood V of � and a neighbourhood W of f inDiff1(M) such that if g ∈ W ∩ R then
�g ≡⋂
n∈N
gn(V )
is an attractor (with V an attracting neighbourhood).
2 C Morales has pointed out that in the two- and three-dimensional cases this result may be obtained more directly,without the need for generic transitivity, by, respectively, using arguments from Mane [M2] and Diaz–Pujals–Ures[DPU].
304 F Abdenur
Moreover, if {gk} ⊂ (W ∩ R) converges to f in the C1 topology, then �gkconverges to �
in the Hausdorff topology.
Finally, we characterize hyperbolicity for �-isolated transitive sets of genericdiffeomorphisms in terms of approximation by homoclinic bifurcations. Let us first definethe type of homoclinic bifurcation we are dealing with.
Definition 4. A diffeomorphism f ∈ Diff1(M) exhibits a heterodimensional cycle if there areperiodic saddles P and Q of different stability indices (i.e. dimensions of the stable subspaces)such that
Ws(P ) ∩ Wu(Q) �= ∅ and Wu(P ) ∩ Ws(Q) �= ∅.
Generic transitivity allows us to obtain
Theorem C. There exists a residual subset R of Diff1(M) such that if � is an �-isolatedtransitive set of f ∈ R then either (1) or (2) holds:
(1) � is a hyperbolic set of f
(2) � is nonhyperbolic and f is C1-approximated by a diffeomorphism which exhibits aheterodimensional cycle
Theorem C above is a ‘setwise’ version of the following (global) conjecture by Palis [P].
Conjecture (Palis). There exists a dense subset D of Diffk(M) such that if f ∈ D, then f iseither hyperbolic or exhibits a homoclinic tangency or a heterodimensional cycle.
We note that the global Palis conjecture is much more difficult than the ‘setwise’ resultgiven by theorem C. Even the solution of the two-dimensional case requires very delicatearguments (see [PS]).
This paper is organized as follows: in section 2 we list some definitions and results thatare used in the remainder of the paper; in section 3 we prove theorem A using the results ofsection 2; in section 4 we deduce theorem B from theorem A; in section 5 we show how toadapt the proof of theorem 4 of [BDP] in order to obtain proposition A; and in section 6 weuse theorem A to prove theorem C.
2. Technical preliminaries
We now state some definitions and results to be used later.
Definition 5. Let f ∈ Diff1(M) and p be a hyperbolic periodic point of f . The homoclinicclass of f relative to p is given by
H(p, f ) = cl[Ws(p) � Wu(p)],
where � denotes points of transverse intersection of the invariant manifolds.
It is well-known that H(p, f ) is a transitive compact f -invariant subset of �(f ), whichis not necessarily hyperbolic. When f is axiom A, its basic sets are hyperbolic homoclinicclasses. In the absence of ambiguity, we may write H(p) for H(p, f ).
Note that by definition the homoclinic class of a periodic sink or source coincides with itsorbit.
Theorem 1 ([Pu]). There exists a residual subset R1 ⊂ Diff1(M) such that if f ∈ R1, then�(f ) = cl(Per(f )) and Per(f ) = Perh(f ), where Perh(f ) denotes the set of hyperbolicperiodic points of f .
Attractors of generic diffeomorphisms 305
Theorem 2 ([CMP]). There exists a residual subset R2 ⊂ Diff1(M) such that if f ∈ R2
and H(p, f ) is a homoclinic class of f , then H(p, f ) has no 1-cycle (i.e. Ws(H(p, f )) ∩Wu(H(p, f )) = H(p, f ))3.
Theorem 3 below is an immediate consequence of theorem B of [BD2] combined withtheorem 1 above.
Theorem 3. There exists a residual subset R3 ⊂ Diff1(M) such that if f ∈ R3 then, givenany �-isolated transitive set � of f , we have � = H(p, f ) for some periodic point p ∈ �.
Definition 6. Let p be a periodic saddle of f ∈ Diff1(M). Then, H(p) varies C1-continuouslyat f if given any sequence gj → f in the C1 topology, we have H(pgj , gj ) → H(p, f ) inthe Hausdorff topology, where pg is the continuation of p relative to g.
The next result is an immediate consequence of the lower semi-continuity of homoclinicclasses.
