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

Background

Let (M, \omega) be a compact symplectic manifold. A vector field V on M is called a Hamiltonian vector field if the 1-form \omega( V, \cdot) is exact (i.e., equals to the differential of a function H. A Hamiltonian diffeomorphism \phi: M \to M is the integration of a 1-parameter family of Hamiltonian vector fields V_t, t \in [0, 1].

In dynamical systems, one would like to understand the distribution of fixed points or periodic points. A periodic point of a Hamiltonian diffeomorphism \phi (of period k) is a point x \in M such that \phi^k(x) = x. 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 M is a torus.[1]

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

History of studies

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

  1. ^ Charles Conley, Lecture at University of Wisconsin, April 6, 1984. [2]
  2. ^ Ginzburg, Viktor L. & Gürel, Başak Z. (2015). "The Conley Conjecture and Beyond". Arnold Mathematical Journal. 1 (3): 299–337. arXiv:1411.7723. Bibcode:2015ArnMJ...1..299G. doi:10.1007/s40598-015-0017-3. S2CID 256398699
  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