Leray functor and cohomological Conley index for discrete dynamical systems

Mrozek, Marian

doi:10.1090/S0002-9947-1990-0968888-1

Leray functor and cohomological Conley index for discrete dynamical systems
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Trans. Amer. Math. Soc. 318 (1990), 149-178 Request permission

Abstract:

We introduce the Leray functor on the category of graded modules equipped with an endomorphism of degree zero and we use this functor to define the cohomological Conley index of an isolated invariant set of a homeomorphism on a locally compact metric space. We prove the homotopy and additivity properties for this index and compute the index in some examples. As one of applications we prove the existence of nonconstant, bounded solutions of the Euler approximation of a certain system of ordinary differential equations.

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
Robert D. Franzosa, The connection matrix theory for Morse decompositions, Trans. Amer. Math. Soc. 311 (1989), no. 2, 561–592. MR 978368, DOI 10.1090/S0002-9947-1989-0978368-7
Henry L. Kurland, The Morse index of an isolated invariant set is a connected simple system, J. Differential Equations 42 (1981), no. 2, 234–259. MR 641650, DOI 10.1016/0022-0396(81)90028-0
Jean Leray, Théorie des points fixes: indice total et nombre de Lefschetz, Bull. Soc. Math. France 87 (1959), 221–233 (French). MR 143202
Marian Mrozek, Index pairs and the fixed point index for semidynamical systems with discrete time, Fund. Math. 133 (1989), no. 3, 179–194. MR 1065901, DOI 10.4064/fm-133-3-179-194

The cohomological index of Conley type for multi-valued admissible flows

The Morse equation in Conley’s index theory for homeomorphisms

A cohomological conley index for discrete time dynamical systems

Zbigniew Nitecki, Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms, The M.I.T. Press, Cambridge, Mass.-London, 1971. MR 0649788
Joel W. Robbin and Dietmar Salamon, Dynamical systems, shape theory and the Conley index, Ergodic Theory Dynam. Systems 8$^*$ (1988), no. Charles Conley Memorial Issue, 375–393. MR 967645, DOI 10.1017/S0143385700009494
Krzysztof P. Rybakowski, On the homotopy index for infinite-dimensional semiflows, Trans. Amer. Math. Soc. 269 (1982), no. 2, 351–382. MR 637695, DOI 10.1090/S0002-9947-1982-0637695-7
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112