Theorem 4. There exists a residual subset R4 ⊂ Diff1(M) such that if f ∈ R4 and p is asaddle of f , then H(p) varies C1-continuously at f .
The following lemma is a well-known fact from general topology (see for instance [Ku]).
Topological lemma. Let X be a Baire topological space and � : X → N a lowersemicontinuous map. Then, there exists a residual subset N of X such that �|N is locallyconstant at each point of N .
3. Proof of theorem A
Given two periodic points p, q of f ∈ Diff1(M), we say that the homoclinic classes of p andq are distinct if they do not coincide as sets; that is, if H(p, f ) �= H(q, f ).
Now let {Uk}k∈N be a (countable) base of open subsets of M . For each k ∈ N, define�k : Diff1(M) → (N ∪ {∞}) by setting
�k(g) ≡ card{distinct homoclinic classes of g that intersect Uk}.That is, �k is the map that ‘counts’ the number of homoclinic classes that intersect the openset Uk .
We will need to use the following ‘local’ version of proposition 2 of [A].
Proposition 1. There exists a residual subset R5 ⊂ Diff1(M) such that, for every k ∈ N andf ∈ R5, if �k(f ) ∈ N, then there exists a neighbourhood V of f in R5 such that �k(g) ∈ N ifg ∈ V .
The proof of proposition 1 is almost identical to that of proposition 2 of [A], so we will notwrite it out in detail. Only two modifications to the proof of proposition 2 of [A] are needed:first, replace ‘homoclinic classes of g’ by ‘homoclinic classes of g that intersect Uk’, therebyobtaining for each k ∈ N a residual subset Sk of Diff1(M); second, take the intersection of allSk , k � 1, in order to obtain R5.
Lemma 1. There exists a residual subset R′ of Diff1(M) such that given any k ∈ N and g ∈ R′
with �k(g) ∈ N, then �k is constant in a neighbourhood Ak of g in R′.3 The results in [CMP] are stated and proved in the context of flows, but they hold also for diffeomorphisms viaanalogous arguments. The Lyapunov stability techniques that constitute the core of the proofs in [CMP] may bereadily adapted for diffeomorphisms. In particular, the connecting lemma with a jump used in [CMP] also hasdiffeomorphism versions, see [Ar] for instance.
306 F Abdenur
Proof. By theorem 4, there exists a residual subset R4 of Diff1(M) where all the homoclinicclasses vary continuously. Let R∗ ≡ R4 ∩ R5, where R5 as in proposition 1. Then R∗ is aresidual subset of Diff1(M) such that, for all k ∈ N, if f ∈ C∗ only has finitely many distincthomoclinic classes that intersect Uk , then there is a neighbourhood V of f in R∗ such thatevery g ∈ V only has finitely many distinct homoclinic classes that intersect Uk .
Set Rkfin ≡ {g ∈ R∗ : �k(g) ∈ N} and Rk∞ ≡ R∗ \ Rk
fin. Then, for each k ∈ N, we havethat R∗ = Rk
fin ∪ Rk∞, where the union is disjoint and Rkfin is open in R∗.
Given k ∈ N and g ∈ Rkfin, then, for all g′ ∈ Diff1(M) sufficiently near g, we have by
their continuity that all the continuations of the ‘original’ distinct homoclinic classes of g thatintersect Uk will still intersect Uk and be pairwise distinct. That is, there is a neighbourhoodA′ of g in Rk
fin such that �k(g′) is greater than or equal to �k(g) for all g′ ∈ A′.
Hence, �k is lower-semicontinuous on Rkfin. Therefore, by the topological lemma, there
exists a residual subset Sk of Rkfin such that �k is locally constant when restricted to Sk . Setting
Rk ≡ Sk ∪Rk∞ we clearly have that Rk is residual in Diff1(M), and moreover �k|Rk is constantin a neighbourhood of any g ∈ Rk with �k(g) ∈ N.
Setting R′ ≡ ⋂k∈N
Rk we obtain a residual subset of Diff1(M) with the desiredproperties. �
We can now prove theorem A.
Proof of theorem A. Let R be the residual subset of Diff1(M) defined by
R = R1 ∩ R3 ∩ R4 ∩ R′,
where R1, R3 and R4 are respectively as in theorems 1, 3 and 4, and R′ is as in lemma 1. Takef ∈ R and let � be an �-isolated transitive set of f with U an open neighbourhood of � suchthat U ∩ �(f ) = �. By theorem 3, we have � = H(p, f ) for some periodic point p ∈ �.
