The Conley conjecture, named after mathematician Charles Conley, is a mathematical conjecture in the field of symplectic geometry, a branch of differential geometry.

Background

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Let   be a compact symplectic manifold. A vector field   on   is called a Hamiltonian vector field if the 1-form   is exact (i.e., equals to the differential of a function  . A Hamiltonian diffeomorphism   is the integration of a 1-parameter family of Hamiltonian vector fields  .

In dynamical systems one would like to understand the distribution of fixed points or periodic points. A periodic point of a Hamiltonian diffeomorphism   (of periodic  ) is a point   such that  . A feature of Hamiltonian dynamics is that Hamiltonian diffeomorphisms tend to have infinitely many periodic points. Conley first made such a conjecture for the case that   is a torus. [2]

The Conley conjecture is false in many simple cases. For example, a rotation of a round sphere   by an angle equal to an irrational multiple of  , which is a Hamiltonian diffeomorphism, has only 2 geometrically different periodic points.[1] On the other hand, it is proved for various types of symplectic manifolds.

History of studies

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The Conley conjecture was proved by Franks and Handel for surfaces with positive genus. [3] The case of higher dimensional torus was proved by Hingston. [4] Hingston's proof inspired the proof of Ginzburg of the Conley conjecture for symplectically aspherical manifolds. Later Ginzburg--Gurel and Hein proved the Conley conjecture for manifolds whose first Chern class vanishes on spherical classes. Finally, Ginzburg--Gurel proved the Conley conjecture for negatively monotone symplectic manifolds.

References

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  1. ^ a b Ginzburg, Viktor L.; Gürel, Başak Z. (2015). "The Conley Conjecture and Beyond". Arnold Mathematical Journal. 1 (3): 299–337. arXiv:1411.7723. doi:10.1007/s40598-015-0017-3. S2CID 256398699.
  2. ^ Charles Conley, Lecture at University of Wisconsin, April 6, 1984. [1]
  3. ^ Franks, John; Handel, Michael (2003). "Periodic points of Hamiltonian surface diffeomorphisms". Geometry & Topology. 7 (2): 713–756. arXiv:math/0303296. doi:10.2140/gt.2003.7.713. S2CID 2140632.
  4. ^ Hingston, Nancy (2009). "Subharmonic solutions of Hamiltonian equations on tori". Annals of Mathematics. 170 (2): 529–560. doi:10.4007/annals.2009.170.529.