On foundations of the Conley index theory

Roman Srzednicki

Banach Center Publications (1999)

  • Volume: 47, Issue: 1, page 21-27
  • ISSN: 0137-6934

Abstract

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The Conley index theory was introduced by Charles C. Conley (1933-1984) in [C1] and a major part of the foundations of the theory was developed in Ph. D. theses of his students, see for example [Ch, Ku, Mon]. The Conley index associates the homotopy type of some pointed space to an isolated invariant set of a flow, just as the fixed point index associates an integer number to an isolated set of fixed points of a continuous map. Examples of isolated invariant sets arise naturally in the critical point theory - each isolated critical point of a functional is also an isolated invariant set of its gradient flow. If the critical point is nondegenerate then its Conley index is equal to the homotopy type of the pointed k-sphere, where k is the Morse index of that point. There are other relations to Morse theory, for example a generalization of Morse inequalities can be achieved. The aim of this note is to describe briefly some basic facts of the Conley index theory for (continuous-time) flows. We refer to [C2, Ry, S1, Smo] for a more detailed presentation. We do not touch more advanced topics of the theory: the Conley index as a connected simple system (see [C2, Ku, McM, S1]), connection and transition matrices (see [F1, F2, FM, McM, Mi1, Moe, Re]]), infinite dimensional Conley indices (see [Be, Ry], the Conley index for multivalued flows (see [KM, Mr2]), Conley-type indices for discrete-time flows (see [Mr3, RS, Sz]), equivariant Conley indices (see [Ba, Ge]), and relations to the Floer homology (see [S2]). (The list of bibliography items is far from completeness.) Moreover, we do not present any applications of the index. For some more recent results we refer to [Mi2]. We also refer to the other articles in this Proceedings.

How to cite

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Srzednicki, Roman. "On foundations of the Conley index theory." Banach Center Publications 47.1 (1999): 21-27. <http://eudml.org/doc/208936>.

@article{Srzednicki1999,
abstract = {The Conley index theory was introduced by Charles C. Conley (1933-1984) in [C1] and a major part of the foundations of the theory was developed in Ph. D. theses of his students, see for example [Ch, Ku, Mon]. The Conley index associates the homotopy type of some pointed space to an isolated invariant set of a flow, just as the fixed point index associates an integer number to an isolated set of fixed points of a continuous map. Examples of isolated invariant sets arise naturally in the critical point theory - each isolated critical point of a functional is also an isolated invariant set of its gradient flow. If the critical point is nondegenerate then its Conley index is equal to the homotopy type of the pointed k-sphere, where k is the Morse index of that point. There are other relations to Morse theory, for example a generalization of Morse inequalities can be achieved. The aim of this note is to describe briefly some basic facts of the Conley index theory for (continuous-time) flows. We refer to [C2, Ry, S1, Smo] for a more detailed presentation. We do not touch more advanced topics of the theory: the Conley index as a connected simple system (see [C2, Ku, McM, S1]), connection and transition matrices (see [F1, F2, FM, McM, Mi1, Moe, Re]]), infinite dimensional Conley indices (see [Be, Ry], the Conley index for multivalued flows (see [KM, Mr2]), Conley-type indices for discrete-time flows (see [Mr3, RS, Sz]), equivariant Conley indices (see [Ba, Ge]), and relations to the Floer homology (see [S2]). (The list of bibliography items is far from completeness.) Moreover, we do not present any applications of the index. For some more recent results we refer to [Mi2]. We also refer to the other articles in this Proceedings.},
author = {Srzednicki, Roman},
journal = {Banach Center Publications},
keywords = {Conley index; multivalued flows; Floer homology},
language = {eng},
number = {1},
pages = {21-27},
title = {On foundations of the Conley index theory},
url = {http://eudml.org/doc/208936},
volume = {47},
year = {1999},
}

TY - JOUR
AU - Srzednicki, Roman
TI - On foundations of the Conley index theory
JO - Banach Center Publications
PY - 1999
VL - 47
IS - 1
SP - 21
EP - 27
AB - The Conley index theory was introduced by Charles C. Conley (1933-1984) in [C1] and a major part of the foundations of the theory was developed in Ph. D. theses of his students, see for example [Ch, Ku, Mon]. The Conley index associates the homotopy type of some pointed space to an isolated invariant set of a flow, just as the fixed point index associates an integer number to an isolated set of fixed points of a continuous map. Examples of isolated invariant sets arise naturally in the critical point theory - each isolated critical point of a functional is also an isolated invariant set of its gradient flow. If the critical point is nondegenerate then its Conley index is equal to the homotopy type of the pointed k-sphere, where k is the Morse index of that point. There are other relations to Morse theory, for example a generalization of Morse inequalities can be achieved. The aim of this note is to describe briefly some basic facts of the Conley index theory for (continuous-time) flows. We refer to [C2, Ry, S1, Smo] for a more detailed presentation. We do not touch more advanced topics of the theory: the Conley index as a connected simple system (see [C2, Ku, McM, S1]), connection and transition matrices (see [F1, F2, FM, McM, Mi1, Moe, Re]]), infinite dimensional Conley indices (see [Be, Ry], the Conley index for multivalued flows (see [KM, Mr2]), Conley-type indices for discrete-time flows (see [Mr3, RS, Sz]), equivariant Conley indices (see [Ba, Ge]), and relations to the Floer homology (see [S2]). (The list of bibliography items is far from completeness.) Moreover, we do not present any applications of the index. For some more recent results we refer to [Mi2]. We also refer to the other articles in this Proceedings.
LA - eng
KW - Conley index; multivalued flows; Floer homology
UR - http://eudml.org/doc/208936
ER -

