Heterodimensional tangencies on cycles leading to strange attractors

In this paper, we study heterodimensional cycles of two-parameter families of 3-dimensional diffeomorphisms some element of which contains nondegenerate heterodimensional tangencies of the stable and unstable manifolds of two saddle points with different indexes, and prove that such diffeomorphisms can be well approximated by another element which has a quadratic homoclinic tangency associated to one of these saddle points. Moreover, it is shown that the tangency unfolds generically with respect to the family. This result together with some theorem in Viana, we detect strange attractors appeared arbitrarily close to the original element with the heterodimensional cycle.


Introduction
When diffeomorphisms act on manifolds of dimension greater than or equal to three, it is well known that nonhyperbolic phenomena are caused by the existence of heteroclinic cycles containing two saddle points with different indexes, called heterodimensional cycles, as well as that of homoclinic tangencies [13,16,10,15,17,11]. Heterodimensional cycles presented a new mechanism in dynamics, which has been studied Díaz et al. [7,5,8,2,9,3,4]. As is suggested in [14], these two classes of nonhyperbolic diffeomorphisms seem to occupy a large part in the complement of the hyperbolic diffeomorphisms. In this paper, we study 3-dimensional diffeomorphisms which have heterodimensional cycles and tangencies of certain type simultaneously.
Let ϕ be a diffeomorphism on a 3-dimensional smooth manifold M which has two saddle fixed points p and q satisfying index(q) = index(p) + 1, where index(·) denotes the dimension of the unstable manifold of the saddle point. A heteroclinic point r of the stable manifold W s (p) and the unstable manifold W u (q) is called a heterodimensional tangency of W s (p) and W u (q) if r satisfies • T r W s (p) + T r W u (q) = T r M ; • dim(T r W s (p)) + dim(T r W u (q)) > dim(M ), where dim(·) denotes the dimension of the space.
Our main theorem in this paper is as follows. We will present some definitions and generic conditions used in the statement of Theorem A in Section 2.
(1) The generic conditions (C2) and (C3) imply the setting such that a heterodimensional tangency and a quasi-transverse intersection unfold generically with respect to the parameters µ and ν, respectively. The point to notice is that the newly detected homoclinic tangency in Theorem A can be also controlled by these parameters. It is not hard to get a one parameter family in the infinite dimensional space Diff 2 (M ) with respect to which the homoclinic tangency unfolds generically. However, in our proof, we need to show that the tangency given in Subsection 4.1 unfolds generically with respect to the ν-parameter family {ϕ µ0,ν } in the two-dimensional subspace {ϕ µ,ν } of Diff 2 (M ).
(2) Theorem A holds for a homoclinic tangency associated to q µ,ν instead of p µ,ν if we replace the conditions in (C1) and (C4) by appropriate ones. (3) We need the C 2 smoothness in the local linearization around q µ,ν to estimate the curvatures of W s (p µ,ν ) and W u (p µ,ν ) at the tangency in Section 3.  The conclusion obtained from Theorem A reminds us of prior works associated with homoclinic tangencies. The one is related to strange attractors and the other C 2 robust tangencies.
First, let us discuss the former one. Viana showed the following theorem.
Theorem 1.2 (Viana [18]). For a generic subset of one-parameter families {ϕ µ } of C ∞ diffeomorphisms on any manifolds of the dimension greater than or equal to two that unfolds a homoclinic tangency at parameter value µ = 0 associated to a sectionally dissipative saddle periodic point, there is a subset S of R such that • S ∩ (−ǫ, ǫ) has a positive Lebesgue measure for every ǫ > 0, • for all µ ∈ S, ϕ µ exhibits nonhyperbolic strange attractors in a µ-dependent neighborhood of the orbit of tangency.
Leal [12] extended this result and showed the existence of infinitely many strange attractors. A saddle periodic point is said to be sectionally dissipative if the product of any two eigenvalues of the derivative at the point has norm less than one. Also, Λ is a strange attractor of ϕ if Λ is a compact, ϕ-invariant, transitive set and the basin W s (Λ) has a nonempty interior and there exists z 1 ∈ Λ such that {ϕ n (z 1 ) : n ≥ 0} is dense in Λ and ||dϕ n (z 1 )|| ≥ e cn for all n ≥ 0 and some c > 0. Note, Viana [18] assumed that for simplicity ϕ µ is C 3 linearizable in a neighborhood of the saddle point for µ is sufficiently close to 0. Combining these extra conditions and our result for the cycle containing the heterodimensional tangency imply the following corollary.
Next, we discuss C 2 robust homoclinic tangencies derived from the heterodimensional cycle having a nondegenerate tangency. A homoclinic tangency of a diffeomorphism ϕ associated with a hyperbolic set Γ is C r robust if there is a C r neighborhood U of ϕ such that every diffeomorphism ψ ∈ U has a homoclinic tangency associated with the continuations of Γ for ψ. Newhouse [13] showed, in the C 2 topology, a homoclinic tangency of surface diffeomorphisms generates C 2 robust homoclinic tangencies. This property yields the so-called C 2 Newhouse phenomenon: there is a non-empty open set of C 2 diffeomorphisms and its residual subset such that every diffeomorphism in the subset has infinitely many sinks. The result is extended by Palis-Viana [15] to the higher dimensional case. Palis and Viana proved the result under the sectional dissipativeness and linearizing conditions as in [18]. Moreover, Romero [17] proved the following theorem without these conditions. Theorem 1.3 (Romero [17]). Let ϕ be a C 2 diffeomorphism on a manifold M , dim(M ) ≥ 3, having a homoclinic tangency associated with a saddle periodic point of ϕ whose index is greater or equal to 2. Then there are diffeomorphisms arbitrarily C 2 close to ϕ having robust homoclinic tangencies.
Therefore, we have the following corollary.
Corollary C. Let {ϕ µ,ν } be the two-parameter family of 3-dimensional C 2 diffeomorphisms having the heterodimensional cycle with the nondegenerate heterodimensional tangency for (µ, ν) = (0, 0) given in Theorem A. Then, for a sufficiently small ε > 0 and any µ in either (0, ε) or (−ε, 0), there are infinitely many ν such that every C 2 neighborhood of ϕ µ,ν contains a diffeomorphism having a C 2 robust homoclinic tangency. Remark 1.4. If we add a weak dissipative condition (see in [17, Theorem A (1.1)]) to the assumptions of Corollary C then, the C 2 Newhouse phenomenon is obtained from our settings.
Remark 1.5. In the C 1 case, Díaz et al. [6] have already proved that the unfolding of heterodimensional tangencies leads to non-dominated dynamics and therefore (by results of [2] and [3]) to the C 1 Newhouse phenomenon (see also [1] for a different approach to the phenomenon). On the other hand, the theory of strange attractors has not been so far developed in the C l category (l = 1, 2).
Outline of the proof of Theorem A: We will finish Introduction by presenting a sketch of the proof of Theorem A.
In Section 2, we give definitions and the generic conditions (C1)-(C4) used in Theorem A. Especially, a nondegenerate heterodimensional tangency, which is one of main ingredients of this paper, is introduced explicitly there. Such tangencies are classified into the elliptic and hyperbolic types, see Definition 2.1-(3). In Section 3, we prove three lemmas. Lemma 3.1 shows the existence of the reparametrization of {ϕ µ,ν } such that, for any (µ, 0) in the new parameter, W s (q µ,0 ) and W u (p u,0 ) still have a quasi-transverse intersection s µ,0 which unfolds generically with respect to the ν-parameter but the tangency r is annihilated. See Fig.1.2 (1) and (2). Lemma 3.2 applies the C 2 inclination lemma to a shorter curve l m0 in W u (p µ,0 ) passing through s µ,0 so that some curve l m in ϕ m µ,0 (l m0 ) containing ϕ m µ,0 (s µ,0 ) C 2 converges to W uu loc (q µ,0 ) as m → ∞. See Fig.1.2 (3). Lemma 3.3 explains a connection between quadratic tangencies and curvatures, which is used to show that the homoclinic tangencies obtained in Section 4 are quadratic.
Assertion 4.2 and Assertion 4.3 in Section 4 show that the generic unfolding of heterodimensional tangencies introduces the existence of a quadratic homoclinic tangency τ m as illustrated in Fig. 1.2-(4). Finally, we prove in Proposition 4.4 that the tangency τ m unfolds generically with respect to the ν-parameter. These results assure the proof of Theorem A.

