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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The continuation theory for Morse decompositions and connection matrices


Author: Robert D. Franzosa
Journal: Trans. Amer. Math. Soc. 310 (1988), 781-803
MSC: Primary 58F25; Secondary 34C35, 58F12, 58F14
MathSciNet review: 973177
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Abstract: The continuation theory for ($ < $-ordered) Morse decompositions and the indices defined on them--the homology index braid and the connection matrices--is established. The equivalence between $ < $-ordered Morse decompositions and $ < $-consistent attractor filtrations is displayed. The spaces of ($ < $-ordered) Morse decompositions for a product parametrization of a local flow are introduced, and the local continuation of ($ < $-ordered) Morse decompositions is obtained via the above-described equivalence and the local continuation of attractors. The homology index braid and the connection matrices of an admissible ordering of a Morse decomposition are shown to be invariant on path components of the corresponding space of $ < $-ordered Morse decompositions. This invariance is used to prove that the collection of connection matrices of a Morse decomposition is upper semicontinuous over the space of Morse decompositions (and over the parameter space) under local continuation.


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DOI: https://doi.org/10.1090/S0002-9947-1988-0973177-6
Keywords: Conley index, Morse decomposition, connection matrix, continuation
Article copyright: © Copyright 1988 American Mathematical Society