Robust Heteroclinic Tangencies

Abstract

We construct diffeomorphisms in dimension \(d\ge 2\) exhibiting \(C^1\)-robust heteroclinic tangencies.

Introduction

An important problem in the modern theory of Dynamical Systems is to describe diffeomorphisms whose qualitative behavior exhibits robustness under (small) perturbations and how abundant these sets of dynamics can be. Motivated by this issue, Smale (1967) introduced the hyperbolic diffeomorphisms as examples of structural stable dynamics (open sets of dynamics which are all of them conjugated). However, the transverse intersection between the invariant manifolds of basic sets was soon observed as a necessary condition (Williams 1970; Palis 1978; Mañé 1987). The main goal of this article is to study the persistence of the non-transverse intersection between those manifolds. Namely, we focus in tangencial heteroclinic orbits.

A diffeomorphism f of a manifold \({\mathcal {M}}\) has a heteroclinic tangency if there are different transitive hyperbolic sets \(\Lambda \) and \(\Gamma \), points \(P\in \Lambda \), \(Q\in \Gamma \) and \(Y\in W^s(P)\cap W^u(Q)\) such that

$$\begin{aligned}&c_T {\mathop {=}\limits ^{\scriptscriptstyle \mathrm def}}\dim \mathcal {M} - \dim [T_Y W^s(P) + T_Y W^u(Q)]> 0\quad \text {and} \\&d_T{\mathop {=}\limits ^{\scriptscriptstyle \mathrm def}}\dim T_Y W^s(P)\cap T_Y W^u(Q)>0. \end{aligned}$$

The number \(c_T\) is called codimension of the tangency and measures how far the tangencial intersection is from a transverse intersection. On the other hand, \(d_T\) indicates the number of linearly independent common tangencial directions. Observe that

$$\begin{aligned} c_T=d_T- k_T \qquad \text {with} \quad {k_T={{\,\mathrm{ind}\,}}(\Lambda )-{{\,\mathrm{ind}\,}}(\Gamma )} \end{aligned}$$

where \({{\,\mathrm{ind}\,}}(\Sigma )\) denotes the stable index of a (transitive) hyperbolic set \(\Sigma \). The integer \(k_T\) is called signed co-index. Notice that when \(k_T>0\) this number coincides with the classical co-index between \(\Lambda \) and \(\Gamma \). Moreover, \(k_T>0\) if and only if

$$\begin{aligned} \dim T_Y W^s(P) + \dim T_Y W^u(Q) > \dim {\mathcal {M}}. \end{aligned}$$

If \(k_T=0\), the heteroclinic tangency is called equidimensional and otherwise heterodimensional. Figure 1 illustrates the different types of heteroclinic tangencies in dimension three.

Fig. 1
figure1

Heteroclinic tangencies in dimension 3: (a) \(c_T=2\), \(d_T=1\) and \(k_T=-1\); (b) \(c_T=1\), \(d_T=1\) and \(k_T=0\); (c) \(c_T=1\), \(d_T=2\) and \(k_T=1\)

Heterodimensional tangencies with signed co-index \(k_T>0\) was introduced in Díaz et al. (2006) where interesting dynamics consequences were obtained. Indeed, the authors showed that the \(C^1\)-unfolding of a three dimensional heterodimensional tangency (with \(k_T=1\)) leads to \(C^1\)-robustly non-dominated dynamics and in some cases to very intermingled dynamics related to universal dynamics, for details see (Díaz et al. 2006; Bonatti and Díaz 2003). In the \(C^r\)-topologies with \(r>1\), the bifurcation of such tangencies leads, for instance, to the existence of blender dynamics (Díaz et al. 2014; Díaz and Pérez 2019). This illustrates the dynamic richness of some configurations involving these type of tangencies. Motivated by the results of Díaz et al. (2006), Kiriki and Soma (2012) obtain the first examples of \(C^2\)-robust heterodimensional tangencies with \(c_T=1\) and positive signed co-index \(k_T=d-2\) in any manifold of dimension \(d\ge 3\). Recently in Barrientos and Raibekas (2017) new examples of \(C^2\)-robust heterodimensional tangencies with \(0<c_T\le \lfloor (d-3)/2 \rfloor \) and \(1\le k_T \le d-2-2c_T\) were also constructed in any manifold of dimension \(d\ge 5\). In the same work, \(C^2\)-robust heteroclinic tangencies with signed co-index \(k_T\le 0\) were also obtained.

Kiriki and Soma in (2012, page 3281) wrote that \(C^1\)-robust heterodimensional tangencies with positive signed co-index could not be possible for generic diffeomorphisms in any dimension greater than 2. The main result of this work answer in the negative direction to this claim.

Theorem A

Every manifold of dimension \(d\ge 2\) admits a diffeomorphism f having a \(C^1\)-robust heteroclinic tangency of codimension \(c_T=1\) and any signed co-index \(0 \le k_T \le d-2\).

By constraints of the dimension, in surfaces, we only get equidimensional tangencies. In higher dimensions, we construct both type of heteroclinic tangencies: equidimensional and heterodimensional with all possible positive signed co-index \(0<k_T<d-1\).

Theorem A will be proved in Sect. 2, by providing a local construction close to the classical examples given by Abraham and Smale (1970), Simon (1972) and Asaoka (2008). In Sect. 3 we will give a different proof of Theorem A using ideas of the recent work (Barrientos and Raibekas 2019) studying the differential cocyle in the tangent space. These new ideas allow us to generalize the construction for large codimension (\(c_T\ge 2\)) in some particular cases. Namely, we get the following result.

Theorem B

Given integers \(c_T \ge 1\) and \(s> c_T\), there are diffeomorphisms f of the d-dimensional torus \(\mathbb {T}^d\) with \(d=c_T \cdot (s+1)\) having a \(C^1\)-robust heterodimensional tangency of codimension \(c_T\) and signed co-index \(k_T=s-c_T>0\).

The proof of the above theorem will be carried on in Sect. 4. Finally, in Sect. 5 we conclude the work with a section of open questions and future directions.

Geometric Construction of \(C^1\)-Robust Heteroclinic Tangencies

Let \(\Lambda \) be a Plykin attractor in a disc with three holes (Robinson 1999). Let Q be a saddle in the complement of this disc as in Fig. 2. To carry out this construction, we need to assume that Q belongs to a Plykin repellor \(\Gamma \). This figure illustrates the two-dimensional version of the Asaoka’s (2008) argument (see also Simon 1972) providing a \(C^1\)-robust equidimensional tangency in any surface between the stable manifold of \(\Lambda \) and the unstable manifold of Q.

