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

DOI:
https://doi.org/10.1090/S0002-9947-1990-0968888-1

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]**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), 193-213. MR**857439 (88a:58121)****[Fr2]**-,*The connection matrix theory for Morse decompositions*, Trans. Amer. Math. Soc.**311**(1989), 561-592. 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), 234-259. MR**641650 (83a:58077)****[Le]**J. Leray,*Théories des points fixes: indice total á nombre de Lefschetz*, Bull. Soc. Math. France**87**(1959), 221-233. 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 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]**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), 375-393. MR**967645 (89h:58094)****[Ry]**K. Rybakowski,*On the homotopy index of infinite dimensional semiflows*, Trans. Amer. Math. Soc.**269**(1982), 351-382. MR**637695 (83h:58084)****[Sp]**E. H. Spanier,*Algebraic topology*, McGraw-Hill, New York, 1966. MR**0210112 (35:1007)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
34C35,
39A10,
55M99,
58F25

Retrieve articles in all journals with MSC: 34C35, 39A10, 55M99, 58F25

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1990-0968888-1

Keywords:
Conley index,
Leray endomorphism,
isolated invariant set

Article copyright:
© Copyright 1990
American Mathematical Society