High dimension diffeomorphisms exhibiting infinitely many strange attractors

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www.elsevier.com/locate/anihpc

High dimension diffeomorphisms exhibiting infinitely many strange attractors

Bladismir Leal

Departamento de Matemáticas, Facultad de Ciencias, ULA, La Hechicera, Mérida 5101, Venezuela Received 21 March 2006; received in revised form 23 March 2007; accepted 23 March 2007

Available online 2 August 2007

Abstract

In this work we show, on a manifold of any dimension, that arbitrarily near any smooth diffeomorphism with a homoclinic tangency associated to a sectionally dissipative fixed or periodic point (i.e. the product of any pair of eigenvalues has norm less than 1), there exists a diffeomorphism exhibiting infinitely many Hénon-like strange attractors. In the two-dimensional case this has been proved in [E. Colli, Infinitely many coexisting strange attractors, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 539–579]. We also show that a parametric version of this result is true.

Résumé

Dans ce travail nous montrons, sur une variété de dimension quelconque, qu’arbitrairement près de chaque difféomorphisme possédant une tangence homocline, associée à un point fixe ou périodique sectionnelement dissipatif (le module du produit de deux valeurs propres quelconques est plus petit que 1) il existe un difféomorphisme qui possède une infinité d’attracteurs étranges du type Hénon. Dans le cas bidimensionnel ceci a été prouvé dans [E. Colli, Infinitely many coexisting strange attractors, Ann. Inst.

H. Poincaré Anal. Non Linéaire 15 (1998) 539–579]. Nous démontrons également une version paramétrique de ce résultat.

Keywords:Diffeomorphisms; Homoclinic tangency; Strange attractors

1. Introduction

The two-parameter Hénon family of transformations of the plane ha,b(x, y)=(1−ax²+y, bx)

was studied by Hénon [3] to show, via a numerical approach, how a simple model of an invertible dynamical system suggests the presence of a nonhyperbolic strange attractor. However, the possibility that the attractor observed by Hénon was just a periodic orbit with very high period could not be excluded. In a remarkable work Benedicks and

E-mail address:bladismir@ula.ve.

doi:10.1016/j.anihpc.2007.03.002

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Carleson [1] showed that this is not the case and they exhibited a positive Lebesgue measure subset of parameters (a, b)for which the maph_a,bhas a nonhyperbolic strange attractor.

An important application of Benedicks–Carleson’s methods [1] was done by Mora and Viana in [4] in the setting of homoclinic bifurcation on surfaces. More precisely, they showed that generic one-parameter families of surfaces diffeomorphisms unfolding a homoclinic tangency always include the presence, for a Lebesgue positive measure set of parameter values, of Hénon-like strange attractors or repellers.

The result in [4] was extended by Viana [15] to homoclinic bifurcations on manifolds of any dimension. Later on, Colli [2] showed that a diffeomorphism of surfaces having a homoclinic tangency can be approximated by diffeomorphisms exhibiting not only a strange attractor, but also by diffeomorphisms displaying infinitely many of such strange attractors.

Our purpose in the present work is to extend the existence of infinitely many strange attractors in [2] to higher dimensions sectionally dissipative homoclinic bifurcations. Our main result is as follows

Theorem A.Letϕ:M→Mbe a smooth diffeomorphism on any manifold with a homoclinic tangency associated to a sectionally dissipative point. Then, there exists an open set U ofDiff^∞(M)containingϕ in its closure, such that everyψ∈U can be approximated by a diffeomorphism exhibiting infinitely many nonhyperbolic strange attractors.

In the statement above,smoothmeans thatϕ:M→MisC^∞, M being an-dimensional manifold. We also recall that a homoclinic tangency is just a tangency between the stable and unstable manifolds of a saddle periodic point.

The saddle is called (codimension-one)sectionally or strongly dissipativeif it has just one expanding eigenvalue and the product of any two eigenvalues has norm less than one. As in [15], we defineattractorof a transformationϕto be a compact,ϕ-invariant and transitive setΛwhose basinW^s(Λ)= {z∈M: dist(ϕⁿ(z), Λ)→0 asn→ ∞}has nonempty interior. We call the attractor strangeif it contains a dense orbit{ϕⁿ(z₁): n0}displaying exponential growth of the derivative, that is,

Dϕⁿ(z1)e^cn for alln0 and somec >0.

We also obtain a one-parameter version of Theorem A. More precisely,

Theorem B.For a generic subset of smooth one-parameter families{ϕ_μ}of diffeomorphisms, on any manifold, that unfolds a homoclinic tangency at parameter value μ=0 associated to a sectionally dissipative fixed(or periodic) point, there exist sequencesI_n→0of intervals and dense subsetsE_n⊂I_nsuch that for allμ∈E_n, the corresponding mapϕ_μdisplays infinitely many nonhyperbolic strange attractors.

