Conley index for set-valued maps: from theory to computation

Tomasz Kaczynski

Banach Center Publications (1999)

  • Volume: 47, Issue: 1, page 57-65
  • ISSN: 0137-6934

Abstract

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Recent results on the Conley index theory for discrete multi-valued dynamical systems with their consequences for the computation of the index for representable maps are recapitulated. The terminology is simplified with respect to previous presentations, some superfluous hypotheses are abandoned and some conclusions are proved in a simpler way.

How to cite

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Kaczynski, Tomasz. "Conley index for set-valued maps: from theory to computation." Banach Center Publications 47.1 (1999): 57-65. <http://eudml.org/doc/208942>.

@article{Kaczynski1999,
abstract = {Recent results on the Conley index theory for discrete multi-valued dynamical systems with their consequences for the computation of the index for representable maps are recapitulated. The terminology is simplified with respect to previous presentations, some superfluous hypotheses are abandoned and some conclusions are proved in a simpler way.},
author = {Kaczynski, Tomasz},
journal = {Banach Center Publications},
keywords = {upper semicontinuous set-valued map; isolated invariant set; index pair; Leray reduction; Conley index},
language = {eng},
number = {1},
pages = {57-65},
title = {Conley index for set-valued maps: from theory to computation},
url = {http://eudml.org/doc/208942},
volume = {47},
year = {1999},
}

TY - JOUR
AU - Kaczynski, Tomasz
TI - Conley index for set-valued maps: from theory to computation
JO - Banach Center Publications
PY - 1999
VL - 47
IS - 1
SP - 57
EP - 65
AB - Recent results on the Conley index theory for discrete multi-valued dynamical systems with their consequences for the computation of the index for representable maps are recapitulated. The terminology is simplified with respect to previous presentations, some superfluous hypotheses are abandoned and some conclusions are proved in a simpler way.
LA - eng
KW - upper semicontinuous set-valued map; isolated invariant set; index pair; Leray reduction; Conley index
UR - http://eudml.org/doc/208942
ER -

References

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  1. [1] M. Allili and T. Kaczynski, An algorithmic approach to the construction of homomorphisms induced by maps in homology, Trans. Amer. Math. Soc., to appear. Zbl0947.37006
  2. [2] R. Bowen, Equilibrium States and The Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math. (470), Springer, 1975. Zbl0308.28010
  3. [3] L. Górniewicz, Homological Methods in Fixed Point Theory of Multi-Valued Maps, Dissertationes Math. (129), PWN, Warsaw, 1976. Zbl0324.55002
  4. [4] T. Kaczynski and M. Mrozek, Conley index for discrete multivalued dynamical systems, Topology Appl. 65 (1995), 83-96. Zbl0843.54042
  5. [5] T. Kaczynski and M. Mrozek, Stable index pairs for discrete dynamical systems, Canad. Math. Bull. 40 (4) (1997), 448-455. Zbl0903.54018
  6. [6] T. Kaczynski and M. Mrozek, Connected simple systems and the Conley functor, Topol. Meth. in Nonlin. Anal. 10 (1) (1997), 183-193. Zbl0909.54033
  7. [7] T. Kaczynski, M. Mrozek and M. Ślusarek, Homology computation by reduction of chain complexes, Computers & Math. Appl. 35 (4) (1998), 59-70. Zbl0904.55004
  8. [8] W. Kalies, K. Mischaikow and G. Watson, Cubical approximation and computation of homology, this volume. Zbl0938.55012
  9. [9] K. Mischaikow and M. Mrozek, Isolating neighborhoods and chaos, Japan J. Ind. and Appl. Math. 12 (1995), 205-236. Zbl0840.58033
  10. [10] K. Mischaikow and M. Mrozek, Chaos in the Lorenz equations: a computer assisted proof, Bull. Amer. Math. Soc. 32 (1995), 66-72. Zbl0820.58042
  11. [11] K. Mischaikow, M. Mrozek, J. Reiss and A. Szymczak, From time series to symbolic dynamics: an algebraic topological approach, preprint. 
  12. [12] M. Mrozek, Leray functor and cohomological index for discrete dynamical systems, Trans. Amer. Math. Soc. 318 (1990), 149-178. Zbl0686.58034
  13. [13] M. Mrozek, An algorithmic approach to the Conley index theory, preprint. 
  14. [14] J. W. Robbin and D. Salamon, Dynamical systems, shape theory and the Conley index, Ergodic Theory Dynamical Systems 8 (1988), 375-393. Zbl0682.58040
  15. [15] E. H. Spanier, Algebraic Topology, McGraw-Hill, 1966. 
  16. [16] A. Szymczak, A combinatorial procedure for finding isolating neighborhoods and index pairs, Proc. Royal Soc. Edinburgh Sect. A 127 (5) (1997), 1075-1088. Zbl0882.54032
  17. [17] A. Szymczak, The Conley index for discrete semidynamical systems, Topology Appl. 66 (3) (1995), 214-240. Zbl0840.34043

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