Stabilization of heterodimensional cycles
SSTABILIZATION OF HETERODIMENSIONAL CYCLES
C. BONATTI, L. J. D´IAZ, AND S. KIRIKI
Abstract.
We consider diffeomorphisms f with heteroclinic cycles associatedto saddles P and Q of different indices. We say that a cycle of this type can bestabilized if there are diffeomorphisms close to f with a robust cycle associatedto hyperbolic sets containing the continuations of P and Q . We focus on thecase where the indices of these two saddles differ by one. We prove that,excluding one particular case (so-called twisted cycles that additionally satisfysome geometrical restrictions), all such cycles can be stabilized. introduction In [17] Palis proposed a program whose main goal is a geometrical description forthe behavior of most dynamical systems. This program pays special attention tothe generation of non-hyperbolic dynamics and to robust dynamical properties (i.e.,properties that hold for open sets of dynamical systems). An important part of thisprogram is the
Density Conjecture olicity versus cycles) : the two main sources ofnon-hyperbolic dynamics are heterodimensional cycles and homoclinic tangencies(shortly, cycles ), see [17, Conjecture 1] . The goal of this paper is to study the generation of robust heterodimensional cycles (see Definition 1.1).Besides Palis’ program, we have the following two motivations for this paper: Motivation I ([16, 21, 22]) . Every C -diffeomorphism having a homoclinic tan-gency associated with a saddle P is in the C -closure of the set of diffeomorphismshaving C -robust homoclinic tangencies. Moreover, these robust homoclinic tan-gencies can be taken associated to hyperbolic sets containing the continuations ofthe saddle P . Using the terminology that will be introduced in this paper this means thathomoclinic tangencies of C -diffeomorphisms can be stabilized, see Definition 1.1.On the other hand, for C -diffeomorphisms of surfaces homoclinic tangencies cannotbe stabilized, see in [14]. This leads to the following motivation. Motivation II ([8]) . Every diffeomorphism with a heterodimensional cycle asso-ciated with a pair of hyperbolic saddles P and Q with dim E s ( P ) = dim E s ( Q ) ± belongs to the C -closure of the set of diffeomorphisms having C -robust heterodi-mensional cycles. Here E s denotes the stable bundle of a saddle. Date : October 30, 2018.2000
Mathematics Subject Classification.
Primary:37C29, 37D20, 37D30.
Key words and phrases. heterodimensional cycle, homoclinic class, hyperbolic set, blender, C robustness. This conjecture was proved by Pujals-Sambarino for C -surface diffeomorphisms in [18] (sinceheterodimensional cycles only can occur in manifolds of dimension n ≥
3, for surface diffeomor-phisms it is enough to consider homoclinic tangencies). a r X i v : . [ m a t h . D S ] A p r C. BONATTI, L. J. D´IAZ, AND S. KIRIKI
One may think of the result in Motivation II as a version of the results in Mo-tivation I for heterodimensional cycles in the C -setting. However, the results in[8] does not provide information about the relation between the hyperbolic setsinvolved in the robust cycles and the saddles in the initial one. Thus, one aims foran extension of [8] giving some information about the hyperbolic sets displayingthe robust cycles, see [8, Question 1.9].In this paper we prove that, with the exception of a special type of heterodi-mensional cycles (so-called twisted cycles , see Definition 4.6), the hyperbolic setsexhibiting the robust cycles can be taken containing the continuations of the sad-dles in the initial cycle. In fact, by [10] our results cannot be improved: there aretwisted cycles that cannot be stabilized, that is, the hyperbolic sets with robustcycles cannot be taken containing the continuations of the saddles in the initialcycle.To state precisely our results we need to introduce some definitions. Recall thatif Λ is a hyperbolic basic set of a diffeomorphism f : M → M then there are aneighborhood U f of f in the space of C -diffeomorphisms and a continuous map U f → M : g (cid:55)→ Λ g , such that Λ f = Λ, Λ g is a hyperbolic basic set, and the dynamicsof f | Λ and g | Λ g are conjugate. The set Λ g is called the continuation of Λ for g .Note that these continuations are uniquely defined. Definition 1.1 (Heterodimensional cycles) . • The s -index (u -index ) of a transitive hyperbolic set is the dimension of itsstable (unstable) bundle. • A diffeomorphism f has a heterodimensional cycle associated to transitivehyperbolic basic sets Λ and Σ of f if these sets have different s-indices andtheir invariant manifolds meet cyclically, that is, if W s (Λ , f ) ∩ W u (Σ , f ) (cid:54) = ∅ and W u (Λ , f ) ∩ W s (Σ , f ) (cid:54) = ∅ . • The heterodimensional cycle has coindex k if s -index(Λ) = s -index(Σ) ± k .In such a case we just write coindex k cycle. • A diffeomorphism f has a C -robust heterodimensional cycle associated toits hyperbolic basic sets Λ and Σ if there is a C -neighborhood U of f suchthat every diffeomorphism g ∈ U has a a heterodimensional cycle associatedto the continuations Λ g and Σ g of Λ and Σ, respectively. • Consider a diffeomorphism f with a heterodimensional cycle associatedto a pair of saddles P and Q . This cycle can be C -stabilized if every C -neighborhood U of f contains a diffeomorphism g with hyperbolic basic setsΛ g (cid:51) P g and Σ g (cid:51) Q g having a robust heterodimensional cycle. Otherwisethe cycle is said to be C -fragile. Remark that, by the Kupka-Smale genericity theorem (invariant manifolds ofhyperbolic periodic points of generic diffeomorphisms are in general position), atleast one of the hyperbolic sets involved in a robust cycle is necessarily non-trivial, that is, not a periodic orbit.
Definition 1.2 (Homoclinic class) . The homoclinic class of a saddle P is theclosure of the transverse intersections of the stable and unstable manifolds W s ( P, f )and W u ( P, f ) of the orbit of P . We denote this class by H ( P, f ). A homoclinicclass is non-trivial if it contains at least two different orbits.A homoclinic class can be also defined as the closure of the set of saddles thatare homoclinically related with P . Here we say that a saddle Q is homoclinically TABILIZATION OF HETERODIMENSIONAL CYCLES 3 related with P if the invariant manifolds of the orbits of P and Q meet cyclicallyand transversely, that is, W s ( P, f ) (cid:116) W u ( Q, f ) (cid:54) = ∅ and W s ( Q, f ) (cid:116) W u ( P, f ) (cid:54) = ∅ .The following is a consequence of our results (see Theorems 2 and 3 below). Theorem 1.
Let f be a C -diffeomorphism with a coindex one cycle associatedto saddles P and Q . Suppose that at least one of the homoclinic classes of thesesaddles is non-trivial. Then the heterodimensional cycle of f associated to P and Q can be C -stabilized. A simple consequence of this result is the following:
Corollary 1.
Let f be a C -diffeomorphism with a heterodimensional cycle asso-ciated to saddles P and Q such that s -index( P ) = s -index( Q ) + 1 . Suppose thatthe intersection W u ( P, f ) ∩ W s ( Q, f ) contains at least two different orbits. Thenthe cycle can be C -stabilized. The question of the stabilization of cycles is relevant for describing the globaldynamics of diffeomorphisms (indeed this is another motivation for this paper).Let us explain this point succinctly. Following [12, 15, 1], this global dynamics isstructured by means of homoclinic or/and chain recurrence classes. The goal is todescribe the dynamics of these classes and their relating cycles. In general, homo-clinic classes are (properly) contained in chain recurrence classes. For C -genericdiffeomorphisms and for hyperbolic periodic points, these two kinds of classes coin-cide, [4]. However, there are non-generic situations where two different homoclinicclasses are “joined” by a cycle. In this case these classes are contained in one com-mon chain recurrence class which hence is strictly larger. We would like to knowunder which conditions after small perturbations these two homoclinic classes ex-plode and fall into the very same homoclinic class C -robustly. Indeed this occursif the cycle can be C -stabilized. Examples where this stabilization is used for de-scribing global dynamics can be found in [5, 23, 24]. See [11, Chapter 10.3-4] and[3] for a broader discussion of these questions.To prove our results we analyze the dynamics associated to different types ofcoindex one cycles. This analysis essentially depends on two factors: the centralmultipliers of the cycle and its unfolding map. Let us now discuss this point briefly,for further details we refer to Section 4.1.1.
Multipliers and unfolding map of a cycle.
Let f be a diffeomorphismwith a coindex one cycle associated to saddles P and Q . In what follows we willassume that s -index( P ) = s -index( Q ) + 1. Denote by π ( R ) the period of a periodicpoint R .We say that the cycle is partially hyperbolic if there are heteroclinic points X ∈ W s ( P, f ) ∩ W u ( Q, f ) and Y ∈ W u ( P, f ) ∩ W s ( Q, f ) such that the closed set formedby the orbits of
P, Q, X, and Y has a partially hyperbolic splitting of the form E ss ⊕ E c ⊕ E uu , where E c is one-dimensional, E ss is uniformly contracting, and E uu is uniformly expanding. We call E c the central bundle . Note that, in particular, thisimplies that X is a transverse intersection and Y is a quasi-tranverse intersectionof the invariant manifolds. Also observe that the bundle E c is necessarily non-hyperbolic. Bearing in mind this property we introduce the following definition. Definition 1.3 (Central multipliers) . The cycle has real central multipliers if thereare a contracting real eigenvalue λ of Df π ( P ) ( P ) and an expanding real eigenvalue β C. BONATTI, L. J. D´IAZ, AND S. KIRIKI of Df π ( Q ) ( Q ) such that: (i) λ and β have multiplicity one, (ii) | λ | > | σ | for everycontracting eigenvalue σ of Df π ( P ) ( P ), and (iii) | β | < | η | for every expandingeigenvalue η of Df π ( Q ) ( Q ). In this case, we say that λ and β are the real centralmultipliers of the cycle. Similarly, the cycle has non-real central multipliers if either (i) there are a pairof non-real (conjugate) contracting eigenvalues λ and ¯ λ of Df π ( P ) ( P ) such that | λ | = | ¯ λ | ≥ | σ | for every contracting eigenvalue σ of Df π ( P ) ( P ), or (ii) there area pair of non-real (conjugate) expanding eigenvalues β and ¯ β of Df π ( Q ) ( Q ) suchthat | β | = | ¯ β | ≤ | η | for every expanding eigenvalue η of Df π ( Q ) ( Q ).Let us note that cycles with central real multipliers can be perturbed to getpartially hyperbolic ones (associated to the continuations of the saddles in theinitial one).In the case of cycles with real central multipliers we will distinguish so-called twisted and non-twisted cycles, see Definition 4.6. An intuitive explanation of thesetwo sorts of cycles goes as follows, see Figure 1.In order to study the dynamics of the cycle we select heteroclinic points X ∈ W s ( P, f ) ∩ W u ( Q, f ) and Y ∈ W u ( P, f ) ∩ W s ( Q, f ). Typically, X is a transverseintersection point and Y is a quasi-transverse intersection point (due to dimensiondeficiency). The next step is to consider a neighborhood of the cycle, that is, anopen set V containing the orbits of P, Q, X , and Y , and study the dynamics ofperturbations of f in such a neighborhood. If the neighborhood V is small enough,possibly after a perturbation of f , the dynamics of f in V is partially hyperbolicwith a splitting of the form E ss ⊕ E c ⊕ E uu (recall the definition above).Replacing Y by some backward iterate, we can assume that the heteroclinic point Y is close to P . We pick some large number k such that f k ( Y ) is nearby Q andconsider the map T = f k defined in a small neighborhood of Y . This map is calledthe unfolding map . If it is possible to pick k in such a way that Df k preservesthe orientation of the central bundle then we say that the cycle is non-twisted .Otherwise, the cycle is twisted . Note that in the previous discussion the choice ofthe heteroclinic point X does not play any relevant role. Figure 1.
