Index filtrations and Morse decompositions for discrete dynamical systems

P. Bartłomiejczyk; Z. Dzedzej

Annales Polonici Mathematici (1999)

  • Volume: 72, Issue: 1, page 51-70
  • ISSN: 0066-2216

Abstract

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On a Morse decomposition of an isolated invariant set of a homeomorphism (discrete dynamical system) there are partial orderings defined by the homeomorphism. These are called admissible orderings of the Morse decomposition. We prove the existence of index filtrations for admissible total orderings of a Morse decomposition and introduce the connection matrix in this case.

How to cite

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Bartłomiejczyk, P., and Dzedzej, Z.. "Index filtrations and Morse decompositions for discrete dynamical systems." Annales Polonici Mathematici 72.1 (1999): 51-70. <http://eudml.org/doc/262787>.

@article{Bartłomiejczyk1999,
abstract = {On a Morse decomposition of an isolated invariant set of a homeomorphism (discrete dynamical system) there are partial orderings defined by the homeomorphism. These are called admissible orderings of the Morse decomposition. We prove the existence of index filtrations for admissible total orderings of a Morse decomposition and introduce the connection matrix in this case.},
author = {Bartłomiejczyk, P., Dzedzej, Z.},
journal = {Annales Polonici Mathematici},
keywords = {Conley index; index filtration; Morse decomposition; connection matrix; index filtrations; index triple},
language = {eng},
number = {1},
pages = {51-70},
title = {Index filtrations and Morse decompositions for discrete dynamical systems},
url = {http://eudml.org/doc/262787},
volume = {72},
year = {1999},
}

TY - JOUR
AU - Bartłomiejczyk, P.
AU - Dzedzej, Z.
TI - Index filtrations and Morse decompositions for discrete dynamical systems
JO - Annales Polonici Mathematici
PY - 1999
VL - 72
IS - 1
SP - 51
EP - 70
AB - On a Morse decomposition of an isolated invariant set of a homeomorphism (discrete dynamical system) there are partial orderings defined by the homeomorphism. These are called admissible orderings of the Morse decomposition. We prove the existence of index filtrations for admissible total orderings of a Morse decomposition and introduce the connection matrix in this case.
LA - eng
KW - Conley index; index filtration; Morse decomposition; connection matrix; index filtrations; index triple
UR - http://eudml.org/doc/262787
ER -

References

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  1. [C] C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Ser. in Math. 38, Amer. Math. Soc., Providence, 1980. 
  2. [CoZ] C. Conley and R. Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian systems, Comm. Pure Appl. Math. 37 (1984), 207-253. Zbl0559.58019
  3. [Fra1] R. Franzosa, Index filtrations and the homology index braid for partially ordered Morse decompositions, Trans. Amer. Math. Soc. 298 (1986), 193-213. Zbl0626.58013
  4. [Fra2] R. Franzosa, The connection matrix theory for Morse decompositions, ibid. 311 (1989), 561-592. Zbl0689.58030
  5. [Mr1] M. Mrozek, Index pairs and the fixed point index for semidynamical systems with discrete time, Fund. Math. 133 (1989), 179-194. Zbl0708.58024
  6. [Mr2] M. Mrozek, Leray functor and cohomological Conley index for discrete dynamical systems, Trans. Amer. Math. Soc. 318 (1990), 149-178. Zbl0686.58034
  7. [Mr3] M. Mrozek, Morse equation in Conley's index theory for homeomorphisms, Topology Appl. 38 (1991), 45-60. Zbl0725.58022
  8. [Re] J. Reineck, The connection matrix in Morse-Smale flows II, Trans. Amer. Math. Soc. 347 (1995), 2097-2110. Zbl0831.58031
  9. [Ri] D. Richeson, Connection matrix pairs for the discrete Conley index, ibid., to appear. 
  10. [Sal] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, ibid. 291 (1985), 1-41. 
  11. [Szy] A. Szymczak, The Conley index for discrete dynamical systems, Topology Appl. 66 (1995), 215-240. Zbl0840.34043

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