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 nontransverse 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
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
where \({{\,\mathrm{ind}\,}}(\Sigma )\) denotes the stable index of a (transitive) hyperbolic set \(\Sigma \). The integer \(k_T\) is called signed coindex. Notice that when \(k_T>0\) this number coincides with the classical coindex between \(\Lambda \) and \(\Gamma \). Moreover, \(k_T>0\) if and only if
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.
Heterodimensional tangencies with signed coindex \(k_T>0\) 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 nondominated 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 coindex \(k_T=d2\) in any manifold of dimension \(d\ge 3\). Recently in Barrientos and Raibekas (2017) new examples of \(C^2\)robust heterodimensional tangencies with \(0<c_T\le \lfloor (d3)/2 \rfloor \) and \(1\le k_T \le d22c_T\) were also constructed in any manifold of dimension \(d\ge 5\). In the same work, \(C^2\)robust heteroclinic tangencies with signed coindex \(k_T\le 0\) were also obtained.
Kiriki and Soma in (2012, page 3281) wrote that \(C^1\)robust heterodimensional tangencies with positive signed coindex 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 coindex \(0 \le k_T \le d2\).
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 coindex \(0<k_T<d1\).
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 ddimensional torus \(\mathbb {T}^d\) with \(d=c_T \cdot (s+1)\) having a \(C^1\)robust heterodimensional tangency of codimension \(c_T\) and signed coindex \(k_T=sc_T>0\).
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 twodimensional 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.
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 \({{\,\mathrm{ind}\,}}(\Lambda )=d1\) and whose attracting region is a ddimensional connected set foliated by \((d1)\)dimensional stable submanifolds. After that, we consider a fixed point P in \(\Lambda \) and another fixed point Q of f of stable index \(d1k\) where \(0\le k\le d2\) creating a heteroclinic tangency between \(W^s(P)\) and \(W^u(Q)\) of signed coindex \(k_T=k\), 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 twodimensional 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 \(D_\varepsilon {\mathop {=}\limits ^{\scriptscriptstyle \mathrm def}}[\varepsilon ,\varepsilon ]^{d2}\times D\) is given by
Thus, the set
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 \(d1\). 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
Consider a hyperbolic fixed point \(Q \not \in D_\varepsilon \) of f with stable index \(d1k\). By means of a homotopic deformation, we force to the \((k+1)\)dimensional unstable manifold \(W^{u}(Q)\) intersects nontransversely 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 DAattractor instance a Plykin attractor.
Thus, this last diffeomorphism, that again we call f, has a heteroclinic tangency of codimension \(c_T=1\) and signed coindex \(k_T=k\) with \(0\le k\le d2\), 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
Since y is a heteroclinic tangency of elliptic type between \(W^{u}(Q)\) and \(W^{s}(P)\) we can assume that (see Fig. 3)
Our conditions imply that for each \(g\in {\mathcal {V}}\), the set
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 coindex \(k_T=k\) with \(0 \le k\le d2\). This completes the proof of Theorem A.
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.
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 onedimensional 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,
is a twodimensional manifold in the threedimensional 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 conefield 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 \(\alpha >0\):
This property allows us to see the set
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 onedimensional 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 \(\Lambda =\{0^{d2}\}\times \Sigma \) 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 \((d1)\)dimensional. Observe that \(E^s\) can be uniquely extended to a continuous Dfinvariant 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 \((d1)\)dimensional (local) stable manifolds of \(\Lambda _g\) which are tangent to the bundle \(E^s_g\) continuation of \(E^s\).
Fix \(s{\mathop {=}\limits ^{\scriptscriptstyle \mathrm def}}d1\). On the set \(D_\varepsilon \) we have defined a stable conefield \(\mathcal {C}^s_\alpha \) of dimension s and size \(\alpha >0\) satisfying
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 sdimensional conefield \(\mathcal {C}^s\) 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(s, d) is set of the splanes 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
Observe that \(f^s\) is a \(C^{r1}\)diffeomorphism of \(G_s(\mathbb {R}^d)\) with \(r\ge 2\). Since \(E^s\) is a repelling point of Df,
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(s, d). On the other hand, the local stable manifold \(W^s_{loc}(\Lambda ^s)\) of \(\Lambda ^s\) contains the set
This is a manifold of codimension the dimension of G(s, d).