Take a smaller neighbourhood V of � such that cl(V ) ⊂ U . Then, we have that
� = �(f ) ∩ cl(V ).
Consider the open base {Uk} of M defined above. We then have
U =⋃
i∈N
Uki
for some subsequence {Uki} of {Uk}. Since
⋃i∈N
Ukicontains the compact set cl(V ), there
exists a finite number of such sets Ukiwhose union covers cl(V ). We rename these sets
U1, . . . , Us in order to simplify the notation.Since f ∈ R, it follows from lemma 1 that for each Ui , i ∈ {1, . . . , s}, there is a
neighbourhood Ai of f in R such that �i is constant in Ai . Note that U intersects only onehomoclinic class (namely, H(p, f )), so we have either �i ≡ 1 or 0 in each neighbourhoodAi of f .
Let W ≡ A1 ∩· · ·∩As . Then, each �1, . . . , �s is constant (equal to 1 or 0) when restrictedto W . It is easily seen that this means that U1 ∪ · · · ∪ Us intersects only one homoclinic class(the continuation H(pg, g) of H(p, f )) for each g ∈ W . From theorem 1, it follows that
(U1 ∪ · · · ∪ Us) ∩ �(g) = H(pg, g)
for each g ∈ W . Therefore,
V ∩ �(g) = H(pg, g)
for each g ∈ W , as desired. The continuity of � with respect to the diffeomorphism followsdirectly from theorem 4. �
Attractors of generic diffeomorphisms 307
4. Proof of theorem B
We now prove theorem B.
Proof of theorem B. Consider the residual set R given by theorem A, and let � be anattractor of f ∈ R with an attracting neighbourhood U . By theorem 3 above we haveU ∩ �(f ) = � = H(p, f ) for some p ∈ Per(f ).
Applying theorem A to �, we obtain a neighbourhood V of � in M and a neighbourhoodW∗ of f in Diff1(M) such that if g ∈ R ∩ W∗, then,
V ∩ �(g) = �g ≡ H(pg, g).
Taking V sufficiently small, we can assume that cl(V ) ∩ �(g) = �g .Since � ⊂ V ⊂ U , we have
� ⊂⋂
k∈N
f k(V ) ⊂⋂
k∈N
f k(U) = �,
and therefore⋂
k∈Nf k(V ) = �. By lemma 2.9 of [S], we may assume that f (cl(V )) ⊂ V by
shrinking V if necessary. We conclude that V is an attracting neighbourhood of � for f .We must now show that V is also an attracting neighbourhood for �g , given any
g ∈ W∗ ∩ R. The condition cl(f (V )) ⊂ V is open: it holds for all g C1-near f . Wetherefore need only prove that⋂
k∈N
gk(V ) = H(pg, g) ≡ �g
for all g ∈ W∗.Taking W∗ sufficiently small, we guarantee, by the continuity of homoclinic classes, that
H(pg, g) ⊂ V for all g ∈ W∗. We therefore have⋂
k∈N
gk(V ) ⊃ H(pg, g) ≡ �g
for all g ∈ W∗, and need only prove the other inclusion.Assume by contradiction that there is some x ∈ ⋂
k∈Ngk(V ) \ H(pg, g). Then, the future
g-orbit θ+(x) of x is contained in V since g(V ) ⊂ V , and the past g-orbit θ−(x) of x iscontained in V since x ∈ ⋂
k∈Ngk(V ). Now, because the future and past g-orbits of every
point of M converge to the nonwandering set of g, and cl(V ) ∩ �(g) = �g ≡ H(pg, g), wehave (taking convergent subsequences if necessary)
limk→+∞
gk(x) ∈ cl(V ) ∩ �(g) = �g and limk→−∞
gk(x) ∈ cl(V ) ∩ �(g) = �g.
But this means that �g = H(pg, g) has a 1-cycle, contradicting theorem 2. Hence wehave that
⋂k∈N
gk(V ) = H(pg, g) ≡ �g , and theorem B is proved. �
5. Proof of proposition A
This section consists of the proof of proposition A. For the sake of brevity, we only pointout the necessary modifications to the proof of theorem 4 of [BDP], freely using notation andconcepts from that paper.