By smooth one-parameter family of diffeomorphismwe mean thatΦ:R×M→M,Φ(μ, x)=(μ, ϕ_μ(x))is a C^∞map andϕ_μis a diffeomorphism for allμ.

It is worth to point out that diffeomorphisms with the homoclinic tangencies are not only approximated by ones displaying the phenomenon described before but also by ones exhibiting different striking phenomena. For instance, it has been shown that homoclinic tangencies are approximated by Newhouse’s infinitely many sinks (attracting periodic orbits) [5,6] and cascades of period doubling bifurcation [16]. Still, it is conjectured that such an important phenomenon concerning infinitely many attractors might be rare, in parameter terms, for parameterized families of diffeomorphisms going through bifurcations of homoclinic tangencies: a conjecture in [7] and [8] states that for most parameter values, the corresponding diffeomorphisms display only finitely many attractors.

It is also worth to point out, that in the direction of proving the existence of infinitely many strange attractors, some particular results have been found. In 1990 [14], Gambaudo and Tresser constructed an example of aC²diffeomorphism in the two-dimensional disk exhibiting infinitely many hyperbolic strange attractors. In 2000 [11], Pumariño and Rodriguez exhibited a C^∞ family of vector fields in R³, related to a saddle-focus connection, which, with a positive Lebesgue measure set in the parameter values, displays infinitely many Henón-like strange attractors.

Among the difficulties to extend the result in [2] from two to higher dimensions, we have that projections along the invariant foliations (in our case unstable foliations) of a basic set may not have a much regular metric behavior:

in general, these projections are not Lipschitz but just Hölder continuous. We follow some ideas presented in [10]

to bypass these difficulties and also to obtain further estimates necessary to prove Theorems A and B. On the other

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hand, to construct strange attractors we need to display a high dimensional renormalization scheme for heteroclinic tangencies in 2-cycles and then apply results in [15].

This work is organized as follows. In Section 2, we review the construction used to prove that infinitely many coexisting attracting periodic orbits for diffeomorphisms in high dimensions as presented in [10]. We take special care with the expansion and contraction rates of the basic sets involved. This chapter finishes with Theorem 1.1, which summarizes the facts established in the previous sections. In Section 3, we prove some preliminary machinery to show the main theorems. In Section 3.1, we describe a higher dimension version of the renormalization scheme in 2-cycles of periodic points with a heteroclinic tangency considered in [2], following ideas in [9] and [15]. This renormalization scheme depends on a delicate relation between the contracting and expanding eigenvalue of periodic points involved. In Section 3.2, we give a brief summary of the main result in [15] and derive several consequences of its proof. In Section 3.3, we make a special perturbation for one-parameter families of diffeomorphisms to obtain new families which have linearizing coordinates in a neighborhood of the periodic points, as in Section 3.1. Such a perturbation is necessary since in the renormalization scheme of Section 3.1, we assume that there exist linearizing coordinates in a neighborhood of the periodic points. In Section 4, we prove Theorems A and B. The proofs are consequence of the a main lemma showed in Section 4.4. The proof of Theorem B is more delicate and we have to be more careful in applying the main lemma.

2. Preliminaries

In this section, we follow ideas and rewrite some results in [10] to create a language which we shall use in the proof of the main theorems. We start by giving a formal definition of stable thickness for a hyperbolic basic set whose stable foliation have codimension one. We use a condition given in [10] to obtain a basic hyperbolic set with

“intrinsically”C¹ unstable foliations. Moreover, the projection along leaves ofW^u(Λ₁)is intrinsicallyC¹. In the next section we give a formal definition of unstable thickness for a hyperbolic basic setΛ₁whose unstable foliation has codimension bigger than one. In this case we assume thatΛ₁has a periodic point displaying a unique weakest contracting eigenvalue. Later on, we show that we can obtain such a condition.

2.1. Cantor sets and thickness

ACantor setinR, is a compact, perfect and totally disconnected set. LetKbe a Cantor set andI itsconvex hull, i.e. the minimal (closed) interval ofRcontainingK. AgapofKis a connected component ofR\K. Apresentation ofK is an orderingU= {U_n}n1 of the bounded gaps. Anordered presentationofKis a presentationU such that (U_n)(U_m)for alln > m, where(U_n)denotes the length ofU_n. Thebridgeatu∈∂U_n,U_n∈U, is the component ofI\(U₁∪ · · · ∪U_n)that containsu. ThethicknessofKis the number

τ (K)=inf

τ (K,U, u): u∈K , whereU is any ordered presentation ofK,

τ (K,U, u)= (C) (U_n),

and whereC is the bridge atu∈∂U_n. This definition of thickness does not depend on the ordered presentation U (see [9]). Letk∈K. Thelocal thicknessofKatkis the number

τ (K, k)=lim

ε→0

sup

τ (K):˜ K˜ ⊂K∩Bε(k)a Cantor set .

