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Lyapunov maps, simplicial complexes and the Stone functor

Published online by Cambridge University Press:  19 September 2008

Joel W. Robbin
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
Dietmar A. Salamon
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

Abstract

Let be an attractor network for a dynamical system ft: MM, indexed by the lower sets of a partially ordered set P. Our main theorem asserts the existence of a Lyapunov map ψ:MK(P) which defines the attractor network. This result is used to prove the existence of connection matrices for discrete-time dynamical systems.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1992

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References

REFERENCES

[1]Cairns, S. S.. A simple triangulation method for smooth manifolds. Bull. Amer. Math. Soc. 67 (1951), 389390.CrossRefGoogle Scholar
[2]Cohen, R. L., Jones, J. D. S. & Segal, G. B.. To appear.Google Scholar
[3]Conley, C. C.. Isolated invariant sets and the Morse index. CBMS Reg. Conf. Series Math., Vol. 38. Amer. Math. Soc: Providence, RI 1978.CrossRefGoogle Scholar
[4]Conley, C. C.. The gradient structure of a flow. Ergod. Th. & Dynam. Sys. 8 (1988), 1126.CrossRefGoogle Scholar
[5]Conley, C. C. & Zehnder, E.. Morse type index theory for flows and periodic solutions for Hamiltonian equations. Commun. Pure Appl. Math. 37 (1984), 207253.CrossRefGoogle Scholar
[6]Eilenberg, S. & Steenrod, N.. Foundations of Algebraic Topology. Princeton, 1952.CrossRefGoogle Scholar
[7]Folkman, J.. The homology groups of a lattice. J. Math. & Mech. 15 (1966), 631636.Google Scholar
[8]Franzosa, R.. Index fitrations and the homology index braid for partially ordered Morse decompositions, Amer. Math. Soc. Trans. 298 (1986), 193213.CrossRefGoogle Scholar
[9]Franzosa, R.. The continuation theory for Morse decompositions and connection matrices. Trans. Amer. Math. Soc. 310 (1988), 781803.CrossRefGoogle Scholar
[10]Franzosa, R.. The connection matrix theory for Morse decompositions. Trans. Amer. Math. Soc. 311 (1989), 561592.CrossRefGoogle Scholar
[11]Johnstone, P.. Stone spaces, Studies in Advanced Mathematics. Cambridge University Press: Cambridge, 1982.Google Scholar
[12]Kelly, J.. General Topology. Van Nostrand, 1955.Google Scholar
[13]Mrozek, M.. The Morse equation in Conley's index theory for homeomorphisms. Preprint, University of Krakow (1988).Google Scholar
[14]Palis, J. & de Melo, W.. Geometric Theory of Dynamical Systems. Springer-Verlag: New York, 1982.CrossRefGoogle Scholar
[15]Robbin, J. W. & Salamon, D. A.. Dynamical systems, shape theory and the Conley index. Ergod. Th. & Dynam. Sys. 8 (1988), 375393.CrossRefGoogle Scholar
[16]Rota, G. C.. On the foundations of combinatorial theory. Z. Warscheinlichkeitstheorie und Verwandte Gebiete 2 (1964), 340368.CrossRefGoogle Scholar
[17]Salamon, D. A.. Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), 141.CrossRefGoogle Scholar
[18]Salamon, D.. Morse theory, the Conley index and Floer homology. Bull. L. M. S. 22 (1990), 113140.Google Scholar
[19]Segal, G.. Classifying spaces and spectral sequences Publ. Math. IHES.Google Scholar
[20]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747817;CrossRefGoogle Scholar
Reprinted in: The Mathematics of Time. Springer-Verlag: New York, 1980.Google Scholar
[21]Smoller, J.. Shock Waves and Reaction Diffusion Equations. Springer-Verlag: New York, 1983.CrossRefGoogle Scholar
[22]Spanier, E.. Algebraic Topology. Springer-Verlag: New York, 1966.Google Scholar
[23]Stone, M.. Topological representation of distributive lattices and Brouwerian logics. Casopis Pest. Mat. Fys. 67 (1937), 127Google Scholar
[24]Witten, E.. Supersymmetry and Morse theory. J. Diff. Geom. 17 (1982), 661692.CrossRefGoogle Scholar
[25]Zeeman, E. C.. On the filtered differential group. Ann. Math. 66 (1957), 557585.CrossRefGoogle Scholar
[26]Wilson, W.. Smoothing derivative of functions and applications. Trans. Amer. Math. Soc. 139 (1969), 413428.CrossRefGoogle Scholar
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