Robust Degenerate Unfoldings of Cycles and Tangencies


We construct open sets of degenerate unfoldings of heterodimensional cycles of any co-index \(c>0\) and homoclinic tangencies of arbitrary codimension \(c>0\). These type of sets are known to be the support of unexpected phenomena in families of diffeomorphisms, such as the Kolmogorov typical co-existence of infinitely many attractors. As a prerequisite, we also construct robust homoclinic tangencies of large codimension which cannot be inside a strong partially hyperbolic set.

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We thank P. Berger who told us the ideais behind his proof in [10]. We are grateful to J. Rojas for helping us to understand P. Berger’s articles and for the initial discussions on this project. We also thank the anonymous referee for his comments and pointing out some problems. Funding was provided by Ministerio de Educación, Cultura y Deporte (Grant No. MTM2017-87697-P) and Conselho Nacional de Desenvolvimento Científico e Tecnológico.

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Appendix A: Estimates for the Folding Manifold of Example 3.2

Appendix A: Estimates for the Folding Manifold of Example 3.2

We consider the \((ss+c)\)-dimensional \(C^\infty \)-manifold of Example 3.2 given by

$$\begin{aligned} \mathcal {S}:[-2,2]^{ss}\times [-\epsilon ,\epsilon ]^c \rightarrow \mathbb {R}^m, \ \ \mathcal {S}(x,t)=(x,(t_1,\dots ,t_u),h(t)) \in \mathbb {R}^{ss}\times \mathbb {R}^u\times \mathbb {R}^c \end{aligned}$$

where \(t=(t_1,\dots ,t_u,\dots ,t_c) \in [-\epsilon ,\epsilon ]^c\) and \(h(t)=(h_1(t),\dots ,h_c(t))\) with

$$\begin{aligned} h_i(t)= \sum _{j=0}^{u-1} t_{j+1} t_{ju+i} \quad \text {for} i=1,\dots ,u \quad \hbox {and} \quad h_i(t)=t_i \quad \text {for} i=u+1,\dots ,c. \end{aligned}$$

Let \(t=t(E)\) be the \(C^{r-1}\)-function on \(\mathcal {C}^{{G}}\) computes in Example 3.2. We have that:

Proposition A.1

For every \(\varepsilon >0\) suffices small there is a neighborhood \(\mathcal {C}^{{G}}\) of \(E^u\) such that

$$\begin{aligned} C=\Vert Dh\Vert _\infty \cdot \max \{1,\Vert Dt\Vert _\infty \} \le 1+\varepsilon \quad \text {over}\,\mathcal {C}^{{G}}. \end{aligned}$$

Thus, \(\mathcal {S}\) is a \(({\alpha ,}\nu ,\delta )\)-folding manifold for any \(\nu >0\) such that \((1+\varepsilon )\nu <\delta \).


First of all, notice that for all \(t\in [-\epsilon ,\epsilon ]^c\), it holds that

$$\begin{aligned} \Vert Dh(t)\Vert _{\infty } {\mathop {=}\limits ^{\scriptscriptstyle \mathrm def}}\max _{i=1,\dots , c} | Dh_i(t)| =\max \bigg \{ \,1, \ \max _{i=1,\dots ,u} \big |\sum _{j=0}^{u-1} t_{j+1}+t_{ju+i} \big |\,\bigg \}. \end{aligned}$$

Hence, taking \(\epsilon >0\) small enough we have that \(\Vert Dh\Vert _\infty =1\). On the other hand, by Cramer’s rule we get that the solution of the linear system \(At= \mathbf {c}\) is given by

$$\begin{aligned} t_i = \frac{\det A_i^*}{\det A} \qquad \text {for}\,i=1,\dots ,c \end{aligned}$$

where \(A^*_{i}\) is the matrix formed by replacing the ith column of A by the column vector \(\mathbf {c}\). In order to compute Dt we write the variable of t by \(E=\langle v_{1},\dots ,v_u\rangle \) with \(v_k=(a_k,b_k,c_k)\in \mathbb {R}^{ss}\times \mathbb {R}^{u}\times \mathbb {R}^c\) and then

$$\begin{aligned} Dt=(\partial _{v_1}t,\dots ,\partial _{v_u}t) \quad \text {with} \ \ \ \partial _{v_k} t=(\partial _{a_k}t,\partial _{b_k}t,\partial _{c_k}t) \quad \text {for}\, k=1,\dots ,u. \end{aligned}$$

Moreover, since the matrix A does not depend on the variables \(a_k\) we get that \(\partial _{a_k}t=0\). In the sequel we will use the symbol \(D_{k}\) to denote any partial derivative of the form \(\partial _{b_{k\iota }}\) or \(\partial _{c_{k\iota }}\). For each \(i=1,\dots ,c\), using Jacobi’s formula it follows that

$$\begin{aligned} D_kt_i =\frac{D_k(\det A^*_i)-\mathrm {tr}(A^{-1} \cdot D_kA ) \cdot \det A^*_i}{\det A}. \end{aligned}$$