References

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  1. [Ba] T. Bartsch, Topological methods for variational problems with symmetries, Lecture Notes 1560, Springer-Verlag, Berlin, 1993, 
  2. [Be] V. Benci, A new approach to the Morse-Conley theory and some applications, Ann. Math. Pura Appl. 4 (1991), 231-305. Zbl0778.58011
  3. [Do] A. Dold, Lectures on Algebraic Topology, Springer-Verlag, Berlin, Heidelberg and New York, 1980. 
  4. [Ch] R. C. Churchill, Isolated invariant sets in compact metric spaces, J. Differential Equations 12 (1972), 330-352. Zbl0238.54044
  5. [C1] C. C. Conley, On a generalization of the Morse index, in: Ordinary Differential Equations, 1971 NRL-MRC Conference (L. Weiss, Ed.), Academic Press, New York and London, 1972, 133-146. 
  6. [C2] C. C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Ser. Math. 38, AMS, Providence, R.I., 1978. 
  7. [F1] R. Franzosa, The continuation theory for Morse decompositions and connection matrices, Trans. Amer. Math. Soc. 310 (1988), 781-803. Zbl0708.58021
  8. [F2] R. Franzosa, The connection matrix theory for Morse decompositions, Trans. Amer. Math. Soc. 311 (1989), 561-592. Zbl0689.58030
  9. [FM] R. Franzosa and K. Mischaikow, The connection matrix theory for semiflows on (not necessarily locally compact) metric spaces, J. Differential Equations 71 (1988), 270-287. Zbl0676.54048
  10. [Ge] K. Gęba, Degree for gradient equivariant maps and equivariant Conley index, Topological nonlinear analysis, II (Frascati, 1995), 247-272, Progr. Nonlinear Differential Equations Appl. 27, Birkhäuser, Boston, 1997. 
  11. [KM] T. Kaczynski and M. Mrozek, Conley index for discrete multi-valued dynamical systems, Topology Appl. 65 (1995), 83-96. Zbl0843.54042
  12. [Ku] H. L. Kurland, The Morse index of an isolated invariant set is a connected simple system, J. Differential Equations 42 (1981), 234-259. Zbl0477.58029
  13. [Mc] C. McCord, On the Hopf index and the Conley index, Trans. Amer. Math. Soc. 313 (1989), 853-860. Zbl0712.34062
  14. [McM] C. McCord and K. Mischaikow, Connected simple systems, transition matrices, and heteroclinic bifurcations, Trans. Amer. Math. Soc. 333 (1992), 379-422. Zbl0763.34028
  15. [Mi1] K. Mischaikow, Transition systems, Proc. Roy. Soc. Edinburgh, Sect. A 112 (1989), 155-175. Zbl0677.34046
  16. [Mi2] K. Mischaikow, Conley index theory, in: Dynamical Systems, CIME-Session, Montecatini Terme 1994, 119-207, Lecture Notes 1609, Springer-Verlag, Berlin, 1995. 
  17. [Moe] R. Moeckel, Morse decompositions and connection matrices, Ergodic Theory Dynamical Systems 8* (1988), 227-249. 
  18. [Mon] J. T. Montgomery, Cohomology of isolated invariant sets under perturbation, J. Differential Equations 13 (1973), 257-299. Zbl0238.58010
  19. [Mr1] M. Mrozek, Periodic and stationary trajectories of flows and ordinary differential equations, Univ. Iagel. Acta Math. 27 (1988), 29-37. Zbl0684.34046
  20. [Mr2] M. Mrozek, A cohomological index of Conley type for multivalued admissible flows, J. Differential Equations 84 (1990), 15-51. Zbl0703.34019
  21. [Mr3] M. Mrozek, Leray functor and cohomological index for discrete dynamical systems, Trans. Amer. Math. Soc. 318 (1990), 149-178. Zbl0686.58034
  22. [Po] M. Poźniak, Lusternik-Schnirelman category of an isolated invariant set, Univ. Iagel. Acta Math. 31 (1994), 129-139. Zbl0834.55006
  23. [Re] J. F. Reineck, The connection matrix in Morse-Smale flows, Trans. Amer. Math. Soc. 322 (1990), 523-545. Zbl0714.58027
  24. [RS] J. Robbin and D. Salamon, Dynamical systems, shape theory and the Conley index, Ergodic Theory Dynamical Systems 8* (1988), 375-393. Zbl0682.58040
  25. [Ry] K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg and New York, 1987. 
  26. [S1] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), 1-41. Zbl0573.58020
  27. [S2] D. Salamon, Morse theory, the Conley index and Floer homology, Bull. London Math. Soc. 22 (1990), 113-140. Zbl0709.58011
  28. [Sma] S. Smale, Morse inequalities for a dynamical system, Bull. Amer. Math. Soc. 66 (1960), 43-49. Zbl0100.29701
  29. [Smo] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York and Berlin, 1983. 
  30. [Sr] R. Srzednicki, Periodic and bounded solutions in blocks for time-periodic non-autonomous ordinary differential equations, Nonlinear Anal. - Theory Meth. Appl. 22 (1994), 707-737. Zbl0801.34041
  31. [Sz] A. Szymczak, The Conley index for discrete semidynamical systems, Topology Appl. 66 (1995), 215-240. Zbl0840.34043

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