Twisted and non-twisted cyclesMore precisely, the dynamics of the unfolding of the cycle mostly depends onthe signs of the central eigenvalues λ (associated to P ) and β (associated to Q ) and TABILIZATION OF HETERODIMENSIONAL CYCLES 5 on the restriction of T to the central bundle. We associate to the cycle the signs ,sign( Q ), sign( P ), and sign( T ) in { + , −} determined by the following rules: • sign( Q ) = + if β > Q ) = − if β < • sign( P ) = + if λ > P ) = − if λ <
0; and • sign( T ) = + if T preserves the orientation in the central direction andsign( T ) = − if the orientation is reversed.A cycle is twisted if sign( Q ) = + , sign( P ) = +, and sign( T ) = − . Otherwise thecycle is non-twisted. For details see Definition 4.6.Let us observe that the discussion above is reminiscent of the one in [19, Sec-tion 2] about bifurcations of homoclinic tangencies of surface diffeomorphisms. Itinvolves similar ingredients to the ones above: the signs of the eigenvalues of thederivatives, the sides of the tangencies, and the connections (homoclinic and hete-roclinic intersections).We are now ready to state our main results. Theorem 2.
Consider a diffeomorphism f having a coindex one cycle associatedto saddles P and Q . Suppose that (A) either the cycle has a non-real central multiplier, (B) or the cycle has real multipliers and is non-twisted.Then the cycle of f associated to P and Q can be C -stabilized. Let us observe that Theorem 2 cannot be improved. Indeed, there are examplesof diffeomorphisms with twisted cycles that cannot be stabilized, see [10]. On theother hand, we prove that cycles with the bi-accumulation property can be C -stabilized. Let us state this result more precisely.Given a periodic point R of f , consider the eigenvalues λ ( R ) , . . . , λ n ( R ) of Df π ( R ) ( R ) ordered in increasing modulus and counted with multiplicity. If R is hyperbolic, has s-index k , and | λ k − ( R ) | < | λ k ( R ) | then there is a uniqueinvariant manifold W ss ( R, f ) (the strong stable manifold of R ) tangent to theeigenspace associated to λ ( R ) , . . . , λ k − ( R ) (the strong stable bundle). The man-ifold W ssloc ( R, f ) has codimension one in W sloc ( R, f ) and W ssloc ( R, f ) splits each com-ponent of W sloc ( R, f ) into two parts.
Definition 1.4 (Bi-accumulation property) . A saddle R of s-index k such that | λ k − ( R ) | < | λ k ( R ) | is s -bi-accumulated (by homoclinic points) if every componentof W sloc ( R, f ) \ W ssloc ( R, f ) contains transverse homoclinic points of R .A heterodimensional cycle associated to saddles P and Q with s -index( P ) =s -index( Q ) + 1 is bi-accumulated is either P is s-bi-accumulated for f or Q is s-bi-accumulated for f − .In the next result we consider cycles with real central multipliers. Theorem 3.(A)
Every non-twisted cycle can be C -stabilized. (B) Every twisted cycle with the bi-accumulation property can be C -stabilized. Indeed, Theorems 1 and 2 are consequence of Theorem 3.Finally, our results can be summarized as follows:
Corollary 2.
Consider a diffeomorphism f with a fragile cycle associated to saddles P and Q with s -index( P ) = s -index( Q ) + 1 . Then C. BONATTI, L. J. D´IAZ, AND S. KIRIKI • the cycle has positive central real multipliers, • the cycle is persistently twisted (i.e., the cycle cannot be perturbed to get anon-twisted cycle associated to P and Q ), • the intersection W u ( P, f ) ∩ W u ( Q, f ) consists of exactly one orbit, and • the homoclinic classes of P and Q are both trivial. Examples of fragile cycles satisfying the four properties in the corollary can befound in [10]. 2.
Ingredients of the proofs
In this section we review some tools of our constructions.2.1.
Reduction to the case of cycles with real multipliers.
A first step isto see that to prove our results it is enough to consider cycles with real centralmultipliers. For that let us recall a result from [8].
Theorem 2.1 (Theorem 2.1 in [8]) . Let f be a diffeomorphism with a coindex onecycle associated to saddles P and Q . Then there are diffeomorphisms g arbitrarily C close to f with a coindex one cycle with real central multipliers associated tosaddles P (cid:48) g and Q (cid:48) g which are homoclinically related to the continuations P g and Q g of P and Q . In this result one may have P = P g and/or Q = Q g . Note that the previous theorem means the following.
Remark 2.2.
Assume that the saddle P in Theorem 2.1 has non-real centralmultipliers. Then the homoclinic class of P (cid:48) g is non-trivial and contains P .There is also the following simple fact: Lemma 2.3.
Consider a diffeomorphism f with a heterodimensional cycle associ-ated to P and Q . Suppose that there are saddles P (cid:48) g and Q (cid:48) g homoclinically relatedto P g and Q g , respectively, with a heterodimensional cycle that can be C -stabilized.Then the initial cycle can also be C -stabilized.Proof. The stabilization of the cycle associated to P (cid:48) g and Q (cid:48) g means that there is h arbitrarily close to g having a pair of basic hyperbolic sets Λ (cid:48) h (cid:51) P (cid:48) h and Σ (cid:48) h (cid:51) Q (cid:48) h with a robust cycle. Since the saddles P h and P (cid:48) h are homoclinically related thereis a basic set Λ h containing Λ (cid:48) h and P h . Similarly, there is a basic set Σ h containingΣ (cid:48) h and Q h . Since W i (Λ h , h ) ⊃ W i (Λ (cid:48) h , h ) and W i (Σ h , h ) ⊃ W i (Σ (cid:48) h , h ), i = s, u , itis immediate that there is a robust cycle associated to Λ h (cid:51) P h and Σ h (cid:51) Q h . (cid:3) Remark 2.4.
Theorem 2.1 and Lemma 2.3 mean that to prove Theorems 1 and 2it is enough to stabilize cycles with real central multipliers (indeed this is the sortof cycles considered in Theorem 3). Thus in what follows we will focus on this typecycles.2.2.
Strong homoclinic intersections and blenders.
A key ingredient for ob-taining robust heterodimensional cycles in [8] is the notion of a blender.
A blenderis a hyperbolic set with some additional geometrical intersection properties thatguarantee some robust intersections, see Section 3.1 and Definition 3.1. The keystep in [8] to obtain robust cycles is that coindex one cycles yield periodic pointsof saddle-node/flip type with strong homoclinic intersections : the strong stable
TABILIZATION OF HETERODIMENSIONAL CYCLES 7 manifold of the saddle-node/flip intersects its strong unstable manifold, see Defi-nition 3.3. These strong homoclinic intersections generate blenders yielding robustcycles, see Proposition 3.4.In [8] the generation of blenders is not controlled and in general the saddle-node/flip has “nothing to do” with the saddles in the initial cycle. This is why in[8] the hyperbolic sets with robust cycles are not related (in general) to the sad-dles in the initial cycle. Here we control the “generation” of the saddle-node/flipwith strong homoclinic intersections, obtaining blenders that contains the continu-ation of a saddle in the initial cycle and intersecting the invariant manifolds of theother saddle in the cycle. This configuration provides robust cycles associated tohyperbolic sets containing the continuation of both initial saddles, see Theorem 3.5.We next explain the “generation” of saddle-node/flip poits with strong homo-clinic intersections.2.3.
Simple cycles and iterated function systems (IFSs).
To analyze thedynamics of cycles with real multipliers we borrow some constructions and thenotion of a simple cycle from [8], see Section 4.In very rough terms, if a diffeomorphism has a simple cycle then its dynamics ina neighborhood of the cycle is affine and preserves a partially hyperbolic splitting E ss ⊕ E c ⊕ E uu , where E ss is uniformly contracting, E uu is uniformly expanding,and E c is one-dimensional and non-hyperbolic, see Proposition 4.1. Following [8],to prove our results it is enough to consider simple cycles and their (suitable)unfoldings.We consider one-parameter families of diffeomorphisms ( f t ) t unfolding a simplecycle at t = 0 and preserving the affine structure associated to the splitting E ss ⊕ E c ⊕ E uu . In particular, the foliation of hyperplanes parallel to E ss ⊕ E uu ispreserved. Considering the central dynamics given by the quotient of the dynamicsof the diffeomorphism f t by these hyperplanes one gets a one-parameter familyof iterated function systems (IFSs). Some properties of these IFSs are translatedto properties of the diffeomorphisms f t , see Proposition 4.9. This IFS providesrelevant information about the dynamics of the the diffeomorphisms f t such as,for example, the existence of saddle-nodes with strong homoclinic intersections.Such IFSs play a role similar to the one of the quadratic family in the setting ofhomoclinic bifurcations, compare [20, Chapter 6.3].2.4. Organization of the paper.
The discussion above corresponds to the con-tents in Sections 3 and 4. The key step is to analyze the dynamics of the IFSs asso-ciated to simple cycles. Using these IFSs, in Section 5 we analyze non-twisted cy-cles (which is the principal case) and explain how they yield saddle-nodes/flips withstrong homoclinic intersections as well as further intersection properties, see Propo-sition 5.3. We study (twisted and non-twisted) cycles with the bi-accumulationproperty in Section 5.3. In Section 6 we prove Theorem 3, which is the main tech-nical step in the paper. Finally, in Section 7 we see how Theorems 1 and 2 can beeasily derived from Theorem 3.3.
Robust cycles and blenders
In this section, we recall the definition and main properties of blenders. We alsostate the tools to get the stabilization of heterodimensional cycles, see Proposi-tion 3.4 and Theorem 3.5.
C. BONATTI, L. J. D´IAZ, AND S. KIRIKI
Blenders.
Let us recall the definition of a cu-blender in [9]. See also theexamples in [6] and the discussion in [11, Chapter 6]:
Definition 3.1 ( cu -blender, Definition 3.1 in [9]) . Let f : M → M be a diffeomor-phism. A transitive hyperbolic compact set Γ of f with u -index(Γ) = k , k ≥
2, isa cu -blender if there are a C -neighborhood U of f and a C -open set D of embed-dings of ( k − D into M such that for every g ∈ U and everydisk D ∈ D the local stable manifold W s loc (Γ g ) of Γ g intersects D . The set D iscalled the superposition region of the blender. Remark 3.2.