Let us first recall that [CMP] proved that, generically, �-isolated transitive sets areisolated (i.e. locally maximal) homoclinic classes. Combining this result with theorem A andproposition 6.2 of [BDP] we easily obtain the following ‘generic version’ of proposition 6.2of [BDP].
Proposition 6.2′. There is some residual subset R of Diff1(M) such that if � is an �-isolatedtransitive set of f ∈ R that admits some dominated splitting E ⊕ F , E ≺ F , then, there is
308 F Abdenur
some neighbourhood U of � and some neighbourhood W of f such that
(a) if g ∈ R ∩ W then �g admits a dominated continuation Eg ⊕Fg of the original splitting;and
(b) either (i) Df |E uniformly contracts volume or (ii) there is some arbitrarily smallC1-perturbation g of f and some hyperbolic periodic point q of g in U such that Dg
expands volume on Eg(q).
We now prove proposition A by closely following the proof of theorem 4 of [BDP].Essentially, the only difficulty here is adapting the original proof (which deals with robustlytransitive sets) to the case of generically transitive sets.
Proof of proposition A. Let � be a non-trivial generically transitive set of f ∈ Diff1(M).That is, there is a neighbourhood U of � in M , a neighbourhood W of f in Diff1(M), and aresidual subset R of W such that if g ∈ R, then
�g ≡ �(g) ∩ U
is a compact transitive continuation of � (indeed, there is some periodic saddle p such that�g = H(pg, g) for all g ∈ R). Since intersections of residual sets are residual, we may bycorollary 0.3 of [BDP] assume that all such continuations �g of � admit dominated splittings(as does � itself). Finally, we assume also that R satisfies the properties of proposition 6.2′
above.Let N be the dimension of M and K > 0 a strict upper bound for the norms of Df and
Df −1. Assume that W is an open ball in Diff1(M), centred at f , with diameter small enoughso that K is an upper bound for all g ∈ W .
Let F1 ⊕ F2 ⊕ · · · ⊕ Fk be the finest dominated splitting of Df over �. Let E ≡ F1 andF ≡ F2 ⊕ · · · ⊕ Fk and fix L so that the splitting E ⊕ F is 2L-dominated. Let l > 0 be theconstant associated with K, N, 2L, and let ε ≡ diam(W)/2 be as in lemma 6.1 of [BDP].
Assume by contradiction that Df |E does not contract volume uniformly. Then, byproposition 6.2′ there are g ∈ W and some periodic hyperbolic saddle q ∈ U such that Dg
expands the volume in Eg(q). By generic transitivity and the persistence of hyperbolic periodicpoints, we may assume after a perturbation that g ∈ R and, therefore, that q ∈ �g = H(pg, g).
Consider the set � of hyperbolic periodic saddles in �g(=H(q, g)) that are homoclinicallyrelated to q. Then, � is dense in �g and g induces the periodic linear system (�, g, T�M, Dg),which has dimension N , is bounded by K , and has transitions. Moreover, for g close to f ,E(g) ⊕ F(g) is an L-dominated splitting for this system and the bundle E(g) does not admitany l-dominated splitting (by lemma 1.4 of [BDP]). Applying lemma 6.1 of [BDP] we obtainan ε-perturbation B of (�, g, T�M, Dg) and some hyperbolic periodic point s ∈ � such thatall the eigenvalues of MB(s) have modulus greater than one. Then using Franks’ lemma weobtain some perturbation g′ ∈ W of g such that s is a periodic hyperbolic source of g′ in�(g′) ∩ U . Again using generic transitivity and the persistence of hyperbolic periodic points,we may assume (after a small perturbation) that g′ ∈ R and that, therefore, the source s belongsto the (non-trivial and transitive) set �g′ , which is a contradiction. We have thus proved thatDf |E uniformly contracts volume.
Applying the same argument to f −1 and to the splitting E′⊕F ′, where E′ ≡ F1⊕· · ·⊕Fk−1
and F ′ ≡ Fk , we obtain that Df |F uniformly expands volume, as desired. �
6. Proof of theorem C
Given � a compact invariant set of f ∈ Diff1(M), let Perf (�) denote the set of periodic pointsof f that belong to �.