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
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 coindex 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 coindex \(k_T=s1=d2\). By means of a similar argument one can also get the other possible coindex 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
or
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 sdimensional 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 \((1^{d1},0)\) and y with \(0^d\),

the local unstable manifold \(W^u_{loc}(P)\) is \(t_1=\dots =t_{d1}=1\),

for each \(z=(1^{d1},t)\in W^u_{loc}(P)\), 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
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 coindex \(k_T=s1=d2\). 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 \(s=d1\). Observe that the codimension of the tangency is given by the formula \(c_T=d_Tk_T\) where \(d_T\) is the number of tangent directions and \(k_T\) is the signed coindex between the hyperbolic set involved. In this case, \(c_T=s(s1)=1\) and \(k_T=s1=d2\).
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
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
As above, we are standing that \(\mathcal {C}^s\) is a small open set in G(s, d) 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,
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
Hence,
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 ith 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 \(n=ds+1\). 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 DAattractor) in the ntorus \(\mathbb {T}^n\), see (Smale 1967). After that, similar as Fig. 4, we will consider a \(C^r\) diffeomorphism f of \(\mathbb {T}^d\) locally defined on \(D_\varepsilon =[\varepsilon ,\varepsilon ]^{s1}\times \mathbb {T}^n\) for a fixed small \(\varepsilon >0\) and \(r\ge 2\) as
Notice that the set \(\Lambda =\{0^{s1}\}\times \Sigma \) is a hyperbolic attractor of f whose basin of attraction contains \(D_\varepsilon \). Moreover, \(E^{s}=\mathbb {R}^{s1}\times {\tilde{E}}^{s}\) is the stable bundle of \(\Lambda \) where \({\tilde{E}}^s\) is the onedimensional stable bundle of \(\Sigma \) for h. Thus, \(s=\dim E^{s}\). Analogously as in previous sections, this bundle can be uniquely extended to a Dfinvariant bundle over \(D_\varepsilon \) which we also denote by \(E^s\). Consequently the set \(D_\varepsilon \) is foliated by sdimensional stable manifolds of \(\Lambda \) which are tangent to \(E^s\). This allows us to consider a stable conefield \(\mathcal {C}^s\) 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(s, d).
Restricting us to a small ball \(B\subset D_\varepsilon \), we can assume that the stable cone is give by
where \(\alpha >0\) is small enough and \(E^s=\mathbb {R}^s\times \{0^{ds}\}\). 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 \(k=dc_T=c_T\cdot s\). Let us consider a kdimensional manifold \(\mathcal {S}\) defined by
where
Hence, for \(x=\mathcal {S}(t_1,\dots ,t_k)\) we have that
where
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
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 ith 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
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 \(k=dc_T=c_T \cdot s\) folded with respect to \(\mathcal {C}^s\). Then \(\mathcal {S}\) has a heterodimensional tangency of codimension \(c_T\) and signed coindex \(k_T=sc_T>0\) with the stable foliation of \(\Lambda \) which persists under small \(C^1\)perturbations of f.
Proof
By assumption if \(x\in B\) then \(E^s(x)=\mathbb {R}^s\times \{0^{ds}\} \in \mathcal {C}^s\). 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 coindex of the tangency is \(k_T=s+kd=k1=sc_T>0\) and the codimension is \(d_Tk_T=s(sc_T)=c_T\). 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 coindex \(k_T=sc_T\) 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 nontransverse (heterodimensional) cycle if besides its cyclic intersections involves some heterodimensional tangency. In order to construct a robust nontransverse 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 nontransverse 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 nontransverse heterodimensional cycle associated with saddles P and Q be \(C^r\)approximated by a diffeomorphism g with a \(C^r\)robust nontransverse 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 coindex \(k_T>0\) 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 nontrivial 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 conefield 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 coindex \(k_T>0\). Thus we address the following questions:
Question 3
Is it possible to build a robust heterodimensional tangency with signed coindex \(k_T>0\) (resp. homoclinic tangency) of codimension \(c_T = 2\) in dimension \(d=5\) (resp. \(d=4\))?
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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 MTM201787697P from Ministerio de Economía y Competividad de España and CNPQBrasil. 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/MATCAL/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/s00574019001856
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Keywords
 Folding manifolds
 Robust equidimensional tangencies
 Robust heterodimensional tangencies