Let Γ be a blender of f . Then for every g close enough to f thecontinuation Γ g of Γ is a blender of g .In fact, the cu-blenders considered in [8] to obtain robust cycles are a specialclass of blenders, called blender-horseshoes, see [9, Definition 3.8]. In this definition,the blender-horseshoe Γ is the maximal invariant set in a “cube” C and has ahyperbolic splitting with three non-trivial bundles T Γ M = E s ⊕ E cu ⊕ E uu , suchthat the unstable bundle of Γ is E u = E cu ⊕ E uu and E cu is one-dimensional.Moreover, the set Γ is conjugate to the complete shift of two symbols. Thus it hasexactly two fixed points, say A and B , called distinguished points of the blender, and that play a special role in the definition of a blender-horseshoe.The definition of a blender-horseshoe involves a Df -invariant strong unstablecone-field C uu corresponding to the strong unstable direction E uu , the local stablemanifolds W sloc ( A, f ) and W sloc ( B, f ) of the distinguished saddles A and B (definedas the connected component of W s ( R, f ) ∩ C containing R , R = A, B ), and thelocal strong unstable manifolds W uuloc ( A, f ) and W uuloc ( B, f ) of A (the component of W uu ( R, f ) ∩ C containing R ). Recall that the strong unstable manifold of R is theonly invariant manifold of dimension dim( E uu ) that is tangent to E uu at R .Let dim( E uu ) = u . One considers vertical disks through the blender, that is,disks ∆ of dimension u tangent to the cone-field C uu joining the “top” and the“bottom” of the cube C . Then there are two isotopy classes of vertical disks thatdo not intersect W sloc ( A, f ) (resp. W sloc ( B, f )), called disks at the right and at theleft of W sloc ( A, f ) (resp. W sloc ( B, f )). For instance, W uuloc ( B, f ) (that is a verticaldisk) is at the right of W sloc ( A, f ). Similarly, W uuloc ( A, f ) is at the left of W sloc ( B, f ).The superposition region D of the blender-horseshoe consists of the vertical disksin between W sloc ( A, f ) and W sloc ( B, f ) (i.e., at the right of W sloc ( A, f ) and at theleft of W sloc ( B, f )). See Figure 2.
Figure 2.
Vertical disks in a blender.
TABILIZATION OF HETERODIMENSIONAL CYCLES 9
Generation of blenders and robust cycles.
To state a criterion for theexistence of robust cycles we need some definitions.
Definition 3.3.
Let S be a periodic point of a diffeomorphism f . • We say that S is a partially hyperbolic saddle-node (resp. flip) of f if thederivative of Df π ( S ) ( S ) has exactly one eigenvalue σ of modulus 1, theeigenvalue σ is equal to 1 (resp., − λ and β of Df π ( S ) ( S ) with | λ | < < | β | . • Consider the strong unstable (resp. stable) invariant direction E uu (resp. E ss ) corresponding to the eigenvalues κ of Df π ( S ) ( S ) with | κ | > | κ | < strong unstable manifold W uu ( S, f ) of S is the unique f -invariant manifold tangent to E uu of the same dimension as E uu . The strong stable manifold W ss ( S, f ) of S is defined similarly considering E ss . • We say that S has a strong homoclinic intersection if W ss ( S, f ) ∩ W uu ( S, f )contains points which do not belong to the orbit of S . Proposition 3.4 (Criterion for robust cycles. Theorem 2.4 in [8]) . Let f be adiffeomorphism having a partially hyperbolic saddle-node/flip S with a strong ho-moclinic intersection. Then there is a diffeomorphism h arbitrarily C -close to f with a robust heterodimensional cycle. Note that this result does not provide information about the sets involved in therobust cycle. We state in Theorem 3.5 a version of this proposition providing someinformation about these sets. Before proving this theorem let us explain the mainsteps of the proof of Proposition 3.4, for further details see [8].
Sketch of the proof of Proposition 3.4.
For simplicity, let us assume that S is asaddle-node of f of period one. After a perturbation, we can suppose that thesaddle-node S splits into two hyperbolic fixed points S − g (contracting in the centraldirection) and S + g (expanding in the central direction), here g is a diffeomorphismobtained by a small the perturbation of f . The saddles S + g and S − g have differ-ent indices and the manifolds W s ( S − g ) and W u ( S + g ) have a transverse intersectionthat contains the interior of a “central” curve joining S − g and S + g . Note that thisintersection property is C -robust. The proof has three steps (see Figure 3): (I) There is a blender-horseshoe Γ g having S + g as a distinguished fixed point. (II) The unstable manifold of S − g contains a vertical disk ∆ in the superpo-sition region D of the blender-horseshoe Γ g . Thus, by the definition ofblender-horseshoe, W s (Γ g , g ) intersects W u ( S − g , g ). Hence, as S + g ∈ Γ g and W u ( S − g , g ) (cid:116) W s ( S + g , g ) (cid:54) = ∅ , there is a heterodimensional cycle associatedto Γ g and S − g . (III) The following properties are open ones: i) the continuation of the hyper-bolic set Γ g to be a blender (the elements in the definition of a blenderdepend continuously on g , see Remark 3.2), ii) W u ( S − g , g ) to contain a ver-tical disk in the superposition region D of the blender, and iii) W s ( S − g , g ) (cid:116) W u ( S + g , g ) (cid:54) = ∅ .Therefore, every diffeomorphism h that is C -close to g has a heterodimensionalcycle associated to S − h and Γ h . Since g can be taken arbitrarily close to f thisconcludes the proof. (cid:3) Figure 3.
Proof of Proposition 3.4.Next result is just a reformulation of the construction above that allows us toget robust cycles associated to sets that contain the continuations of a given saddle.This theorem will be the main tool for stabilizing cycles.
Theorem 3.5.
Let f be a diffeomorphism, P a saddle of f , and S a partiallyhyperbolic saddle-node/flip of f such that:(1) s -index( P ) = dim( W ss ( S )) + 1 = s + 1 ,(2) S has a strong homoclinic intersection,(3) W u ( P, f ) ∩ W ss ( S, f ) (cid:54) = ∅ , and(4) W s ( P, f ) (cid:116) W uu ( S, f ) (cid:54) = ∅ . Then there is a diffeomorphism h arbitrarily C -close to f with a robust heterodi-mensional cycle associated to the continuation P h of P and a transitive hyperbolicset Γ h containing a hyperbolic continuation S + h of S of s -index s .Proof. One proceeds as in the proof of Proposition 3.4, considering a perturbation h of g with saddles S ± h satisfying conditions (I) and (II) above and such that W u ( P h , h ) (cid:116) W s ( S − h , h ) (cid:54) = ∅ . Since W u ( S − h , h ) (cid:116) W s ( S + h , h ) (cid:54) = ∅ , the inclination lemma now implies that W s ( P h , h ) (cid:116) W u ( S + h , h ) (cid:54) = ∅ , see Figure 4. Figure 4.
Proof of Theorem 3.5.
TABILIZATION OF HETERODIMENSIONAL CYCLES 11
Recall that W u ( S − h , h ) contains a vertical disk in the superposition region ofthe blender Γ h . Since W u ( P h , h ) (cid:116) W s ( S − h , h ) (cid:54) = ∅ , the inclination lemma im-plies that the same holds for W u ( P h , h ). Thus we can repeat the construction inProposition 3.4 replacing S − h by P h . Hence W u ( P ϕ , ϕ ) intersects W s (Γ ϕ , ϕ ) for anydiffeomorphism ϕ close to h . Since W s ( P ϕ , h ) ∩ W u ( S + ϕ , ϕ ) (cid:54) = ∅ and S + ϕ ∈ Γ ϕ forevery ϕ close to h , there is a robust heterodimensional cycles associated to P ϕ andΓ ϕ , ending the proof of the theorem. (cid:3) Simple cycles and systems of iterated funtions
In this section, following [8], we introduce simple cycles (Section 4.1) and theirassociated one-dimensional dynamics (Section 4.3). We see that given any diffeo-morphism f with a co-index one cycle with real central multipliers (associated tosaddles P and Q ) there is a diffeomorphism g arbitrarily C -close to f with a cycleassociated to P and Q whose dynamics in a neighborhood of the cycle is affine, seeProposition 4.1. In such a case we say that this cycle of g is simple.In fact, for a diffeomorphism g with a simple cycle there is a one-parameterfamily of diffeomorphisms ( g t ) t , g = g , preserving a (semi-local) partially hyper-bolic splitting E ss ⊕ E c ⊕ E uu such that the bundles E ss and E uu are non-trivialand hyperbolic (uniformly contracting and uniformly expanding, respectively) andthe bundle E c is not hyperbolic and one-dimensional. We consider the quotientdynamics by the hyperplanes E ss ⊕ E uu , obtaining a one-parameter family of one-dimensional iteration function systems (IFSs) which describe the central dynamicsof the maps g t . Properties of these IFSs are translated to properties of the diffeo-morphisms g t , see Proposition 4.9.In Section 5 we will write intersection properties implying the existence of robustcycles (similar to the ones in Theorem 3.5) in terms of properties of the IFSsassociated to simple cycles. We now discuss simple cycles and their IFSs.4.1. Simple cycles.
Next proposition summarizes the results in [8] about simplecycles and their unfoldings.
This proposition means that if ( f t ) is a “model arc”unfolding a simple cycle then the dynamics of the maps f t in a neighborhoodof the cycle is given by suitable compositions of two linear maps (the dynamicsnearby the saddles in the cycle) and two affine maps (iterations corresponding tothe “transition” and the “unfolding maps”). Proposition 4.1 (Proposition 3.5 and Section 3.2 in [8]) . Let f be a diffeomorphismhaving a co-index one cycle with real central multipliers associated to saddles P and Q such that s -index( Q ) + 1 = s -index( P ) . Then there is a one-parameter family of diffeomorphisms ( f t ) t ∈ [ − (cid:15),(cid:15) ] , (cid:15) > , suchthat it satisfies properties (C1)–(C3) below and f is arbitrarily close to f .Let s and u be the dimensions of W s ( Q, f ) and of W u ( P, f ) , respectively. Thereare linear maps • φ λ , ψ β : R → R , φ λ ( x ) = λ x and ψ β ( x ) = β ( x ) , • A s , B s , T s1 , T s2 : R s → R s , which are contractions (i.e., their norms arestrictly less than one), • A u , B u , T u1 , T u2 : R u → R u , which are expansions (i.e., their inverse mapsare contractions), such that: (C1) There are local charts U P and U Q centered at P and Q such that in thesecoordinates we have, for all t , f π ( P ) t ( x s , x c , x u ) = ( A s ( x s ) , φ λ ( x c ) , A u ( x u )) ,f π ( Q ) t ( x s , x c , x u ) = ( B s ( x s ) , ψ β ( x c ) , B u ( x u )) , where | λ | ∈ (0 , and | β | > , x s ∈ R s , x c ∈ R , and x u ∈ R u , and π ( P ) and π ( Q ) are the periods of P and Q , respectively. (C2) There is a quasi-transverse heteroclinic point Y P ∈ W s ( Q, f ) ∩ W u ( P, f ) in U P such that, in the coordinates in the chart U P , it holds:(1) For every t , Y P = (0 s , , a u ) ∈ W uloc ( P, f t ) , a u ∈ R u .(2) There is a neighborhood C s ( Y P ) of Y P in W s ( Q, f ) ∩ U P of the form ( − , s × { (0 , a u ) } .(3) There is τ p,q ∈ N such that for all tY Q,t = ( a s , t, u ) = f τ p,q t ( Y P ) ∈ U Q ∩ W u ( P, f t ) , a s ∈ R s , and Y Q,t ∈ C u ( Y Q,t ) = { ( a s , t ) } × ( − , u ⊂ W u ( P, f t ) ∩ U Q . (4) There is a neighborhood U Y P of Y P , U Y P ⊂ U P , such that T ,t = f τ p,q t : U Y P → f τ p,q ( U Y P ) ⊂ U Q is an affine map of the form T ,t ( x s , x c , x u ) = T ( x s , x c , x u ) + (0 , t, (cid:0) T s1 ( x s ) , ± x c , T u1 ( x u ) (cid:1) + (cid:0) a s , t, − T u1 ( a u ) (cid:1) = (cid:0) T s1 ( x s ) + a st , θ ,t ( x c ) , T u1 ( x u ) − T u1 ( a u ) (cid:1) . Figure 5. (C3)
For every t , there is a point X Q ∈ U Q in W u ( Q, f t ) (cid:116) W s ( P, f t ) (independentof t ) such that, in the coordinates in the chart U Q , it holds:(1) X Q = (0 s , , u ) and there is δ > such that X Q ⊂ I = { s } × [1 − δ, δ ] × { u } ⊂ W u ( Q, f t ) (cid:116) W s ( P, f t ) . (2) There is τ q,p ∈ N such that X P = f τ q,p t ( X Q ) = (0 , − , ∈ U P and X P ∈ J = f τ q,p t ( I ) = { s } × [ − − δ, − δ ] × { u } ⊂ U P . TABILIZATION OF HETERODIMENSIONAL CYCLES 13 (3) There is a neighborhood U X Q of X Q , U X Q ⊂ U Q , such that T ,t = T = f τ q,p t : U X Q → f τ q,p ( U X Q ) ⊂ U P is an affine map of the form T ( x s , x c , x u ) = (cid:0) T s2 ( x s ) , ± ( x c − , T u2 ( x u ) (cid:1) + (0 s , − , u )= (cid:0) T s2 ( x s ) , θ ( x c ) , T u2 ( x u ) (cid:1) . According to [8, Sections 3.1-2], we give the following definition.