In particular, since \(\mathbf {c}(E^u)=0\) then \(\det A^*_i(E^u)=0\), and \(\det A(E^u)=2\), we obtain that

$$\begin{aligned} D_kt_i(E^u)=\frac{1}{2} \, D\det A^*_i|_{E^u} = \frac{1}{2} \, \mathrm {tr}(\mathrm {Adj}(A^*_i) \cdot D_kA^*_i)|_{E^u} \end{aligned}$$

where \(\mathrm {Adj}(A^*_i)\) is the adjugate matrix of \(A^*_i\). Notice that

$$\begin{aligned}&\mathrm {Adj}(A^*_i(E^u))=(C_{\ell j})^T \quad \text {with} \ \ C_{\ell j} =0 \\&\quad \mathrm{if} \ j\not =i \quad \mathrm{and} \quad C_{\ell i}=(-1)^{\ell +i}\cdot \sigma _i \quad \mathrm{for}\,\ell =1,\dots ,c \end{aligned}$$

where \(\sigma _i=2\) if \(i\not =1\) and \(\sigma _i=1\) otherwise (\(i=1\)). From this follows that

$$\begin{aligned} \mathrm {tr}(\mathrm {Adj}(A^*_i) \cdot D_kA^*_i)|_{\,(E^u)}= (C_{1i},\dots ,C_{ci})\cdot D_k\mathbf {c}(E^u). \end{aligned}$$

If \(D_k\) is either, \(\partial _{b_{k\iota }}\) for \(\iota =1,\dots ,u\) or \(\partial _{c_{k\iota }}\) for \(\iota =u+1,\dots ,c\) then \(D_k\mathbf {c}=0\) and thus \(D_kt_i(E^u)=0\). Otherwise,

$$\begin{aligned} (C_{1i},\dots ,C_{ci})\cdot D_k\mathbf {c}(E^u) = C_{(k-1)u+\iota \, i} \quad \text {and hence} \quad D_kt_i(E^u)=\frac{1}{2} \, C_{(k-1)u+\iota \, i}. \end{aligned}$$


$$\begin{aligned}&\Vert Dt(E^u)\Vert _{\infty }=\frac{1}{2} \ \max _{i=1,\dots ,c} \ \max _{k=1,\dots ,u} \\&\quad \big |\sum _{\iota =1}^{u} C_{(k-1)u+\iota \, i} \big |= \frac{1}{2} \ \max _{i=1,\dots ,c} \ \max _{k=1,\dots ,u} \ \big |\sum _{\iota =1}^u (-1)^{i+(k-1)u+\iota } \sigma _i \big | \le 1. \end{aligned}$$

Since Dt varies continuously with respect to E we have that \(\Vert Dt(E)\Vert _\infty \) is close to \(\Vert Dt(E^u)\Vert _\infty \) for any E close enough to \(E^u\). Thus, shrinking \(\mathcal {C}^{{G}}\) if necessary, this implies that \(\Vert Dt\Vert _\infty \le 1+\varepsilon \) over \( \mathcal {C}^{{G}}_\alpha \). Hence, \(C=\Vert Dh\Vert _\infty \cdot \max \{1,\Vert Dt\Vert _\infty \} \le 1+\varepsilon \) for a fixed but arbitrarily small \(\varepsilon >0\).

This \(({\alpha ,}\nu ,\delta )\)-folding \(C^r\)-manifold \(\mathcal {S}\) induces a \(C^{r-1}\)-disc

$$\begin{aligned} \mathcal {H}^{{G}}: [-2,2]^{ss}\times \mathcal {C}^{{G}}_\alpha \longrightarrow G_u(\mathbb {R}^m), \qquad \mathcal {H}^{{G}}(x,E)=(\mathcal {S}(x,t), E) \in \overline{\mathcal {B}}\times \mathcal {C}^{{G}}= \overline{\mathcal {B}^{{G}}} \end{aligned}$$

Set \(h^{{G}}=\mathscr {P}\circ \mathcal {H}^{{G}}\). Here \(\mathscr {P}\) denotes the standard projection onto \(\mathbb {R}^c\). We consider

$$\begin{aligned} \widehat{h}^{{G}}(J(\xi ^{{G}}))=J(h^{{G}}\circ \xi ^{{G}}) \quad \text {for}\, \ \ \xi ^{{G}}=(\xi ^{{G}}_a)_a\in C^d(\mathbb {R}^k,\mathbb {R}^{ss}\times G(u,m)) \end{aligned}$$


$$\begin{aligned} J(\xi ^{{G}})=((\xi _0,E_0),\partial ^1\xi ^{{G}},\dots ,\partial ^d\xi ^{{G}}) \in \big ([-2,2]^{ss}\times \mathcal {C}^{{G}}\big ) \times \overline{B}_\rho (0) \end{aligned}$$

where \(\overline{B}_\rho (0)\) denotes a closed ball of radius \(\rho >0\) at 0 velocity of the jets over \(\mathbb {R}^{ss}\times G(u,m)\).

