On the two definitions of the Conley index
Author:
Henry L. Kurland
Journal:
Proc. Amer. Math. Soc. 106 (1989), 11171130
MSC:
Primary 58F25
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 twopoint problem for a class of ordinary differential equations with a small parameter coefficient of the derivative, USSR Comput. Math.Math. Phys. 10 (1970), 190204.
 [C]
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)
 [K1]
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
 [K2]
Henry
L. Kurland, Homotopy invariants of repellerattractor pairs. I. The
Puppe sequence of an RA pair, J. Differential Equations
46 (1982), no. 1, 1–31. MR 677580
(84d:58049), http://dx.doi.org/10.1016/00220396(82)901061
 [K3]
Henry
L. Kurland, Homotopy invariants of repellerattractor pairs. II.
Continuation of RA pairs, J. Differential Equations
49 (1983), no. 2, 281–329. MR 708647
(85i:58095), http://dx.doi.org/10.1016/00220396(83)900165
 [K4]
Henry
L. Kurland, Following homology in singularly perturbed
systems, J. Differential Equations 62 (1986),
no. 1, 1–72. MR 830047
(87h:34080), http://dx.doi.org/10.1016/00220396(86)901051
 [K5]
, Layers in singularly perturbed systems via homology contintuation (to appear).
 [S]
Dietmar
Salamon, Connected simple systems and the
Conley index of isolated invariant sets, Trans.
Amer. Math. Soc. 291 (1985), no. 1, 1–41. MR 797044
(87e:58182), http://dx.doi.org/10.1090/S00029947198507970443
 [B]
 Yu. P. Boglaev, The twopoint problem for a class of ordinary differential equations with a small parameter coefficient of the derivative, USSR Comput. Math.Math. Phys. 10 (1970), 190204.
 [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), 234259. MR 641650 (83a:58077)
 [K2]
 , Homotopy invariants of a repellerattractor pair, I: the Püppe sequence of an RA pair, J. Differential Equations 46 (1982), 131. MR 677580 (84d:58049)
 [K3]
 , Homotopy invariants of a repellerattractor pair, II: continuation of an RA pair, J. Differential Equations 49 (1983), 281329. MR 708647 (85i:58095)
 [K4]
 , Following homology in singularly perturbed systems, J. Differential Equations 62 (1986), 172. 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), 141. MR 797044 (87e:58182)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198909824059
PII:
S 00029939(1989)09824059
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