Definition 4.2 (Simple cycles) . The map f in Proposition 4.1 has a simple cycle and ( f t ) t ∈ [ − (cid:15),(cid:15) ] is a model unfolding family of f . • T ,t and T are the unfolding and the transition maps , • θ ,t and θ are the central unfolding and the central transition maps, • τ p,q and τ q,p are the unfolding and the transition times, • λ and β are the central multipliers, and • φ λ ( x ) = λ x and ψ β ( x ) = β x are the linear central maps of the cycle. Remark 4.3.
Since we are only interested in the dynamics in the central di-rection of the simple cycle, we denote the simple cycle and its unfolding modelby sc ( f, Q, P, β, λ, ± , ± ), where the symbols ± and ± refer to the orientationpreservation or reversion of the maps T and T , respectively. These symbols coin-cide with the choices of ± in (C2)(4) and (C3)(3). To emphasize the unfolding andthe transition times τ p,q and τ q,p we will write sc ( f, Q, P, β, λ, ± , ± , τ p,q , τ q,p ).We now state some generalizations of the simple cycles above.4.1.1. Simple cycles with homoclinic intersections and semi-simple cycles.
In ourconstructions we will consider cycles associated to saddles with non-trivial homo-clinic classes. We want that some of these homoclinic intersections associated tothis saddle were “detected” by the cycle and “well posed” in relation to it. Thisleads to the next definition.
Definition 4.4 (Simple cycles with adapted homoclinic intersections) . Consider asimple cycle sc ( f, Q, P, β, λ, ± , ± ). Write f = f and let ( f t ) t ∈ [ − ε,ε ] be a modelunfolding family of f . The family ( f t ) t ∈ [ − ε,ε ] has adapted homoclinic intersections (associated to P ) if it satisfies conditions (C1)–(C3) in Proposition 4.1 and (C4) In the local coordinates in U Q , there is ¯ a s ∈ ( − , s such that∆ = { (¯ a s , } × [ − , u ⊂ W u ( P, f t ) , for every t close to 0.This implies that (¯ a s , ,
0) is a transverse homoclinic point of P of f t for all t closeto 0.The family ( f t ) t ∈ [ − ε,ε ] has a sequence of adapted homoclinic intersections (asso-ciated to P ) if it satisfies conditions (C1)–(C4) and (C5) In the local coordinates in U Q , for every t close to 0 there are sequences¯ a si → ¯ a s and x i → , ¯ a si ∈ ( − , s and x i ∈ (1 − δ, δ ) , such that ∆ i = { (¯ a si , x i ) } × [ − , u ⊂ W u ( P, f t ) for every t close to 0.Moreover, the orbits by f t of the disks ∆ i , i ≥
0, are pairwise disjoint.
As above, this implies that (¯ a si , x i ,
0) is a transverse homoclinic point of P of f t .In these cases, we say that f has a simple cycle with an adapted (sequence of )homoclinic intersection(s). Since we will consider perturbations of simple cycles, in some cases we will needto consider diffeomorphisms with “simple cycles” such that the maps ψ β and φ λ inProposition 4.1 are not linear. Definition 4.5 (Semi-simple cycles) . A diffeomorphism f has a semi-simple cycle associated to saddles P and Q if it satisfies Proposition 4.1 where the linear centralmaps φ λ and ψ β in (C1) are replaced by maps ˜ φ λ , ˜ ψ β : R → R with˜ φ λ (0) = ˜ ψ β (0) = 0 , ˜ φ (cid:48) λ (0) = λ, ˜ ψ (cid:48) β (0) = β. For such a semi-simple cycle we use the notation ssc ( f, Q, P, ˜ ψ β , ˜ φ λ , ± , ± ).4.2. Twisted and non-twisted cycles.
To a simple cycle sc ( f, Q, P, β, λ, ± , ± )we associate signs sign( Q ), sign( P ), and sign( T ) in { + , −} by the following rules: • sign( Q ) = + if β > Q ) = − if β < • sign( P ) = + if λ > P ) = − if λ <
0, and • sign( T ) = + if ± = + (i.e., θ , ( x c ) = x c ) and sign( T ) = − if ± = − (i.e., θ , ( x c ) = − x c ). Definition 4.6 (Twisted and non-twisted cycles) . We say that a simple cycle sc ( f, Q, P, β, λ, ± , ± ) is twisted if (sign( Q ) , sign( P ) , sign( T )) = (+ , + , − ). Oth-erwise the cycle is non-twisted. A diffeomorphism f with a co-index one cycle with real central multipliers (as-sociated to P and Q ) is twisted (resp. non-twisted ) if there is a diffeomorphism h arbitrarily C -close to f with a twisted (resp. non-twisted) simple cycle associatedto P and Q .Next lemma means that after a perturbation non-twisted cycles can be chosensatisfying (sign( Q ) , sign( P ) , sign( T )) = ( ± , ± , +) (i.e., the case ( − , − , − ) can bediscarded). Lemma 4.7.
Consider a non-twisted simple cycle sc ( f, Q, P, β, λ, ± , ± ) . Thenthere is a diffeomorphism g arbitrarily close to f with a simple cycle associated to P and Q of such that (sign( Q ) , sign( P ) , sign( T g )) = ( ± , ± , +) . This notation emphasizes that T g is the unfolding map of the cycle associated to g .Proof. If sign( T ) = + we are done. If sign( T ) = − then the definition of non-twisted cycle implies that at least one of the central multipliers λ and β of thecycle is negative. To prove the lemma we fix a constant K >
K > | β | and K − < | λ | ) and replace the unfolding map T , by a composition of the form( Df π ( Q ) ) m ◦ T , ◦ ( Df π ( P ) ) n , where n and m are arbitrarily large and λ n β m < K − < | λ n β m | < K. In this way, we get a new “unfolding map” ¯ T , = f m ◦ T , ◦ f n , defined on a smallneighborhood of f − n ( Y P ), where Y P ∈ W s ( Q, f ) ∩ W u ( P, f ) is the heteroclinic
TABILIZATION OF HETERODIMENSIONAL CYCLES 15 point in (C2) in Proposition 4.1. By construction, the central component ¯ θ , of¯ T , satisfies ¯ θ , ( x c ) = − λ n β m x c = | λ n β m | x c . Consider now the segment of orbit { f − n ( Y P ) , . . . , Y P , . . . , f τ p,q ( Y P ) , . . . , f τ p,q + m ( Y P ) } . Since n and m are arbitrarily big and K − < | λ n β m | < K , we can modify the map f along this segment of orbit to get ¯ θ , ( x c ) = x c . This perturbation can be takenarbitrarily small if n and m are arbitrarily large. Therefore the new simple cycle isof type ( ± , ± , +). This completes the sketch of the proof of the lemma. For furtherdetails see [8, Proposition 3.5]. (cid:3) Quotient dynamics. Families of iterated function systems.
In what fol-lows, ( f t ) t ∈ [ − (cid:15),(cid:15) ] is a model unfolding family associated to a diffeomorphism f = f with a semi-simple cycle. We use the notation in Proposition 4.1. Next remarkallows us to consider (in a neighborhood of a semi-simple cycle) the quotient dy-namics by the strong stable/unstable hyperplanes. Remark 4.8.
Consider a semi-simple cycle ssc ( f, Q, P, ˜ ψ β , ˜ φ λ , ± , ± ) and its mo-del unfolding map ( f t ) t ∈ [ − ε,ε ] , where f = f . Consider the partially hyperbolicsplitting E ss ⊕ E c ⊕ E uu , defined over the orbits of P and Q , that in the localcharts U P and U Q is of the form E ss = R s × { (0 , u ) } , E c = { s } × R × { u } , E uu = { (0 s , } × R u . This splitting is extended to U P ∪ U Q as constant bundles. Proposition 4.1 impliesthat the maps T ,t and T are affine maps preserving E ss ⊕ E c ⊕ E uu .The open set V defined by(4.1) V = U P ∪ U Q ∪ (cid:32) τ q,p (cid:91) i =0 f i ( U X Q ) (cid:33) ∪ (cid:32) τ p,q (cid:91) i =0 f i ( U Y P ) (cid:33) is the neighborhood associated to the cycle. For small t , we consider the maximalinvariant set Λ t ( V ) of f t in V , Λ t ( V ) = (cid:92) i ∈ Z f it ( V ) . By construction, for f t there is a partially hyperbolic extension of the splitting E ss ⊕ E c ⊕ E uu over the set Λ t ( V ). With a slight abuse of notation, we also denotethis extension by E ss ⊕ E c ⊕ E uu .This remark implies that the returns of points X ∈ U X Q ∩ Λ t ( V ) to U X Q , X ∈ U X Q ∩ Λ t ( V ) (cid:55)→ f it ( X ) ∈ U X Q , preserve the codimension one foliation R s × { x c } × R u tangent to E ss ⊕ E uu . Weconsider the “quotient dynamics” by these hyperplanes, obtaining a one parameterfamily of iterated function systems (IFS) defined on the interval I = [1 − δ, δ ] (seeitem (1) in (C3) in Proposition 4.1). This family describes the “central” dynamicsof these returns. We will provide in Proposition 4.9 a “dictionary” translatingproperties of this IFS to properties of the diffeomorphisms f t . These properties areabout the existence of periodic orbits, homoclinic and heteroclinic intersections,and cycles. Families of IFSs induced by the quotient dynamics.
Consider a semi-simplecycle ssc ( f, Q, P, ψ β , φ λ , ± , ± ) and its model unfolding family ( f t ) t ∈ [ − (cid:15),(cid:15) ] , here f = f . Consider the segment I in condition (C3)(1) in Proposition 4.1. For eachpair ( k, n ) of large natural numbers and small t , define the map(4.2) Γ k,nt : I k,nt → I , Γ k,nt ( x ) = ( ψ kβ ◦ θ ,t ◦ φ nλ ◦ θ )( x ) , where I k,nt is the maximal subinterval of I where the map Γ k,nt is defined. Note thatthere are choices of k, n, t such that the set I k,nt is empty.The one-parameter family (Γ k,nt ) t ∈ [ − (cid:15),(cid:15) ] is the IFS associated to ( f t ) t ∈ [ − (cid:15),(cid:15) ] .4.3.2. Dictionary IFS – Global dynamics.
