Robust Degenerate Unfoldings of Cycles and Tangencies

Abstract

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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References

  1. 1.

    Avila, A., Crovisier, S., Wilkinson, A.: \({C}^1\) density of stable ergodicity. arXiv preprint arXiv:1709.04983 (2017)

  2. 2.

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

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bochi, J., Bonatti, C., Díaz, L.J.: Robust criterion for the existence of nonhyperbolic ergodic measures. Commun. Math. Phys. 344(3), 751–795 (2016)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Berger, P., Crovisier, S., Pujals, E.: Iterated functions systems, blenders and parablenders. In: Fractals and Related Fields III: Proceedings of the Conference, Ile de Porquerolles (France), 2015 (in press) (2016)

  5. 5.

    Bonatti, C., Díaz, L.J.: Persistent nonhyperbolic transitive diffeomorphisms. Ann. Math. (2) 43(2), 357–396 (1996)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bonatti, C., Díaz, L.J.: Robust heterodimensional cycles and \(C^1\)-generic dynamics. J. Inst. Math. Jussieu 7(3), 469–525 (2008)

    MathSciNet  Article  Google Scholar 

  7. 7.

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

    Article  Google Scholar 

  8. 8.

    Bonatti, C., Díaz, L.J., Viana, M.: Dynamics Beyond Uniform Hyperbolicity, volume 102 of Encyclopaedia of Mathematical Sciences. Springer, Berlin; A Global Geometric and Probabilistic Perspective. Mathematical Physics, III (2005)

  9. 9.

    Berger, P.: Generic family with robustly infinitely many sinks. Inventiones Mathematicae 205(1), 121–172 (2016)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Berger, P.: Emergence and non-typicality of the finiteness of the attractors in many topologies. Proc. Steklov Inst. Math. 297(1), 1–27 (2017)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Berger, P.: Generic family displaying robustly a fast growth of the number of periodic points. ArXiv e-prints (2017)

  12. 12.

    Biebler, S.: Persistent Homoclinic Tangencies and Infinitely Many Sinks for Residual Sets of Automorphisms of Low Degree in \({C}^{3}\) (2016)

  13. 13.

    Barrientos, P.G., Ki, Y., Raibekas, A.: Symbolic blender-horseshoes and applications. Nonlinearity 27(12), 2805 (2014)

    MathSciNet  Article  Google Scholar 

  14. 14.

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

    MathSciNet  Article  Google Scholar 

  15. 15.

    Barrientos, P.G., Raibekas, A.: Robustly non-hyperbolic transitive symplectic dynamics. Discrete Contin. Dyn. Syst. A 38(12), 5993 (2018)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Dujardin, R.: Non-density of stability for holomorphic mappings on \({\mathbb{P}} ^k\). Journal de l’École polytechnique-Mathématiques 4, 813–843 (2017)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Gavrilov, N.K., Šil’nikov, L.P.: On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve. I. Sb. Math. 17(4), 467–485 (1972)

    Article  Google Scholar 

  18. 18.

    Gonchenko, S.V., Shilnikov, L.P., Turaev, D.V.: On dynamical properties of multidimensional diffeomorphisms from newhouse regions: I. Nonlinearity 21(5), 923 (2008)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Gonchenko, S.V., Turaev, D.V., Shil’nikov, L.P.: On the existence of Newhouse regions in a neighborhood of systems with a structurally unstable homoclinic Poincaré curve (the multidimensional case). Dokl. Akad. Nauk 329(4), 404–407 (1993)

    MATH  Google Scholar 

  20. 20.

    Krupka, M., Krupka, D.: Jets and contact elements. In: Proceedings of the Seminar on Differential Geometry, Vol. 2, pp. 39–85. Mathematical Publications (2000)

  21. 21.

    Michor, P.W.: Manifolds of Differentiable Mappings. Shiva Mathematics Series. Shiva Pub, Jawali (1980)

    Google Scholar 

  22. 22.

    Newhouse, S.E.: Nondensity of axiom A(a) on S2. In: Global Analysis (Proceedings of the Symposium Pure Mathematics, Vol. XIV, Berkeley, CA 1968), pp. 191–202. American Mathematical Society, Providence (1970)

  23. 23.

    Newhouse, S.E.: Diffeomorphisms with infinitely many sinks. Topology 13, 9–18 (1974)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Newhouse, S.E.: The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms. Inst. Hautes Études Sci. Publ. Math. 50, 101–151 (1979)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Nassiri, M., Pujals, E.R.: Robust transitivity in Hamiltonian dynamics. Ann. Sci. Éc. Norm. Supér. 45(2), 191–239 (2012)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Pugh, C., Shub, M.: Stably ergodic dynamical systems and partial hyperbolicity. In: International Conference on Dynamical Systems (Montevideo, 1995), pp. 182–187, Longman, Harlow (1996). Pitman Research Notes Mathematics Series

  27. 27.

    Palis, J., Viana, M.: High dimension diffeomorphisms displaying infinitely many periodic attractors. Ann. Math. (2) 140(1), 207–250 (1994)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Rojas, J.D.: Generic Amilies Exhibiting Infinitely Many Non-uniform Hyperbolic Attractors for a Set of Total Measure of Parameters. PhD thesis, Instituto de Matemática Pura e Aplicada (IMPA) (2017)

  29. 29.

    Romero, N.: Persistence of homoclinic tangencies in higher dimensions. Ergod. Theory Dyn. Syst. 15(4), 735–757 (1995)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Taflin, J.: Blenders Near Polynomial Product Maps of \({\mathbb{C}}^2\) (2017)

Download references

Acknowledgements

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 \((1+\varepsilon )\nu <\delta \).

Proof

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,

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

Therefore

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

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

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

with

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

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 \((1+\varepsilon )\nu <\delta \).

Proof

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

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

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,

$$\begin{aligned} D\widehat{h}^{{G}}= \frac{d\widehat{h}^{{G}}}{dJ(t)} \cdot \frac{dJ(t)}{dJ(E)}= \frac{d\widehat{h}^{{G}}}{dJ(t)} \cdot \frac{d \widehat{t}}{dJ(E)} \end{aligned}$$
(A.2)

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

$$\begin{aligned} \big \Vert \frac{d\widehat{t}}{dJ(E)}(J(E^u))\big \Vert _{\infty } = \Vert Dt(E^u_0)\Vert _\infty \le 1. \end{aligned}$$
(A.3)

By means of a similar computation we can show that

$$\begin{aligned} \big \Vert \frac{d\widehat{h}^{G}}{dJ(t)}(J(t^{u}))\big \Vert _{\infty } = \Vert Dh(0)\Vert _\infty = 1 \end{aligned}$$
(A.4)

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). https://doi.org/10.1007/s10884-020-09857-0

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Keywords

  • Homoclinic tangencies
  • Heterodimensional cycles
  • Blenders
  • Parablenders

Mathematics Subject Classification

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