Definitions and generic conditions
In this section, we present some definitions needed in later sections and generic conditions adopted as hypotheses in Theorem A.
(1) Suppose that l ν0 and m ν0 intersect at a point s for some ν 0 ∈ J and some open neighborhood U of s in M has a C 2 change of coordinates with respect to which m ν = {(0, 0, z) ∈ U } for any ν ∈ J near ν 0 . We say that s is a quasi-transverse intersection of l ν0 and m ν0 if dim(T s (l ν0 ) + T s (m ν0 )) = 2.
(3) Suppose that S ν0 and Y ν0 intersect at a point r for some ν 0 ∈ J. We say that r is a nondegenerate heterodimensional tangency of S ν0 and Y ν0 if there ex- Remark 2.2. It is easy to see that the property (1) does not depend on the coordinates used to set l ν in the z-axis. Similarly, the properties (2) and (3) do not depend on the coordinates used to set S ν in the xy-plane.  2.2. Generic conditions. Throughout the remainder of this paper, we suppose that ϕ is a 3-dimensional C 2 diffeomorphism with saddle fixed points p of index(p) = 1 and q of index(q) = 2, and such that W s (p) and W u (q) have a nondegenerate heterodimensional tangency r, W u (p) and W s (q) have a quasi-transverse intersection s. The ϕ is locally C 2 linearizable in a neighborhood U (q) of q if there exists a C 2 linearizing coordinate (x, y, z) on U (q), that is, for any (x, y, z) ∈ U (q) with ϕ(x, y, z) ∈ U (q), where α, β and γ are eigenvalues of (dϕ) q . One can take a local unstable manifold W u loc (q) so that it is an open disk in the plane {z = 0} centered at (x, y) = (0, 0). We may assume that the both points r, s are contained in U (q) if necessary replacing r (resp. s) by ϕ −n (r) (resp. ϕ n (s)) with sufficiently large n ∈ N. We set with respect to the linearizing coordinate on U (q).
We will put the following generic conditions (C1)-(C4) as the hypotheses in Theorem A. (C1) (Generic condition for q) The ϕ is locally C 2 linearizable at q given as in (2.4). For simplicity, we suppose that every eigenvalues of (dϕ) q is positive, that is, 0 < γ < 1 < β < α. (C2) (Generic unfolding property for r) The nondegenerate heterodimensional tangency r of W u (q) and W s (p) unfolds generically with respect to the µparameter families {W u (q µ,0 )} and {W s (p µ,0 )}. (C3) (Generic unfolding property for s) The quasi-transverse intersection s of W s (q) and W u (p) unfolds generically with respect to the ν-parameter families {W s (q 0,ν )} and {W u (p 0,ν )}. (C4) (Additional generic conditions) The tangency r is not on the x-axis W uu loc (q), that is, There exists a regular parametrization l(t) = (x(t), y(t), z(t)) (t ∈ I) of a small curve in W u (p)∩U (q) with respect to the linearizing coordinate (x, y, z) on U (q) with s = l(0) and ∂ 2 f ∂x 2 (u 0 , v 0 ) = 0. Note that the condition (2.7) is automatically satisfied when r is of elliptic type.

