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A Morse equation in Conley’s index theory for semiflows on metric spaces. (English) Zbl 0581.54026

Consider a continuous two-sided flow on a locally compact metric space. A compact invariant subset S is called isolated if it is a maximal compact invariant set with respect to a suitable compact neighbourhood. With every such set there is associated a formal power series p(t,h(S)) the coefficients of which are the ranks of the Alexander-Spanier cohomology modules of a certain pointed topological space defined by means of C. Conley and the second author [Commun. Pure Appl. Math. 37, 207-253 (1984; Zbl 0559.58019)] have shown that there is a formula relating p(t,h(S)) with the \(p(t,h(M_ i))\), where \((M_ 1,...,M_ n)\) is a Morse decomposition of S (i.e. the \(M_ i\) are compact invariant subsets of S such that for every \(x\in S\) which is contained in no \(M_ i\) there are indices \(i<j\) such that the orbit through x has its limit sets for \(t\to -\infty\) and \(t\to +\infty\) in \(M_ i\) and \(M_ j\), respectively). In the present paper these results are remarkably generalized. Only very weak compactness conditions are needed, semiflows instead of flows can be treated and even general cohomology theories are considered.
Reviewer: E.Behrends

MSC:

54H20 Topological dynamics (MSC2010)
37A99 Ergodic theory
55N35 Other homology theories in algebraic topology
37C10 Dynamics induced by flows and semiflows

Citations:

Zbl 0559.58019
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References:

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