Proposition A.2

For every \(\varepsilon >0\) suffices small there are \(\rho >0\) and a neighborhood \(\mathcal {C}^{{G}}\) of \(E^u\) so that

$$\begin{aligned} \Vert D\widehat{h}^{{G}}\Vert _\infty \le 1+\varepsilon \quad \text {over} \ \ \big ([-2,2]^{ss}\times \mathcal {C}^{{G}}\big ) \times \overline{B}_\rho (0). \end{aligned}$$

Thus, \(\widehat{\mathcal {H}}^{{G}}\) is a \(({\alpha ,}\nu ,\delta )\)-horizontal disc for any \(\nu >0\) such that \((1+\varepsilon )\nu <\delta \).


Observe that \(h^{{G}}(E)=h(t(E))\). In this way,

$$\begin{aligned} \widehat{h}^{{G}}(J(E))=J(h^{{G}}\circ E) =(h(t(E_a)),\partial ^1_a h(t(E_a)),\dots ,\partial ^d_a h(t(E_a)))_{|_{\,a=0}} \end{aligned}$$

for \(E=(E_a)_a \in C^d(\mathbb {I}^k,G(u,m))\) with \(E_0\in \mathcal {C}^{{G}}_\alpha \). Denoting \(t_a=t(E_a)\) for all \(a\in \mathbb {I}^k\), we can rewrite (A.1) as

$$\begin{aligned} \widehat{h}^{{G}}(J(t))=(h(t_a),\partial ^1_ah(t_a),\dots ,\partial ^d_ah(t_a))_{|_{\,a=0}} \quad \text {where}\,t=(t_a)_a \in C^d(\mathbb {I}^k,\mathbb {R}^c) \end{aligned}$$

with \(t_0\) small enough in norm. Therefore,

$$\begin{aligned} D\widehat{h}^{{G}}= \frac{d\widehat{h}^{{G}}}{dJ(t)} \cdot \frac{dJ(t)}{dJ(E)}= \frac{d\widehat{h}^{{G}}}{dJ(t)} \cdot \frac{d \widehat{t}}{dJ(E)} \end{aligned}$$

where \(\widehat{t}(J(E))=J(t\circ E)\). We want to compute \(\Vert D\widehat{h}^{{G}}(J(E^u))\Vert _{\infty }\) where \(E^u=(E^u_a)_a\) with \(E^u_a=E^u_0=\{0^{ss}\}\times \mathbb {R}^u\times \{0^c\}\) for all \(a\in \mathbb {I}^k\). Hence \(J(E^u)=(E^u_0,0)\). By a straightforward calculation using Faà di Bruno’s formula, we have

$$\begin{aligned} \big \Vert \frac{d\widehat{t}}{dJ(E)}(J(E^u))\big \Vert _{\infty } = \Vert Dt(E^u_0)\Vert _\infty \le 1. \end{aligned}$$

By means of a similar computation we can show that

$$\begin{aligned} \big \Vert \frac{d\widehat{h}^{G}}{dJ(t)}(J(t^{u}))\big \Vert _{\infty } = \Vert Dh(0)\Vert _\infty = 1 \end{aligned}$$

where \(t^{u}=(t^{u}_a)_a\) with \(t^{u}_a=t(E^u_a)=t(E^u_0)=0\) for all \(a\in \mathbb {I}^k\) and hence \(J(t^{u})=0\). Finally putting together (A.2)–(A.4) we get that

$$\begin{aligned} \Vert D\widehat{h}^{{G}}(J(E^u))\Vert _\infty \le \big \Vert \frac{d\widehat{h}^{{G}}}{dJ(t)}(J(t^{{G}})) \big \Vert \cdot \big \Vert \frac{dJ(t)}{dJ(E)} (J(E^u)) \big \Vert \le 1. \end{aligned}$$

By continuity with respect to J(E), shrinking \(\mathcal {C}^{{G}}\) if necessary and taking \(\rho >0\) small enough, we have \(\Vert D\widehat{h}^{{G}}\Vert _{\infty } \le 1+\varepsilon \) over \(\big ([-2,2]^{ss}\times \mathcal {C}^{{G}}_\alpha \big ) \times \overline{B}_\rho (0)\) for a fixed but arbitrarily small \(\varepsilon >0\). This completes the proof.

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Barrientos, P.G., Raibekas, A. Robust Degenerate Unfoldings of Cycles and Tangencies. J Dyn Diff Equat (2020).

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  • Homoclinic tangencies
  • Heterodimensional cycles
  • Blenders
  • Parablenders

Mathematics Subject Classification

  • Primary: 58F15
  • 58F17
  • Secondary: 53C35