On the two definitions of the Conley index

Author:
Henry L. Kurland

Journal:
Proc. Amer. Math. Soc. **106** (1989), 1117-1130

MSC:
Primary 58F25

DOI:
https://doi.org/10.1090/S0002-9939-1989-0982405-9

MathSciNet review:
982405

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Abstract: The two definitions of the homotopy equivalences between Conley index spaces of an isolated invariant set, the original one of Conley [**C**] as completed by the author in [**K1**] and the more recent definition of Salamon [**S**], are shown to define the same homotopy classes without reference to the difficult proof of [**K1**] showing the Conley index to be a connected simple system. The equivalences of the original definition are useful in describing certain geometric situations in terms of the index; examples are given.

**[B]**Yu. P. Boglaev,*The two-point problem for a class of ordinary differential equations with a small parameter coefficient of the derivative*, USSR Comput. Math.-Math. Phys.**10**(1970), 190-204.**[C]**C.C. Conley,*Isolated Invariant Sets and the Morse Index*, CBMS Conference Proceedings, Amer. Math. Soc., Providence, R.I., 1978. MR**511133 (80c:58009)****[K1]**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)****[K2]**-,*Homotopy invariants of a repeller-attractor pair*, I:*the Püppe sequence of an R-A pair*, J. Differential Equations**46**(1982), 1-31. MR**677580 (84d:58049)****[K3]**-,*Homotopy invariants of a repeller-attractor pair*, II:*continuation of an R-A pair*, J. Differential Equations**49**(1983), 281-329. MR**708647 (85i:58095)****[K4]**-,*Following homology in singularly perturbed systems*, J. Differential Equations**62**(1986), 1-72. MR**830047 (87h:34080)****[K5]**-,*Layers in singularly perturbed systems via homology contintuation*(to appear).**[S]**D. Salamon,*Connected simple systems and the Conley index of isolated invariant sets*, Trans. Amer. Math. Soc.**291**(1985), 1-41. MR**797044 (87e:58182)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1989-0982405-9

Keywords:
Isolated invariant set,
Conley index,
connected simple system,
homology of the Conley index,
boundary layer,
common squeeze time,
flow map

Article copyright:
© Copyright 1989
American Mathematical Society