LetK₁,K₂be Cantor sets andI₁,I₂their convex hulls. We say that the pair K₁, K₂islinkedifI₁∩I₂= ∅.I₁is not inside a gap ofK₂andI₂is not inside a gap ofK₁. The link is calledstableif the same condition is verified by the interiors int(I1), int(I2)ofI₁,I₂.

LetΛbe a nontrivial basic set of aC²diffeomorphismϕ: 5xM→M, whose stable foliation is of codimension one, i.e., such that dimW^s(x)=n−1,n=dimM, for allx∈Λ. Letz∈W^s(Λ)andφ:[−a, a] →Mbe aC¹embedding transverse to W^s(Λ)at z=φ(0). The local stable thickness of Λatz isτ^s(Λ, z)=τ (φ⁻¹(W^s(Λ)),0). This is independent of the choice ofφ, as a consequence of the fact that (under codimension-one assumption) the holonomy maps (i.e., the projections along the leaves) of the stable foliation ofΛcan be extended toC¹maps. Actually, this

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smoothness of the holonomy ofW^s(Λ), together with the transitivity ofϕ|Λ, also implies thatτ^s(Λ, z)has the same value at everyz∈W^s(Λ). We denote byτ^s(Λ)this constant value and call it thelocal stable thicknessofΛ. This is a strictly positive (finite) number and depends continuously on the diffeomorphism, in the sense that ifΛ_ψ denotes the smooth continuation ofΛfor a diffeomorphismψwhich isC²-close toϕ, thenτ^s(Λ_ψ)is close toτ^s(Λ). The Local unstable thicknessτû(Λ, z)andτû(Λ)are defined in a similar way, whenWû(Λ)has codimension one. In particular, both the stable thickness and unstable thickness are well-defined ifMis a surface.

In the proof of the main theorems we will use the following two important results involving thick Cantor sets, Proposition 2.1(Newhouse’s Gap Lemma). LetK₁,K₂be Cantor sets inRsuch thatτ (K₁)·τ (K₂) >1and K₁, K₂ is linked. Then,K₁∩K₂= ∅.

The next result is used by Colli [2] to show the existence of infinitely many Hénon-like strange attractors for diffeomorphisms on a manifold of dimension two.

Proposition 2.2(Linking Lemma). LetK1,K2be Cantor sets inR, withτ (K1)·τ (K2) >1, andI1,I2the convex hull ofK1,K2, respectively. Letϑβ:I1→Randϑ˜β:I2→Rbe such that

(a) ϑ_β andϑ˜_β are topological embeddings, for allβ∈R;

(b) ϑ_β(x)andϑ˜_β(y)are differentiable with respect toβ, for allx∈K₁andy∈K₂; (c) ∂_β[υ_β(x)− ˜υ_β(y)]c >0, for allx∈K₁andy∈K₂;

(d) ifK˜₁⊂K₁andK˜₂⊂K₂are Cantor subsets and letβ₀∈Rbe such that the pair ϑ_β₀(K˜₁),ϑ˜_β₀(K˜₂)is linked.

Then, for anyε >0, there isβsuch that (i) |β−β₀|< ε;

(ii) the pair ϑ_β(K˜₁),ϑ˜_β(K˜₂)has two stable sublinks.

2.2. Intrinsically smooth foliations of hyperbolic sets

LetX⊂R^mbe a compact set andϕ:X→Rⁿbe continuous. We say thatϕisintrinsicallyC¹onXif there exists a continuous mapϕ:X×X→L(R^m,Rⁿ)such that

ϕ(x)−ϕ(z)=ϕ(x, z)·(x−z) for allx, z∈X.

Such aϕ(which is, in general, far from unique) is called an intrinsic derivativeofϕ. We say thatϕisintrinsically C¹⁺^γ onXif it admits someγ-Hölder continuous intrinsic derivative.

Remark 1.Letϕ:X→Rⁿbe Lipschitz continuous andU⊂X×Xbe such that{x−z: (x, z)∈U}is bounded away from zero. Then, there is a Lipschitz continuous map:U→L(R^m,Rⁿ)such thatϕ(x)−ϕ(z)=(x, z)· (x−z)for every(x, z)∈U.

