# Conley conjecture

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In [mathematics](/source/Mathematics), the **Conley conjecture**, named after mathematician [Charles Conley](/source/Charles_C._Conley), is a conjecture in the field of [symplectic geometry](/source/Symplectic_geometry), a branch of [differential geometry](/source/Differential_geometry).

## Background

Let (M, \omega) be a compact [symplectic manifold](/source/Symplectic_manifold). A [vector field](/source/Vector_field) V on M is called a [Hamiltonian vector field](/source/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](/source/Dynamical_system), 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](/source/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](/source/Irrational_number) multiple of \pi, which is a Hamiltonian diffeomorphism, has only 2 geometrically different periodic points.[2] On the other hand, it has been [proved](/source/Mathematical_proof) 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](/source/Viktor_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]

1. Ginzburg, Viktor L. & Gürel, Başak Z. (2015). "The Conley Conjecture and Beyond". *Arnold Mathematical Journal*. **1** (3): 299–337. [arXiv:1411.7723](https://arxiv.org/abs/1411.7723). [Bibcode:2015ArnMJ...1..299G](https://ui.adsabs.harvard.edu/abs/2015ArnMJ...1..299G). [doi:10.1007/s40598-015-0017-3](https://doi.org/10.1007/s40598-015-0017-3). [S2CID 256398699](https://api.semanticscholar.org/CorpusID:256398699)

1. Franks, John & Handel, Michael (2003). ["Periodic points of Hamiltonian surface diffeomorphisms"](http://eudml.org/doc/123400). *Geometry & Topology*. **7** (2): 713–756. [arXiv:math/0303296](https://arxiv.org/abs/math/0303296). [doi:10.2140/gt.2003.7.713](https://doi.org/10.2140/gt.2003.7.713). [S2CID 2140632](https://api.semanticscholar.org/CorpusID:2140632)

1. Hingston, Nancy (2009). "Subharmonic solutions of Hamiltonian equations on tori". *Annals of Mathematics*. **170** (2): 529–560. [doi:10.4007/annals.2009.170.529](https://doi.org/10.4007/annals.2009.170.529)

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