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Robust Heteroclinic Tangencies

  • Pablo G. BarrientosEmail author
  • Sebastián A. Pérez
Article
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Abstract

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

Keywords

Folding manifolds Robust equidimensional tangencies Robust heterodimensional tangencies 

Notes

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

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)Google Scholar
  2. Asaoka, M.: Hyperbolic sets exhibiting \(C^1\)-persistent homoclinic tangency for higher dimensions. Proc. Amer. Math. Soc. 136, 677–686 (2008)MathSciNetCrossRefGoogle Scholar
  3. Barrientos, P.G., Raibekas, A.: Robust tangencies of large codimension. Nonlinearity 30, 4369–4409 (2017)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  6. Bonatti, C., Díaz, L.J.: On maximal transitive sets of generic diffeomorphisms. Publ. Math., Inst. Hautes Étud. Sci. 96, 171–197 (2003)MathSciNetCrossRefGoogle Scholar
  7. Bonatti, C., Díaz, L.J.: Abundance of \(C^1\)-homoclinic tangencies. Trans. Amer. Math. Soc. 264, 5111–5148 (2012)CrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  9. Díaz, L.J., Nogueira, A., Pujals, E.R.: Heterodimensional tangencies. Nonlinearity 19, 2543 (2006)MathSciNetCrossRefGoogle 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)CrossRefGoogle Scholar
  11. Kiriki, S., Soma, T.: \(C^2\)-robust heterodimensional tangencies. Nonlinearity 25, 3277 (2012)MathSciNetCrossRefGoogle Scholar
  12. Mañé, R.: A proof of the \(C^1\) stability conjecture. Publications Mathématiques de L’Institut des Hautes Scientifiques 66, 161–210 (1987)CrossRefGoogle 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)CrossRefGoogle Scholar
  14. Palis, J.: A differentiable invariant of topological conjugacies and moduli of stability. Asterisque 51, 335–346 (1978)MathSciNetzbMATHGoogle Scholar
  15. Robinson, C.: Stability, Symbolic Dynamics, and Chaos, 2nd edn. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1999) ISBN: 0-8493-8495-837-01Google Scholar
  16. Simon, C.P.: A 3-dimensional Abraham–Smale example. Proc. Amer. Math. Soc. 34, 629–630 (1972)MathSciNetzbMATHGoogle Scholar
  17. Smale, S.: Differentiable dynamical systems. Bulletin of the American MaAthematical Society 73, 747–817 (1967)MathSciNetCrossRefGoogle 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)Google Scholar

Copyright information

© Sociedade Brasileira de Matemática 2019

Authors and Affiliations

  1. 1.Instituto de Matemática e EstatísticaUFFNiteróiBrazil
  2. 2.Centro de Matematica da Universidade do PortoPortoPortugal

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