Using the invariance of the spitting E ss ⊕ E c ⊕ E uu above one gets the following extension of [8, Proposition 3.8]: Proposition 4.9 (Quotient dynamics – Global dynamics) . Consider a semi-simplecycle ssc ( f, Q, P, ψ β , φ λ , ± , ± , τ p,q , τ q,p ) , its model unfolding family ( f t ) t ∈ [ − ε,ε ] ,here f = f , and its associated IFS (Γ n,mt ) t ∈ [ − (cid:15),(cid:15) ] . Suppose that the saddles P and Q have s -indices ( s + 1) and s , respectively. (A) Periodic points: Suppose that there is r ∈ I k,nt such that Γ k,nt ( r ) = r. Then there are r s ∈ R s and r u ∈ R u such that R = ( r s , r, r u ) ∈ U Q ∩ Λ t ( V ) is a periodic point of f t of period π ( R ) = k π ( Q ) + n π ( P ) + τ p,q + τ q,p . The eigenvalue of Df π ( R ) t ( R ) corresponding to central direction { s } × R × { u } is (cid:16) Γ k,nt (cid:17) (cid:48) ( r ) = (cid:16) ψ kβ (cid:17) (cid:48) (cid:0) θ ,t ( φ nλ ( θ ( r ))) (cid:1) (cid:16) φ nλ (cid:17) (cid:48) (cid:0) θ ( r ) (cid:1) . In particular, if (cid:12)(cid:12)(cid:0) Γ k,nt (cid:1) (cid:48) ( r ) (cid:12)(cid:12) > (resp. < ) the periodic point R has s -index s (resp. s -index s + 1 ).Moreover, the periodic point R also satisfies (4.3) W ss ( R, f t ) (cid:116) W u ( Q, f t ) (cid:54) = ∅ and W uu ( R, f t ) (cid:116) W s ( P, f t ) (cid:54) = ∅ . In what follows, let r , R , and ( k, n ) be as in item (A). (B) Strong homoclinic intersections: Suppose that there is a pair (¯ k, ¯ n ) (cid:54) =( k, n ) such that Γ ¯ k, ¯ nt ( r ) = r. Then W ss ( R, f t ) ∩ W uu ( R, f t ) contains points that do not belong to the orbit of R . (C) Heterodimensional cycles: Suppose that there are d ∈ I and d s ∈ R s suchthat (in the coordinates in U Q ) Υ = Υ( d s , d ) = { ( d s , d ) } × [ − , u ⊂ W u ( P, f t ) . If there is i ∈ N such that θ ,t ◦ φ iλ ◦ θ ( d ) = 0 then W u ( P, f t ) ∩ W s ( Q, f t ) (cid:54) = ∅ . TABILIZATION OF HETERODIMENSIONAL CYCLES 17
Thus, as W s ( P, f t ) ∩ W u ( Q, f t ) (cid:54) = ∅ , the diffeomorphism f t has a heterodimensionalcycle associated to P and Q .In particular, if there are i, h ∈ N such that θ ,t ◦ φ iλ ◦ θ ◦ ψ hβ ( t ) = θ ,t ◦ φ iλ ◦ θ ◦ ψ hβ ◦ θ ,t (0) = 0 then f t has a heterodimensional cycle associated to P and Q . (D) Heteroclinic intersections (I): Suppose that there are i, ˜ k, ˜ n ∈ N such that θ ,t ◦ φ iλ ◦ θ ◦ Γ ˜ k, ˜ nt ( r ) = 0 . Then W uu ( R, f t ) ∩ W s ( Q, f t ) (cid:54) = ∅ . If (˜ k, ˜ n ) = (0 , the previous identity just means θ ,t ◦ φ iλ ◦ θ ( r ) = 0 . (E) Heteroclinic intersections (II): Let ( d s , d ) be as in item (C) (i.e., Υ( d s , d ) ⊂ W u ( P, f t ) ). If there are i, j ∈ N such that Γ i,jt ( d ) = r then W u ( P, f t ) ∩ W ss ( R, f t ) (cid:54) = 0 . In particular, if(1) either r = d and ( i, j ) = (0 , ,(2) or there is i such that ψ iβ ◦ θ ,t (0) = ψ iβ ( t ) = r then W u ( P, f t ) ∩ W ss ( R, f t ) (cid:54) = ∅ . (F) Homoclinic points: Suppose that there is i such that ψ iβ ◦ θ ,t (0) = ψ iβ ( t ) = ˆ h ∈ [1 − δ, δ ] . Then there is ˆ h s ∈ ( − , s such that ˆ H = (ˆ h s , ˆ h, u ) ∈ U Q is a transverse homo-clinic point of P for f t and { (ˆ h s , ˆ h ) } × [ − , u ⊂ W u ( P, f t ) . Proof.
For notational simplicity, let us assume that P and Q are fixed points.Items (A) and (B) are stated in [8, Proposition 3.8]. To prove item (A) it isenough to observe that the definition of the pair ( k, n ) and the product structureprovide a pair of cubes ∆ u ⊂ [ − , u and ∆ s ⊂ [ − , s such that f (cid:96)t (cid:0) [ − , s × { r } × ∆ u (cid:1) = ∆ s × { r } × [ − , u , (cid:96) = k + n + τ p,q + τ q,p , if k and n are large enough (note that k, n → ∞ as t → Df (cid:96)t uniformlycontracts vectors parallel to R s × { (0 , u ) } and uniformly expands vectors parallelto { (0 s , } × R u . This gives the periodic point R = ( r s , r, r u ) of period (cid:96) . Notethat our arguments also imply that(4.4) W uu ( R, f t ) ⊃ { ( r s , r ) } × [ − , u , W ss ( R, f t ) ⊃ [ − , s × { ( r, r u ) } . In the previous expression one implicitly assumes that ψ hβ ( t ) ∈ [1 − δ, δ ], otherwise onecannot apply θ . Note also that from (C3)(1) in Proposition 4.1, in the coordinates in U Q , one hasthat { s } × [1 − δ, δ ] × [ − , u ⊂ W u ( Q, f t ) , and[ − , s × [1 − δ, δ ] × { u } ⊂ W s ( P, f t ) . (4.5)The intersection properties between the invariant manifolds of R, P, and Q in item(A) follow immediately from equations (4.4) and (4.5) and r ∈ [1 − δ, δ ].To prove item (B) one argues exactly as in item (A). Note that the choice of(¯ k, ¯ n ) (large ¯ k, ¯ n ) provides a cube ˜∆ u ⊂ [ − , u and a point ˜ r s ∈ [ − , s such that f mt (cid:0) { ( r s , r ) } × ˜∆ u (cid:1) = { (˜ r s , r ) } × [ − , u , m = ¯ k + ¯ n + τ p,q + τ q,p . Since { ( r s , r ) } × ˜∆ u ⊂ W uu ( R, f t ) and (˜ r s , r, r u ) ∈ W ss ( R, f t ) there is a stronghomoclinic intersection associated to R .To prove the first part of item (C) note that if t is small then i is large and thus f τ p,q + i + τ q,p t (Υ) ∩ U Q = f τ p,q + i + τ q,p t (cid:0) { ( d s , d ) } × [ − , u (cid:1) ∩ U Q ⊃ { ( ¯ d s , θ ,t ◦ φ iλ ◦ θ ( d )) } × [ − , u = { ( ¯ d s , } × [ − , u , for some ¯ d s ∈ ( − , s . Since [ − , s × { (0 , u ) } ⊂ W s ( Q, f t ) and Υ ⊂ W u ( P, f t )we get W u ( P, f t ) ∩ W s ( Q, f t ) (cid:54) = ∅ .To prove the second part of item (C) consider a s ∈ R s and the linear map B s asin (C2)(3) and (C1) in Proposition 4.1, respectively. Note that (cid:0) ( B s ) h ( a s ) , ψ hβ ( t ) , u (cid:1) = ( ˜ d s , ˜ d, u ) , ˜ d = ψ hβ ( t ) = ψ hβ ◦ θ ,t (0) ∈ [1 − δ, δ ]is a transverse homoclinic point of P such that˜Υ = { ( ˜ d s , ˜ d ) } × [ − , u ⊂ W u ( P, f t ) ∩ U Q . The intersection between W u ( P, f t ) and W s ( Q, f t ) now follows applying the firstpart of item (C) to the disk ˜Υ: just note that by hypothesis and the definition of˜ d = ψ hβ ◦ θ ,t (0) one has θ ,t ◦ φ iλ ◦ θ ( ˜ d ) = 0.Item (D) follows similarly. Let (in the coordinates in U Q )∆ = { ( r s , r ) } × [ − , u ⊂ W uu ( R, f t ) . In the coordinates in U Q , we have f τ p,q + i + τ q,p +˜ n + τ p,q +˜ k + τ q,p t (∆) ⊃ { (˜ r s , θ ,t ◦ φ iλ ◦ θ ◦ Γ ˜ n, ˜ kt ( r )) } × [ − , u = { (˜ r s , } × [ − , u , for some ˜ r s . As [ − , s × { (0 , u ) } ⊂ W s ( Q, f t ) we get W uu ( R, f t ) ∩ W s ( Q, f t ) (cid:54) = ∅ .The remainder assertions (E) and (F) in the proposition follow analogously, sowe omit their proofs. (cid:3) TABILIZATION OF HETERODIMENSIONAL CYCLES 19 Simple non-twisted cycles
In this section we first consider non-twisted cycles and explain how these cyclesyield partially hyperbolic saddle-node/flip points with strong homoclinic intersec-tions as well as further intersection properties, see Proposition 5.3. Using Proposi-tion 4.9 we will write these properties in terms of the IFSs associated to the cycle.We also see how these intersections are realized by perturbations (model families)of the initial cycle. These intersection properties are the main ingredient for thestabilization of cycles. Finally, in Section 5.3 we consider cycles involving a saddlewith a non-trivial homoclinic class and introduce the bi-accumulation property.5.1.
Non-twisted simple cycles with adapted homoclinic intersections.
The first step is to see that non-twisted simple cycles yield simple cycles withadapted homoclinic intersections.
Lemma 5.1.
Consider a non-twisted cycle sc ( f, Q, P, β, λ, ± , ± ) . There is g arbitrarily C -close to f having a non-twisted simple cycle (associated to Q and P )with a sequence of adapted homoclinic intersections (associated to P ).Proof. Note that by Lemma 4.7 we can assume that θ ,t ( x ) = x + t . The proof hastwo steps. We first perturb the cycle to get a cycle with one adapted homoclinic in-tersection. In the second step we perturb this new cycle with an adapted homoclinicintersection to get a cycle with a sequence of adapted homoclinic intersections. A cycle with one adapted homoclinic intersection.