Letq₀be a transverse homoclinic point associated to some hyperbolic fixed (or periodic) saddle pointpof aC² diffeomorphismϕ:M→M. We assumeq₀∈/W^ss(p)and another mild (open and dense) transversally condition to be stated in (1) below. Then, our goal, in this section, is to prove that there exists a hyperbolic basic setΛ₁containing pandq₀and whose unstable foliation is intrinsicallyC¹. We assume thatϕisC²linearizable on a neighborhoodU ofp.

Let us denote byσ₁, . . . , σ_u, λ₁, . . . , λ_s,u+s=m, the eigenvalues ofDϕ(p), with|σ_u|· · ·|σ₁|>1>|λ₁|

· · ·|λ_s|. We define 1ws by|λ₁| = · · · = |λ_w| and letE^s =E^w⊕E^ss be the invariant splitting such that Dϕ(p)|E^whas eigenvaluesλ₁, . . . , λ_w andDϕ(p)|E^sshas eigenvaluesλ_w₊₁, . . . , λ_s. We chooseC²linearizing coordinates(ξ₁, . . . , ξ_u, ζ₁, . . . , ζ_s)in a neighborhoodUofpand, furthermore, we may assume that

(C1) W^u(p)⊂ {ζ₁= · · · =ζ_s=0}andW^s(p)⊂ {ξ₁= · · · =ξ_u=0};

(C2) E^w= {0^u} ×R^w× {0^s⁻^w}and the strong manifold (tangent toE^ssatp) satisfiesW_loc^ss(p)⊂ {ξ1= · · · =ξ_u= ζ₁= · · · =ζ_w=0}.

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Up to a convenient choice of Riemannian metric we have, forσ= |σ₁|,λ=λ₁| = |λ_w|andθ= |λ_w₊₁|, (C3) (σ−ε)vDϕ(p)v, for allv∈E^u;

(C4) (λ−ε)vDϕ(p)v(λ+ε)v, for allv∈E^w; (C5) Dϕ(p)v(θ+ε)v, for allv∈E^ss,

whereε >0 is fixed small enough so thatθ+2ε < λ−2ε < λ+2ε < σ−2ε. (In the casew=s, i.e., if all contracting eigenvalues have the same norm,E^ss= {0},W^ss(p)= {p}and we leaveθundefined.)

Now, we will construct a hyperbolic set whose unstable foliation is intrinsicallyC¹using the transversality between W^s(p)andW^u(p)atq₀. Fixq, r∈Uin the orbit ofq₀in such a way thatq∈W^s(p)_locandr=ϕ⁻^N(q)∈W^u(p)_loc. Take

V =V_δ=(ξ₁, . . . , ξ_u)δ

×(ζ₁, . . . , ζ_s)ρ ,

whereδ >0 is small andρ >0 is fixed in such a way that{q, r} ⊂int(V )⊂V ⊂U. Letn=n(δ)be minimum such thatr∈int(ϕⁿ(V )). (We suppose thatδis conveniently adjusted so thatϕ^N⁺ⁿ(V )cutsV in two cylinders.

We define Λ=

k∈Z

ϕ^(N⁺^n)k(V ) and Λ₁=

N+n i=1

ϕⁱ(V ).

It is well know thatΛ1is a nontrivial hyperbolic basic set, see [12]. We assume the following generic (open and dense) condition

D_uwφ_uw(r) is an isomorphism. (1)

HereDdenotes the usual derivative and (1) means thatunstable/weak-stabledirections are not sent tostrong-stable directions byφ=ϕ^N. With this condition it is shown in [10] that for every pointx∈Λ₁, theintrinsic tangent space toΛ₁

ITxΛ₁=span v: there is(x_n)_n∈Λ^N₁ so thatx_n→xand xn−x xn−x→v

(for simplicity we consider here M=Rⁿ) is contained in an(u+w)-dimensional space. Moreover the intrinsic tangent space to W_locû (Λ₁) at every point x∈W_locû (Λ₁) is contained in (u+w)-dimensional space. In particular, ITpWû(Λ₁)⊂Eû⊕E^w. The fact that we have a good property for the unstable foliation is showed in the following result.

Proposition 2.3.

(1) Suppose thatϕ^N satisfies condition(1)above and considerΛ1also as above. Then, the mapF:Wû(Λ1)x→ TxWû(x)is intrinsicallyC¹on compact parts ofWû(Λ1).

(2) Let Σ0, Σ1 be (small) C¹ sections transverse to Wû(x)for some x ∈Wû(Λ1)and let π:Σ0∩Wû(Λ1)→ Σ∩Wû(Λ1)denote the projection along the leaves ofWû(Λ1). Then,π is intrinsicallyC¹.

