### Morse decompositions in the absence of uniqueness, II

DOI: http://dx.doi.org/10.12775/TMNA.2003.026

#### 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> ].

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> ].

#### Keywords

Attractor-repeller pairs; partially ordered Morse decompositions; singular perturbations; continuation properties; thin domains

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