Fig. 2
figure2

Equidimensional tangency in dimension \(d=2\)

Using this idea, we built a diffeomorphism f on any manifold \(\mathcal {M}\) of dimension \(d\ge 2\) having a hyperbolic attractor \(\Lambda \) with index \({{\,\mathrm{ind}\,}}(\Lambda )=d-1\) and whose attracting region is a d-dimensional connected set foliated by \((d-1)\)-dimensional stable submanifolds. After that, we consider a fixed point P in \(\Lambda \) and another fixed point Q of f of stable index \(d-1-k\) where \(0\le k\le d-2\) creating a heteroclinic tangency between \(W^s(P)\) and \(W^u(Q)\) of signed co-index \(k_T=k\), so that \(W^u(Q)\) and \(W^s(\Lambda )\) meet transversely. The \(C^1\)- persistence of this last intersection provides a \(C^1\)-robust heteroclinic tangency associated with \(\Lambda \) and Q.

Construction

We now give the details of our construction. Since our argument is local, we can put \({\mathcal {M}}=\mathbb {R}^d\) with \(d \ge 2\). First, we take a two-dimensional diffeomorphism h with a Plykin attractor \(\Sigma \) constructed in local coordinates inside a disk \(D\subset \mathbb {R}^2\) with three holes. We consider a \(C^r\)-diffeomorphism \(f:{\mathcal {M}} \rightarrow {\mathcal {M}}\) with \(r\ge 1\) such that for a small \(\varepsilon >0\), the restriction of f to the set \(D_\varepsilon {\mathop {=}\limits ^{\scriptscriptstyle \mathrm def}}[-\varepsilon ,\varepsilon ]^{d-2}\times D\) is given by

$$\begin{aligned} f=g\times h \quad \text {where} \quad g(t)=\lambda t \ \ \text {for} \ \ t\in [-\varepsilon ,\varepsilon ]^{d-2} \ \ \text {and} \ \ 0<\lambda <1. \end{aligned}$$
(1)

Thus, the set

$$\begin{aligned} \Lambda {\mathop {=}\limits ^{\scriptscriptstyle \mathrm def}}\{0^{d-2}\}\times \Sigma =\bigcap _{n\geqslant 1}f^n(D_{\varepsilon }) \end{aligned}$$
(2)

is a hyperbolic attractor of f and \(D_\varepsilon \) is a trapping region of f, i.e. \(f(D_\varepsilon )\subset \mathrm {interior}(D_\varepsilon )\). The structural stability of \(\Lambda \) provides the existence of a \(C^r\)-neighborhood \(\mathcal {V}\) of f such that for each \(g\in \mathcal {V}\), the continuation \(\Lambda _g\) of \(\Lambda \) has by trapping region the set \(D_\varepsilon \). We remark that the local stable manifolds \(W^s_{loc}(x) =W^s(x)\cap D_\varepsilon \) for \(x\in \Lambda \) provide a foliation of the set \(D_\varepsilon \) by leaves (plaques) of dimension \(d-1\). It is not hard to verify that this property also holds for any diffeomorphism g in \(\mathcal {V}\). We will denote by \(W^s_{loc}(x,g)\) the stable local manifold at x for g.

Now we build the robust heteroclinc tangency of elliptic type. Recall that a heteroclinic tangency \(y\in W^u(Q)\cap W^s(P)\), is of elliptic type if there is a neighborhood U of y contained in either, \(W^u(Q)\) or \(W^s(P)\), say \(W^u(Q)\), such that any point in \(U-\{y\}\) belongs to the same side of the tangent space \(T_y W^u(Q)\). We consider a fixed point \(P\in \Lambda \) and a small open ball B centered at P such that \(\overline{B}\) is contained in \(D_{\varepsilon }\). We observe that for every \(g\in \mathcal {V}\), B is foliated by

$$\begin{aligned} \mathcal {F}_g(x){\mathop {=}\limits ^{\scriptscriptstyle \mathrm def}}W^s_{loc}(x,g)\cap B,\quad x\in \Lambda _g. \end{aligned}$$

Consider a hyperbolic fixed point \(Q \not \in D_\varepsilon \) of f with stable index \(d-1-k\). By means of a homotopic deformation, we force to the \((k+1)\)-dimensional unstable manifold \(W^{u}(Q)\) intersects non-transversely the stable manifold \(W^{s}(P)\) in a heteroclinic tangency of elliptic type, namely y. Taking a suitable iterated if necessary, we can assume that y is in B. A similar deformation is showed in Fig. 4 from a similar construction using a DA-attractor instance a Plykin attractor.

Fig. 3
figure3

a \(C^1\)-robust equidimensional tangency. b \(C^1\)-robust heterodimensional tangency

Thus, this last diffeomorphism, that again we call f, has a heteroclinic tangency of codimension \(c_T=1\) and signed co-index \(k_T=k\) with \(0\le k\le d-2\), associated with the saddles \(P\in \Lambda \) and Q.

On the other hand, by definition, there exists a neighborhood U of y contained in \(W^{u}(Q)\) such that \(U-\{y\}\) is contained in \(W^{s}(\Lambda )\pitchfork W^{u}(Q)\). See Fig. 3. We will see that the \(C^1\)-persistence of these last transverse intersections provides a \(C^1\)-robust heteroclinic tangency associated with \(\Lambda \) and Q. Besides, for each \(g\in \mathcal V\), we consider a small curve \(\gamma _g:t\in (-r,r)\mapsto \gamma _g(t)\in \Lambda _g\subset D_{\varepsilon }\) parameterizing a small local unstable manifold of the continuation \(P_g=\gamma _g(0)\) of P such that

$$\begin{aligned} B=\bigcup _{t\in (-r,r)}{\mathcal {F}}_g(t) \quad \text {where} \quad \mathcal {F}_g(t)=\mathcal {F}_g(\gamma _g(t)). \end{aligned}$$

Since y is a heteroclinic tangency of elliptic type between \(W^{u}(Q)\) and \(W^{s}(P)\) we can assume that (see Fig. 3)

$$\begin{aligned} \begin{aligned}&U \cap {\mathcal {F}}_f(0)=\{y\},\\&U \cap {\mathcal {F}}_f(t)\subset W^{u}(Q)\pitchfork W^{s}(\gamma _f(t)) \quad \text{ for } \text{ all } t\in (0,r),\\&U\cap {\mathcal {F}}_f(t)= \emptyset \quad \text{ for } \text{ all } t\in (-r,0). \end{aligned} \end{aligned}$$

Our conditions imply that for each \(g\in {\mathcal {V}}\), the set

$$\begin{aligned} I_g=\{t\in (-r,r): U_g\pitchfork {\mathcal {F}}_g(t) \ne \emptyset \} \end{aligned}$$

is inferiorly bounded where \(U_g\) is a continuation in \(W^s_{loc}(Q_g)\) of the neighborhood U. Thus, if \(\bar{t}\) is the infimum of \(I_g\) then \(U_g\) and \({\mathcal {F}}_g(\bar{t})\) meet in a heteroclinic tangency \(y_g\) of codimension \(c_T=1\) and signed co-index \(k_T=k\) with \(0 \le k\le d-2\). This completes the proof of Theorem A.

