Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T05:22:32.116Z Has data issue: false hasContentIssue false
  • English
  • Français

A Morse equation in Conley's index theory for semiflows on metric spaces

Published online by Cambridge University Press:  19 September 2008

Krzysztof P. Rybakowski  and
Eduard Zehnder
Krzysztof P. Rybakowski
Affiliation:
Albert-Ludwigs Universität, Institut für Angewandte Mathematik, Hermann-Herder-Str. 10, 7800 Freiburg i. Br., West, German
Eduard Zehnder
Affiliation:
Ruhr-Universität, Bochum, Institut für Mathematik, Universitätsstr. 150, D-4630 Bochum 1, West, Germany
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Given a compact (two-sided) flow, an isolated invariant set S and a Morse-decomposition (M1, …, Mn) of S, there is a generalized Morse equation, proved by Conley and Zehnder, which relates the Alexander-Spanier cohomology groups of the Conley indices of the sets Mi and S with each other. Recently, Rybakowski developed the technique of isolating blocks and extended Conley's index theory to a class of one-sided semiflows on non-necessarily compact spaces, including e.g. semiflows generated by parabolic equations. Using these results, we discuss in this paper Morse decompositions and prove the above-mentioned Morse equation not only for arbitrary homology and cohomology groups, but also in this more general semiflow setting.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 1 , March 1985 , pp. 123 - 143
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[1]Amann, H.. Invariant sets and existence theorems for semi-linear parabolic and elliptic systems. J. Math. Anal Appl. 65 (1978), 432–467.CrossRefGoogle Scholar
[2]Amann, H. & Zehnder, E.. Non-trivial solutions for a class of non-resonance problems and applications to non-linear differential equations. Ann. Scuola Norm. sup. Pisa Cl. Sci., (IV) VII, Nr. 4 (1980), 539–603.Google Scholar
[3]Amann, H. & Zehnder, E.. Periodic solutions of asymptotically linear Hamiltonian systems. Manuscripta Math. 32 (1980), 149–189.CrossRefGoogle Scholar
[4]Bott, R.. Marston Morse and his mathematical works. Bull. Amer. Math. Soc. 3 (1980), 907–950.CrossRefGoogle Scholar
[5]Conley, C. C.. Isolated invariant sets and the Morse index. CBMS Regional Conf. Ser. in Math. 38, Amer. Math. Soc.: Providence, 1978.CrossRefGoogle Scholar
[6]Conley, C. C. & Zehnder, E.. Morse type index theory for flows and periodic solutions for Hamiltonian systems. Comm. Pure and Appl. Math. XXXVII (1984), 207–253.CrossRefGoogle Scholar
[7]Henry, Dan. Geometric Theory of Semilinear Parabolic Equations. Springer Lecture Notes in Math. 840. Springer Verlag: Berlin-Heidelberg-New York, 1981.CrossRefGoogle Scholar
[8]Massey, W. S.. Homology and Cohomology Theory. M. Dekker Inc: New York and Basel, 1978.Google Scholar
[9]Rybakowski, K. P.. On the homotopy index for infinite-dimensional semiflows. Trans. Amer. Math. Soc. 269 (1982), 351–382.CrossRefGoogle Scholar
[10]Rybakowski, K. P.. The Morse-index, repeller-attractor pairs and the connection index for semiflows on noncompact spaces. J. Diff. Eq. 47 (1983), 66–98.CrossRefGoogle Scholar
[11]Rybakowski, K. P.. Trajectories joining points of nonlinear parabolic and hyperbolic partial differential equations, J. Diff. Eq. 51 (1984), 182–212.CrossRefGoogle Scholar
[12]Smoller, J. A. & Conley, C. C.. On the structure of magneto-hydrodynamic shock waves. Comm. Pure and Appl. Math. 27 (1974), 367–375.Google Scholar
[13]Spanier, E. H.. Algebraic Topology. McGraw Hill: New York, 1966.Google Scholar