2.3. Thickness in higher dimension

In this subsection we want to define thelocal unstable thicknessof a basic set with unstable foliation of codimension greater than one.

ConsiderΛ₁as it was constructed in the previous section. We suppose that for a periodic pointp,Dϕ(p)has a unique (necessarily real) weakest contracting eigenvalueλ=λ₁, andϕisC²linearizable nearp. Then, we consider π:Λ₁∩W_loc^s (p)→Rto be an arbitrary intrinsicallyC¹ map such that ker(π(p, p))does not contain IT(Λ1∩ W_loc^s (p))=E^w (i.e.,π(p, p)|E^wis bijective). Define

τ^u(Λ₁, p)=τ π

Λ₁∩W_loc^s (p) , π(p)

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as the local unstable thicknessof Λ₁ atp. It is shown in [10] that the definition does not depends onπ as taken above, also it is strictly positive and varies continuously with the diffeomorphism: ifψis aC²-small perturbation of ϕ,τ^u(Λ₁(ψ ), p)is a small variation ofτ^u(Λ₁, p).

Letπw:Λ1∩W_loc^s (p)→Rbe the restriction toΛ1∩W_loc^s (p)⊂ {0^u} ×R^sof the projection(ξ1, . . . , ξu, ζ1, . . . , ζs)

→ζ₁.π_w is a homeomorphism onto its imageK^w and moreoverπ_w⁻¹is intrinsicallyC¹⁺^γ onK^w, see [10]. The fact thatπ◦π_w⁻¹is an intrinsicallyC¹map with

π◦π_w⁻¹

(0,0)=π(p, p)·π_w⁻¹(0,0)=0.

Then, τ (π(Λ₁∩W_loc^s (p)), π(p))=τ (K^w,0)as a consequence ofπ(Λ₁∩W_loc^s (p))=(π◦π_w⁻¹)(K^w)and the following result (see [10]):

Lemma 2.1.LetK⊂Rbe a Cantor set,y∈Kandg:K→Rbe an intrinsicallyC¹map withg(y, y)=0. Then, τ (g(K), g(y))=τ (K, y).

It is also shown that K^w is a dynamically defined Cantor set, in the same sense as in ([9], Chapter IV), i.e., τ (K^w) >0. Moreover, ifψis a diffeomorphismC²-close toϕ,τ (K^w(ψ ),0)is close toτ (K^w,0).

The following result shows that the definition of unstable thickness does not depend on transverse sections to W^u(Λ₁). We will use such fact in Section 4.

Proposition 2.4.

(a) Letq∈Wû(p),Σbe aC¹section transverse toWû(p)at the pointqandπ:Wû(Λ₁)∩Σ→Rbe an intrinsi- callyC¹map such thatITq(Wû(Λ₁)∩Σ )is not contained inker(π(q, q)). Then,τ (π(Wû(Λ₁)∩Σ ), π(q))= τû(Λ₁, p).

(b) More generality, givenz∈Wû(Λ1),Σa transverse section toWû(Λ1)atzandπ:Wû(Λ1)Σ→Ra submersion withITz(Wû(Λ₁)∩Σ )ker(π(z, z)). Then,τ (π(Wû(Λ₁)∩), π(z))=τû(Λ₁, p).

2.4. Unique least contracting eigenvalue

Let {ϕμ}be aC^∞one-parameter family of diffeomorphisms generically unfolding atμ=0 a quadratic homoclinic tangency associated to saddle fixed (or periodic) point pofϕ₀. We also assume once more that there areC² μ-dependent coordinates(ξ₁, . . . , ξ_u, ζ₁, . . . , ζ_s)linearizing theϕ_μ, forμnear zero, on a neighborhoodUof the analytic continuationp_μofp. Moreover, these coordinates can be taken to satisfy conditions (C1)–(C5) of Section 2.2.

We assume in this section that Dϕ0(p)has exactly two weakest contracting eigenvalues and these are complex conjugate numbers, which means that w=2,λ1=λe⁻îγ,λ2=λeîγ withλ >|λ3|andγ ∈R\ {kπ: k∈Z}. Here we may even assume thatDϕ_μ(p_μ)|E^w is conformal with respect to the Euclidean metric introduced by coordinates ζ₁,ζ₂. On the other hand, we may take, say forμ0, pointsq_μW_loc^s (p_μ),r_μ∈W_locû (p_μ)depending continuously onμ, such thatϕ_μ^N(r_μ)=q_μfor some fixedN1,r0,q0 belong to the orbit of tangency andr_μ,q_μ are points of the transverse intersection ofWû(pμ)andW^s(pμ)for everyμ >0. Moreover, recall that there exists a sequence of parameter valueμ_j→0 such thatWû(p_μ)andW^s(p_μ)also have a point of tangential intersection.

