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Transactions of the American Mathematical Society

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3-manifolds that admit knotted solenoids as attractors


Authors: Boju Jiang, Yi Ni and Shicheng Wang
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 4371-4382
MSC (2000): Primary 57N10, 58K05, 37E99, 37D45
DOI: https://doi.org/10.1090/S0002-9947-04-03503-2
Published electronically: February 27, 2004
MathSciNet review: 2067124
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Abstract: Motivated by the study in Morse theory and Smale's work in dynamics, the following questions are studied and answered: (1) When does a 3-manifold admit an automorphism having a knotted Smale solenoid as an attractor? (2) When does a 3-manifold admit an automorphism whose non-wandering set consists of Smale solenoids? The result presents some intrinsic symmetries for a class of 3-manifolds.


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

  • [EK] J. Eells, Jr., and N.H. Kuiper, Manifolds which are like projective planes, IHES Publ. Math. 14(1962), 5-46. MR 26:3075
  • [F] J. M. Franks, Knots, links and symbolic dynamics, Ann. of Math. 113(1981), 529-552. MR 83h:58074
  • [Ha] S. Hayashi, Hyperbolicity, stability, and creation of homoclinic points, Proceedings of the International Congress of Mathematicians, vol. II (Berlin, 1998). Doc. Math. 1998, Extra vol. II, 789-796 (electronic). MR 99f:58115
  • [He] J. Hempel, $3$-manifolds, Ann. of Math. Studies vol. 86, Princeton University Press, 1976. MR 54:3702
  • [Ni] Z. Nitecki, Differentiable dynamics, An introduction to the orbit structure of diffeomorphisms, MIT Press, 1971. MR 58:31210
  • [Pe] Y. B. Pesin, Dimension theory in dynamical systems, Contemporary views and applications, University of Chicago Press, 1997. MR 99b:58003
  • [Po] H. L. Porteous, Anosov diffeomorphisms of flat manifolds, Topology 11(1972), 307-315. MR 45:6035
  • [R] G. Reeb, Sur certaines propriétés topologiques des variétés feuilletées, Publ. Inst. Math. Univ. Strasbourg 11, pp. 5-89, 155-156. Actualités Sci. Ind., no. 1183, Hermann & Cie., 1952. MR 14:1113a
  • [S] S. Smale, Differentiable dynamical systems, Bull. AMS 73(1967), 747-817. MR 37:3598
  • [Su] M. C. Sullivan, Visually building Smale flows in $S^3$, Topology Appl. 106(2000), 1-19. MR 2001g:37025
  • [T] E. S. Thomas, Jr., One-dimensional minimal sets, Topology 12(1973), 233-242. MR 48:5123
  • [W] R. F. Williams, One-dimensional non-wandering sets, Topology 6(1967), 473-487. MR 36:897

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Additional Information

Boju Jiang
Affiliation: Department of Mathematics, Peking University, Beijing 100871, People’s Republic of China
Email: jiangbj@math.pku.edu.cn

Yi Ni
Affiliation: Department of Mathematics, Peking University, Beijing 100871, People’s Republic of China
Address at time of publication: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: yni@princeton.edu

Shicheng Wang
Affiliation: Department of Mathematics, Peking University, Beijing 100871, People’s Republic of China
Email: wangsc@math.pku.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-04-03503-2
Keywords: $3$-manifolds, homeomorphisms, attractors, solenoids, lens spaces
Received by editor(s): February 20, 2003
Received by editor(s) in revised form: April 18, 2003
Published electronically: February 27, 2004
Additional Notes: This work was partially supported by a MOSTC grant and a MOEC grant
Article copyright: © Copyright 2004 American Mathematical Society

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