Attractors of generic diffeomorphisms 309
Let U be an open subset of M . Then, we set �1(U) ≡ {f ∈ Diff1(M): there exists anopen neighbourhood W of f in Diff1(M) such that if g ∈ W , then all periodic points of g in U
are hyperbolic}. Hayashi proved in [H1] that the set �1(M) coincides with the set of �-stablediffeomorphisms. It is easy to see that the arguments in his proof also yield the following‘local’ version of that result.
Theorem 5 ([H1]). Let � be a transitive compact �-isolated set of f ∈ Diff1(M) such thatPerf (�) is dense in � and such that there is a neighbourhood U of � such that f ∈ �1(U).Then, � is a hyperbolic set of f .
We will also need the following version of Hayashi’s connecting lemma [H2]:
Connecting lemma ([WX]). Consider f ∈ Diff1(M) with hyperbolic periodic points p and q.Assume that there are sequences of points (xj ) and natural numbers (kj ) such that
(xj ) → ps ∈ Ws(p) \ {p} and f kj (xj ) → qu ∈ Wu(q) \ {q}Then, given any ε > 0 there exists an ε-perturbation g of f such that Ws(pg) ∩ Wu(qg) �= ∅.
We now prove a lemma, whose proof follows that of lemma 3.3 of [DPU], and then use itto prove theorem C.
Lemma 2. There exists a residual subset R of Diff1(M) such that if p and q are periodicsaddles of f ∈ R with ind(q) �= ind(p) and such that there is some transitive set � thatcontains both p and q, then, for every ε > 0, there is some g ∈ diff1(M) which is ε-C1-closeto f and exhibits a heterodimensional cycle.
We note that Gan and Wen have actually proved a more general result than the lemma above,see [GW]. In their theorem they impose on � a weaker form of transitivity than topologicaltransitivity. Moreover, in order to create a heterodimensional cycle they require only that �
contain two hyperbolic subsets (not necessarily periodic orbits) of distinct indices. We includethe proof of lemma 2, which is shorter than the proof in [GW], for completeness.
Proof. Let R be the residual set given by theorem B of [BD2] and f ∈ R, p, q, and � be asin the statement. Then, p and q are persistently connected; i.e. there is a neighbourhood Wof f in Diff1(M) and a dense subset D of W such that given g ∈ D then the continuations pg
and qg of p and q belong to some common transitive set.By hypothesis, q and p have different indices. Assume ind(p) > ind(q); the other case
is identical if we switch the roles of p and q. We first want to create an intersection betweenWs(p) and Wu(q).
Let ε > 0. By transitivity, there exists some point x in � whose future orbit is densein �. This means that there are sequences of increasing natural numbers (mj ) and (dj ) withdj > mj such that
f mj (x) → p and f dj (x) → q.
By considering fundamental domains Ds of p in Wsloc(p) and Du of q in Wu
loc(q), we can suppose(after changing the values of the sequences (mj ) and (dj )) that we have instead
f mj (x) → ps ∈ Ws(p) \ {p} and f dj (x) → qu ∈ Wu(q) \ {q}.By setting xj = f mj (x) and kj = dj − mj we can now apply the connecting lemma to
obtain an ε/4-perturbation g of f with Ws(pg) ∩ Wu(qg) �= ∅.Since
dim(Ws(p)) + dim(Wu(q)) = ind(p) + (n − ind(q)) > ind(q) + n − ind(q) = n,
310 F Abdenur
where n is the dimension of M , it follows that after another ε/4-perturbation we obtain g′ ∈ Wsuch that the intersection Ws(pg′
) ∩ Wu(qg′) is transversal at some point, and therefore
persistent. Now, making another ε/4-perturbation if necessary, we can assume that g′ belongsto D, and therefore that qg′
and pg′belong to some common transitive set.
Finally, to obtain an intersection between Wu(pf ′) and Ws(qf ′
) after an ε/4-perturbationf ′ of g′, we use the same argument as above. The end is an ε-perturbation of f that has aheterodimensional cycle between the continuations of p and q, as desired. �
Proof of theorem C. Let R be the residual subset of Diff1(M) obtained by intersecting theresidual sets given by theorem A and lemma 2, and let f ∈ R and � be as in the statementof theorem C. Then, theorem A applies, so take V and W as in the statement of theorem A.Choose now some hyperbolic periodic point p ∈ � with stability index ind(p) = s; shrinkingW if necessary, we can assume that the continuation pg of p persists and belongs to V for allg ∈ W .