Observe that, after an arbitrarilysmall perturbation, we can assume that the central multipliers of the cycle satisfy λ k = β − m > k and m . We fix small t k > t k = λ k = β − m . This choice gives ψ mβ (cid:0) θ ,t k (0) (cid:1) = ψ mβ ( t k ) = 1 . Therefore, by (F) in Proposition 4.9, the point H = ( h s , , ∈ U Q is a transversehomoclinic point of P such that { ( h s , } × [ − , u ⊂ W u ( P, f t k ) . The point H will provide the adapted homoclinic point in Definition 4.4.To see that f t k has a cycle associated to P and Q just note that(5.2) θ ,t k ◦ φ kλ ◦ θ ◦ ψ mβ ◦ θ ,t k (0) = θ ,t k ◦ φ kλ ◦ θ (1) = − λ k + t k = 0 . Item (C) in Proposition 4.9 implies that W u ( P, f t k ) ∩ W s ( Q, f t k ) (cid:54) = ∅ .Let ˜ Y P be the heteroclinic point in W u ( P, f t k ) ∩ W s ( Q, f t k ) corresponding tothe condition in (5.2). This implies that f t k has a cycle associated to P and Q and that the points X Q ∈ W s ( P, f t k ) ∩ W u ( Q, f t k ) (in condition (C3)(1)) and˜ Y P ∈ W u ( P, f t k ) ∩ W s ( Q, f t k ) are heteroclinic points associated to this cycle. Usingthe transverse homoclinic point H of P and arguing as in Lemma 4.7, we will geta cycle with an adapted homoclinic intersection.Indeed, repeating the previous argument we can assume that the cycle has two“adapted homoclinic points”. The additional one is of the form V = ( v s , v, v u ),where 1 + v ∈ [1 − δ, δ ] (in principle v (cid:54) = 0) and ∆ V = { ( v s , v ) } × [ − , u ⊂ W u ( P, f t ). We also have that the disks ∆ V and ∆ H = { ( h s , } × [ − , u ⊂ W u ( P, f t ) have disjoint orbits. We use the disk ∆ V to get the sequence of adaptedhomoclinic intersections. A cycle with a sequence of adapted homoclinic intersections.
To get a cycle with asequence of adapted homoclinic intersections we argue as above, but now startingwith a cycle with “two adapted homoclinic intersections”, say H and V as above.Let us assume that θ (1+ x ) = ( − x ). The case θ (1+ x ) = ( − − x ) is analogous.As above we can assume that equation (5.1) holds for infinitely many m and k .To get a sequence of homoclinic points H i accumulating to H write δ i = β m λ i (1 − v ) > ψ mβ ◦ θ ,t k ◦ φ iλ ◦ θ (1 + v ) = ψ mβ (cid:0) t k − λ i (1 − v ) (cid:1) = 1 − β m λ i (1 − v ) = 1 − δ i . Item (F) in Proposition 4.9 implies that for each i there is h si such that(5.3) H i = ( h si , − δ i , ∈ U Q , δ i > , is a transverse homoclinic point of P and∆ i = { ( h si , − δ i ) } × [ − , u ⊂ W u ( P, f t k ) . This sequence accumulates to ∆ H and the disks ∆ i and ∆ H have disjoint orbits byconstruction.Finally, arguing exactly as above we have that f t k has a heterodimensional cycleassociated to P and Q .Write ˜ f = f t k . We perturb ˜ f to get a simple cycle with a sequence of adaptedhomoclinic intersections. Note that ˜ f preserves the partially hyperbolic splitting E ss ⊕ E c ⊕ E uu in the neighborhood V of the initial simple cycle (recall (4.1)). Forthis new cycle we have “transition maps” say ˜ T ,t k and ˜ T (in principle, these mapsdo not satisfy all the properties of “true” transitions). These new “transitions” ˜ T ,t k and ˜ T are obtained considering compositions of the maps T ,t k , T , Df π ( P ) ( P ),and Df π ( Q ) ( Q ) defined for the initial cycle and replacing the heteroclinic points X Q and ˜ Y P by some backward iterates of them. Note that the central maps ˜ θ , and ˜ θ associated to the “new transitions” may fail to be isometries.Now, exactly as in the proof of Lemma 4.7, we consider an arbitrarily small per-turbation of ˜ f obtained taking multiplications (in the central direction) by numbersclose to one throughout long segments of the orbits of X Q and ˜ Y P . This is possiblesince t k can be taken arbitrarily small and k and m arbitrarily big. The resultingdiffeomorphism has a simple cycle with a sequence of adapted homoclinic intersec-tions associated to P (obtained considering appropriate iterations of the points H i and H ). This completes the proof of the lemma. (cid:3) Remark 5.2.
Using equation (5.3), we can assume that in the coordinates in U Q ,the adapted transverse homoclinic points of P are such that H = ( h s , , u ) and { ( h s , } × [ − , u ⊂ W u ( P, f ) ,H i = ( h si , ζ i , u ) and { ( h si , ζ i ) } × [ − , u ⊂ W u ( P, f ) , where ( ζ i ) is an increasing sequence converging to 1. TABILIZATION OF HETERODIMENSIONAL CYCLES 21
Dynamics generated by non-twisted cycles.
Consider a diffeomorphism f with a simple cycle and its associated neighborhood V in (4.1). For g close to f letΛ g ( V ) = ∩ i ∈ Z g i ( V ) be the maximal invariant set of g in V . Note that the set Λ g ( V )has a partially hyperbolic splitting of the form E ss g ⊕ E c g ⊕ E uu g , where E c g is one-dimensional and E ss g and E uu uniformly contracting and expanding, respectively. Proposition 5.3.
Consider a non-twisted cycle sc ( f, Q, P, β, λ, + , ± ) with a se-quence of adapted homoclinic intersections (associated to P ). Then there is a dif-feomorphism g arbitrarily C -close to f with a partially hyperbolic saddle-node/flip S g ∈ Λ g ( V ) of arbitrarily large period satisfying the following properties:(1) W ss ( S g , g ) (cid:116) W u ( Q, g ) (cid:54) = ∅ ,(2) W uu ( S g , g ) (cid:116) W s ( P, g ) (cid:54) = ∅ ,(3) W u ( P, g ) ∩ W ss ( S g , g ) (cid:54) = ∅ ,(4) W u ( P, g ) ∩ W s ( Q, g ) (cid:54) = ∅ and this intersection is quasi-transverse, and(5) the homoclinic class of P for g is non-trivial. Remark 5.4.
Indeed, the proof of this proposition will imply that the strongunstable manifold of S g transversely intersects the disk [ − , s × I ×{ u } containedin W s ( P, g ) in (C3)-(1) in Proposition 4.1. Now item (3) in Proposition 5.3 impliesthat W u ( P, g ) accumulates to W uu ( S g , g ) (may be after a perturbation). Thus aftera perturbation we can assume that W u ( P, g ) (cid:116) (cid:0) [ − , s × I × { u } (cid:1) (cid:54) = ∅ .5.2.1. Proof of Proposition 5.3.
The main step in the proof of the proposition isthe next lemma about the IFS associated to a simple cycle.
Lemma 5.5.
Consider a non-twisted cycle sc ( f, Q, P, β, λ, + , ± ) with an increas-ing sequence of adapted homoclinic intersections ( h si , ζ i , u ) as in Remark 5.2.Then there are sequences of parameters ( t i ) i , t i → , and of perturbations ψ β,i of ψ β ( x ) = β x , ψ β,i → ψ β , such that the IFS ˜Γ n,kt i associated to φ λ , ψ β,i , θ ,t i , and θ in equation (4.2) satisfies the following properties:(1) There is a sequence of pairs ( v i , w i ) , v i , w i → ∞ , such that ˜Γ v i ,w i t i (1) = 1 ,λ − | λ | ) < | (˜Γ v i ,w i t i ) (cid:48) (1)) | < | λ | − | λ | . (2) There are large j and (cid:96) ∈ N such that θ ,t i ◦ φ (cid:96)λ ◦ θ ( ζ j ) = 0 . (3) There are j ∈ { j − , j + 1 } ( j as in item (2)) and ¯ n, ¯ (cid:96) ∈ N such that Γ ¯ n, ¯ (cid:96)t i ( ζ j ) = 1 . We postpone the proof of this lemma to the next subsection.
Proof of Proposition 5.3.
Note that for each t i there is a perturbation f i of f , f i → f as i → ∞ , having a semi-simple cycle ssc ( f i , Q, P, ψ β,i , λ, + , ± ) “close”to the initial cycle sc ( f, Q, P, β, λ, + , ± ) (i.e., we replace the linear map ψ β by itsperturbation ψ β,i , while preserving the cycle configuration).For large i , write g = f i and select the pair ( v i , w i ) in item (1) of Lemma 5.5.Let S g = ( s s , , s u ) be the saddle associated to this pair and the central coordinate “1” given by (A) in Proposition 4.9. By construction, the eigenvalue λ c ( S g ) of Dg π ( S g ) ( S g ) corresponding to the central direction E c g satisfies | λ | − | λ | ) < | λ c ( S g ) | < | λ | − | λ | . We claim that S g also satisfies the intersection properties in the proposition (notethat in principle S g is not yet a saddle-node/flip). • Itens (1) and (2) in the proposition follow from equation (4.3) in item (A)of Proposition 4.9. • Item (3) in the proposition follows from (3) in Lemma 5.5 and (E) in Propo-sition 4.9, where d = ζ j ± corresponds to adapted homoclinic points (recallalso Remark 5.2). Note that using these points we also get that W u ( P, g )transversely intersects [ − , s × I × { u } , proving Remark 5.4. • Item (4) in the proposition follows from (2) in Lemma 5.5 and (C) in Propo-sition 4.9, where d = ζ j corresponds to an adapted homoclinic point. • Since transverse homoclinic intersections persist and the saddle P has trans-verse homoclinic points for the diffeomorphism f , we get (5) in the propo-sition.It remains to see that we can take S g with λ c ( S g ) = ±
1. Observe that theperiod π ( S g ) of S g can be taken arbitrarily large and | λ c ( S g ) | is uniformly bounded(independent of the period). Arguing as in Lemma 4.7, we perturb g along the orbitof S g in order to transform this point into a saddle-node (if λ c ( S g ) >
0) or a flip(if λ c ( S g ) < (cid:3) Proof of Lemma 5.5.
Let us first consider the case β > λ > Positive central multipliers:
As above, after an arbitrarily small perturbation of thecentral multipliers of cycle, we can assume that there are arbitrarily large m and k with(5.4) β − m = λ k (1 − λ ) . Consider the parameter t k = λ k . This choice givesΓ m,k +1 t k (1) = ψ mβ ◦ θ ,t k ◦ φ k +1 λ ◦ θ (1) = ψ mβ ◦ θ ,t k ( − λ k +1 )= β m ( λ k − λ k +1 ) = β m λ k (1 − λ ) = 1 . Take ( v k , w k ) = ( m, k + 1) and note that(Γ v k ,w k t k ) (cid:48) (1) = ± β m λ k +1 = ± λ − λ . This gives (1) in the lemma. To obtain the other conditions we consider pertur-bations ˜ ψ β of ψ β preserving the condition Γ v k ,w k t k (1) = 1. From now on we fix theparameter t k . We first consider the case where θ has derivative +1. Case θ (1 + x ) = − x : For every small enough µ , define β ( µ ) by β ( µ ) m ( λ k (1 − λ ) + µ ) = 1 TABILIZATION OF HETERODIMENSIONAL CYCLES 23 and consider its associated linear map ψ β ( µ ) ( x ) = β ( µ ) x . Write φ λ ( x ) = λ x . Notethat the IFS ˜Γ i,jt k + µ associated to φ λ , ψ β ( µ ) , θ t k + µ , and θ satisfies(5.5) ˜Γ m,k +1 t k + µ (1) = ψ mβ ( µ ) ◦ θ ,t k + µ ◦ φ k +1 λ ◦ θ (1) = 1 , for all small µ .Thus, for ( v k , w k ) = ( m, k + 1),(5.6) (˜Γ v k ,w k t k + µ ) (cid:48) (1) = β ( µ ) m λ k +1 = λ − λ + µ . Thus, for small µ , these derivatives also satisfy (1).Consider ζ i as in Remark 5.2, that is ζ i = 1 − δ i , δ i → + and δ i > δ i +1 . Forlarge i define(5.7) ω i ( µ ) = θ ,t k + µ ◦ φ kλ ◦ θ ( ζ i ) = θ ,t k + µ ( − λ k − λ k δ i ) = µ − λ k δ i . Note that(5.8) ω i +1 ( µ ) − ω i ( µ ) = λ k ( δ i − δ i +1 ) . Define small µ j > ω j ( µ j ) = 0 , µ j = λ k δ j , lim j →∞ µ j → . By the choice of µ j and (5.8) one has ω j +1 ( µ j ) = λ k ( δ j − δ j +1 ) , lim j →∞ ω j +1 ( µ j ) → + . In particular, ω j +1 ( µ j ) can be taken arbitrarily small in comparison with β ( µ j ) − m = λ k (1 − λ ) + µ j . This immediately implies the following: Fact 5.6.