Fig. 4
figure4

Heterodimensional tangency constructed from a deformation of diffeomorphism f of \(\mathbb {T}^3\) locally defined as the product \(g\times h\) where h has a DA-attractor in \(\mathbb {T}^2\) and g is a contraction

Differential Construction of \(C^1\)-Robust Heteroclinic Tangencies

In this section we will prove again Theorem A but now using a different argument. This different approach allows us to generalize the result to get robust heterodimensional tangencies of large codimension in the next section. In order to explain the idea behind of this new approach we will consider again the situation described in Fig. 2.

Fig. 5
figure5

Figure on the left shows the folding manifold \(\mathcal {S}\) on \(\mathcal {M}\) and how cover the cone \(\mathcal {C}^s_\alpha \) on \(\mathbb {R}^d\). The other shows the transverse intersection between \(\mathcal {S}^s\) and the stable manifold of \(\Lambda ^s\) on \(\mathbb {R}^d\times G(s,d)\)

By considering local coordinates around of the point P, define the projective cocycle \(f^s(x,E)=(f(x),Df(x)E)\) where \(x \in \mathbb {R}^2\) and E belongs to the space G(1, 2) of one-dimensional vector space in \(\mathbb {R}^2\). Recall that the set \(\Lambda \) is a hyperbolic attractor of f with splitting \(E^s\oplus E^u\). Hence, the set \(\Lambda ^s=\Lambda \ltimes E^s =\{(x,E): x\in \Lambda , \, E=E^s(x)\}\) is a hyperbolic set of \(f^s\) where the direction corresponding to the variable in G(1, 2) is uniformly expanding. Thus,

$$\begin{aligned} W^s(\Lambda ^s)=W^s(\Lambda )\ltimes E^s=\{(x,E): x\in W^s(\Lambda ), E=E^s(x)\} \end{aligned}$$

is a two-dimensional manifold in the three-dimensional space \(\mathbb {R}^2\times G(1,2)\) as it is showed in Fig. 5. On the other hand, the unstable manifold of Q contains a folding manifold that we denote by \(\mathcal {S}\). That is, a small piece of the unstable manifold of Q, tangent to the stable manifold of P at a point y. This manifold folds with respect to the stable cone-field of f as it is represented Fig. 5. Thus, identifying the tangent spaces \(T_x\mathbb {R}^2\) with \(\mathbb {R}^2\), we get that the union of tangent spaces \(T_x \mathcal {S}\) where \(x\in \mathcal {S}\) cover the cone \(\mathcal {C}^s_\alpha =\{(x,y): |y|< \alpha |x| \}\cup \{(0,0)\}\) for some small \(\alpha >0\):

$$\begin{aligned} \mathcal {C}^s_\alpha \subset \bigcup _{x\in \mathcal {S}} T_x\mathcal {S}. \end{aligned}$$

This property allows us to see the set

$$\begin{aligned} \mathcal {S}^s{\mathop {=}\limits ^{\scriptscriptstyle \mathrm def}}\mathcal {S}\ltimes T\mathcal {S}=\{(x,E): x\in \mathcal {S}, E=T_x\mathcal {S}\} \end{aligned}$$

as a graph of a function \(E\in \mathcal {C}^s_\alpha \mapsto x=x(E)\in \mathcal {S}\). In other words, \(\mathcal {S}^s\) is a one-dimensional manifold in \(\mathbb {R}^2\times G(1,2)\) which is a graph over \(\mathcal {C}^s_\alpha \) and thus it transversally intersecting \(W^s(\Lambda ^s)\) at the point \((y,E^s)\) where \(E^s=E^s(P)=\mathbb {R}\times \{0\}\). Since this intersection is transversal, it persists for any small perturbation. In particular, for any small perturbation g of f, we get a intersection point between \(\mathcal {S}^s\) and \(W^s(\Lambda _g^s)\) where \(\Lambda _g^s\) is the continuation of \(\Lambda ^s\) for cocycle \(g^s\) induced by g in \(\mathbb {R}^2\times G(1,2)\). Notice that this intersection point between \(\mathcal {S}^s\) and \(W^s(\Lambda ^s_g)\) provides the tangency point and direction between \(\mathcal {S}\) and a stable manifold \(W^s(z)\) for some \(z\in \Lambda _g\). Therefore, we get a robust tangency.

Construction

Now we will give the formal details. Recall the \(C^r\)-diffeomorphism \(f:\mathbb {R}^d \rightarrow \mathbb {R}^d\) in (1) and the attractor \(\Lambda =\{0^{d-2}\}\times \Sigma \) in (2). In what follows we assume that \(r\ge 2\). This set has a well defined hyperbolic structure \(T_\Lambda \mathbb {R}^d=E^s\oplus E^u\) where the stable bundle \(E^s\) of \(\Lambda \) is \((d-1)\)-dimensional. Observe that \(E^s\) can be uniquely extended to a continuous Df-invariant fiber bundle, which we also denote by \(E^s\), over each leaf \(W^s_{loc}(x)\), \(x\in \Lambda \), and so to the whole set \(D_\varepsilon \). Moreover, from the hyperbolicity of \(\Lambda \), we have that \(E^{s}\) varies continuously with respect to the point \(x\in D_{\varepsilon }\) and the diffeomorphism g in a small \(C^1\) neighborhood \(\mathcal {V}\) of f. Thus, for each \(g\in \mathcal {V}\) the set \(D_\varepsilon \) is foliated by \((d-1)\)-dimensional (local) stable manifolds of \(\Lambda _g\) which are tangent to the bundle \(E^s_g\) continuation of \(E^s\).

Fix \(s{\mathop {=}\limits ^{\scriptscriptstyle \mathrm def}}d-1\). On the set \(D_\varepsilon \) we have defined a stable cone-field \(\mathcal {C}^s_\alpha \) of dimension s and size \(\alpha >0\) satisfying

$$\begin{aligned} E^s(x) \in \mathcal {C}^s_\alpha (x)\subset T_x\mathbb {R}^d \quad \text {and} \quad Df^{-1}(x)\mathcal {C}^s_\alpha (f(x))\subset \mathcal {C}^s_\alpha (x)\quad \text{ for } \text{ all } x\in D_\varepsilon . \end{aligned}$$

In what follows, for notational simplicity, we omit the subscript \(\alpha \) in the notation \( \mathcal {C}^s_\alpha \).

