Leray functor and cohomological Conley index for discrete dynamical systems

Mrozek, Marian

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

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6850(online) ISSN 0002-9947(print)

Leray functor and cohomological Conley index for discrete dynamical systems

Author: Marian Mrozek
Journal: Trans. Amer. Math. Soc. 318 (1990), 149-178
MSC: Primary 34C35; Secondary 39A10, 55M99, 58F25
MathSciNet review: 968888
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

[Co] 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 (80c:58009)
[Fr1] 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 (88a:58121), http://dx.doi.org/10.1090/S0002-9947-1986-0857439-7
[Fr2] Robert D. Franzosa, The connection matrix theory for Morse decompositions, Trans. Amer. Math. Soc. 311 (1989), no. 2, 561–592. MR 978368 (90a:58149), http://dx.doi.org/10.1090/S0002-9947-1989-0978368-7
[Ku] 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 (83a:58077), http://dx.doi.org/10.1016/0022-0396(81)90028-0
[Le] Jean Leray, Théorie des points fixes: indice total et nombre de Lefschetz, Bull. Soc. Math. France 87 (1959), 221–233 (French). MR 0143202 (26 #762)
[Mr1] 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 (91i:58026)
[Mr2] -, The cohomological index of Conley type for multi-valued admissible flows, J. Differential Equations (in press).
[Mr3] -, The Morse equation in Conley's index theory for homeomorphisms, Topology Appl. (to appear).
[MR] M. Mrozek and K. P. Rybakowski, A cohomological conley index for discrete time dynamical systems (to appear).
[Ni] Zbigniew Nitecki, Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms, The M.I.T. Press, Cambridge, Mass.-London, 1971. MR 0649788 (58 #31210)
[RS] 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 (89h:58094), http://dx.doi.org/10.1017/S0143385700009494
[Ry] Krzysztof P. Rybakowski, On the homotopy index for infinite-dimensional semiflows, Trans. Amer. Math. Soc. 269 (1982), no. 2, 351–382. MR 637695 (83h:58084), http://dx.doi.org/10.1090/S0002-9947-1982-0637695-7
[Sp] Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York, 1966. MR 0210112 (35 #1007)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|