Advertisement

Inventiones mathematicae

, Volume 205, Issue 1, pp 121–172 | Cite as

Generic family with robustly infinitely many sinks

  • Pierre BergerEmail author
Article

Abstract

We show, for every \(r>d\ge 0\) or \(r=d\ge 2\), the existence of a Baire generic set of \(C^d\)-families of \(C^r\)-maps \((f_a)_{a\in {\mathbb {R}}^k}\) of a manifold M of dimension \(\ge \)2, so that for every a small the map \(f_a\) has infinitely many sinks. When the dimension of the manifold is \(\ge \)3, the generic set is formed by families of diffeomorphisms. When M is the annulus, this generic set is formed by local diffeomorphisms. This is a counter example to a conjecture of Pugh and Shub.

Notes

Acknowledgments

I thanks the referees for their advices and comments. I am very grateful to Enrique Pujals and Sylvain Crovisier for important conversations and their comments on the first version of this work. I am thankful to Jean-Christophe Yoccoz for his encouragements. This research was partially supported by the Balzan project of J. Palis, the French-Brazilian network and the project BRNUH of Université Sorbonne Paris Cité.

References

  1. 1.
    Avila, A., Lyubich, M., de Melo, W.: Regular or stochastic dynamics in real analytic families of unimodal maps. Invent. Math. 154(3), 451–550 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Avila, A., Moreira, C.G.: Hausdorff dimension and the quadratic family (2002). (Manuscript)Google Scholar
  3. 3.
    Asaoka, M.: Hyperbolic sets exhibiting \(C^1\)-persistent homoclinic tangency for higher dimensions. Proc. Am. Math. Soc. 136(2), 677–686 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Benedicks, M., Carleson, L.: The dynamics of the Hénon map. Ann. Math. 133, 73–169 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Berger, P., Dujardin, R.: On stability and hyperbolicity for polynomial automorphisms of \({\bf C}^2\). arXiv:1409.4449
  6. 6.
    Bonatti, C., Díaz, L.J.: Persistent nonhyperbolic transitive diffeomorphisms. Ann. Math. (2) 143(2), 357–396 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bonatti, C., Díaz, L.: Connexions hétéroclines et généricité d’une infinité de puits et de sources. Ann. Sci. École Norm. Sup. (4) 32(1), 135–150 (1999)Google Scholar
  8. 8.
    Berger, P., de Simoi, J.: On the hausdorff dimension of newhouse phenomena. Ann. Henri Poincaré 1–23 (published online 2015)Google Scholar
  9. 9.
    Berger, P.: Abundance of non-uniformly hyperbolic Hénon like endomorphisms. arXiv:0903.1473v2
  10. 10.
    Berger, P.: Normal forms and Misiurewicz renormalization for dissipative surface diffeomorphisms. arXiv:1404.2235
  11. 11.
    Berger, P.: Properties of the maximal entropy measure and geometry of Hénon attractors. arXiv:1202.2822
  12. 12.
    Berger, P.: Persistence of laminations. Bull. Braz. Math. Soc. (N. S.) 41(2), 259–319 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Berger, P., Fave, P.-Y.: Parablender (2015). http://www.math.univ-paris13.fr/~berger. (Videos)
  14. 14.
    Berger, P., Rovella, A.: On the inverse limit stability of endomorphisms. Ann. Inst. H. Poincaré Anal. Non Linéaire 30(3), 463–475 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Benedicks, M., Viana, M.: Solution of the basin problem for Hénon-like attractors. Invent. Math. 143(2), 375–434 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Benedicks, M., Young, L.-S.: Sinaĭ–Bowen–Ruelle measures for certain Hénon maps. Invent. Math. 112(3), 541–576 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Moreira, C.G., Yoccoz, J.-C.: Stable intersections of regular Cantor sets with large Hausdorff dimensions. Ann. Math. (2) 154(1), 45–96 (2001)Google Scholar
  18. 18.
    Dujardin, R., Lyubich, M.: Stability and bifurcations for dissipative polynomial automorphisms of \({\bf C}^2\). Invent. Math. 200(2), 439–511 (2015)Google Scholar
  19. 19.
    Díaz, L.J., Nogueira, A., Pujals, E.R.: Heterodimensional tangencies. Nonlinearity 19(11), 2543–2566 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Gorodetski, A., Kaloshin, V.: How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency. Adv. Math. 208(2), 710–797 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Hofbauer, F., Keller, G.: Quadratic maps without asymptotic measure. Commun. Math. Phys. 127(2), 319–337 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Hunt, B.R., Kaloshin, V.Y.: Prevalence. In: Handbook of Dynamical Systems, vol. 3, pp. 43–87 (2010)Google Scholar
  23. 23.
    Johnson, S.D.: Singular measures without restrictive intervals. Commun. Math. Phys. 110(2), 185–190 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Kozlovski, O., Shen, W., van Strien, S.: Density of hyperbolicity in dimension one. Ann. Math. (2) 166(1), 145–182 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Lyubich, M.: Almost every real quadratic map is either regular or stochastic. Ann. Math. (2) 156(1), 1–78 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Newhouse, S.E.: Diffeomorphisms with infinitely many sinks. Topology 12, 9–18 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Newhouse, S.E.: Lectures on dynamical systems. In: Dynamical Systems (C.I.M.E. Summer School, Bressanone, 1978). Progr. Math., vol. 8, pp. 1–114. Birkhäuser, Boston (1980)Google Scholar
  28. 28.
    Palis, J.: A global view of dynamics and a conjecture on the denseness of finitude of attractors. Astérisque 261(xiii–xiv), 335–347 (2000). [Géométrie complexe et systèmes dynamiques (Orsay, 1995)]zbMATHMathSciNetGoogle Scholar
  29. 29.
    Palis, J.: A global perspective for non-conservative dynamics. Ann Inst H Poincaré Anal Non linéaire 22(4), 485–507 (2005)Google Scholar
  30. 30.
    Palis, J.: Open questions leading to a global perspective in dynamics. Nonlinearity 21(4), T37–T43 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Pugh, C., Shub, M.: Stable ergodicity and partial hyperbolicity. In: International Conference on Dynamical Systems (Montevideo, 1995). Pitman Res. Notes Math. Ser., vol. 362, pp. 182–187. Longman, Harlow (1996)Google Scholar
  32. 32.
    Pujals, E.R., Sambarino, M.: Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Ann. Math. (2) 151(3), 961–1023 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Palis, J., Takens, F.: Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations. In: Cambridge Studies in Advanced Mathematics, vol. 35. Cambridge University Press, Cambridge (1993). (Fractal dimensions and infinitely many attractors)Google Scholar
  34. 34.
    Palis, J., Viana, M.: High dimension diffeomorphisms displaying infinitely many periodic attractors. Ann. Math. (2) 140(1), 207–250 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Palis, J., Yoccoz, J.-C.: Non-uniformly hyperbolic horseshoes arising from bifurcations of Poincaré heteroclinic cycles. Publ. Math. Inst. Hautes Études Sci. 110, 1–217 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Tedeschini-Lalli, Laura, Yorke, James A.: How often do simple dynamical processes have infinitely many coexisting sinks? Commun. Math. Phys. 106(4), 635–657 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Turaev, D.: On the genericity of the newhouse phenomenon. In: EQUADIFF 2003. World Sci. Publ., Hackensack (2005)Google Scholar
  38. 38.
    Yoccoz, J.-C.: Introduction to hyperbolic dynamics. In: Real and Complex Dynamical Systems (Hillerød, 1993). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 464, pp. 265–291 (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Université Paris 13VilletaneuseFrance

Personalised recommendations