Remark 3.1

As usual, by means of the identification of \(T_x\mathbb {R}^d\) with \(\mathbb {R}^d\), we see simultaneously the s-dimensional cone-field \(\mathcal {C}^s\) as family of subset of the Euclidian space \(\mathbb {R}^d\) and an open set of the Grassmannian manifold \(G_{s}(\mathbb {R}^d)=\mathbb {R}^d\times G(s,d)\) where G(sd) is set of the s-planes in \(\mathbb {R}^d\). Observe that in the case \(d=2\), this Grassmannian manifold is the projective space.

Now consider the differential cocycle induced by f on \(G_{s}(\mathbb {R}^d)\) given by

$$\begin{aligned} f^s: G_s(\mathbb {R}^d)\rightarrow G_s(\mathbb {R}^d), \quad f^s(x,E)=(f(x),Df(x)E). \end{aligned}$$

Observe that \(f^s\) is a \(C^{r-1}\)-diffeomorphism of \(G_s(\mathbb {R}^d)\) with \(r\ge 2\). Since \(E^s\) is a repelling point of Df,

$$\begin{aligned} \Lambda ^s=\Lambda \ltimes E^s {\mathop {=}\limits ^{\scriptscriptstyle \mathrm def}}\{(x,E): x\in \Lambda , \ E=E^s(x) \} \end{aligned}$$

is hyperbolic set of \(f^s\) with stable index equals to \(\dim E^s=s\). Namely, the splitting of \(\Lambda ^s\) is of the form \(E^s\oplus E^u\oplus E^{uu}\) where \(E^s\oplus E^u\) corresponds with the splitting of \(\Lambda \) for f and \(E^{uu}\) with the directions over G(sd). On the other hand, the local stable manifold \(W^s_{loc}(\Lambda ^s)\) of \(\Lambda ^s\) contains the set

$$\begin{aligned} D^s_\varepsilon =D_\varepsilon \ltimes E^s {\mathop {=}\limits ^{\scriptscriptstyle \mathrm def}}\{(x,E): x\in D_\varepsilon , \ E=E^s(x) \}. \end{aligned}$$

This is a manifold of codimension the dimension of G(sd).

We now construct the heteroclinic tangency. First, we give a more formal notion of heterodimensional tangency between any two manifolds.

Definition 3.2

Let \(\mathcal {L}\) and \(\mathcal {N}\) be two submanifolds of \(\mathcal {M}\). We say that \(\mathcal {L}\) and \(\mathcal {N}\) has a heteroclinic tangency at \(x\in \mathcal {L}\cap \mathcal {N}\) if \( c_T= d_T - k_T >0 \) where

$$\begin{aligned} d_T=\dim T_x \mathcal {L} \cap T_x \mathcal {N} \quad \text {and} \quad k_T=\dim \mathcal {L} + \dim \mathcal {N} - \dim \mathcal {M}. \end{aligned}$$

The numbers \(d_T=d_T(x,\mathcal {L},\mathcal {N})\), \(c_T=c_T(x,\mathcal {L},\mathcal {N})\) and \(k_T=k_T(x,\mathcal {L},\mathcal {N})\) are called, respectively, dimension, codimension and signed co-index of the tangency between \(\mathcal {L}\) and \(\mathcal {N}\) at x. The tangency is said to be heterodimensional if \(k_T\not = 0\) and equidimensional if \(k_T=0\).

For simplicity and clarity of the exposition we restrict the construction to the case of signed co-index \(k_T=s-1=d-2\). By means of a similar argument one can also get the other possible co-index in Theorem A. We will consider two types of tangencies: elliptical (see Sect. 2) and of saddle type. We recall that a tangency \(y\in W^u(Q)\cap W^s(P)\) is of saddle type if every neighborhood U of y contained in either, \(W^u(Q)\) or \(W^s(P)\), say \(W^u(Q)\), intersects each connected component of \(\mathbb {R}^d\setminus T_y W^u(Q)\).

Example 3.3

Consider a diffeomorphism having two periodic saddles P and Q such that \(0^d\in W^s_{loc}(P)\subset \mathbb {R}^{s}\times \{0\}\), with \(0^d\ne P\) and \(\dim W^u(Q)=s\). Assume that, \(\mathcal {S}([-1,1]^{s})\subset W^u(Q)\), where

$$\begin{aligned} \mathcal {S}:(t_1,\dots ,t_s)\mapsto (t_1,\dots ,t_s, t_1^2+\dots +t_ s^2) \end{aligned}$$
(3)

or

$$\begin{aligned} \mathcal {S}:(t_1,\dots ,t_s)\mapsto (t_1,\dots ,t_s, t_1 t_2+\dots +t_{s-1} t_s). \end{aligned}$$
(4)

Then \(0^d\) is a heteroclinic tangency between \(W^s(P)\) and \(W^s(Q)\) of elliptic type if \(\mathcal {S}\) is as in (3) and the saddle type if \(\mathcal {S}\) is as in (4).

As it is usual, we identified the embedding \(\mathcal {S}\) (as those described above) with its image. Now, using the s-dimensional manifold \(\mathcal {S}\) in (3) and (4) we create a tangency between the leaves the foliation of \(D_\varepsilon \) by stable manifold (of dimension s) of \(\Lambda \). Fix a fixed point \(P\in \Lambda \) and consider \(y\in W^s_{loc}(P)\) with \(y\ne P\). Modifying slightly the construction of the attractor if necessary, we can consider coordinates \((t_1,\dots , t_d)\) in neighborhood of P such that

  • P is identified with \((1^{d-1},0)\) and y with \(0^d\),

  • the local unstable manifold \(W^u_{loc}(P)\) is \(t_1=\dots =t_{d-1}=1\),

  • for each \(z=(1^{d-1},t)\in W^u_{loc}(P)\), the local stable manifold \(W^s_{loc}(z)\) is \(t_d=t\); and

  • the bundle \(E^s\) is constant on this neighborhood.

Hence, in this local coordinates we can assume that \(E^s=\mathbb {R}^s\times \{0\}\) and

$$\begin{aligned} \mathcal {C}^s=\{(u,v)\in \mathbb {R}^s\oplus \mathbb {R}: \Vert v\Vert < \alpha \, \Vert u\Vert \}\cup \{0^d\} \end{aligned}$$

where \(\alpha >0\) is a small constant.

