Conley conjecture

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

Background

Let be a compact symplectic manifold. A vector field on is called a Hamiltonian vector field if the 1-form is exact 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.
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. On the other hand, it is proved for various types of symplectic manifolds.

History of studies

The Conley conjecture was proved by Franks and Handel for surfaces with positive genus. The case of higher dimensional torus was proved by Hingston. 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.