For each fixedμ=μ_j and every sufficiently largen1, there is a neighborhood ofV =V (j, n)of{p_μ, q_μ}, as in Section 2.2, such that

Λ(j, n)=

k∈Z

ϕ⁽ⁿ_μ⁺^{N )k}(V )

is aϕ_μ^N⁺ⁿ-invariant hyperbolic set andϕ_μ^N⁺ⁿ|Λ(j, n)is conjugate to the 2-shift. Moreover, given any periodic point

˜

p∈Λ(j, n), there are parameter valuesμ˜ arbitrarily close toμ_j for whichϕ_μ_˜ has homoclinic tangencies associated to (the analytic continuation of) p. We consider˜ p˜= ˜p(j, n) to be the uniqueϕ_μⁿ⁺^N-fixed point inΛ(j, n)\ {p_μ}.

Clearly, the orbit ofp˜passes arbitrarily close top_μifj andnare sufficiently large. The following result displays our goal in this subsection,

Proposition 2.5. Suppose thatϕ₀satisfies the condition (2)below. Given j sufficiently large, there exist values of n=n(j )arbitrarily large such thatDϕ_μ^N⁺ⁿ(p)˜ has a unique weakest contracting eigenvalue.

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Consider Dϕ_μ^N=

A_uu A_uw A_us A_wu A_ww A_ws A_su A_sw A_ss

, _μ=

A_uu A_uw A_wu A_ww

as the expression ofDϕ_μ^Nwith respect to the splittingE^u×E^w×E^s=R^u×R²×R^s⁻². We also denote Dϕ_μ⁻^N=

A⁻_uu A⁻_uw A⁻_us A⁻_wu A⁻_ww A⁻_ws A⁻_su A⁻_sw A⁻_ss

.

The generic assumption in proposition above is (cf. (1))

_μ₌₀(r₀), and so alsoA⁻_ss(μ=0, q0)is an isomorphism. (2)

From the proof of the Proposition 2.5 above, we can obtain that dimW^u(p)˜ =dimW^u(p), dimW^s(p)˜ = dimW^s(p)and thatDϕ_μⁿ⁺^N(p)˜ is sectionally dissipative ifDϕ₀(p)is.

We conclude that there exist a sequence of parameter valuesμ˜_j→0 such thatϕ_μ_˜_j exhibits homoclinic tangencies associated top˜j→pandDϕ_μ^k_˜^j

j(p˜j), wherekjis the period ofp˜j, has a unique weakest contracting eigenvalue.

2.5. Thick invariant Cantor sets

Letϕ be a C^∞ diffeomorphism with a quadratic homoclinic tangency atq₀ associated to a fixed (or periodic) pointp. We suppose that dimW^u(p)=1 andDϕ(p)is sectionally dissipative.

Let{ϕμ}be aC^∞one-parameter family of diffeomorphisms withϕ0=ϕ, that generically unfolds the homoclinic tangency. We suppose once more that theϕμ,μnear zero, admitsC² μ-dependent linearizing coordinates(ξ, Z)∈ R×Rⁿ⁻¹in a neighborhoodUofp. We fix these coordinates in such a way thatW_loc^u (pμ)⊂ {Z=0}andW_loc^s (pμ)⊂ {ξ=0}. The assumption on the eigenvalues ofDϕ0(p)means that we may choose a norm inRⁿsuch that

|σ_μ| · S_μ<1 for everyμnear zero,

whereσ_μis the expanding eigenvalue ofDϕ_μ(p_μ)andS_μ=Dϕ_μ|E^s(p_μ).

It is shown in [10] that, there are a constantN(positive integer) and, for each positive integern, a reparametrization μ=M_n(ν)of the variableμand(μ, n)-dependent coordinates transformation

(ν, x, Y )→

Mn(ν), Θn,ν(x, Y ) such that the map

(ν, x, Y )→

ν, Θ_n,ν⁻¹◦ϕ_Mⁿ⁺^N

n(ν)◦Θ_n,ν(x, Y ) ,

converge, inC²-topology, to the map(ν, x, Y )→(ν, x²+ν, Ax), whereA∈L(Rⁿ⁻¹).

The existence of a hyperbolic basic setΛ₂with arbitrarily large thickness follows from the fact that for the map x→x²+ν, and also forψ₋₂:(x, Y )→(x²−2, Ax), there exist invariant expanding Cantor setsK_jwith thickness τ (K_j)→ +∞asj → +∞. Moreover, theseK_j are transitive and have a dense subset of periodic orbits. It follows that eachK_jhas, fornlarge,μ=M_n(ν)andνclose to−2, an analytic continuation as a hyperbolic basic setK_j(n, μ)

of

Θ_n,ν⁻¹◦ϕ_Mⁿ⁺^N

n(ν)◦Θ_n,ν(x, Y ) .

