The continuation theory for Morse decompositions and connection matrices

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.