Glasnik Matematicki, Vol. 42, No.1 (2007), 117-130.

SHAPE PROPERTIES OF THE BOUNDARY OF ATTRACTORS

J. J. Sánchez-Gabites and José M. R. Sanjurjo

Departamento de Geometria y Topologia, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
e-mail: jajsanch@mat.ucm.es
e-mail: jose_sanjurjo@mat.ucm.es


Abstract.   Let M be a locally compact metric space endowed with a continuous flow φ : M × RM. Assume that K is a stable attractor for φ and P subset A(K) is a compact positively invariant neighbourhood of K contained in its basin of attraction. Then it is known that the inclusion K --> P is a shape equivalence and the question we address here is whether there exists some relation between the shapes of K and P. The general answer is negative, as shown by example, but under certain hypotheses on K the shape domination Sh(∂K) ≥ Sh(∂P) or even the equality Sh(∂K) = Sh(∂P) hold. However we also put under study interesting situations where those hypotheses are not satisfied, albeit other techniques such as Lefschetz's duality render results relevant to our question.

2000 Mathematics Subject Classification.   54H20, 55P55, 58F12.

Key words and phrases.   Dynamical systems, attractors, boundary of attractors, shape.


Full text (PDF) (free access)

DOI: 10.3336/gm.42.1.10


References:

  1. N. P. Bhatia and G. P. Szegö, Stability Theory of Dynamical Systems, Die Grundlehren der mathematischen Wissenschaften, Band 161, Springer--Verlag, New York-Berlin, 1970.
    MathSciNet

  2. S. A. Bogatyi and V. I. Gutsu, On the structure of attracting compacta, Differentsialnye Uravneniya 25 (1989), 907-909.
    MathSciNet

  3. K. Borsuk, Theory of Retracts, PWN-Polish Scientific Publishers, Warszawa, 1967.
    MathSciNet

  4. K. Borsuk, Concerning homotopy properties of compacta, Fund. Math. 62 (1968), 223-254.
    MathSciNet

  5. K. Borsuk, Theory of Shape, PWN-Polish Scientific Publishers, Warszawa, 1975.
    MathSciNet

  6. M. Brown, A proof of the generalized Schönflies theorem, Bull. Amer. Math. Soc. 66 (1960), 74-76.
    MathSciNet

  7. W. C. Chewning and R. S. Owen, Local sections of flows on manifolds, Proc. Amer. Math. Soc. 49 (1975), 71-77.
    MathSciNet     CrossRef

  8. R. C. Churchill, Isolated invariant sets in compact metric spaces, J. Differential Equations 12 (1972), 330-352.
    MathSciNet     CrossRef

  9. C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics 38, American Mathematical Society, Providence, 1978.
    MathSciNet

  10. C. Conley and R. Easton, Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc. 158 (1971), 35-61.
    MathSciNet     CrossRef

  11. J. Dydak and J. Segal, Shape Theory. An Introduction, Lecture Notes in Mathematics 688, Springer, Berlin, 1978.
    MathSciNet

  12. 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), 837-847.
    MathSciNet     CrossRef

  13. A. Giraldo and J. M. R. Sanjurjo, On the global structure of invariant regions of flows with asymptotically stable attractors, Math. Z. 232 (1999), 739-746.
    MathSciNet     CrossRef

  14. 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), 321-329.
    MathSciNet     CrossRef

  15. H. M. Hastings, A higher-dimensional Poincaré-Bendixson theorem, Glas. Mat. Ser. III 14(34) (1979), 263-268.
    MathSciNet

  16. S. Hu, Theory of Retracts, Wayne State University Press, Detroit, 1965.
    MathSciNet

  17. S. Mardesic, Strong Shape and Homology, Springer-Verlag, Berlin-Heidelberg-New York 2000.
    MathSciNet

  18. S. Mardesic and J. Segal, Equivalence of the Borsuk and the ANR-system approach to shapes, Fund. Math. 72 (1971), 61-68.
    MathSciNet

  19. S. Mardesic and J. Segal, Shapes of compacta and ANR-systems, Fund. Math. 72 (1971), 41-59.
    MathSciNet

  20. S. Mardesic and J. Segal, Shape Theory, North-Holland Publishing Co., Amsterdam-New York-Oxford 1982.
    MathSciNet

  21. F. Raymond, Separation and union theorems for generalized manifolds with boundary, Michigan Math. J. 7 (1960), 7-21.
    MathSciNet     CrossRef

  22. J. C. Robinson and O. M. Tearne, Boundaries of attractors as omega limit sets, Stoch. Dyn. 5 (2005), 97-109.
    MathSciNet     CrossRef

  23. D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), 1-41.
    MathSciNet     CrossRef

  24. J. J. Sánchez-Gabites and J. M. R. Sanjurjo, On the topology of the boundary of a basin of attraction, Proc. Amer. Math. Soc., to appear.

  25. J. M. R. Sanjurjo, Multihomotopy, Cech spaces of loops and shape groups, Proc. London Math. Soc. (3) 69 (1994), 330-344.
    MathSciNet     CrossRef

  26. J. M. R. Sanjurjo, On the structure of uniform attractors, J. Math. Anal. Appl. 192 (1995), 519--528.
    MathSciNet     CrossRef

  27. E. H. Spanier, Algebraic Topology, McGraw-Hill Book Co., New York-Toronto-London, 1966.
    MathSciNet

  28. J. E. West, Mapping Hilbert cube manifolds to ANR's: a solution of a conjecture of Borsuk, Ann. Math. (2) 106 (1977), 1-18.
    MathSciNet     CrossRef

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