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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
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Abstract

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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

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