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

In dynamical systems one would like to understand the distribution of fixed points or periodic points. A periodic point of a Hamiltonian diffeomorphism ${\displaystyle \phi }$ (of periodic ${\displaystyle k}$) is a point ${\displaystyle x\in M}$ such that ${\displaystyle \phi ^{k}(x)=x}$. 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 ${\displaystyle M}$ is a torus. [2]

The Conley conjecture is false in many simple cases. For example, a rotation of a round sphere ${\displaystyle S^{2}}$ by an angle equal to an irrational multiple of ${\displaystyle \pi }$, 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

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. 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.