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Every attractor of a flow on a manifold has the shape of a finite polyhedron

Authors: Bernd Günther and Jack Segal
Journal: Proc. Amer. Math. Soc. 119 (1993), 321-329
MSC: Primary 54C56; Secondary 54H20, 57N25, 58F12
MathSciNet review: 1170545
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Abstract: It is shown that the class of compacta which can occur as attractors of continuous flows on topological manifolds coincides with the class of finite dimensional compacta having the shape of a finite polyhedron.

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  • [1] D. V. Anosov and V. I. Arnold, Dynamical systems. I, Encyclopedia Math. Sci., vol. 1, Springer, New York, 1988. MR 970793 (89g:58060)
  • [2] R. H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math. 1 (1951), 43-51. MR 0043451 (13:265b)
  • [3] T. A. Chapman, Shapes of finite-dimensional compacta, Fund. Math. 76 (1972), 261-276. MR 0320998 (47:9531)
  • [4] A. Dold, Lectures on algebraic topology, Grundlehren Math. Wiss., vol. 200, Springer, New York and Berlin, 1980. MR 606196 (82c:55001)
  • [5] R. Geoghegan and R. R. Summerhill, Concerning the shapes of finite-dimensional compacta, Trans. Amer. Math. Soc. 179 (1973), 281-292. MR 0324637 (48:2987)
  • [6] H. M. Hastings, A higher dimensional Poincaré-Bendixson theorem, Glas. Mat. 14 (1979), 263-268. MR 646352 (83e:34041)
  • [7] M. W. Hirsch and S. Smale, Differential equations, dynamical systems, and linear algebra, Academic Press, New York, 1974. MR 0486784 (58:6484)
  • [8] S. Mardešić and J. Segal, Shape theory, Mathematical Library, vol. 26, North-Holland, Amsterdam, 1982. MR 676973 (84b:55020)
  • [9] P. Mrozik, Mapping cylinders of approaching maps and strong shape, J. London Math. Soc. (2) 41 (1990), 159-174. MR 1063553 (91j:55013)
  • [10] J. W. Robbin and D. Salamon, Dynamical systems, shape theory and the Conley index, Ergodic Theory Dynamical Systems 8$ ^{{\ast}}$ (1988), 375-393. MR 967645 (89h:58094)
  • [11] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), 1-41. MR 797044 (87e:58182)
  • [12] R. M. Schori, Chaos: An introduction to some topological aspects, Continuum Theory and Dynamical Systems (Morton Brown, ed.), Contemp. Math., vol. 117, Amer. Math. Soc., Providence, RI, 1991, pp. 149-161. MR 1112812 (92f:58113)
  • [13] E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 0210112 (35:1007)
  • [14] J. E. West, Mapping Hilbert cube manifolds to $ ANR$'s: A solution to a conjecture of Borsuk, Ann. of Math. (2) 106 (1977), 1-18. MR 0451247 (56:9534)
  • [15] R. L. Wilder, Topology of manifolds, Colloq. Publ., vol. 32, Amer. Math. Soc., Providence, RI, 1949. MR 0029491 (10:614c)

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Keywords: Dynamical systems, attractors, shape, complement theorem
Article copyright: © Copyright 1993 American Mathematical Society

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