Duke Mathematical Journal

Homotopical dynamics, III: Real singularities and Hamiltonian flows

Octavian Cornea

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Abstract

On the space of nondepraved (see [8]) real, isolated singularities, we consider the stable equivalence relation induced by smooth deformations whose asymptotic behaviour is controlled by the Palais-Smale condition. It is shown that the resulting space of equivalence classes admits a canonical semiring structure and is isomorphic to the semiring of stable homotopy classes of CW-complexes.

In an application to Hamiltonian dynamics, we relate the existence of bounded and periodic orbits on noncompact level hypersurfaces of Palais-Smale Hamiltonians with just one singularity that is nondepraved to the lack of self-duality (in the sense of E. Spanier and J. Whitehead) of the sublink of the singularity.

Article information

Source
Duke Math. J., Volume 109, Number 1 (2001), 183-204.

Dates
First available in Project Euclid: 5 August 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1091737226

Digital Object Identifier
doi:10.1215/S0012-7094-01-10917-4

Mathematical Reviews number (MathSciNet)
MR1844209

Zentralblatt MATH identifier
1107.37300

Subjects
Primary: 37B30: Index theory, Morse-Conley indices
Secondary: 37J45: Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods 57R45: Singularities of differentiable mappings 58K30: Global theory

Citation

Cornea, Octavian. Homotopical dynamics, III: Real singularities and Hamiltonian flows. Duke Math. J. 109 (2001), no. 1, 183--204. doi:10.1215/S0012-7094-01-10917-4. https://projecteuclid.org/euclid.dmj/1091737226


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See also

  • See also: Octavian Cornea. Homotopical dynamics: suspension and duality. Ergodic Theory Dynam. Systems Vol. 20, No. 2 (2000), pp. 379-391.
  • See also: Octavian Cornea. Homotopical dynamics. II. Hopf invariants, smoothings and the Morse complex. Ann. Sci. École Norm. Sup. (4) Vol. 35, No. 4 (2002), pp. 549-573.
  • See also: Octavian Cornea. Homotopical dynamics. IV. Hopf invariants and Hamiltonian flows. Comm. Pure Appl. Math. Vol. 55, No. 8 (2002), pp. 1033-1088.