Given any
N > there is large j such that [ ω j +1 ( µ j ) , β ( µ j ) − m ] containsat least N consecutive fundamental domains of ψ β ( µ j ) . Using this fact, we get that for every large j there is a small perturbation ˜ ψ β ( µ j ) of the linear map ψ β ( µ j ) such that: • ˜ ψ β ( µ j ) ( x ) = ψ β ( µ j ) ( x ) if x ∈ [ β ( µ j ) − m − , • There is large n j such that ˜ ψ n j β ( µ j ) ( ω j +1 ( µ j )) = β ( µ j ) − m . • The maps ˜ ψ β ( µ j ) and ψ β ( µ j ) coincide in a small neighborhood of 0. • The size of the perturbation goes to 0 as j → ∞ . Remark 5.7.
Note that the first two conditions above imply that(5.9) ˜ ψ n j + mβ ( µ j ) ( ω j +1 ( µ j )) = 1 . Also important, note that this perturbation can be done (and we do) in such a wayprevious conditions (5.5), (5.6), and (5.7) are preserved.The previous construction can be summarized as follows. Fix large k and thesequence of parameters t k,j = t k + µ j . For each large j , consider the perturbation˜ ψ β ( µ j ) of ψ β and the IFS ˜Γ (cid:96),nt k,j corresponding to ˜ ψ β ( µ j ) , φ λ , θ ,t k,j , and θ . Then (i) ˜Γ v k ,w k t k,j (1) = 1, (recall (5.5)), (ii) (˜Γ v k ,w k t k,j ) (cid:48) (1) = λ − λ + µ j , (recall (5.6)), (iii) θ ,t k,j ◦ φ kλ ◦ θ ( ζ j ) = ω j ( µ j ) = 0, (recall the choice of µ j and (5.7)), and (iv) ˜Γ n j + m,kt k,j ( ζ j +1 ) = ˜ ψ n j + mβ ( µ j ) ◦ θ ,t k,j ◦ φ kλ ◦ θ ( ζ j +1 ) = ˜ ψ n j + mβ ( µ j ) ( ω j +1 ( µ j )) = 1, (recall(5.9)).To conclude the proof the lemma in this first case (positive multipliers and θ (1 + x ) = − x ) just note that (i)–(ii) correspond to (1) in the lemma, (iii) to(2) in the lemma, and (iv) to (3) in the lemma. Case θ (1+ x ) = − − x : We proceed as in the previous case and define the sequence ω i ( µ ) similarly. In this case, instead equation (5.7) we get ω i ( µ ) = θ ,t k + µ ◦ φ kλ ◦ θ ( ζ i ) = θ ,t k + µ ( − λ k + λ k δ i ) = µ + λ k δ i . We define µ j as above, ω j ( µ j ) = 0, and consider ω j − ( µ j ) > ω j +1 ( µ j ).The proof now follows as above. Non-positive central multipliers:
In this case, after an arbitrarily small perturbationof the central multipliers of cycle, we can assume that there are arbitrarily large m and k with(5.10) β − m = λ k (1 − λ ) . We consider the parameter t k = λ k . The proof now follows exactly as in the casewhere the multipliers are both positive considering the sequences ω i ( µ ) = θ ,t k + µ ◦ φ kλ ◦ θ ( ζ i ) = θ ,t k + µ ( − λ k ± λ k δ i ) = µ ± λ k δ i . This completes the proof of Lemma 5.5. (cid:3)
Cycles associated to a bi-accumulated saddles.
Given a periodic point R of f , consider the eigenvalues λ ( R ) , . . . , λ n ( R ) of Df π ( R ) ( R ) ordered in increasingmodulus and counted with multiplicity. Denote by Per k ( f ) the set of (hyperbolic)saddles R of f of s-index k satisfying | λ k − ( R ) | < | λ k ( R ) | <
1. Given such a saddle R ∈ Per k ( f ), its local strong stable manifold W ssloc ( R, f ) is well defined (recall that W ss ( R, f ) is the unique invariant manifold tangent to the eigenspace associated to λ ( R ) , . . . , λ k − ( R )). Moreover, W ssloc ( R, f ) has codimension one in W sloc ( R, f ) and W sloc ( R, f ) \ W ssloc ( R, f ) has 2 π ( R ) connected components (indeed W ssloc ( R, f ) splitseach component of W sloc ( R, f ) into two parts).Given a saddle P of s-index s + 1, we consider the following subsets of H ( P, f ): • Per h ( H ( P, f )) is the subset of H ( P, f ) of hyperbolic periodic points R whichare homoclinically related to P (thus R also has index ( s + 1)), • Per s +1 h ( H ( P, f )) = Per h ( H ( P, f )) ∩ Per s +1 ( f ). Definition 5.8 (Bi-accumulation property) . A saddle R ∈ Per s +1 ( f ) is s -bi-accumulated (by homoclinic points) if every component of ( W sloc ( R, f ) \ W ssloc ( R, f ))contains transverse homoclinic points of R .We have the following result. Lemma 5.9.
Let f be a diffeomorphism with a coindex one cycle associated to P and Q such that H ( P, f ) is non-trivial. Let s -index( P ) = s + 1 . Then there is g arbitrarily C -close to f such that • there is a saddle ¯ P g ∈ Per s +1 h ( H ( P g , g )) that is s -bi-accumulated and • the diffeomorphism g has a cycle associated to ¯ P g and Q g . TABILIZATION OF HETERODIMENSIONAL CYCLES 25
Proof.
The lemma follows from [2, 13]. From [2, Proposition 2.3], if H ( P, f ) isnon-trivial then there is g arbitrarily C -close to f with a cycle associated to P g and Q g and such that Per s +1 h ( H ( P g , g )) is infinite.By [13, Lemma 3.4], if the set Per s +1 h ( H ( P, f )) is infinite then there is a diffeo-morphism g arbitrarily C -close to f with a cycle associated to P g and Q g and suchthat Per s +1 h ( H ( P g , g )) contains infinitely many s-bi-accumulated saddles. Pick oneof these saddles ¯ P g and note that to be bi-accumulated is a property that persistsunder perturbations. We can now perturb g to get h with a cycle associated to ¯ P h and Q h , ending the proof of the lemma. (cid:3) Stabilization of cycles. Proof of Theorem 3
Stabilization of non-twisted cycles.
Next proposition is the main step toprove the stabilization of non-twisted cycles.
Proposition 6.1.
Let f be a diffeomorphism with a non-twisted cycle associatedto saddles P and Q such that s -index( P ) = s -index( Q ) + 1 . Then there is a dif-feomorphism g arbitrarily C -close to f with a partially hyperbolic saddle-node/flip S g such that:(1) W ss ( S g , g ) (cid:116) W u ( Q g , g ) (cid:54) = ∅ ,(2) W uu ( S g , g ) (cid:116) W s ( P g , g ) (cid:54) = ∅ ,(3) W uu ( S g , g ) ∩ W ss ( S g , g ) contains a point that is not in the orbit of S g (stronghomoclinic intersection),(4) W ss ( S g , g ) ∩ W u ( P g , g ) (cid:54) = ∅ , and(5) W uu ( S g , g ) ∩ W s ( Q g , g ) (cid:54) = ∅ . The dynamical configuration in the proposition is depicted in Figure 6.We postpone the proof of this proposition to Section 6.1.1. We now prove (A)in Theorem 3.
Figure 6.
Dynamical configuration in Proposition 6.1.6.1.1.
Proposition 6.1 implies (A) in Theorem 3.
Note that the transverse inter-section conditions immediately imply that s -index( P ) = dim( W ss ( S )) + 1 = s + 1(condition (1) in Theorem 3.5). Moreover, conditions (2)–(4) in Proposition 6.1imply that S and P satisfy (2)–(4) in Theorem 3.5. Thus the diffeomorphism g satisfies all conditions in Theorem 3.5 and hence there is h arbitrarily C -closeto g having a robust heterodimensional cycle associated to P h and a (transitive)hyperbolic set Γ h containing a continuation S + h of s-index s of S g . Observe that items (1) and (5) in Proposition 6.1 imply that the saddle S + h of h can be chosen such that W s ( S + h , h ) (cid:116) W u ( Q h , h ) (cid:54) = ∅ and W u ( S + h , h ) ∩ W s ( Q h , h ) (cid:54) = ∅ . Thus the saddles S + h and Q h are homoclinically related and then there is a transitivehyperbolic set Σ h containing Q h and Γ h . In particular, for every diffeomorphism ϕ close to h it holds W s , u (Γ ϕ , ϕ ) ⊂ W s , u (Σ ϕ , ϕ ). Thus, by the first step of the proof,the diffeomorphism h has a robust cycle associated to Σ h and P h , ending the proofof (A) in Theorem 3. (cid:3) Proof of Proposition 6.1.
This proposition follows from Proposition 5.3. Firstnote that by Lemma 5.1, after a small perturbation, we can assume that the cy-cle (associated to P and Q ) has a sequence of adapted homoclinic intersectionsassociated to the saddle P . Thus applying Proposition 5.3 we obtain g close to f with a partially hyperbolic saddle-node/flip satisfying conditions (1), (2), and (4)in Proposition 6.1. It remains to obtain conditions (3) ( W uu ( S g , g ) ∩ W ss ( S g , g )contains a point that is not in the orbit of S g ) and (5) ( W uu ( S g , g ) ∩ W s ( P g , g ) (cid:54) = ∅ )in Proposition 6.1. To get these two properties we use arguments analogous to theones in Lemmas 5.1 and 5.5.Since in what follows we do not modify the orbits of P g , Q g , and S g let us omitthe dependence on g . Note that since W uu ( S, g ) (cid:116) W s ( P, g ) (condition (2) inProposition 5.3) we have that W uu ( S, g ) accumulate to W u ( P, g ). Since by condi-tion (4) in Proposition 5.3 we have that W u ( P, g ) ∩ W s ( Q, g ) (cid:54) = ∅ , thus W uu ( S, g )also accumulates to W s ( Q, g ). In particular there are segments of W uu ( S, g ) (withdisjoint orbits) arbitrarily close to W sloc ( Q, g ). We use one of these segments to get W uu ( S, h ) ∩ W s ( Q, h ) (cid:54) = ∅ for some h close to g (condition (5) in Proposition 6.1).Moreover, the previous perturbation can be done in such a way there are seg-ments of W uu ( S, h ) close to W s ( Q, h ) in the “same side” of W s ( Q, h ) as W ss ( S, h ).See Figures 7 and 8. Thus modifying the derivative of Q in the central directionwe get that W uu ( S, h ) intersects W ss ( S, h ) (condition (3) in Proposition 6.1). Notethat these perturbations can be done preserving the saddle-node/flip S and theintersections properties (1), (2), and (4) in Proposition 6.1. (cid:3) Figure 7.