In particular, the setKj(n, μ)has codimension-1 stable foliation and stable thicknessτ (Kj(n, μ))close toτ (Kj)1.

Then, we just takeΛ2=Λ2(μ)=Θn,ν(Kj(n, μ))withj andnlarge andμ=Mn(μ),νclose to−2. It is also shown that parameter valuesν_n→ −2 can be taken in such a way that

f_ν⁽ⁿ⁾

n =Θ_n,ν⁻¹

n◦ϕ_Mⁿ⁺^N

n(νn)◦Θ_n,ν_n

have periodic pointsP (n, ν_n)andQ(n, ν_n)∈K_j(n, μ),μ=M_n(ν_n), which are heteroclinic related andW^u(Q(n, ν_n)) also has nontransverse intersections withW^s(P (n, ν_n)).

Now, forf =f_ν⁽ⁿ⁾_n , there are 0< λ=λ(n) <λ¯= ¯λ(n) <1, 1< σ <σ¯ andc=c(n) >0 such that

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(1) c⁻¹σⁱuDfⁱ(x)·ucσ¯ⁱu; (2) c⁻¹λⁱvDfⁱ(x)·vcλ¯ⁱv,

for allx ∈K_j(n, μ),u∈E_x^u,v∈E_x^s andi0. If Λ₂=Λ(n, μ)=Θ_n,ν_n(K_j(n, μ)) is a hyperbolic basic set for ϕ_μⁿ⁺^N, whereμ=M_n(ν_n)andz∈Λ₂is a periodic point ofϕ_μⁿ⁺^Nof periodk=(n+N )j, then,z=Θ_n,ν_n(x), where x is a periodic point off of periodj. We conclude that ifσ2is the expanding eigenvalue ofDϕ_μ^k(z), then,

σ₂^k=σ₂⁽ⁿ⁺^{N )j}σ¯^jn+√N

¯ σj

, and, therefore,σ₂ ⁿ⁺√^N

¯

σ →1, asn→ +∞. Forμnear zero andy nearϕ_μ⁻^N(q₀)inU, we haveDϕ_μ^N(z)k,¯ for a large constantk, and if¯ S^k_2μ=Dϕ^k_μ(z)|E_z^s, we have

S_2μ^k =Dϕ_μ^k|E^s(z)Dϕ_μ^Nj|E^s ϕ_μ^nj(z)

◦Dϕ_μ^nj|E^s(z) (k)¯ ^jS_μ^nj<1,

fornsufficiently large, which means that,S_2μ< λ₀<1 fornlarge, whereλ₀does not depend onn.

From the discussion above, together with Sections 2.2–2.4 we can conclude this section with the following result which is a summary of this section.

Theorem 2.1.Letϕ₀be a smooth diffeomorphism having a homoclinic tangency associated to a sectionally dissipative saddle fixed(or periodic)point. Then, there exists a smooth diffeomorphismϕarbitrarily nearϕ₀such that

(a) ϕhas hyperbolic basic setsΛ₁andΛ₂withτ_loc^s (Λ₂)·τ_loc^u (Λ₁) >1;

(b) there are periodic pointsp1∈Λ1andp2∈Λ2such thatW^u(p2)has a transversal intersection withW^s(p1)and W^u(p1)has a quadratic tangency withW^s(p2)at a pointq;

(c) the hyperbolic basic setΛ1has intrinsicallyC¹ unstable foliation andp1∈Λ1 has a unique least contracting eigenvalue;

(d) there existsc >0such that ifQ1∈Λ1andQ2∈Λ2are periodic points of periodk1andk2, respectively. Denote λ_i= S_i=Dϕ|E_Qi^s andσ_i^kⁱ the unstable eigenvalue ofDϕ^kⁱ,i=1,2. Then,

(d1) |λ₂·σ₂|<1;

(d2) |σ₁^2c·λ₂|<1;

(d3) σ₂is so small that|σ₂·(λ₁σ₁)^c/2|<1.

3. Renormalization scheme and quadratic-like families

In this section we describe a higher-dimensional version of the renormalization scheme in 2-cycles of periodic points with a heteroclinic tangency. We follow ideas from [15] and [9]. We also state and comment aboutquadratic- likefamilies as considered in [15]. Finally, we make a delicate discussion on how to perturb a one-parameter families of diffeomorphisms to obtain linearizability.