Assume that � is nonhyperbolic. Then, by theorem 5, we have that f does not belong to�1(V ). Choose some ε > 0. To prove the theorem we must create a heterodimensional cycleafter an ε-perturbation of f .
Since f /∈ �1(V ), it follows that there exists some g in W , ε/4-close to f , with somenonhyperbolic periodic point q ∈ V . Since q is nonhyperbolic (i.e. it has some eigenvaluewith norm 1), we can ε/4-perturb g along the orbit of q so as to obtain some f ′ ∈ W witha hyperbolic periodic point q ′ such that ind(q ′) �= s and whose f ′-orbit coincides with thef -orbit of q; thus, we have q ′ ∈ V .
Clearly q ′ cannot be a sink or source, because otherwise we would contradict the generictransitivity of � given by theorem A. Therefore, q ′ is a hyperbolic periodic saddle whoseindex differs from that of the continuation pf ′
of p. The saddle q ′ is hyperbolic and thereforepersistent, so after another ε/4-perturbation we obtain g′ ∈ R such that the continuationq ′g′
of q ′ still belongs to V . By theorem A, we have that both q ′g′and pg′
belong to�g′ = V ∩ �(g′), which is a transitive set. So, by lemma 2, we get a heterodimensionalcycle after an ε/4-perturbation of g′ as desired. �
Acknowledgments
Thanks to F Rodriguez-Hertz for pointing out a gap in an early version of the proof of theorem B,and also to both anonymous referees for their suggestions and corrections. I would also liketo thank my thesis advisor M Viana for his support and encouragement, as well as IMPA foran exciting research environment.
References
[A] Abdenur F 2002 Generic robustness of spectral decompositions Ann. Scient. Ecole Nor. Sup. Paris at press[Ar1] Arnaud M-C 2001 Creation de connexions en topologie C1 Erg. Th. Dyn. Sys. 21 339–81[BD1] Bonatti Ch and Diaz L J 1996 Persistent nonhyperbolic transitive diffeomorphisms Ann. Math. 143 357–96[BD2] Bonatti Ch and Diaz L J 1999 Connexions heteroclines et genericite d’une infinite de puits ou de sources
Ann. Scient. Ecole Nor. Sup. Paris 32 135–50[BDP] Bonatti Ch, Diaz L J and Pujals E A 2002 C1-generic dichotomy for diffeomorphisms: weak forms of
hyperbolicity or infinitely many sinks or sources Ann. Math. at press[CMP] Carballo C M, Morales C A and Pacifico M J 2002 Homoclinic classes for generic C1 vector fields Erg. Th.
Dyn. Sys. at press[DPU] Diaz L J, Pujals E and Ures R 1999 Partial hyperbolicity and robust transitivity Acta Math. 183 1–43
Attractors of generic diffeomorphisms 311
[GW] Gan G and Wen L 2002 Heteroclinic cycles and homoclinic closures for generic diffeomorphisms PekingUniversity Preprint
[H1] Hayashi S 1992 Diffeomorphisms in �1(M) satisfy Axiom A Erg. Th. Dyn. Sys. 12 233–53[H2] Hayashi S 1997 Connecting invariant manifolds and the solution of the C1 stability and �-stability
conjectures for flows Ann. Math. 145 81–137[Ku] Kuratowski K 1968 Topology II (Warsaw: Academic Press–PWN–Polish Sci. Publishers)[M1] Mane R 1978 Contributions to the C1-stability conjecture Topology 17 386–96[M2] Mane R 1982 An ergodic closing lemma Ann. Math. 116 503–40[P] Palis J 2000 A global view of dynamics and a conjecture on the denseness of finitude of attractors Asterisque
261 335–47[Pu] Pugh C 1967 An improved closing lemma and a general density theorem Am. J. Math. 89 1010–21[PS] Pujals E and Sambarino M 2000 Homoclinic tangencies and hyperbolicity for surface diffeomorphisms: a
conjecture of Palis Ann. Math. 151 961–1023[S] Shub M 1986 Global Stability of Dynamical Systems (New York: Springer)[WX] Wen L and Xia Z 2000 C1 connecting lemmas Trans. Am. Math. Soc. 352 5213–30