At this coordinates, the folding manifolds \(\mathcal {S}\) in (3) and (4) intersect \(W^s(P)\) at y in a heteroclinic tangency of codimension \(c_T=1\) and signed co-index \(k_T=s-1=d-2\). The next result state that this tangency persist under perturbations.

Proposition 3.4

The folding manifold \(\mathcal {S}\) has a heteroclinic tangency with the stable foliation of \(\Lambda \) which persists under small \(C^1\)-perturbations of f.

The proof of this proposition makes use of the following result:

Lemma 3.5

The set \(\mathcal {S}^s= \mathcal {S}\ltimes T\mathcal {S} {\mathop {=}\limits ^{\scriptscriptstyle \mathrm def}}\{(x,E): x\in \mathcal {S}, \ E= T_x \mathcal {S} \}\) is a manifold of dimension \(\dim G(s,d)\) embedded as a disc in \(\mathbb {R}^d\times G(s,d)\). Namely it is a graph of a function of the form \(E\in \mathcal {C}^s \mapsto x=x(E)\in \mathcal {S}\).

Let us postpone for a while the proof of lemma, to conclude the proof of the proposition.

Proof of Proposition 3.4

Since f is a \(C^r\)-diffeomorphism with \(r\ge 2\) then \(W^s(\Lambda ^s)\) is a \(C^1\)-manifold. Since \(\mathcal {S}\) and \(W^s_{loc}(P)\) has a quadratic tangency at y, we get that \(\mathcal {S}^s\) transversally intersect \(W^s_{loc}(\Lambda ^s)\) at \(y^s=(y,E^s)\). The transversality follows from Lemma 3.5 since \(\mathcal {S}^s\) is a graph function over \(\mathcal {C}^s\) and thus, a \(C^1\)-disc of dimension \(\dim G(s,d)\). On the other hand \(W^{s}_{loc}(\Lambda ^s)\) is a manifold of codimension \(\dim G(s,d)\) which locally near \(y^s\) is contained in \(\mathbb {R}^d\times \{E^s\}\). See Fig. 5. Moreover, notice these type of nonempty intersections are preserved even by \(C^0\)-perturbations of the submanifolds (that is, intersection but not necessarily transverse is preserved), cf. Abraham and Smale (1970, Pag. 737).

Now consider a diffeomorphism g \(C^1\)-close to f. Observe that the cocycle \(g^s\) is a homeomorphism of \({G}_s(\mathbb {R}^d)\) only \(C^0\)-close to \(f^s\). However, \(\Lambda ^s_g=\Lambda _g\ltimes E^s_g\) is still a topological hyperbolic set for \(g^s\) where \(\Lambda _g\) and \(E^s_g\) are the continuations of \(\Lambda \) and \(E^s\) for g. Thus, the set \(W^s(\Lambda ^s_g)\) contains a manifold \(C^0\)-close to \(W^s_{loc}(\Lambda ^s)\) of codimension \(\dim G(s,d)\). Thus we still have an intersection between \(\mathcal {S}^s\) and \(W^s_{loc}(\Lambda ^s_g)\). Observe that if \((x,E)\in \mathcal {S}^s \cap W^s_{loc}(\Lambda ^s_g)\) then \(x\in \mathcal {S}\cap W^s_{loc}(\Lambda _g)\), \(E=E^s_g(x)\) and \(E=T_x\mathcal {S}\). Thus \(E^s_g(x)=T_x\mathcal {S}\). This provides a tangency between \(\mathcal {S}\) and the stable foliation of \(\Lambda _g\) concluding the proof of the proposition. \(\square \)

Remark 3.6

Proposition 3.4 also holds for any small enough \(C^1\)-perturbation of \(\mathcal {S}\). To see this, if the perturbation is \(C^1\)-close then we have a change of variable \(C^1\)-close to the identity sending the perturbed manifold to the folding manifold \(\mathcal {S}\). Hence we get a new diffeomorphism g which is \(C^1\)-close to f. Thus, applying Proposition 3.4 we get a tangency.

Theorem A follows from the above proposition and remark by considering that the folding manifold is contained in the unstable manifold of a hyperbolic fixed point of f of unstable index \(s=d-1\). Observe that the codimension of the tangency is given by the formula \(c_T=d_T-k_T\) where \(d_T\) is the number of tangent directions and \(k_T\) is the signed co-index between the hyperbolic set involved. In this case, \(c_T=s-(s-1)=1\) and \(k_T=s-1=d-2\).

To complete our construction we give the proof of Lemma 3.5.

Proof of Lemma 3.5

To prove that \(\mathcal {S}^s\) is an embedded disc in \(\mathbb {R}^d\times G(s,d)\) we need to show that \(\mathcal {S}^s\) is a graph of a function of the form

$$\begin{aligned} \mathcal {S}^s: E \in \mathcal {C}^s \mapsto x=x(E)\in G(s,d). \end{aligned}$$

To do this, we must associate to E a point \(x\in \mathcal {S}\) such that \(E=T_x\mathcal {S}\). In other words, we need to show that

$$\begin{aligned} \mathcal {C}^s \subset \bigcup _{x\in \mathcal {S}} T_x\mathcal {S}. \end{aligned}$$

As above, we are standing that \(\mathcal {C}^s\) is a small open set in G(sd) centered at \(E^s=\mathbb {R}^s\times \{0\}\) and the tangent space \(T_x\mathcal {S}\) as a vector space of \(\mathbb {R}^d\). Analytically, we need to solve the following problem: given \(E \in \mathcal {C}^s\) we look for \(t=(t_1,\dots ,t_s)\) such that \(E=T_x\mathcal {S}\) where \(x=\mathcal {S}(t)\).

In order to do the calculation, we choose the elliptic form of the folding manifold given in (3). For folding manifold of saddle type in (4) the argument is similar. Hence,

$$\begin{aligned}&T_x\mathcal {S}: (t'_1,\dots ,t'_s) \in \mathbb {R}^s \mapsto (t'_1,\dots ,t'_s,2t_1t'_1+\dots +2t_st'_s) \in \mathbb {R}^d,\\&\quad \text {where }x=\mathcal {S}(t_1,\dots ,t_s). \end{aligned}$$

We write \(E={{\,\mathrm{span}\,}}\langle v_1,\dots ,v_s\rangle \) where \(v_i=(a_{1i},\dots ,a_{di})\) for \(i=1,\dots ,s\). Hence \(E=T_x\mathcal {S}\) if, and only if, \(v_i \in T_x\mathcal {S}\) for all \(i=1,\dots ,s\). Equivalently, if

$$\begin{aligned} t'_{ji}=a^{}_{ji} \quad \text {for }j=1,\dots ,s \text { and} \ \ \ 2t^{}_{1}t'_{1i} + \dots + 2 t^{}_{s}t'_{si}= a^{}_{di} \quad \text {for all }i=1,\dots ,s. \end{aligned}$$

