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Noncompact chain recurrence and attraction. (English) Zbl 0759.58031
The author intends to extend a theorem of C. Conley from the setting of dynamical systems on compact metric spaces to metric spaces that are only locally compact, although the basic results are analogous to Conley’s theorem that characterizes the chain recurrent set of \(f\) in terms of the attractors of \(f\) and their basins of attraction. According to the results by the author the given metric is of primary importance rather than the topology it generated. The author gives the results in this note that depend on the topology induced by a metric rather than on the particular choice of the metric.
Reviewer: Y.Kozai (Tokyo)

MSC:
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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[1] Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. · Zbl 0397.34056
[2] James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. · Zbl 0144.21501
[3] John Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dynam. Systems 8* (1988), no. Charles Conley Memorial Issue, 99 – 107. · Zbl 0634.58023
[4] John Franks, A variation on the PoincarĂ©-Birkhoff theorem, Hamiltonian dynamical systems (Boulder, CO, 1987) Contemp. Math., vol. 81, Amer. Math. Soc., Providence, RI, 1988, pp. 111 – 117.
[5] -, A new proof of the Brouwer plane translation theorem, preprint. · Zbl 0767.58025
[6] Barnabas M. Garay, Uniform persistence and chain recurrence, J. Math. Anal. Appl. 139 (1989), no. 2, 372 – 381. · Zbl 0677.54033
[7] Mike Hurley, Chain recurrence and attraction in noncompact spaces, Ergodic Theory Dynam. Systems 11 (1991), no. 4, 709 – 729. · Zbl 0785.58033
[8] Mike Hurley, Attractors in restricted cellular automata, Proc. Amer. Math. Soc. 115 (1992), no. 2, 563 – 571. · Zbl 0761.58024
[9] John L. Kelley, General topology, D. Van Nostrand Company, Inc., Toronto-New York-London, 1955. · Zbl 0066.16604
[10] R. McGehee, Some metric proprties of attractors with applications to computer simulations of dynamical systems, Univ. of Minnesota, preprint, 1988.
[11] V. V. Nemytskii and V. V. Stepanov, Qualitative theory of differential equations, Princeton Mathematical Series, No. 22, Princeton University Press, Princeton, N.J., 1960. · Zbl 0089.29502
[12] Clark Robinson, Stability theorems and hyperbolicity in dynamical systems, Proceedings of the Regional Conference on the Application of Topological Methods in Differential Equations (Boulder, Colo., 1976), 1977, pp. 425 – 437. · Zbl 0375.58016
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