The connection matrix theory for Morse decompositions
Author:
Robert D. Franzosa
Journal:
Trans. Amer. Math. Soc. 311 (1989), 561592
MSC:
Primary 58F25; Secondary 58E05, 58F09
MathSciNet review:
978368
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Abstract: The connection matrix theory for Morse decompositions is introduced. The connection matrices are matrices of maps between the homology indices of the sets in the Morse decomposition. The connection matrices cover, in a natural way, the homology index braid of the Morse decomposition and provide information about the structure of the Morse decomposition. The existence of connection matrices of Morse decompositions is established, and examples illustrating applications of the connection matrix are provided.
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 C. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conf. Ser. in Math., no. 38, Amer. Math. Soc., Providence, R.I., 1980. MR 511133 (80c:58009)
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 C. Conley and E. Zehnder, Morsetype index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37 (1984). MR 733717 (86b:58021)
 [3]
 R. Franzosa, Index filtrations and connection matrices for partially ordered Morse decompositions, Ph. D. dissertation, Univ. of WisconsinMadison, 1984.
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 , Index filtrations and the homology index braid for partially ordered Morse decompositions, Trans. Amer. Math. Soc. 298 (1986). MR 857439 (88a:58121)
 [5]
 , The continuation theory for Morse decompositions and connection matrices, Trans. Amer. Math. Soc. 310 (1988). MR 973177 (90g:58111)
 [6]
 R. Franzosa and K. Mischaikow, The connection matrix theory for semiflows on (not necessarily locally compact) metric spaces, J. Differential Equations 71 (1988). MR 927003 (89c:54078)
 [7]
 H. Kurland, The Morse index of an isolated invariant set is a connected simple system, J. Differential Equations 42 (1981). MR 641650 (83a:58077)
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 , Homotopy invariants of repellerattractor pairs. I, The Püppe sequence of an  pair, J. Differential Equations 46 (1982).
 [9]
 , Homotopy invariants of repellerattractor pairs. II, Continuation of  pairs, J. Differential Equations 49 (1983).
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 C. McCord, Mappings and homological properties in the Conley index theory, Ph. D. dissertation, Univ. of WisconsinMadison, 1986.
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 K. Mischaikow, Classification of traveling wave solutions of reactiondiffusion systems, Brown Univ., LCDS #865, 1985.
 [12]
 J. Reineck, The connection matrix and the classification of flows arising from ecological models, Ph. D. dissertation, Univ. of WisconsinMadison, 1985.
 [13]
 , Connecting orbits in one parameter families of flows, preprint.
 [14]
 D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985). MR 797044 (87e:58182)
 [15]
 J. Smoller, Shock waves and reactiondiffusion equations, SpringerVerlag, Berlin and New York, 1983. MR 688146 (84d:35002)
 [16]
 E. Spanier, Algebraic topology, McGrawHill, New York, 1966. MR 0210112 (35:1007)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198909783687
PII:
S 00029947(1989)09783687
Keywords:
Conley index,
Morse decomposition,
connection matrix
Article copyright:
© Copyright 1989
American Mathematical Society
