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Perturbing a product of stable flows


Author: Anthony Manning
Journal: Proc. Amer. Math. Soc. 133 (2005), 1693-1697
MSC (2000): Primary 37C70, 37C75, 37D20; Secondary 37D10, 37C27
DOI: https://doi.org/10.1090/S0002-9939-05-07872-X
Published electronically: January 13, 2005
MathSciNet review: 2120252
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Abstract | References | Similar Articles | Additional Information

Abstract: Suppose that $f$ and $f'$ are axiom A flows with attractors $A$ and $A'$. Then the attractor $A \times A'$ for the product flow $ g_t =f_t \times f'_t$ on the product manifold is no longer hyperbolic (although there is a hyperbolic action of $\mathbb{R}^2$).

It is easy to see that the attractor cannot explode but we show here that it cannot implode: for any flow $(h_t)$ sufficiently close to $(g_t)$ any attractor whose basin is not too thin is $\varepsilon$-dense in $A \times A'$.


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  • 1. A A Andronov, A A Vitt and S E Khaikin, Theory of oscillators, Pergamon Press, Oxford, 1966. MR 0198734 (33:6888)
  • 2. D V Anosov, Geodesic flows on compact Riemannian manifolds with negative curvature, Proc. Steklov Inst. Math. 90 (1967). MR 0224110 (36:7157)
  • 3. R Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972) 1-30. MR 0298700 (45:7749)
  • 4. R Bowen and D Ruelle, The ergodic theory of axiom A flows, Inv. Math., 29 (1975) 181-202. MR 0380889 (52:1786)
  • 5. K Falconer, Fractal geometry: mathematical foundations and applications, Wiley, Chichester, 1990. MR 1102677 (92j:28008)
  • 6. S Hayashi, Connecting invariant manifolds and the solution of the $C\sp 1$stability and $\Omega$-stability conjectures for flows. Ann. of Math. (2) 145 (1997), no. 1, 81-137. MR 1432037 (98b:58096)
  • 7. M Hirsch, C Pugh, M Shub, Invariant manifolds. Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977. MR 0501173 (58:18595)
  • 8. M Hurley, Attractors: persistence, and density of their basins, Trans. Amer. Math. Soc., 269 (1982) 247-271. MR 0637037 (83c:58049)
  • 9. A Katok and B Hasselblatt, Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995. MR 1326374 (96c:58055)
  • 10. R Mañé, An ergodic closing lemma. Ann. of Math. (2) 116 (1982), no. 3, 503-540. MR 0678479 (84f:58070)
  • 11. M B Moreva, The stability of attractors in the metric $R_0$ with respect to $C^0$ perturbations, Vestnik Leningrad Univ. Math., 21 no. 3 (1988) 44-49. MR 0974791 (90b:58136)
  • 12. C Robinson, Structural stability of vector fields, Ann. Math., 99 (1974) 154-75. MR 0334283 (48:12602)
  • 13. C Robinson, Structural stability of $C^1$ flows, Lect. Notes in Math. 468 (1975) Springer, 262-277. MR 0650640 (58:31251)
  • 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)
  • 15. S Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967) 747-817. MR 0228014 (37:3598)
  • 16. S Smale, The mathematics of time, Springer, New York, 1980. MR 0607330 (83a:01068)
  • 17. S Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Springer, New York, 1990. MR 1056699 (92a:58041)

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Additional Information

Anthony Manning
Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
Email: akm@maths.warwick.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-05-07872-X
Received by editor(s): January 24, 2004
Published electronically: January 13, 2005
Communicated by: Michael Handel
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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