Shape index and other indices of Conley type for local maps on locally compact Hausdorff spaces

Marian Mrozek

Fundamenta Mathematicae (1994)

  • Volume: 145, Issue: 1, page 15-37
  • ISSN: 0016-2736

Abstract

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We present a scheme for constructing various Conley indices for locally defined maps. In particular, we extend the shape index of Robbin and Salamon to the case of a locally defined map in a locally compact Hausdorff space. We compare the shape index with the cohomological Conley index for maps. We also prove the commutativity property of the Conley index, which is analogous to the commutativity property of the fixed point index.

How to cite

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Mrozek, Marian. "Shape index and other indices of Conley type for local maps on locally compact Hausdorff spaces." Fundamenta Mathematicae 145.1 (1994): 15-37. <http://eudml.org/doc/212031>.

@article{Mrozek1994,
abstract = {We present a scheme for constructing various Conley indices for locally defined maps. In particular, we extend the shape index of Robbin and Salamon to the case of a locally defined map in a locally compact Hausdorff space. We compare the shape index with the cohomological Conley index for maps. We also prove the commutativity property of the Conley index, which is analogous to the commutativity property of the fixed point index.},
author = {Mrozek, Marian},
journal = {Fundamenta Mathematicae},
keywords = {shape index; Conley index},
language = {eng},
number = {1},
pages = {15-37},
title = {Shape index and other indices of Conley type for local maps on locally compact Hausdorff spaces},
url = {http://eudml.org/doc/212031},
volume = {145},
year = {1994},
}

TY - JOUR
AU - Mrozek, Marian
TI - Shape index and other indices of Conley type for local maps on locally compact Hausdorff spaces
JO - Fundamenta Mathematicae
PY - 1994
VL - 145
IS - 1
SP - 15
EP - 37
AB - We present a scheme for constructing various Conley indices for locally defined maps. In particular, we extend the shape index of Robbin and Salamon to the case of a locally defined map in a locally compact Hausdorff space. We compare the shape index with the cohomological Conley index for maps. We also prove the commutativity property of the Conley index, which is analogous to the commutativity property of the fixed point index.
LA - eng
KW - shape index; Conley index
UR - http://eudml.org/doc/212031
ER -

References

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  1. [Bo1] K. Borsuk, Concerning homotopy properties of compacta, Fund. Math. 62 (1968), 223-254. Zbl0159.24603
  2. [Bo2] K. Borsuk, Concerning the notion of the shape of compacta, in: Proc. Internat. Sympos. on Topology and its Applications (Herceg-Novi, 1968), Savez Društava Mat. Fiz. i Astronom., Belgrade, 1969, 98-104. 
  3. [Bo3] K. Borsuk, Theory of Shape, PWN-Polish Scientific Publishers, Warszawa, 1975. 
  4. [Co] C. C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Ser. in Math. 38, Amer. Math. Soc., Providence, R.I., 1978. 
  5. [Do] A. Dold, Lectures on Algebraic Topology, Springer, Berlin, 1972. 
  6. [Le] J. Leray, Théorie des points fixes: indice total et nombre de Lefschetz, Bull. Soc. Math. France 87 (1959), 221-223. Zbl0093.36702
  7. [Ma] S. Mardešić, A survey of the shape theory of compacta, in: General Topology and its Relations to Modern Analysis and Algebra III, Proc. of the Third Prague Topological Symposium, 1971, Academia, Prague, 1972, 291-300. 
  8. [MS] S. Mardešić and J. Segal, Shape Theory, North-Holland, Amsterdam, 1982. 
  9. [Mr1] M. Mrozek, Leray functor and the cohomological Conley index for discrete dynamical systems, Trans. Amer. Math. Soc. 318 (1990), 149-178. Zbl0686.58034
  10. [Mr2] M. Mrozek, Normal functors and retractors in categories of endomorphisms, Univ. Iagell. Acta Math. 29 (1991), 57-75. 
  11. [MR] M. Mrozek and K. P. Rybakowski, A cohomological Conley index for maps on metric spaces, J. Differential Equations 90 (1991), 143-171. Zbl0721.58040
  12. [RS] J. W. Robbin and D. Salamon, Dynamical systems, shape theory and the Conley index, Ergodic Theory Dynamical Systems 8* (1988), 375-393. Zbl0682.58040
  13. [Wa] T. Ważewski, Sur un principe topologique pour l'examen de l'allure asymptotique des intégrales des équations différentielles ordinaires, Ann. Soc. Polon. Math. 20 (1947), 279-313. Zbl0032.35001

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