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2012 | 10 | 6 | 2138-2159
Tytuł artykułu

Equivariant Morse equation

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The paper is concerned with the Morse equation for flows in a representation of a compact Lie group. As a consequence of this equation we give a relationship between the equivariant Conley index of an isolated invariant set of the flow given by .x = −∇f(x) and the gradient equivariant degree of ∇f. Some multiplicity results are also presented.
Wydawca
Czasopismo
Rocznik
Tom
10
Numer
6
Strony
2138-2159
Opis fizyczny
Daty
wydano
2012-12-01
online
2012-10-12
Twórcy
  • Faculty of Technical Physics and Applied Mathematics, Gdańsk University of Technology, Narutowicza 11/12, 80-233, Gdańsk, Poland, marcins@mif.pg.gda.pl
Bibliografia
  • [1] Bredon G.E., Introduction to Compact Transformation Groups, Pure Appl. Math., 46, Academic Press, New York-London, 1972
  • [2] Conley C., Isolated Invariant Sets and the Morse Index, CBMS Reg. Conf. Ser. Math., 38, American Mathematical Society, Providence, 1978 [Crossref]
  • [3] Conley C., Zehnder E., Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math., 1984, 37(2), 207–253 http://dx.doi.org/10.1002/cpa.3160370204[Crossref]
  • [4] tom Dieck T., Transformation Groups, de Gruyter Stud. Math., 8, Walter de Gruyter, Berlin, 1987 http://dx.doi.org/10.1515/9783110858372[Crossref]
  • [5] Field M.J., Dynamics and Symmetry, ICP Adv. Texts Math., 3, Imperial College Press, London, 2007
  • [6] Floer A., A refinement of the Conley index and an application to the stability of hyperbolic invariant sets, Ergodic Theory Dynam. Systems, 1987, 7(1), 93–103 http://dx.doi.org/10.1017/S0143385700003825[Crossref]
  • [7] Floer A., Zehnder E., The equivariant Conley index and bifurcations of periodic solutions of Hamiltonian systems, Ergodic Theory Dynam. Systems, 1988, 8, 87–97 http://dx.doi.org/10.1017/S0143385700009354[Crossref]
  • [8] Gęba K., Degree for gradient equivariant maps and equivariant Conley index, In: Topological Nonlinear Analysis II, Frascati, June, 1995, Progr. Nonlinear Differential Equations Appl., 27, Birkhäuser, Boston, 1997, 247–272
  • [9] Gęba K., Rybicki S., Some remarks on the Euler ring U(G), J. Fixed Point Theory Appl., 2008, 3(1), 143–158 http://dx.doi.org/10.1007/s11784-007-0043-4[Crossref]
  • [10] Goębiewska A., Rybicki S., Global bifurcations of critical orbits of G-invariant strongly indefinite functionals, Nonlinear Anal., 2011, 74(5), 1823–1834 http://dx.doi.org/10.1016/j.na.2010.10.055[Crossref][WoS]
  • [11] Hatcher A., Algebraic Topology, Cambridge University Press, Cambridge, 2002
  • [12] Illman S., Equivariant singular homology and cohomology for actions of compact Lie groups, Proceedings of the Second Conference on Compact Transformation Groups, Amherst, June 7–18, 1971, Lecture Notes in Math., 298, Springer, Berlin, 1972, 403–415 http://dx.doi.org/10.1007/BFb0070055[Crossref]
  • [13] Izydorek M., Equivariant Conley index in Hilbert spaces and applications to strongly indefinite problems, Nonlinear Anal., 2002, 51(1), Ser. A: Theory Methods, 33–66
  • [14] Izydorek M., Styborski M., Morse inequalities via Conley index theory, In: Topological Methods in Nonlinear Analysis, Torun, February 9–13, 2009, Lect. Notes Nonlinear Anal., 12, Juliusz Schauder Center for Nonlinear Studies, Torun, 2011
  • [15] Kawakubo K., The Theory of Transformation Groups, Clarendon Press, Oxford University Press, New York, 1991
  • [16] Razvan M.R., On Conley’s fundamental theorem of dynamical systems, Int. J. Math. Math. Sci., 2004, 25–28, 1397–1401 http://dx.doi.org/10.1155/S0161171204202125[Crossref]
  • [17] Ruan H., Rybicki S., Applications of equivariant degree for gradient maps to symmetric Newtonian systems, Nonlinear Anal., 2008, 68(6), 1479–1516 http://dx.doi.org/10.1016/j.na.2006.12.039[Crossref][WoS]
  • [18] Rybakowski K.P., The Homotopy Index and Partial Differential Equations, Universitext, Springer, Berlin, 1987 http://dx.doi.org/10.1007/978-3-642-72833-4[Crossref]
  • [19] Rybicki S., A degree for S 1-equivariant orthogonal maps and its applications to bifurcation theory, Nonlinear Anal., 1994, 23(1), 83–102 http://dx.doi.org/10.1016/0362-546X(94)90253-4[Crossref]
  • [20] Rybicki S., Degree for equivariant gradient maps, Milan J. Math., 2005, 73, 103–144 http://dx.doi.org/10.1007/s00032-005-0040-2[Crossref]
  • [21] Spanier E.H., Algebraic Topology, McGraw-Hill, New York-Toronto, 1966
  • [22] Styborski M., Topological Invariants for Equivariant Flows: Conley Index and Degree, PhD thesis, Polish Academy of Sciences, Warsaw, 2009
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-012-0124-5
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