Some lemmas about parametrization and curvatures
The goal of this section is to prove three lemmas needed for the proof of Theorem A. These play important roles in Section 4.
• Lemma 3.1 presents a new parameter (μ,ν) such that, for anyμ near 0, there exists a quasi-transverse intersection sμ ,0 of W s (qμ ,0 ) and W u (pμ ,0 ) which unfolds generically atν = 0 with respect to theν-parameter. After Lemma 3.1, we denote the new parameter (μ,ν) again by (µ, ν) for simplicity. • In Lemma 3.2, we show that, for any µ 0 near 0, there exists a regular curve l m in W u (p µ0,0 ) containing the quasi-transverse intersection ϕ m µ0,0 (s µ0,0 ) and arbitrarily C 2 close to W uu loc (q µ0,0 ). In particular, this implies that the curvature of l m can be taken arbitrarily close to 0 with respect to the linearizing coordinate (2.4) on U (q µ0,0 ). • Lemma 3.3 gives a connection between the curvature and quadratic tangencies. In fact, we show that a tangency τ of a regular curve l and a regular surface S in R 3 is quadratic if the curvature of l at τ is different from the normal curvature of S at τ along the direction tangent to l.
Lemma 3.2. The sequence {l m } C 2 converges uniformly to W uu loc (q µ0,0 ) as m → ∞. In particular, for any ε > 0, there existsm 0 ≥ m 0 such that the curvature at any point of l m is less than ε with respect to the standard Euclidean metric on U (q µ0,0 ) = D(δ) if m ≥m 0 .
Proof. By (2.4), for any m ≥ m 0 , l m (t) = (t, β n y m0 (α −n t), γ n z m0 (α −n t)), as m → ∞. Since {l m (0)} ∞ m=m0 converges to q µ0,0 = (0, 0, 0), it follows from (3.2) that {l m } C 2 converges uniformly to the x-axis in D(δ). Lemma 3.3. Let S be a regular surface in the Euclidean 3-space R 3 and l a regular curve tangent to S at τ . Suppose that the curvature κ l (τ ) of l at τ is less than the absolute value of the normal curvature κ S (τ, w) of S at τ along a non-zero vector w tangent to l. Then tangency of S and l at τ is quadratic.

Proof of Theorem A
In this section, we give the proof of Theorem A.
When b µ,ν > 0 for any (µ, ν) near (0, 0), one can prove the existence of a homoclinic tangency τ m near r associated to p µ0,νm by arguments quite similar to those as above for any µ 0 with µ 0 < 0.