Hence,

$$\begin{aligned} 2 \cdot \begin{pmatrix} a_{11} &{} \dots &{} {a_{s1}} \\ &{} \ddots &{} \\ {a_{1s}} &{} \dots &{} a_{ss} \end{pmatrix} \cdot \begin{pmatrix} t_{1} \\ \vdots \\ t_{s} \end{pmatrix} = \begin{pmatrix} a_{d1} \\ \vdots \\ a_{ds} \end{pmatrix} \end{aligned}$$
(5)

That is, we have a square linear system \(A t = b\) where \(A=A(E)\) and \(b=b(E)\) depends on the vector space E. To find t we need to show that A is an invertible matrix. To do this, we will take as the vector space E the center \(E^s=\mathbb {R}^s\times \{0\}={{\,\mathrm{span}\,}}\langle e_1,\dots e_s \rangle \) of \(\mathcal {C}^s\) where \(e_i\) denotes the vector with 1 in the i-th coordinate and 0’s elsewhere. We get in this case that \(A(E^s)=2 \cdot I_s\) where \(I_s\) is the identity square matrix of order s. Thus \(\det A(E^s)\not = 0\). Then by the continuity for all \(E\in \mathcal {C}^s\) close to \(E^s\) we uniquely solve (5) and thus we find \(t=(t_1,\dots ,t_s)\) such that \(E=T_x\mathcal {S}\) where \(x=\mathcal {S}(t)\). This completes the proof of the lemma. \(\square \)

\(C^1\)-Robust Heterodimensional Tangencies of Large Codimension

Fix \(c_T\ge 1\) and \(s> c_T\). Set \(d=c_T\cdot (s+1)\). A hyperbolic set \(\Lambda \) of a diffeomorphism of a manifold M is said to be a codimension one expanding attractor if for every \(x\in \Lambda \), holds that \(W^u(x) \subset \Lambda \) and \(\dim W^u(x)=\dim M -1\). Let us take a codimension one expanding hyperbolic attractor \(\Lambda \) of a diffeomorphism h on a manifold of dimension \(n=d-s+1\). In order to avoid the problem of classifying the manifold that support these kind of attractors, we set \(\Sigma \) as the Derived from Anosov (by short DA-attractor) in the n-torus \(\mathbb {T}^n\), see (Smale 1967). After that, similar as Fig. 4, we will consider a \(C^r\) diffeomorphism f of \(\mathbb {T}^d\) locally defined on \(D_\varepsilon =[-\varepsilon ,\varepsilon ]^{s-1}\times \mathbb {T}^n\) for a fixed small \(\varepsilon >0\) and \(r\ge 2\) as

$$\begin{aligned} f=g\times h \quad \text {where} \quad g(t)=\lambda t \ \ \text {for} \ \ t\in [-\varepsilon ,\varepsilon ]^{s-1} \ \ \text {and} \ \ 0<\lambda <1. \end{aligned}$$

Notice that the set \(\Lambda =\{0^{s-1}\}\times \Sigma \) is a hyperbolic attractor of f whose basin of attraction contains \(D_\varepsilon \). Moreover, \(E^{s}=\mathbb {R}^{s-1}\times {\tilde{E}}^{s}\) is the stable bundle of \(\Lambda \) where \({\tilde{E}}^s\) is the one-dimensional stable bundle of \(\Sigma \) for h. Thus, \(s=\dim E^{s}\). Analogously as in previous sections, this bundle can be uniquely extended to a Df-invariant bundle over \(D_\varepsilon \) which we also denote by \(E^s\). Consequently the set \(D_\varepsilon \) is foliated by s-dimensional stable manifolds of \(\Lambda \) which are tangent to \(E^s\). This allows us to consider a stable cone-field \(\mathcal {C}^s\) of dimension s defined in whole \(D_\varepsilon \). As in Sect. 3, we defined the differential cocycle \(f^s\) induced by f on \(G_s(\mathbb {R}^d)\). Similarly, we have that the set \(\Lambda ^s_f=\Lambda \ltimes E^s\) is also a hyperbolic set of \(f^s\) with stable index equals to s and whose local stable manifold \(W^s_{loc}(\Lambda ^s_f)\) contains the set \(D^s_\varepsilon =D_\varepsilon \ltimes E^s\). Thus, this manifold has by codimension the dimension of G(sd).

Restricting us to a small ball \(B\subset D_\varepsilon \), we can assume that the stable cone is give by

$$\begin{aligned} \mathcal {C}^s=\{(u,v)\in {\mathbb {R}^s\times \mathbb {R}^{d-s}}: \Vert v\Vert <\alpha \Vert u\Vert \}\cup \{0^d\} \end{aligned}$$
(6)

where \(\alpha >0\) is small enough and \(E^s=\mathbb {R}^s\times \{0^{d-s}\}\). We will consider a folding manifold \(\mathcal {S}\) in B folded with respect to the cone \(\mathcal {C}^s\) which we introduce formally as follows:

Definition 4.1

A manifold \(\mathcal {S}\) of dimension \(k\ge \dim (E^s)\) is called folding manifold in an open ball B folded with respect to the the cone \(\mathcal {C}^s\) if \(\mathcal {S}\subset B\) and the tangent space of \(\mathcal {S}\) covers \(\mathcal {C}^s\): for every \(E\in \mathcal {C}^s\) there is \(x=x(E)\in \mathcal {S}\), varying continuously with respect to E, such that \(E \le T_x\mathcal {S}\), that is, E is a subspace of \(T_x\mathcal {S}\).

