Connection matrix theory for discrete dynamical systems

Piotr Bartłomiejczyk; Zdzisław Dzedzej

Banach Center Publications (1999)

  • Volume: 47, Issue: 1, page 67-78
  • ISSN: 0137-6934

Abstract

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In [C] and [F1] the connection matrix theory for Morse decomposition is developed in the case of continuous dynamical systems. Our purpose is to study the case of discrete time dynamical systems. The connection matrices are matrices between the homology indices of the sets in the Morse decomposition. They provide information about the structure of the Morse decomposition; in particular, they give an algebraic condition for the existence of connecting orbit set between different Morse sets.

How to cite

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Bartłomiejczyk, Piotr, and Dzedzej, Zdzisław. "Connection matrix theory for discrete dynamical systems." Banach Center Publications 47.1 (1999): 67-78. <http://eudml.org/doc/208943>.

@article{Bartłomiejczyk1999,
abstract = {In [C] and [F1] the connection matrix theory for Morse decomposition is developed in the case of continuous dynamical systems. Our purpose is to study the case of discrete time dynamical systems. The connection matrices are matrices between the homology indices of the sets in the Morse decomposition. They provide information about the structure of the Morse decomposition; in particular, they give an algebraic condition for the existence of connecting orbit set between different Morse sets.},
author = {Bartłomiejczyk, Piotr, Dzedzej, Zdzisław},
journal = {Banach Center Publications},
keywords = {Conley index; Morse decomposition; connection matrix; index filtration},
language = {eng},
number = {1},
pages = {67-78},
title = {Connection matrix theory for discrete dynamical systems},
url = {http://eudml.org/doc/208943},
volume = {47},
year = {1999},
}

TY - JOUR
AU - Bartłomiejczyk, Piotr
AU - Dzedzej, Zdzisław
TI - Connection matrix theory for discrete dynamical systems
JO - Banach Center Publications
PY - 1999
VL - 47
IS - 1
SP - 67
EP - 78
AB - In [C] and [F1] the connection matrix theory for Morse decomposition is developed in the case of continuous dynamical systems. Our purpose is to study the case of discrete time dynamical systems. The connection matrices are matrices between the homology indices of the sets in the Morse decomposition. They provide information about the structure of the Morse decomposition; in particular, they give an algebraic condition for the existence of connecting orbit set between different Morse sets.
LA - eng
KW - Conley index; Morse decomposition; connection matrix; index filtration
UR - http://eudml.org/doc/208943
ER -

References

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  1. [BD] P. Bartłomiejczyk and Z. Dzedzej, Index filtrations and Morse decompositions for discrete dynamical systems, submitted to Annales Polonici Math. Zbl0985.37011
  2. [C] C. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conf. Ser. in Math. 38, AMS, Providence, R.I., 1980. 
  3. [CoZ] C. Conley and R. Zehnder, Morse-type index theory for flows and periodic solutions for Hamoltonian systems, Comm. Pure Appl. Math. 37 (1984), 207-253. Zbl0559.58019
  4. [D] A. Dold, Lectures on algebraic topology, Springer Verlag, Berlin Heidelberg New York 1972. Zbl0234.55001
  5. [F1] R. Franzosa, Index filtrations and connection matrices for partially ordered Morse decompositions, Trans. Amer. Math. Soc. 298 (1986), 193-213. Zbl0626.58013
  6. [F2] R. Franzosa, The connection matrix theory for Morse decompositions, Trans. Amer. Math. Soc. 311 (1989), 561-592. Zbl0689.58030
  7. [M1] M. Mrozek, Index pairs and the fixed point index for semidynamical systems with discrete time, Fund. Math. 133 (1989), 179-194. Zbl0708.58024
  8. [M2] M. Mrozek, Leray functor and cohomological Conley index for discrete dynamical systems, Trans. Amer. Math. Soc. 318 (1990), 149-178. Zbl0686.58034
  9. [M3] M. Mrozek, Morse equation in Conley's index theory for homeomorphisms, Topology Appl. 38 (1991), 45-60. Zbl0725.58022
  10. [R] D. Richeson, Connection matrix pairs for the discrete Conley index, to appear in Trans. Amer. Math. Soc. 
  11. [Sal] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), 1-41. Zbl0573.58020
  12. [Szy] A. Szymczak, The Conley index for discrete dynamical systems, Topology Appl. 66 (1995), 215-240. Zbl0840.34043

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