Morse decompositions in the absence of uniqueness, II

Abstract

This paper is a sequel to our previous work [< i> Morse decompositions in the absence of uniqueness< /i> ,
Topol. Methods Nonlinear Anal. < b> 18< /b> (2001), 205–242]. We
first extend the concept of $\mathcal{T}$-Morse decompositions to the partially
ordered case and prove a generalization of a result by Franzosa
and Mischaikow characterizing partially ordered $\mathcal{T}$-Morse
decompositions by the so-called $\mathcal{T}$-attractor semifiltrations.
Then we extend the (regular) continuation result for Morse
decompositions from [< i> Morse decompositions in the absence of uniqueness< /i> ,
Topol. Methods Nonlinear Anal. < b> 18< /b> (2001), 205–242] to the partially ordered case.
We also define singular convergence of families of ``solution''
sets in the spirit of our previous paper [< i> On a general Conley index continuation principle for singular perturbation
problems< /i> , Ergodic Theory Dynam. Systems < b> 22< /b> (2002), 729–755] and
prove various singular continuation results for attractor-repeller
pairs and Morse decompositions. We give a few applications of
our results, e.g. to thin domain problems.
The results of this paper are a main ingredient in the proof of
regular and singular continuation results for the homology braid and the connection matrix
in infinite dimensional Conley index theory. These topics are
considered in the forthcoming publications [< i> Continuation of the connection matrix in infinite-dimensional Conley index
theory< /i> ] and
[< i> Continuation of the connection matrix in singular perturbation problems< /i> ].