Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin


Author: Ittai Kan
Journal: Bull. Amer. Math. Soc. 31 (1994), 68-74
MSC: Primary 58F12; Secondary 58F30
DOI: https://doi.org/10.1090/S0273-0979-1994-00507-5
MathSciNet review: 1254075
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We announce the discovery of a diffeomorphism of a three-dimensional manifold with boundary which has two disjoint attractors. Each attractor attracts a set of positive 3-dimensional Lebesgue measure whose points of Lebesgue density are dense in the whole manifold. This situation is stable under small perturbations.


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

  • [1] S. W. McDonald, C. Grebogi, E. Ott, and J. A. Yorke, Fractal basin boundaries, Phys. D 17 (1985), 125-153. MR 815280 (87k:58170)
  • [2] R. Bowen, On Axiom A diffeomorphisms, CBMS Regional Conf. Ser. in Math., vol. 35, Amer. Math. Soc., Providence, RI, 1978. MR 0482842 (58:2888)
  • [3] J. C. Alexander, I. Kan, J. A. Yorke, and Zhiping You, Riddled basins, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 2 (1992), 795-813. MR 1206103 (93k:58140)
  • [4] J. Milnor, On the concept of attractor, Comm. Math. Phys. 99 (1985), 177-195. MR 790735 (87i:58109a)
  • [5] Ju. S. Il'yashenko, The concept of minimal attractor and maximal attractors of partial differential equations of the Kuramoto-Sivashinsky type, Chaos 1 (1991), 168-173. MR 1135904 (92k:58165)
  • [6] I. Kan, Intermingled basins, Ergodic Theory Dynamical Systems (to appear).
  • [7] M. Hirsch, C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Math., vol. 583, Springer-Verlag, New York, 1977. MR 0501173 (58:18595)
  • [8] Y. B. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Russian Math. Surveys 32 (1977), 55-114. MR 0466791 (57:6667)
  • [9] A. B. Katok and J. M. Strelcyn, Invariant manifolds, entropy and billiards: Smooth maps with singularities, Lecture Notes in Math., vol. 1222, Springer-Verlag, New York, 1986. MR 872698 (88k:58075)
  • [10] C. Pugh and M. Shub, Ergodic attractors, Trans. Amer. Math. Soc. 312 (1989), 1-54. MR 983869 (90h:58057)

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC: 58F12, 58F30

Retrieve articles in all journals with MSC: 58F12, 58F30


Additional Information

DOI: https://doi.org/10.1090/S0273-0979-1994-00507-5
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society