Leray functor and cohomological Conley index for discrete dynamical systems
Author:
Marian Mrozek
Journal:
Trans. Amer. Math. Soc. 318 (1990), 149178
MSC:
Primary 34C35; Secondary 39A10, 55M99, 58F25
MathSciNet review:
968888
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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/S00029947198608574397
 [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/S00029947198909783687
 [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/10.1016/00220396(81)900280
 [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 multivalued 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
infinitedimensional semiflows, Trans. Amer.
Math. Soc. 269 (1982), no. 2, 351–382. MR 637695
(83h:58084), http://dx.doi.org/10.1090/S00029947198206376957
 [Sp]
Edwin
H. Spanier, Algebraic topology, McGrawHill Book Co., New
YorkToronto, Ont.London, 1966. MR 0210112
(35 #1007)
 [Co]
 C. C. Conley Isolated invariant sets and the Morse index, CBMS Regional Conf. Ser. in Math., no. 38, Amer. Math. Soc., Providence, R.I., 1978. MR 511133 (80c:58009)
 [Fr1]
 R. Franzosa, Index filtrations and the homology index braid for partially ordered Morse decompositions, Trans. Amer. Math. Soc. 298 (1986), 193213. MR 857439 (88a:58121)
 [Fr2]
 , The connection matrix theory for Morse decompositions, Trans. Amer. Math. Soc. 311 (1989), 561592. MR 978368 (90a:58149)
 [Ku]
 H. L. Kurland, The Morse index of an isolated invariant set is a connected simple system, J. Differential Equations 42 (1981), 234259. MR 641650 (83a:58077)
 [Le]
 J. Leray, Théories des points fixes: indice total á nombre de Lefschetz, Bull. Soc. Math. France 87 (1959), 221233. MR 0143202 (26:762)
 [Mr1]
 M. Mrozek, Index pairs and the fixed point index for semidynamical systems with discrete time, Fund. Math. (in press). MR 1065901 (91i:58026)
 [Mr2]
 , The cohomological index of Conley type for multivalued 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]
 Z. Nitecki, Differentiable dynamics, an introduction to the orbit structure of diffeomorphisms, MIT Press, Cambridge, Mass., and London, 1971. MR 0649788 (58:31210)
 [RS]
 J. W. Robbin and D. Salamon, Dynamical systems, shape theory and the Conley index, Ergodic Theory Dynamical Systems 8* (1988), 375393. MR 967645 (89h:58094)
 [Ry]
 K. Rybakowski, On the homotopy index of infinite dimensional semiflows, Trans. Amer. Math. Soc. 269 (1982), 351382. MR 637695 (83h:58084)
 [Sp]
 E. H. Spanier, Algebraic topology, McGrawHill, New York, 1966. MR 0210112 (35:1007)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199009688881
PII:
S 00029947(1990)09688881
Keywords:
Conley index,
Leray endomorphism,
isolated invariant set
Article copyright:
© Copyright 1990
American Mathematical Society
