Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T16:06:36.541Z Has data issue: false hasContentIssue false
  • English
  • Français

LUSTERNIK–SCHNIRELMANN CATEGORY BASED ON THE DISCRETE CONLEY INDEX THEORY

Part of: Classical Topics in Algebraic Topology Topological dynamics

Published online by Cambridge University Press:  30 October 2018

KATSUYA YOKOI
KATSUYA YOKOI*
Affiliation:
Department of Mathematics, Jikei University School of Medicine, Chofu, Tokyo 182-8570, Japan e-mail: yokoi@jikei.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study Lusternik–Schnirelmann type categories for isolated invariant sets by the use of the discrete Conley index.

MSC classification

Primary: 37B30: Index theory, Morse-Conley indices
Secondary: 37B35: Gradient-like and recurrent behavior; isolated (locally maximal) invariant sets 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirel'man) category of a space
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 61 , Issue 3 , September 2019 , pp. 693 - 704
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

REFERENCES

Anderson, R. D., On topological infinite deficiency, Michigan Math. J. 14 (1967), 365–383.Google Scholar
Borsuk, K., Theory of retracts, Monografie Matematyczne, Tom 44 (Państwowe Wydawnictwo Naukowe, Warsaw, 1967).Google Scholar
Bott, R., Lectures on Morse theory, old and new, Bull. Amer. Math. Soc. 7 (2) (1982), 331–358.CrossRefGoogle Scholar
Brown, M., Locally flat imbeddings of topological manifolds, Ann. Math. (2) 75 (1962), 331–341.CrossRefGoogle Scholar
Case, J. H. and Chamberlin, R. E., Characterizations of tree-like continua, Pac. J. Math. 10 (1960), 73–84.CrossRefGoogle Scholar
Chapman, T. A., Lectures on Hilbert cube manifolds, Expository lectures from the CBMS Regional Conference held at Guilford College, October 11–15, 1975 Regional Conference Series in Mathematics, vol. 28 (American Mathematical Society, Providence, R. I., 1976).CrossRefGoogle Scholar
Churchill, R. C., Isolated invariant sets in compact metric spaces, J. Differ. Equ. 12 (1972), 330–352.CrossRefGoogle Scholar
Conley, C., Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38 (American Mathematical Society, Providence, R.I., 1978).CrossRefGoogle Scholar
Conley, C. and Zehnder, E., Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37 (2), (1984), 207–253.CrossRefGoogle Scholar
Easton, R., Isolating blocks and epsilon chains for maps, Phys. D 39 (1) (1989), 95–110.CrossRefGoogle Scholar
Easton, R., Geometric methods for discrete dynamical systems, Oxford Engineering Science Series vol. 50 (Oxford University Press, New York, 1998).Google Scholar
Franks, J. and Richeson, D., Shift equivalence and the Conley index, Trans. Amer. Math. Soc. 352 (7) (2000), 3305–3322.CrossRefGoogle Scholar
Hu, S.-T., Theory of retracts (Wayne State University Press, Detroit, 1965).Google Scholar
James, I. M., On category, in the sense of Lusternik–Schnirelmann, Topology 17 (4) (1978), 331–348.CrossRefGoogle Scholar
Kato, H., On expansiveness of shift homeomorphisms of inverse limits of graphs, Fund. Math. 137 (1991), 201–210.CrossRefGoogle Scholar
Lusternik, L. and Schnirelmann, L. Méthodes Topoligiques dans les Problémes Variationnels, Herman, Paris (1934).Google Scholar
Moeckel, R., Morse decompositions and connection matrices, Ergodic Theory Dynam. Systems 8 (1988),227–249.CrossRefGoogle Scholar
Mrozek, M., Index pairs and the fixed point index for semidynamical systems with discrete time, Fund. Math. 133 (3), (1989), 179–194.CrossRefGoogle Scholar
Mrozek, M., Leray functor and cohomological Conley index for discrete dynamical systems, Trans. Amer. Math. Soc. 318 (1) (1990), 149–178.CrossRefGoogle Scholar
Poźniak, M., Lusternik-Schnirelmann category of an isolated invariant set, Univ. Iagel. Acta Math. 31 (1994), 129–139.Google Scholar
Razvan, M. R., Lusternik-Schnirelmann theory for a Morse decomposition, Southeast Asian Bull. Math. 29 (6) (2005), 1131–1138.Google Scholar
Richeson, D., Connection matrix pairs. Conley index theory (Warsaw, 1997), 219–232, Banach Center Publ., 47, Polish Acad. Sci., Warsaw, 1999.Google Scholar
Robbin, J. W. and Salamon, D., Dynamical systems, shape theory and the Conley index, Ergodic Theory Dynam. Syst. 8* (1988), 375–393.CrossRefGoogle Scholar
Salamon, D., Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1) (1985), 1–41.CrossRefGoogle Scholar
Sanjurjo, J. M. R., Lusternik-Schnirelmann category and Morse decompositions, Mathematika, 47 (1–2) (2000), 299–305.CrossRefGoogle Scholar
Schnirelmann, L., Über eine neue kombinatorische Invariante, Monatsh. Math. Phys. 37 (1), (1930), 131–134.CrossRefGoogle Scholar
Srinivasan, T., The Lusternik-Schnirelmann category of metric spaces, Topol. Appl. 167 (2014), 87–95.CrossRefGoogle Scholar
Szymczak, A., The Conley index for discrete semidynamical systems, Topol. Appl. 66 (3) (1995), 215–240.CrossRefGoogle Scholar
Szymczak, A., A combinatorial procedure for finding isolating neighbourhoods and index pairs, Proc. Roy. Soc. Edinburgh Sect. A 127 (5) (1997), 1075–1088.CrossRefGoogle Scholar
Szymczak, A., Wójcik, K. and Zgliczyński, P., On the discrete Conley index in the invariant subspace, Topol. Appl. 87 (2)(1998), 105–115.CrossRefGoogle Scholar
Walsh, J. J., Dimension, cohomological dimension, and cell-like mappings, Shape theory and geometric topology (Dubrovnik, 1981), Lecture Notes in Math., vol. 870 (Springer, Berlin-New York, 1981), 105–118.CrossRefGoogle Scholar
Williams, R. F., Classification of one dimensional attractors, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) (Amer. Math. Soc., Providence, R.I., 1970), 341–361.Google Scholar