Homology index braids in infinite-dimensional Conley index theory

Maria C. Carbinatto, Krzysztof P. Rybakowski

DOI: http://dx.doi.org/10.12775/TMNA.2005.024

Abstract


We extend the notion of a categorial Conley-Morse
index, as defined
in [K. P. rybakowski, < i> The Morse index, repeller-attractor pairs and the connection index
for semiflows on noncompact spaces< /i> , J. Differential Equations < b> 47< /b> (1987), 66–98],
to the case based on a more general
concept of an index pair introduced in [R. D. Franzosa and K. Mischaikow,
< i> The connection matrix theory for semiflows
on (not necessarily locally compact) metric spaces< /i> , J. Differential Equations
< b> 71< /b> (1988), 270–287]. We
also establish a naturality result of the long exact
sequence of attractor-repeller pairs with respect to the
choice of index triples. In particular, these results
immediately give a complete and rigorous existence result
for homology index braids in infinite dimensional Conley
index theory.

Finally, we describe some general regular and singular
continuation results for homology index braids obtained in
our recent papers [M. C. Carbinatto and K. P. Rybakowski,
< i> Nested sequences of index filtrations and continuation of the connection matrix< /i> ,
J. Differential Equations < b> 207< /b> (2004), 458–488] and
[M. C. Carbinatto and K. P. Rybakowski,
< i> Continuation of the connection matrix in singular perturbation problems< /i> ].

Keywords


Morse-Conley index theory; homology index braid; continuation properties; singular perturbations

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