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Unstable attractors in manifolds


Author: J. J. Sánchez-Gabites
Journal: Trans. Amer. Math. Soc. 362 (2010), 3563-3589
MSC (2000): Primary 54H20, 57N65, 34D45
DOI: https://doi.org/10.1090/S0002-9947-10-05061-0
Published electronically: February 12, 2010
MathSciNet review: 2601600
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Abstract: Assume that $ K$ is a compact attractor with basin of attraction $ \mathcal{A}(K)$ for some continuous flow $ \varphi$ in a space $ M$. Stable attractors are very well known, but otherwise (without the stability assumption) the situation can be extremely wild. In this paper we consider the class of attractors with no external explosions, where a mild form of instability is allowed.

After obtaining a simple description of the trajectories in $ \mathcal{A}(K) - K$ we study how $ K$ sits in $ \mathcal{A}(K)$ by performing an analysis of the Poincaré polynomial of the pair $ (\mathcal{A}(K),K)$. In case $ M$ is a surface we obtain a nice geometric characterization of attractors with no external explosions, as well as a converse to the well known fact that the inclusion of a stable attractor in its basin of attraction is a shape equivalence. Finally, we explore the strong relations which exist between the shape (in the sense of Borsuk) of $ K$ and the shape (in the intuitive sense) of the whole phase space $ M$, much in the spirit of the Morse-Conley theory.


