A dynamical interpretation of the connection map of an attractor-repeller decomposition
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911817124544512 |
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| author | Sánchez-Gabites, J. J. |
| author_facet | Sánchez-Gabites, J. J. |
| contents | In Conley index theory one may study an invariant set $S$ by decomposing it into an attractor $A$, a repeller $R$, and the orbits connecting the two. The Conley indices of $S$, $A$ and $R$ fit into an exact sequence where a certain connection homomorphism $Γ$ plays an important role. In this paper we provide a dynamical interpretation of this map. Roughly, $R$ "emits" an element of its Conley index as a "wavefront", part of which intersects the connecting orbits in $S$. This subset of the wavefront evolves towards $A$ and is then "received" by it to produce an element in its Conley index. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18815 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A dynamical interpretation of the connection map of an attractor-repeller decomposition Sánchez-Gabites, J. J. Dynamical Systems 37B30, 37B35, 55N99 In Conley index theory one may study an invariant set $S$ by decomposing it into an attractor $A$, a repeller $R$, and the orbits connecting the two. The Conley indices of $S$, $A$ and $R$ fit into an exact sequence where a certain connection homomorphism $Γ$ plays an important role. In this paper we provide a dynamical interpretation of this map. Roughly, $R$ "emits" an element of its Conley index as a "wavefront", part of which intersects the connecting orbits in $S$. This subset of the wavefront evolves towards $A$ and is then "received" by it to produce an element in its Conley index. |
| title | A dynamical interpretation of the connection map of an attractor-repeller decomposition |
| topic | Dynamical Systems 37B30, 37B35, 55N99 |
| url | https://arxiv.org/abs/2403.18815 |