An index for periodic orbits of local semidynamical systems

Fenske, Christian

doi:10.1090/S0002-9947-98-02114-X

An index for periodic orbits of local semidynamical systems
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Trans. Amer. Math. Soc. 350 (1998), 4973-4991 Request permission

Abstract:

We define an index of Fuller type counting the number of periodic orbits of a semiflow on an ANR by a suitable approximation process.

References

Shui Nee Chow and John Mallet-Paret, The Fuller index and global Hopf bifurcation, J. Differential Equations 29 (1978), no. 1, 66–85. MR 492560, DOI 10.1016/0022-0396(78)90041-4
E. N. Dancer and J. F. Toland, Degree theory for orbits of prescribed period of flows with a first integral, Proc. London Math. Soc. (3) 60 (1990), no. 3, 549–580. MR 1044311, DOI 10.1112/plms/s3-60.3.549
James Dugundji, Topology, Allyn and Bacon Series in Advanced Mathematics, Allyn and Bacon, Inc., Boston, Mass.-London-Sydney, 1978. Reprinting of the 1966 original. MR 0478089
Doug Curtis, Tadeusz Dobrowolski, and Jerzy Mogilski, Some applications of the topological characterizations of the sigma-compact spaces $l^{2}_{f}$ and $\Sigma$, Trans. Amer. Math. Soc. 284 (1984), no. 2, 837–846. MR 743748, DOI 10.1090/S0002-9947-1984-0743748-7
Hideaki Kaneko, Generalized contractive multivalued mappings and their fixed points, Math. Japon. 33 (1988), no. 1, 57–64. MR 935499
Christian C. Fenske, An index for periodic orbits of functional-differential equations, Math. Ann. 285 (1989), no. 3, 381–392. MR 1019708, DOI 10.1007/BF01455063
F. Brock Fuller, An index of fixed point type for periodic orbits, Amer. J. Math. 89 (1967), 133–148. MR 209600, DOI 10.2307/2373103
Ross Geoghegan and Andrew Nicas, Trace and torsion in the theory of flows, Topology 33 (1994), no. 4, 683–719. MR 1293306, DOI 10.1016/0040-9383(94)90004-3
Jack Girolo, Approximating compact sets in normed linear spaces, Pacific J. Math. 98 (1982), no. 1, 81–89. MR 644940
Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977
William S. Massey, Homology and cohomology theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 46, Marcel Dekker, Inc., New York-Basel, 1978. An approach based on Alexander-Spanier cochains. MR 0488016
A. J. B. Potter, On a generalization of the Fuller index, Nonlinear functional analysis and its applications, Part 2 (Berkeley, Calif., 1983) Proc. Sympos. Pure Math., vol. 45, Amer. Math. Soc., Providence, RI, 1986, pp. 283–286. MR 843615
A. J. B. Potter, Approximation methods and the generalised Fuller index for semiflows in Banach spaces, Proc. Edinburgh Math. Soc. (2) 29 (1986), no. 3, 299–308. MR 865263, DOI 10.1017/S0013091500017740
Roman Srzednicki, Periodic orbits indices, Fund. Math. 135 (1990), no. 3, 147–173. MR 1077507, DOI 10.4064/fm-135-3-147-173
Roman Srzednicki, Topological invariants and detection of periodic orbits, J. Differential Equations 111 (1994), no. 2, 283–298. MR 1284415, DOI 10.1006/jdeq.1994.1083