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  • 1. K. Athanassopoulos, Explosions near isolated unstable attractors, Pacific J. Math. 210 (2003), no. 2, 201-214. MR 1988531 (2004c:37030)
  • 2. -, Remarks on the region of attraction of an isolated invariant set, Colloq. Math. 104 (2006), 157-167. MR 2197074 (2006i:37033)
  • 3. A. Beck, On invariant sets, Ann. of Math. 67 (1958), no. 1, 99-103. MR 0092106 (19:1064c)
  • 4. N. P. Bhatia and G. P. Szegö, Stability theory of dynamical systems, Die Grundlehren der mathematischen Wissenschaften, Band 161, Springer-Verlag, 1970. MR 0289890 (44:7077)
  • 5. S. A. Bogatyĭ and V. I. Gutsu, On the structure of attracting compacta, Differentsialnye Uravneniya 25 (1989), no. 5, 907-909, 920. MR 1003051 (90j:58076)
  • 6. K. Borsuk, Theory of retracts, Monografie Matematyczne, Tom 44, Państwowe Wydawnictwo Naukowe, 1967. MR 0216473 (35:7306)
  • 7. -, Concerning homotopy properties of compacta, Fund. Math. 62 (1968), 223-254. MR 0229237 (37:4811)
  • 8. -, Theory of shape, Monografie Matematyczne, Tom 59, Państwowe Wydawnictwo Naukowe, 1975. MR 0418088 (54:6132)
  • 9. G. E. Bredon, Wilder manifolds are locally orientable, Proc. N. A. S. 63 (1969), no. 4, 1079-1081. MR 0286109 (44:3325)
  • 10. M. Brown and H. Gluck, Stable structures on manifolds. I. Homeomorphisms of $ \mathbb{S}^n$, Ann. of Math. (2) 79 (1964), 1-17. MR 0158383 (28:1608a)
  • 11. R. C. Churchill, Isolated invariant sets in compact metric spaces, J. Diff. Eq. 12 (1972), 330-352. MR 0336763 (49:1536)
  • 12. C. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, 38, American Mathematical Society, 1978. MR 511133 (80c:58009)
  • 13. C. Conley and R. Easton, Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc. 158 (1971), 35-61. MR 0279830 (43:5551)
  • 14. J. Dydak and J. Segal, Shape theory. An introduction, Lecture Notes in Mathematics, 688, Springer, 1978. MR 520227 (80h:54020)
  • 15. R. D. Edwards, The solution of the $ 4$-dimensional annulus conjecture (after Frank Quinn), Four-manifold theory (C. Gordon and R. Kirby, eds.), Contemp. Math., vol. 35, Amer. Math. Soc., 1984, pp. 211-264. MR 780581 (86j:57006)
  • 16. A. Giraldo, M. A. Morón, F. R. Ruiz del Portal, and J. M. R. Sanjurjo, Shape of global attractors in topological spaces, Nonlinear Anal. 60 (2005), no. 5, 837-847. MR 2113160 (2006k:37024)
  • 17. B. Günther and J. Segal, Every attractor of a flow on a manifold has the shape of a finite polyhedron, Proc. Amer. Math. Soc. 119 (1993), no. 1, 321-329. MR 1170545 (93k:54044)
  • 18. C. Gutiérrez, Smoothing continuous flows on two-manifolds and recurrences, Ergodic Theory Dynam. Systems 6 (1986), no. 1, 17-44. MR 837974 (87k:58222)
  • 19. H. M. Hastings, A higher-dimensional Poincaré-Bendixson theorem, Glas. Mat. Ser. III 14(34) (1979), no. 2, 263-268. MR 646352 (83e:34041)
  • 20. A. Hatcher, Algebraic topology, Cambridge University Press, 2002. MR 1867354 (2002k:55001)
  • 21. M. W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, 1976. MR 0448362 (56:6669)
  • 22. S. Hu, Theory of retracts, Wayne State University Press, 1965. MR 0181977 (31:6202)
  • 23. L. Kapitanski and I. Rodnianski, Shape and Morse theory of attractors, Comm. Pure Appl. Math. 53 (2000), no. 2, 218-242. MR 1721374 (2000h:37019)
  • 24. R. C. Kirby, Stable homeomorphisms and the annulus conjecture, Ann. of Math. 89 (1969), 575-582. MR 0242165 (39:3499)
  • 25. S. Mardešić and J. Segal, Shape theory. The inverse system approach, North-Holland Mathematical Library, 26, North-Holland Publishing Co., 1982. MR 676973 (84b:55020)
  • 26. C. K. McCord, On the Hopf index and the Conley index, Trans. Amer. Math. Soc. 313 (1989), no. 2, 853-860. MR 961594 (90a:58151)
  • 27. M. A. Morón, J. J. Sánchez-Gabites, and J. M. R. Sanjurjo, Topology and dynamics of unstable attractors, Fund. Math. 197 (2007), 239-252. MR 2365890 (2008j:37032)
  • 28. F. Quinn, Ends of maps. III. Dimensions $ 4$ and $ 5$, J. Diff. Geom 17 (1982), no. 3, 503-521. MR 679069 (84j:57012)
  • 29. F. Raymond, Separation and union theorems for generalized manifolds with boundary, Michigan Math. J. 7 (1960), 7-21. MR 0120638 (22:11388)
  • 30. D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), no. 1, 1-41. MR 797044 (87e:58182)
  • 31. J. W. Robbin and D. Salamon, Dynamical systems and shape theory, Ergod. Th. and Dynam. Sys. 8* (1988), 375-393. MR 967645 (89h:58094)
  • 32. J. M. R. Sanjurjo, Multihomotopy, Čech spaces of loops and shape groups, Proc. London Math. Soc. (3) 69 (1994), no. 2, 330-344. MR 1281968 (95h:55011)
  • 33. -, On the structure of uniform attractors, J. Math. Anal. Appl. 192 (1995), no. 2, 519-528. MR 1332224 (96c:58109)
  • 34. -, Morse equations and unstable manifolds of isolated invariant sets, Nonlinearity 16 (2003), 1435-1448. MR 1986304 (2004d:37022)
  • 35. E. H. Spanier, Algebraic topology, McGraw-Hill Book Co., 1966. MR 0210112 (35:1007)
  • 36. R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17-86. MR 0061823 (15:890a)
  • 37. R. L. Wilder, Topology of manifolds, American Mathematical Society Colloquium Publications, vol. 32, American Mathematical Society, 1949. MR 0029491 (10:614c)

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Additional Information

J. J. Sánchez-Gabites
Affiliation: Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Address at time of publication: Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, United Kingdom
Email: jaigabites@mat.ucm.es

DOI: https://doi.org/10.1090/S0002-9947-10-05061-0
Received by editor(s): April 9, 2008
Published electronically: February 12, 2010
Additional Notes: This paper was written under partial support by Direccioń General de Investigacioń
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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