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Lusternik-Schnirelmann category and Morse decompositions

Part of: Classical Topics in Algebraic Topology

Published online by Cambridge University Press:  26 February 2010

José M. R. Sanjurjo
José M. R. Sanjurjo
Affiliation:
Facultad de Matemáticas, Universidad Complutense, 28040 Madrid, Spain. E-mail: [email protected]
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Extract

§1. Introduction. We study in this paper some properties of the Lusternik-Schnirelmann category of isolated invariant sets of continuous dynamical systems. There are several different definitions of this coefficient, although most of them agree in the important case of ANR's (Absolute Neighbourhood Retracts). We refer to the review articles [10] by R. H. Fox and [15, 16] by I. M. James for general information about this topological invariant. We shall use in this paper the definition of the Lusternik-Schnirelmann category of a compactum introduced by K. Borsuk in [4].

MSC classification

Secondary: 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirel'man) category of a space
Type
Research Article
Information
Mathematika , Volume 47 , Issue 1-2 , December 2000 , pp. 299 - 305
Copyright
Copyright © University College London 2000

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References

1.Bhatia, N. P.. Attraction and nonsaddle sets in dynamical systems. J. Differential Equations, 8 (1970), 229–249.CrossRefGoogle Scholar
2.Bhatia, N. P. and Szego, G. P.. Stability Theory of Dynamical Systems (Springer, Berlin, 1970).CrossRefGoogle Scholar
3.Borsuk, K.. Theory of Shape, Monografie Matematyczne 59, Warszawa, 1975.Google Scholar
4.Borsuk, K.. On the Lusternik-Schnirelmann category in the theory of shape. Fund. Math., 99 (1978), 35–42.CrossRefGoogle Scholar
5.Cartwright, M. L. and Littlewood, J. E.. Some fixed point theorems. Ann. Math., 54 (1951), 1–37.CrossRefGoogle Scholar
6.Conley, C. C.. Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Ser. in Math., 38 (Amer. Math. Soc., Providence, R.I., 1976).Google Scholar
7.Conley, C. C. and Zehnder, E.. The Birkhoff-Lewis Fixed Point Theorem and a conjecture of V. I. Arnold. Inventions Math., (1983), 33–49.CrossRefGoogle Scholar
8.Cordier, J. M. and Porter, T.. Shape Theory. Categorical Methods of Approximation, Ellis Horwood Series: Mathematics and its Applications, Chichester, 1989.Google Scholar
9.Dydak, J. and Segal, J.. Shape Theory: An Introduction, Lecture Notes in Math. 688, (Springer-Verlag, Berlin, 1978).CrossRefGoogle Scholar
10.Fox, R. H.. On the Lusternik Schnirelmann category. Ann. Math. 42 (1941), 333–370.CrossRefGoogle Scholar
11.Giraldo, A. and Sanjurjo, J. M. R.. On the global structure of invariant regions of flows with asymptotically stable attractors, Math. Zeitschrift, to appear.Google Scholar
12.Gunther, B. and Segal, J.. Every attractor of a flow on a manifold has the shape of a finite polyhedron. Proc. Amer. Math. Soc., 119 (1993), 321–329.CrossRefGoogle Scholar
13.Hastings, H. M.. Shape theory and dynamical systems. In eds. Markley, N. G. and Perizzo, W.: The Structure of Attractors in Dynamical Systems, Lecture Notes in Math. 668 (Springer-Verlag, Berlin 1978), 150–160.CrossRefGoogle Scholar
14.Hu, S. T.. Theory of Retracts (Wayne State University Press, Detroit, 1967).Google Scholar
15.James, I. M.. On category in the sense of Lusternik-Schnirelmann, Topology 17 (1978), 331–348.CrossRefGoogle Scholar
16.James, I. M.. Lusternik-Schnirelmann category. In Handbook of Algebraic Topology (Elsevier, 1995), 1293–1310.CrossRefGoogle Scholar
17.Mardešić, S. and Segal, J.. Shape Theory (North Holland, Amsterdam, 1982).Google Scholar
18.Pozniak, M.. Lusternik-Schnirelmann category of an isolated invariant set. Univ. Iagellonicae Ada Math., 31 (1994), 129–139.Google Scholar
19.Robbin, J. W. and Salamon, D.. Dynamical systems, shape theory and the Conley index. Ergod. Th. and Dynam. Sys., 8* (1998), 375–393.Google Scholar
20.Salamon, D.. Connected simple systems and the Conley index of isolated invariant sets. Trans. Amer. Math. Soc., 291 (1985), 1–41.CrossRefGoogle Scholar
21.Sanjurjo, J. M. R.. Multihomotopy, Cech. spaces of loops and shape groups. Proc. London Math. Soc., 69 (1994), 330–344.CrossRefGoogle Scholar
22.Sanjurjo, J. M. R.. On the structure of uniform attractors. J. Math. Anal. Appl., 192 (1995), 519–528.CrossRefGoogle Scholar