3.1. Renormalization scheme in 2-cycles

Letϕbe aC^∞diffeomorphism having basic setsΛ₁,Λ₂and fixed (or periodic) pointsp₁∈Λ₁andp₂∈Λ₂, such that dimWû(p₁)=W^s(p₂)=1;W^s(p₁)andWû(p₂)have a transverse intersection in a pointr₀andWû(p₁)have a nontransverse contact (i.e. tangency) withW^s(p₂)in a pointq, see Fig. 1. We suppose thatDϕ(p₁)is sectionally dissipative, (i.e. the product of any two of its eigenvalues has norm less than one). We also suppose that the tangency is quadratic.

Let{ϕ_μ}be aC^∞one-parameter family of diffeomorphisms withϕ₀=ϕand generically unfolding the tangency.

We assume thatϕ₀isC⁴linearizable nearp₁andp₂. AsC^k-linearizable is an open condition (see [13]), we assume that theϕ_μ,μ close to zero, admitC⁴ μ-dependent linearizing coordinates in a neighborhood ofp₁andp₂. That is, there are neighborhoods U₁ of p₁ andU₂ of p₂ such that the expression ofϕ_μ, μ small, in U₁ is (ξ, H )→ (σ_1μξ, S_1μH ), inU₂is(η, J )→(σ_1μ2η, S2μJ )whereσ_1μandσ_2μare the expanding eigenvalue ofDϕ_μ(p₁)and

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Fig. 1. Renormalization scheme.

Dϕμ(p2), respectively, andSiμ=Dϕμ|E^s(p_i),i=1,2. We may suppose thatq=(1,0ⁿ⁻¹)∈U1and, therefore, there existsN >1 such thatϕ_μ^N(q)=(0, J0)∈U2, see Fig. 1. We assume that for(μ, ξ, H )close to(0,1,0ⁿ⁻¹), we may writeϕ_μ^N(ξ, H )as

α(ξ−1)²+β·H+aμ+r(μ, ξ−1, H ), J0+γ (ξ−1)+R(μ, ξ−1, H ) , where we haveα,a∈R,β∈L(Rⁿ⁻¹,R),γ∈L(R,Rⁿ⁻¹)and

r, R, Dr, DR, ∂_{ξ ξ}r, ∂_μξr and ∂μμr vanish at

0,1,0ⁿ⁻¹

. (3)

The hypothesis of nondegeneracy of the tangency amounts to having α=0 and a =0. Moreover, using a μ-reparametrization and μ-dependent linear changes of the space of coordinates, we may even assume a =1, r(μ,0,0)=0,R(μ,0,0)=0ⁿ⁻¹and∂_ξ(μ,0,0)=0.

We still have to consider the transition map among the neighborhoodsU₂ofp₂andU₁ofp₁and their “transverse”

intersection. We may suppose thatr₀=(1,0)∈U₂, then there isN₁>0 such thatϕ_μ^N¹(r₀)=(0, R0)∈U₁, forμsmall.

Suppose thatϕ_μ^N¹, for(η, J )near(1,0), has the form ϕ_μ^N¹(η, J )=(0, R0)+

a_μ B_μ c_μ D_μ

η−1 J

+

θ (μ, η−1, J ), Θ(μ, η−1, J ) ,

whereaμ∈R,Bμ∈L(Rⁿ⁻¹,R),cμ∈L(R,Rⁿ⁻¹),Dμ∈L(Rⁿ⁻¹),θ (μ,0,0)=0,Θ(μ,0,0)=0 and

Dθ, DΘvanish at(μ, η−1, J )=(μ,0,0). (4)

From the transversality betweenW^u(p₂)andW^s(p₁), we havea_μ=0, forμsmall.

Now we fixA₀3 a being a real constant. FixNandN₁as above. We denoteΦ:R×M→R×M, aC^∞map such that,Φ(μ, x)=(μ, ϕ_μ).

Theorem 3.1.LetN,N1positive integer as above and let0< c <1 be a small constant such that the following hold|σ₁^2c·λ₂|<1 and |σ₂·(λ₁·σ₁)^c/2|<1. Choosen=n(m)such that (c/2)·mn(m)c·m. Then, there exists a sequenceΘ_n,m:[1/A0, A₀] × [−A₀, A₀] →R×MofC^k diffeomorphisms such that the sequencef_n,m= Θ_n,m⁻¹ ◦Φ^N⁺ⁿ⁺^N¹⁺^m◦Θ_n,mconverges to the map

φ(a, x, y₁, . . . , y_n₋₁)=

a,1−ax²,0ⁿ⁻¹ in theC^ktopology, asn, m→ ∞.