On the shape Conley index theory of semiflows on complete metric spaces (Q256176)

A shape version of Conley index theory is developed for local semiflows on complete metric spaces. Instead of compactness, the authors assume that the semiflow \(\Phi\) is asymptotically compact. That means that each bounded set \(B\) in \(X\) is admissible, i.e., for any sequences \(x_n\in B\) and \(t_n\to +\infty\) with \( \Phi ([0,t_n])x_n \subset B\) for all \(n\), the sequence of end points \(\Phi (t_n)x_n\) has a convergent subsequence.NEWLINENEWLINEThe shape equivalence between index pairs is a bit more flexible than the homotopy equivalence. In locally compact spaces, the idea goes back to \textit{J. W. Robbin} and \textit{D. Salamon} [Ergodic Theory Dyn. Syst. 8, 375--393 (1988; Zbl 0682.58040)] (see also [\textit{M. Mrozek}, Fundam. Math. 145, No. 1, 15--37 (1994; Zbl 0870.54043)] and [\textit{J. J. Sánchez-Gabites}, Rev. Mat. Complut. 24, No. 1, 95--114 (2011; Zbl 1218.37021)]).NEWLINENEWLINEAttractors and Morse decompositions are also treated by use of unstable manifolds; see [\textit{J. M. R. Sanjurjo}, Nonlinearity 16, No. 4, 1435--1448 (2003; Zbl 1050.37007)].