Example 4.2

Take \(k=d-c_T=c_T\cdot s\). Let us consider a k-dimensional manifold \(\mathcal {S}\) defined by

$$\begin{aligned} \mathcal {S}:(t_1,\dots ,t_k)\mapsto (t_1,\dots ,t_k,\, \phi _{1},\, \phi _{2}, \dots , \,\phi _{c_T}) \end{aligned}$$

where

$$\begin{aligned} \phi _\ell = \sum _{j=1}^s t_j \,t_{\,(\ell -1)\,s+j} \qquad \text {for}\quad \ell =1,\dots ,c_T \end{aligned}$$
(7)

Hence, for \(x=\mathcal {S}(t_1,\dots ,t_k)\) we have that

$$\begin{aligned} T_x\mathcal {S}: (t'_1,\dots ,t'_k) \mapsto (t'_1,\dots ,t'_k, \phi '_{1},\, \phi '_{2}, \dots , \,\phi '_{c_T}) \end{aligned}$$

where

$$\begin{aligned} \phi '_\ell = \sum _{j=1}^s t'_{j}\, t_{\,(\ell -1)s+j}+t_j \,t'_{\,(\ell -1)\,s+j} \qquad \text {for}\quad \ell =1,\dots ,c_T. \end{aligned}$$

We write \(E={{\,\mathrm{span}\,}}\langle v_1,\dots ,v_s\rangle \in \mathcal {C}^s\) where \(v_i=(a_{1i},\dots ,a_{di})\) for \(i=1,\dots ,s\). Hence \(E\le T_x\mathcal {S}\) if, and only if, \(v_i \in T_x\mathcal {S}\) for all \(i=1,\dots ,s\). Equivalently, if for every \(i=1,\dots ,s\) it holds

$$\begin{aligned} \begin{aligned} a_{ji}&=t'_{j,i}\qquad \text {for}\quad j=1,\dots ,k \\ a_{k+\ell ,i}&=\sum _{j=1}^s t'_{j,\,i} \, t_{(\ell -1)s+j}+t_j t'_{(\ell -1)s+j,\,i} \qquad \text {for}\quad \ell =1,\dots ,c_T. \end{aligned} \end{aligned}$$

This defines a linear system of \(c_T \cdot s\) equations and k variable. Since \(k=c_T \cdot s\) we can write the system in the form \(At=b\) where \(A=A(E)\) is a square matrix of order k and \(b=b(E)\) is a vector in \(\mathbb {R}^k\) depending on the vector space E. To find \(t=(t_1,\dots ,t_k)\) we need to show that A is an invertible matrix. To do this, we will take as the vector space E the center \(E^s=\mathbb {R}^s\times \{0\}={{\,\mathrm{span}\,}}\langle e_1,\dots e_s \rangle \) of \(\mathcal {C}^s\) where \(e_i\) denotes the vector with 1 in the i-th coordinate and 0’s elsewhere. In this case we have that \(\det A(E^s)=2^{s}\). Then, by continuity, for all \(E\in \mathcal {C}^s\) close to \(E^s\) we uniquely solve the equation \(At=b\) and thus we find \(t=(t_1,\dots ,t_s)\) such that \(E\le T_x\mathcal {S}\) where \(x=\mathcal {S}(t)\). Therefore \(\mathcal {S}\) is folding manifold with respect to \(\mathcal {C}^s\).

As a consequence of the definition of folding manifold we get the following lemma:

Lemma 4.3

Let \(\mathcal {S}\) be a folding manifold folded with respect to \(\mathcal {C}^s\). Then the set

$$\begin{aligned} \mathcal {S}^s {\mathop {=}\limits ^{\scriptscriptstyle \mathrm def}}\{(x,E): x\in \mathcal {S}, \ E \le T_x \mathcal {S} \ \text {with} \ \dim E=s \} \end{aligned}$$

contains a manifold of dimension \(\dim G(s,d)\) embedded as a disc in \(\mathbb {R}^d\times G(s,d)\).

Proof

From the definition of folding manifold, we have a continuous function \(E \in \mathcal {C}^s \mapsto x\in \mathcal {S}\) such that \(E\le T_x\mathcal {S}\). This defines a subset of \(\mathcal {S}^s\) which is an embedding given by \(E \in \mathcal {C}^s \mapsto (x,E) \in \mathcal {S}\times G(s,d)\) proving the lemma. \(\square \)

The following result is the analogous to Proposition 3.4 (see also Remark 3.6).

Proposition 4.4

Let \(\mathcal {S}\) be a folding manifold in B of dimension \(k=d-c_T=c_T \cdot s\) folded with respect to \(\mathcal {C}^s\). Then \(\mathcal {S}\) has a heterodimensional tangency of codimension \(c_T\) and signed co-index \(k_T=s-c_T>0\) with the stable foliation of \(\Lambda \) which persists under small \(C^1\)-perturbations of f.

Proof

By assumption if \(x\in B\) then \(E^s(x)=\mathbb {R}^s\times \{0^{d-s}\} \in \mathcal {C}^s\). Thus, we have that \(\mathcal {S}\) has a heterodimensional tangency of codimension \(c_T\) with \(W^s(z)\) for some \(z\in \Lambda \). Indeed, by definition of the folding manifold \(\mathcal {S}\) and the stable bundle \(E^s\) we find \(x\in \mathcal {S}\subset B\) and \(z\in \Lambda \) such that \(E^s(x) \le T_x\mathcal {S}\) and \(E^s(x)=T_xW^s(z)\). Furthermore, the signed co-index of the tangency is \(k_T=s+k-d=k-1=s-c_T>0\) and the codimension is \(d_T-k_T=s-(s-c_T)=c_T\). On the other hand, the point \((x,E^s(x))\) belongs to \(\mathcal {S}^s \cap W^s(\Lambda ^s)\). Moreover, from Lemma 4.3, we have that \(\mathcal {S}^s\) contains a disc of dimension \(\dim G(s,d)\). Additionally, the manifold \(W^s(\Lambda ^s)\) has codimension \(\dim G(s,d)\). Hence \(\mathcal {S}^s\) contains a submanifold which transversally intersects \(W^s(\Lambda ^s)\).

Arguing as in Proposition 3.4, we still have an intersection between \(\mathcal {S}^s\) and \(W^s(\Lambda ^s_g)\) for any \(C^1\)-close diffeomorphism g to f. Thus there is \((x,E)\in \mathcal {S}^s \cap W^s(\Lambda ^s_g)\). Then \(x\in \mathcal {S}\cap W^s(\Lambda _g)\), \(E\le T_x\mathcal {S}\) and \(E=T_xW^s(z)\) for some \(z\in \Lambda _g\). Similar as above, this implies that \(\mathcal {S}\) and \(W^s(z)\) has a heterodimensional tangency of codimension \(c_T\) and signed co-index \(k_T=s-c_T\) concluding the proof of the proposition. \(\square \)

Proof of Theorem B

It suffices to consider that the folding manifold in Proposition 4.4 is contained in the unstable manifold of a hyperbolic fixed point of f of unstable index k. \(\square \)

Discussion and Open Questions

The goal of this paper was to construct heteroclinic tangencies which are robust under \(C^1\) perturbations. This question was proposed in Kiriki and Soma (2012, pag. 3281) where the authors showed the existence of \(C^2\)-robust heterodimensional tangencies. To approach this problem we have constructed \(C^1\)-robust tangencies where one of the hyperbolic sets involved is an attractor. This limitation prevents that our construction could be carried on a heterodimensional cycle. A diffeomorphism has a heterodimensional cycle associated with two transitive hyperbolic sets if these sets have different indices (dimension of the stable bundle) and their invariant manifolds meet cyclically. This cycle is called non-transverse (heterodimensional) cycle if besides its cyclic intersections involves some heterodimensional tangency. In order to construct a robust non-transverse heterodimensional cycle one must construct the tangency involving hyperbolic sets which are not attractors. This leads to our first question:

Question 1

Is it possible to construct \(C^1\)-robust non-transverse heterodimensional cycles?

