Arnold conjecture

The Arnold conjecture, named after mathematician Vladimir Arnold, is a mathematical conjecture in the field of symplectic geometry, a branch of differential geometry.[1]

Statement

Let be a compact symplectic manifold. For any smooth function , the symplectic form induces a Hamiltonian vector field on , defined by the identity:

The function is called a Hamiltonian function.

Suppose there is a 1-parameter family of Hamiltonian functions , inducing a 1-parameter family of Hamiltonian vector fields on . The family of vector fields integrates to a 1-parameter family of diffeomorphisms . Each individual is a Hamiltonian diffeomorphism of .

The Arnold conjecture says that for each Hamiltonian diffeomorphism of , it possesses at least as many fixed points as a smooth function on possesses critical points.[2]

Nondegenerate Hamiltonian and weak Arnold conjecture

A Hamiltonian diffeomorphism is called nondegenerate if its graph intersects the diagonal of transversely. For nondegenerate Hamiltonian diffeomorphisms, a variant of the Arnold conjecture says that the number of fixed points is at least equal to the minimal number of critical points of a Morse function on , called the Morse number of .

In view of the Morse inequality, the Morse number is also greater than or equal to a homological invariant of , for example, the sum of Betti numbers over a field :

The weak Arnold conjecture says that for a nondegenerate Hamiltonian diffeomorphism on the above integer is a lower bound of its number of fixed points.

See also

References

  1. Asselle, L.; Izydorek, M.; Starostka, M. (2022). "The Arnold conjecture in and the Conley index". arXiv:2202.00422 [math.DS].
  2. Buhovsky, Lev; Humilière, Vincent; Seyfaddini, Sobhan (2018-04-11). "A C0 counterexample to the Arnold conjecture". Inventiones Mathematicae. Springer Science and Business Media LLC. 213 (2): 759–809. arXiv:1609.09192. doi:10.1007/s00222-018-0797-x. ISSN 0020-9910. S2CID 46900145.
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