2010
DOI: 10.1007/s13163-010-0031-x
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An approach to the shape Conley index without index pairs

Abstract: Robbin and Salamon proved that the shape of the homotopy Conley index of an isolated invariant set K coincides with the shape of the one-point compactification of the unstable region W u (K) of K endowed with a certain topology which they called the intrinsic topology. In this paper an equivalent, simplified definition of the latter is given in elementary terms, without resorting to index pairs whatsoever. We then show how our approach allows for a development of the shape index and its basic properties in an … Show more

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Cited by 10 publications
(8 citation statements)
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“…Second, it allows to calculate shape indices and Morse equations of invariant sets by using only the unstable manifolds of the invariant sets and their Morse sets. Robbin and Salamon's approach to the shape index theory was further developed in the works of Mrozek [12] and Sánchez-Gabites [15] for dynamical systems on locally compact spaces.…”
mentioning
confidence: 99%
“…Second, it allows to calculate shape indices and Morse equations of invariant sets by using only the unstable manifolds of the invariant sets and their Morse sets. Robbin and Salamon's approach to the shape index theory was further developed in the works of Mrozek [12] and Sánchez-Gabites [15] for dynamical systems on locally compact spaces.…”
mentioning
confidence: 99%
“…. , γ c independent non-torsion cohomology classes in im i * n−1 and the result follows.A direct consequence of Theorem 3.6 is the following result that generalizes[24, Theorem 4.6].…”
mentioning
confidence: 80%
“…In particular, we see that phase spaces with little cohomology do not support isolated non-saddle sets with overly complicated regions influence. This is motivated by some results from [23,24] about certain unstable attractors. In addition, motivated by the results in [6] and [3] about the continuation properties of isolated non-saddle sets we find necessary and sufficient conditions for the property of being non-saddle to be robust for families of smooth flows defined on smooth manifolds without further assumptions about the dimension or cohomology of the phase space.…”
mentioning
confidence: 99%
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“…Remark 1.1. There is another approach to developing Conley index theory without using index pairs by Sánchez-Gabites [Sán11].…”
mentioning
confidence: 99%