Bearing in mind the classic constructions of robust homoclinic tangencies and heterodimensional cycles (Newhouse 1979; Bonatti and Díaz 2008) via the unfolding of tangencies and cycles associated with saddles, we ask the following:

Question 2

Can a diffeomorphism f having a non-transverse heterodimensional cycle associated with saddles P and Q be \(C^r\)-approximated by a diffeomorphism g with a \(C^r\)-robust non-transverse heterodimensional cycle associated with hyperbolic sets containing the continuations \(P_g\) and \(Q_g\) of P and Q?

On the other hand, we also deal in this paper with the construction of heterodimensional tangencies with signed co-index \(k_T>0\) of large codimension. Robust tangencies of large codimension were discovered in Barrientos and Raibekas (2017). Namely, the authors provided a method to construct \(C^2\)-robust bundle tangencies which are non-trivial intersection between different fiber bundles. Bundle tangencies include homoclinic, heterodimensional and equidimensional tangencies. Recently in Barrientos and Raibekas (2019), using ideas similar to this paper, we have constructed new examples of robust homoclinic tangencies of large codimension. The construction also uses an abstract notion of folding manifold with respect to a cone-field extending previous approach on robust homoclinic tangencies in Bonatti and Díaz (2012). However, as in the case of this work, the construction are limited to consider high dimensional manifolds. The lower possible dimension that allows to have a homoclinic tangency of large codimension is \(d=4\). Similarly, \(d=5\) is the lower dimension to construct a large heterodimensional tangency with signed co-index \(k_T>0\). Thus we address the following questions:

Question 3

Is it possible to build a robust heterodimensional tangency with signed co-index \(k_T>0\) (resp. homoclinic tangency) of codimension \(c_T = 2\) in dimension \(d=5\) (resp. \(d=4\))?

References

  1. Abraham, R., Smale, S.: Nongenericity of \(\Omega \)-stability. In: Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., pp. 5–8 (1970)

  2. Asaoka, M.: Hyperbolic sets exhibiting \(C^1\)-persistent homoclinic tangency for higher dimensions. Proc. Amer. Math. Soc. 136, 677–686 (2008)

    MathSciNet  Article  Google Scholar 

  3. Barrientos, P.G., Raibekas, A.: Robust tangencies of large codimension. Nonlinearity 30, 4369–4409 (2017)

    MathSciNet  Article  Google Scholar 

  4. Barrientos, P.G., Raibekas, A.: Robust nongeneric unfoldings of cycles and tangencies. arXiv:1907.01089 (2019)

  5. Bonatti, C., Díaz, L.: Robust heterodimensional cycles and \(C^1\)-generic dynamics. J. Inst. Math. Jussieu 7, 469–525 (2008)

    MathSciNet  Article  Google Scholar 

  6. Bonatti, C., Díaz, L.J.: On maximal transitive sets of generic diffeomorphisms. Publ. Math., Inst. Hautes Étud. Sci. 96, 171–197 (2003)

    MathSciNet  Article  Google Scholar 

  7. Bonatti, C., Díaz, L.J.: Abundance of \(C^1\)-homoclinic tangencies. Trans. Amer. Math. Soc. 264, 5111–5148 (2012)

    Article  Google Scholar 

  8. Díaz, L.J., Kiriki, S., Shinohara, K.: Blenders in centre unstable Hénon-like families: with an application to heterodimensional bifurcations. Nonlinearity 27, 353–378 (2014)

    MathSciNet  Article  Google Scholar 

  9. Díaz, L.J., Nogueira, A., Pujals, E.R.: Heterodimensional tangencies. Nonlinearity 19, 2543 (2006)

    MathSciNet  Article  Google Scholar 

  10. Díaz, L.J., Pérez, S.A.: Hénon-like families and blender-horseshoes at non-transverse heterodimensional cycles. Int. J. Bifurc. Chaos Appl. Sci. Eng. 29(3), 1930006 (2019)

    Article  Google Scholar 

  11. Kiriki, S., Soma, T.: \(C^2\)-robust heterodimensional tangencies. Nonlinearity 25, 3277 (2012)

    MathSciNet  Article  Google Scholar 

  12. Mañé, R.: A proof of the \(C^1\) stability conjecture. Publications Mathématiques de L’Institut des Hautes Scientifiques 66, 161–210 (1987)

    Article  Google Scholar 

  13. Newhouse, S.E.: The abundance of wild hyperbolic sets for diffeomorphisms. Publications Mathématiques de L’Institut des Hautes Scientifiques 50, 101–151 (1979)

    Article  Google Scholar 

  14. Palis, J.: A differentiable invariant of topological conjugacies and moduli of stability. Asterisque 51, 335–346 (1978)

    MathSciNet  MATH  Google Scholar 

  15. Robinson, C.: Stability, Symbolic Dynamics, and Chaos, 2nd edn. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1999) ISBN: 0-8493-8495-837-01

  16. Simon, C.P.: A 3-dimensional Abraham–Smale example. Proc. Amer. Math. Soc. 34, 629–630 (1972)

    MathSciNet  MATH  Google Scholar 

  17. Smale, S.: Differentiable dynamical systems. Bulletin of the American MaAthematical Society 73, 747–817 (1967)

    MathSciNet  Article  Google Scholar 

  18. Williams, R.F.: The “\({\rm DA}\)” maps of Smale and structural stability. In: Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., pp. 329–334 (1970)

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Acknowledgements

We are grateful to Artem Raibekas for discussions and helpful suggestions. During the preparation of this article PB was supported by MTM2017-87697-P from Ministerio de Economía y Competividad de España and CNPQ-Brasil. SP were partially supported by CMUP (UID/MAT/00144/2019), which is funded by FCT with national (MCTES) and European structural funds through the programs FEDER, under the partnership agreement PT2020. SP also acknowledges financial support from a postdoctoral grant of the project PTDC/MAT-CAL/3884/2014

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Barrientos, P.G., Pérez, S.A. Robust Heteroclinic Tangencies. Bull Braz Math Soc, New Series 51, 1041–1056 (2020). https://doi.org/10.1007/s00574-019-00185-6

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Keywords

  • Folding manifolds
  • Robust equidimensional tangencies
  • Robust heterodimensional tangencies