The connection map for attractor-repeller pairs
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- by Christopher McCord PDF
- Trans. Amer. Math. Soc. 307 (1988), 195-203 Request permission
Abstract:
In the Conley index theory, the connection map of the homology attractor-repeller sequence provides a means of detecting connecting orbits between a repeller and attractor in an isolated invariant set. In this work, the connection map is shown to be additive: under suitable decompositions of the connecting orbit set, the connection map of the invariant set equals the sum of the connection maps of the decomposition elements. This refines the information provided by the homology attractor-repeller sequence. In particular, the properties of the connection map lead to a characterization of isolated invariant sets with hyperbolic critical points as an attractor-repeller pair.References
- Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
- Robert Franzosa, Index filtrations and the homology index braid for partially ordered Morse decompositions, Trans. Amer. Math. Soc. 298 (1986), no. 1, 193–213. MR 857439, DOI 10.1090/S0002-9947-1986-0857439-7 —, The connection matrix theory for the Conley index, Trans. Amer. Math. Soc. (to appear).
- Henry L. Kurland, Homotopy invariants of repeller-attractor pairs. I. The Puppe sequence of an R-A pair, J. Differential Equations 46 (1982), no. 1, 1–31. MR 677580, DOI 10.1016/0022-0396(82)90106-1 —, Isolated connections and chain recurrence, preprint.
- Christopher McCord, Mappings and homological properties in the Conley index theory, Ergodic Theory Dynam. Systems 8$^*$ (1988), no. Charles Conley Memorial Issue, 175–198. MR 967637, DOI 10.1017/S014338570000941X —, Mappings and Morse decompositions in the Conley index theory, preprint. K. Mischaikow, Classification of traveling wave solutions of reaction-diffusion systems, Lefschetz Center for Dynamical Systems, no. 86-5, 1985. —, Homoclinic orbits in Hamiltonian systems and heteroclinic orbits in gradient and gradient-like systems, Lefschetz Center for Dynamical Systems, no 86-33, 1986.
- James F. Reineck, Connecting orbits in one-parameter families of flows, Ergodic Theory Dynam. Systems 8$^*$ (1988), no. Charles Conley Memorial Issue, 359–374. MR 967644, DOI 10.1017/S0143385700009482
- Dietmar Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), no. 1, 1–41. MR 797044, DOI 10.1090/S0002-9947-1985-0797044-3
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 307 (1988), 195-203
- MSC: Primary 58F12; Secondary 34C35
- DOI: https://doi.org/10.1090/S0002-9947-1988-0936812-4
- MathSciNet review: 936812