# 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

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topBartłomiejczyk, 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

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

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