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An approach to the shape Conley index without index pairs. (English) Zbl 1218.37021

Let \(K\) be an isolated invariant set of a continuous flow \(\varphi: M \times {\mathbb R} \to M\) on a locally compact metrizable space \(M\). The unstable region \(W^u(K)\) of \(K\) supports an intrinsic topology described by J. W. Robbin and D. Salamon [Ergodic Theory Dyn. Syst. (Charles Conley Mem. Vol.) 8, 375–393 (1988; Zbl 0682.58040)] who used the language of sequences of index pairs.
In the paper under review the author gives an alternative simplified definition of the intrinsic topology. On the basis of this description he develops the theory of the shape index and simplifies some classical constructions. The author also presents some new results concerning the geometry of \(W^u(K)/K\) and complexity of the overall dynamics.

MSC:

37B30 Index theory for dynamical systems, Morse-Conley indices
54H20 Topological dynamics (MSC2010)
55P55 Shape theory

Citations:

Zbl 0682.58040

Software:

conley
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Full Text: DOI

References:

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