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Spectral sequences in Conley’s theory

Published online by Cambridge University Press:  13 October 2009

O. CORNEA ,
K. A. DE REZENDE  and
M. R. DA SILVEIRA
O. CORNEA
Affiliation:
Département de mathématiques et de statistique, Université de Montréal, Montréal, Québec, Canada (email: cornea@dms.umontreal.ca)
K. A. DE REZENDE
Affiliation:
Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas-UNICAMP, 13083-970, Campinas, SP, Brazil (email: ketty@ime.unicamp.br, silveira@ime.unicamp.br)
M. R. DA SILVEIRA
Affiliation:
Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas-UNICAMP, 13083-970, Campinas, SP, Brazil (email: ketty@ime.unicamp.br, silveira@ime.unicamp.br)
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Abstract

In this paper, we analyse the dynamics encoded in the spectral sequence (Er,dr) associated with certain Conley theory connection maps in the presence of an ‘action’ type filtration. More specifically, we present an algorithm for finding a chain complex C and its differential; the method uses a connection matrix Δ to provide a system that spans Er in terms of the original basis of C and to identify all of the differentials drp:Erp→Erp−r. In exploring the dynamical implications of a non-zero differential, we prove the existence of a path that joins the singularities generating E0p and E0p−r in the case where a direct connection by a flow line does not exist. This path is made up of juxtaposed orbits of the flow and of the reverse flow, and proves to be important in some applications.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 4 , August 2010 , pp. 1009 - 1054
Copyright
Copyright © Cambridge University Press 2009

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