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Robust Heteroclinic Tangencies

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Abstract

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

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References

  • 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)

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • 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 

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

    Article  MathSciNet  Google Scholar 

  • 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 

  • 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 

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

    MathSciNet  MATH  Google Scholar 

  • 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

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • 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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Authors and Affiliations

  1. Instituto de Matemática e Estatística, UFF, Rua Prof. Marcos Waldemar de Freitas Reis, s/n, Niterói, Brazil

    Pablo G. Barrientos

  2. Centro de Matematica da Universidade do Porto, Rua do Campo Alegre 687, 4169-007, Porto, Portugal

    Sebastián A. Pérez

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  1. Pablo G. Barrientos
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  2. Sebastián A. Pérez
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Correspondence to Pablo G. Barrientos.

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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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  • DOI: https://doi.org/10.1007/s00574-019-00185-6

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