Accumulation of W uu ( S ) to W s loc ( Q ). TABILIZATION OF HETERODIMENSIONAL CYCLES 27
Figure 8.
Accumulation of W uu ( S ) to W s loc ( Q ).6.2. Stabilization of bi-accumulated twisted cycles.
In this section we proveitem (B) in Theorem 3.
Proposition 6.2 (Generation of non-twisted cycles) . Let f be a diffeomorphismwith a twisted cycle associated to saddles P and Q with s -index( P ) = s -index( Q ) +1 . Assume that P is s -bi-accumulated. Then there is g arbitrarily C -close to f with a non-twisted cycle associated to Q g and a saddle R g that is homoclinicallyrelated to P g . Item (A) in Theorem 3 implies that the cycle associated to R g and Q g can bestabilized. Since R g is homoclinically related to P g , Lemma 2.3 implies that thecycle associated to P f and Q f can also be stabilized. Thus Proposition 6.2 implies(B) in Theorem 3.6.2.1. Proof of Proposition 6.2.
The proposition is an immediate consequence ofthe following two lemmas:
Lemma 6.3.
Under the hypotheses of Proposition 6.2, there is g arbitrarily C -close to f with a twisted simple cycle associated to P and Q and with an adaptedhomoclinic point of P . Lemma 6.4.
Consider a twisted cycle sc ( f, Q, P, β, λ, − , ± ) , λ, β > , with anadapted homoclinic intersection (associated to P ). Then there is g arbitrarily C -close to f with a saddle R g such that • R g is homoclinically related to P g and • g has a non-twisted cycle associated to R g and Q g . Proof of Lemma 6.3.
We claim that (in the coordinates in U Q in Proposi-tion 4.1) there are sequences of points ( x i ) i and ( a si ) i , x i ∈ R and a si ∈ R s , and ofdisks ∆ i of dimension u such that • ( a si , x i , ∈ ∆ i where x i → + and a si → a s , and • ∆ i → { ( a s , } × [ − , u and ∆ i ⊂ W u ( P, f ),here ( a s , , u ) is the heteroclinic intersection between W u ( P, f ) and W s ( Q, f ) in(C2) in Proposition 4.1.To see why this assertion is so just note that, by the bi-accumulation property,there is as sequence of unstable disks ˜∆ i ⊂ W u ( P, f ) of dimension u approachingto W uloc ( P, f ) from the “negative side”, see Figure 9. Since the cycle is twistedthe map T , reverses the ordering in the central direction. Thus these disks are Figure 9.
The disks ∆ i .mapped by T , into a disks ∆ i that approaches ( a s , , u ) from the “positive side”.See Figure 9. We need to perform a perturbation in order to put these disks in“vertical” position.Arguing exactly as in the proof of Lemma 5.5, after an arbitrarily small pertur-bation we can assume that β is such that ψ k i β ( x i ) = 1 for some arbitrarily large i and k i . This provides a transverse homoclinic point of P of the form ( h s , , P and Q .Finally, using this transverse homoclinic point and after an arbitrarily smallperturbation, we get the simple cycle with an adapted homoclinic intersection as-sociated to P and Q (the argument is similar to the one in Lemma 4.7.) (cid:3) Proof of Lemma 6.4.
The lemma follows arguing as in [8, Lemma 3.13] andusing Proposition 4.9. Note that we can assume (after a small modification of β and λ ) that β − m = λ k . Noting that the cycle is twisted (i.e., θ ,t ( x ) = t − x ) wehave that this equality implies thatΓ m,k (1) = ψ mβ ◦ θ , ◦ φ kλ ◦ θ (1) = ψ mβ (cid:0) − φ kλ ( − (cid:1) = 1 . In this case we also have,(Γ m,k ) (cid:48) (1) = ( ψ mβ ) (cid:48) (cid:0) − φ kλ ( − (cid:1) ( φ kλ ) (cid:48) ( −
1) = ± β m λ k = ± . Thus modifying the central derivatives at P and Q , we can assume that the cycleis semi-simple with central maps ˜ ψ β and ˜ φ λ such that there are large m, k, and (cid:96) ,with (cid:96) >> k , satisfying(6.1) ˜ ψ mβ (cid:0) − ˜ φ kλ ( −
1) + ˜ φ (cid:96)λ ( − (cid:1) = 1and(6.2) | (cid:0) ˜ ψ mβ (cid:1) (cid:48) (cid:0) − ˜ φ kλ ( −
1) + ˜ φ (cid:96)λ ( − (cid:1) (cid:0) ˜ φ kλ (cid:1) (cid:48) ( − | < . For that note that ˜ φ (cid:96)λ ( −
1) is arbitrarily small in comparison with ˜ φ kλ ( − t = ˜ φ (cid:96) ( − <
0. By equation (6.1) one has˜Γ m,kt (1) = ˜ ψ mβ ◦ θ ,t ◦ ˜ φ kλ ◦ θ (1) = ˜ ψ mβ ◦ θ ,t (cid:0) ˜ φ kλ ( − (cid:1) = ˜ ψ mβ (cid:0) − ˜ φ kλ ( −
1) + ˜ φ (cid:96)λ ( − (cid:1) = 1 . TABILIZATION OF HETERODIMENSIONAL CYCLES 29
Let R = ( r s , , r u ) ∈ U Q be the saddle of f t associated to 1 and the itinerary ( m, k )given by (A) in Proposition 4.9. Note that | (˜Γ m,kt ) (cid:48) (1) | = | (cid:0) ˜ ψ mβ (cid:1) (cid:48) (cid:0) θ ,t (cid:0) ˜ φ kλ ( θ (1)) (cid:1)(cid:1) (cid:0) ˜ φ kλ (cid:1) (cid:48) ( θ (1)) | = | (cid:0) ˜ ψ mβ (cid:1) (cid:48) (cid:0) ˜ φ kλ ( −
1) + ˜ φ (cid:96)λ ( − (cid:1) (cid:0) ˜ φ kλ (cid:1) (cid:48) ( − | < , where the inequality follows from (6.2). By (A) in Proposition 4.9 the saddle R hasindex s + 1. Indeed, since θ ,t ( x ) = − x + t , the central multiplier of R is positiveif θ reverses the orientation and negative otherwise.We claim that the saddle R is homoclinically related to P and has a cycle asso-ciated to Q . Note that W uu ( R, f t ) = W u ( R, f t ).By equation (4.3) in Proposition 4.9 we have that(6.3) W s ( R, f t ) (cid:116) W u ( Q, f t ) (cid:54) = ∅ and W u ( R, f t ) (cid:116) W s ( P, f t ) (cid:54) = ∅ . From the existence of an adapted homoclinic intersection and item (E)(1) inProposition 4.9: • H = ( h s , ,
0) is a transverse homoclinic point of P , • { ( h s , } × [ − , u ⊂ W u ( P, f t ) ∩ U Q , and • { [ − , s × { (1 , r u ) } ⊂ W ss ( R, f t ).This implies that W u ( P, f t ) (cid:116) W s ( R, f t ). Thus, by the second part of (6.3), thesaddles P and R are homoclinically related for f t .To get cycle associated to R and Q note that the choice of t implies that θ ,t ◦ ˜ φ (cid:96)λ ◦ θ (1) = − ˜ φ (cid:96)λ ( −
1) + t = 0 . Since R = ( r s , , r u ), condition (D) in Proposition 4.9 implies that W u ( R, f t ) ∩ W s ( Q, f t ) (cid:54) = ∅ . Thus by the first part of (6.3) the diffeomorphism f t has a cycleassociated to R and Q .We claim that this cycle is non-twisted. If θ reverses the orientation then thecentral multiplier of R is negative and the cycle is non-twisted. Otherwise, we havea cycle whose central “unfolding map” is obtained considering the composition θ ,t ◦ ˜ φ λ ◦ θ . This map preserves the central orientation: just note that θ ,t and θ both reverse the orientation and ˜ φ λ preserves this orientation (recall that λ > (cid:3) Proof of Theorems 1 and 2
Proof of Theorem 2.
Note that (B) in Theorem 2 is an immediate conse-quence from (A) in Theorem 3.To prove item (A) let us assume that, for instance, the saddle P has non-realcentral multipliers. By Theorem 2.1 (see also Remark 2.2) there is g close to f having saddles P (cid:48) g and Q (cid:48) g such that • there is a cycle with real central multipliers associated to P (cid:48) g and Q (cid:48) g , • P (cid:48) g and Q (cid:48) g are homoclinically related to P g and Q g , • the homoclinic class of P (cid:48) g is non-trivial (note that we may have Q (cid:48) g = Q g and a trivial homoclinic class H ( Q g , g )).By Lemma 2.3 it is enough to prove that this new cycle can be stabilized.If the cycle associated to P (cid:48) g and Q (cid:48) g is non-twisted the stabilization follows from(A) in Theorem 3. Otherwise, if the cycle is twisted, by Lemma 5.9 there is adiffeomorphism h close to g having a saddle ¯ P h such that • ¯ P h is homoclinically related to P (cid:48) h and has the bi-accumulation property, • there is a cycle associated to Q (cid:48) h and ¯ P h . Note that this cycle has realcentral multipliers.As above, it is enough to prove that this cycle can be stabilized. The stabilizationof this cycle follows from Theorem 3. This ends the proof of the theorem. (cid:3) Proof of Theorem 1.
By Theorem 2.1 and Lemma 2.3 we can assume thatthe cycle associated to the saddles P and Q has real central multipliers and that,for instance, the homoclinic class of P is non-trivial. If the cycle is non-twisted theresult follows from (A) in Theorem 3.Otherwise, if the cycle is twisted, arguing as in the proof of Theorem 2, thereis a diffeomorphism g close to f having a cycle associated to Q g and to a saddle¯ P g that is homoclinically related to P g and satisfies the s-bi-accumulation property.By (B) in Theorem 3 this cycle can be stabilized. Since ¯ P g is homoclinically relatedto P g the initial cycle also can be stabilized, ending the proof of the theorem. (cid:3) Proof of Corollary 1.
This result follows immediately from Theorem 1 con-sidering the following perturbation of the initial cycle. First, we preserve one of theheterocinic orbits in W u ( P, f ) ∩ W s ( Q, f ). We can also assume that W s ( P, f ) trans-versely intersects W u ( Q, f ) and thus accumulates to W s ( Q, f ). We can now usethe second heteroclinic orbit in W u ( P, f ) ∩ W s ( Q, f ) to get a transverse homoclinicpoint of P . In this way we obtain a cycle satisfying Theorem 1. Acknowledgments
The authors would like to express their gratitude to H. Kokubu, M. C. Li, andM. Tsujii for their hospitality and financial support during their visits to RIMS(Japan) and NCTU (Taiwan) where a substantial part of this paper was developed.This paper is also partially supported by CNPq, FAPERJ, and Pronex (Brazil),“Brazil-France Cooperation in Mathematics”The authors also thank S. Crovisier, K. Shinohara, and T. Soma for useful con-versations in this subject.
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