On a generalization of the Conley index

Marian Mrozek; James Reineck; Roman Srzednicki

Banach Center Publications (1999)

  • Volume: 47, Issue: 1, page 157-171
  • ISSN: 0137-6934

Abstract

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In this note we present the main ideas of the theory of the Conley index over a base space introduced in the papers [7, 8]. The theory arised as an attempt to solve two questions concerning the classical Conley index. In the definition of the index, the exit set of an isolating neighborhood is collapsed to a point. Some information is lost on this collapse. In particular, topological information about how a set sits in the phase space is lost. The first question addressed is how to retain some of the information which is lost on collapse. As an example, is it possible that two repelling periodic orbits which are not homotopic in the punctured plane are related by continuation? Clearly one cannot be continued to the other as periodic orbits, but the index of such a periodic orbit is the same as the index of the disjoint union of two rest points, so the question of continuation as isolated invariant sets is far less obvious.

How to cite

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Mrozek, Marian, Reineck, James, and Srzednicki, Roman. "On a generalization of the Conley index." Banach Center Publications 47.1 (1999): 157-171. <http://eudml.org/doc/208931>.

@article{Mrozek1999,
abstract = {In this note we present the main ideas of the theory of the Conley index over a base space introduced in the papers [7, 8]. The theory arised as an attempt to solve two questions concerning the classical Conley index. In the definition of the index, the exit set of an isolating neighborhood is collapsed to a point. Some information is lost on this collapse. In particular, topological information about how a set sits in the phase space is lost. The first question addressed is how to retain some of the information which is lost on collapse. As an example, is it possible that two repelling periodic orbits which are not homotopic in the punctured plane are related by continuation? Clearly one cannot be continued to the other as periodic orbits, but the index of such a periodic orbit is the same as the index of the disjoint union of two rest points, so the question of continuation as isolated invariant sets is far less obvious.},
author = {Mrozek, Marian, Reineck, James, Srzednicki, Roman},
journal = {Banach Center Publications},
keywords = {Conley index; fiberwise pointed space; fiberwise deforming homotopy type; Szymczak category},
language = {eng},
number = {1},
pages = {157-171},
title = {On a generalization of the Conley index},
url = {http://eudml.org/doc/208931},
volume = {47},
year = {1999},
}

TY - JOUR
AU - Mrozek, Marian
AU - Reineck, James
AU - Srzednicki, Roman
TI - On a generalization of the Conley index
JO - Banach Center Publications
PY - 1999
VL - 47
IS - 1
SP - 157
EP - 171
AB - In this note we present the main ideas of the theory of the Conley index over a base space introduced in the papers [7, 8]. The theory arised as an attempt to solve two questions concerning the classical Conley index. In the definition of the index, the exit set of an isolating neighborhood is collapsed to a point. Some information is lost on this collapse. In particular, topological information about how a set sits in the phase space is lost. The first question addressed is how to retain some of the information which is lost on collapse. As an example, is it possible that two repelling periodic orbits which are not homotopic in the punctured plane are related by continuation? Clearly one cannot be continued to the other as periodic orbits, but the index of such a periodic orbit is the same as the index of the disjoint union of two rest points, so the question of continuation as isolated invariant sets is far less obvious.
LA - eng
KW - Conley index; fiberwise pointed space; fiberwise deforming homotopy type; Szymczak category
UR - http://eudml.org/doc/208931
ER -

References

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  1. [Bo] K. Borsuk, Theory of Retracts, PWN, Warszawa, 1967. Zbl0153.52905
  2. [1] I. M. James, General topology over a base, in: I. M. James, E. H. Kronheimer (editors), Aspects of Topology, Cambridge University Press, Cambridge, 1985. Zbl0586.54016
  3. [2] I. M. James, General Topology and Homotopy Theory, Springer-Verlag, New York, 1984. Zbl0562.54001
  4. [3] I. M. James, Fibrewise Topology, Cambridge University Press, Cambridge, 1989. 
  5. [4] T. Kaczynski and M. Mrozek, Connected simple systems and the Conley functor, Topol. Methods Nonlinear Anal. 10 (1997), 183-193. Zbl0909.54033
  6. [5] Ch. McCord, K. Mischaikow, and M. Mrozek, Zeta functions, periodic trajectories and the Conley index, J. Differential Equations 121 (1995), 258-292. Zbl0833.34045
  7. [6] M. Mrozek, Leray functor and cohomological index for discrete dynamical systems, Trans. Amer. Math. Soc. 318 (1990), 149-178. Zbl0686.58034
  8. [7] M. Mrozek, J. F. Reineck, and R. Srzednicki, The Conley index over a base, Trans. Amer. Math. Soc., to appear. Zbl0967.37011
  9. [8] M. Mrozek, J. F. Reineck, and R. Srzednicki, The Conley index over the circle, J. Dynamics Differential Equations, to appear. Zbl0967.37010
  10. [9] M. Mrozek and K. P. Rybakowski, A cohomological Conley index for maps on metric spaces, J. Differential Equations 90 (1991), 143-171. Zbl0721.58040
  11. [10] J. W. Robbin and D. Salamon, Dynamical systems, shape theory and the Conley index, Ergodic Theory Dynamical Systems 8* (1988), 375-393. Zbl0682.58040
  12. [11] A. Szymczak, The Conley index for discrete semidynamical systems, Top. & Appl. 66 (1995), 215-240. Zbl0840.34043
  13. [12] A. Szymczak, A combinatorial procedure for finding isolating neighborhoods and index pairs, Proc. Royal Soc. Edinburgh Sect. A 127 (1997). Zbl0882.54032

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