Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Hyperbolic sets exhibiting $ C^1$-persistent homoclinic tangency for higher dimensions

Author: Masayuki Asaoka
Journal: Proc. Amer. Math. Soc. 136 (2008), 677-686
MSC (2000): Primary 37C29; Secondary 37C20, 37B10
Published electronically: October 18, 2007
Erratum: Proc. Amer. Math. Soc. 138 (2010), 1533
MathSciNet review: 2358509
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For any manifold of dimension at least three, we give a simple construction of a hyperbolic invariant set that exhibits $ C^1$-persistent homoclinic tangency. It provides an open subset of the space of $ C^1$-diffeomorphisms in which generic diffeomorphisms have arbitrary given growth of the number of attracting periodic orbits and admit no symbolic extensions.

References [Enhancements On Off] (What's this?)

  • 1. R. Abraham and S. Smale, Nongenericity of $ \Omega$-stability, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), pp. 5-8, Amer. Math. Soc., Providence, R.I., 1970. MR 0271986 (42:6867)
  • 2. C. Bonatti and L. Díaz, Connexions hétéroclines et généricité d'une infinité de puits et de sources. Ann. Sci. École Norm. Sup. (4) 32(1999), no. 1, 135-150. MR 1670524 (2000e:37015)
  • 3. C. Bonatti and L. Díaz, On maximal transitive sets of generic diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 96(2002), 171-197 (2003). MR 1985032 (2007d:37017)
  • 4. C. Bonatti, L. Díaz, and M. Viana, Dynamics beyond uniform hyperbolicity, Encyclopedia of Mathematical Sciences, 102. Springer-Verlag, Berlin, 2004. MR 2105774 (2005g:37001)
  • 5. T. Downarowicz and S. Newhouse, Symbolic extensions and smooth dynamical systems. Invent. Math. 160(2005), no. 3, 453-499. MR 2178700 (2006j:37021)
  • 6. M.W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds. Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977. MR 0501173 (58:18595)
  • 7. V.Y. Kaloshin, An extension of the Artin-Mazur theorem. Ann. Math. (2) 150(1999), no. 2, 729-741. MR 1726706 (2000j:37020)
  • 8. V.Y. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits. Comm. Math. Phys. 211(2000), no. 1, 253-271. MR 1757015 (2001e:37035)
  • 9. S.E. Newhouse, Nondensity of axiom $ {\rm A}({\rm a})$ on $ S^2$. Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), pp. 191-202, Amer. Math. Soc., Providence, R.I., 1970. MR 0277005 (43:2742)
  • 10. S.E. Newhouse, Diffeomorphisms with infinitely many sinks. Topology 13(1974), 9-18. MR 0339291 (49:4051)
  • 11. S.E. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms. Inst. Hautes Études Sci. Publ. Math. 50(1979), 101-151. MR 556584 (82e:58067)
  • 12. J. Palis and F. Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations. Cambridge Studies in Advanced Mathematics, 35. Cambridge University Press, Cambridge, 1993. MR 1237641 (94h:58129)
  • 13. J. Palis and M. Viana, High dimension diffeomorphisms displaying infinitely many periodic attractors. Ann. Math. (2) 140(1994), no. 1, 207-250. MR 1289496 (95g:58140)
  • 14. C. Robinson, Dynamical systems. Stability, symbolic dynamics, and chaos. Second edition. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1999. MR 1792240 (2001k:37003)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37C29, 37C20, 37B10

Retrieve articles in all journals with MSC (2000): 37C29, 37C20, 37B10

Additional Information

Masayuki Asaoka
Affiliation: Department of Mathematics, Kyoto University, 606-8502 Kyoto, Japan

Keywords: Newhouse phenomena, wild dynamics, symbolic extensions
Received by editor(s): October 17, 2006
Received by editor(s) in revised form: February 1, 2007
Published electronically: October 18, 2007
Additional Notes: The author was supported by JSPS PostDoctoral Fellowships for Research Abroad.
Communicated by: Jane M. Hawkins
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society