---
tag: HY7VRFBAP9
kind: conjecture
name: Arnold conjecture
msc: [53D40, 37J11, 37J12]
uses: [CAN490EPRT, C5EH3B4BEP, A8TS2SNX9Z, ST71ZV3FES, KPQ96640ZJ]
about: [A8TS2SNX9Z, CAN490EPRT]
sources:
  - url: https://en.wikipedia.org/wiki/Arnold_conjecture
same_as:
  wikidata: Q115803233
  wikipedia: Arnold conjecture
history:
  posed_by: "Vladimir Arnold"
  first_stated: "1965"
---

## Statement

Let $(M, \omega)$ be a closed [[symplectic manifold|CAN490EPRT]].
Let $H \colon [0, 1] \times M \to \R$ be smooth, let $X_{H_t}$ be the [[Hamiltonian vector field|C5EH3B4BEP]] of $H_t = H(t, \cdot)$, and let $\phi \colon M \to M$ be the time-$1$ map of the flow of $X_{H_t}$.
Such a [[symplectomorphism|A8TS2SNX9Z]] $\phi$ is called a Hamiltonian diffeomorphism.
Assume that every fixed point $x$ of $\phi$ is nondegenerate: $1$ is not an eigenvalue of $d\phi_x$.
Then the number of fixed points of $\phi$ is at least the minimal number of critical points of a [[Morse function|KPQ96640ZJ]] on $M$.

## Remarks

The homological version of the conjecture replaces the right side by the sum of the [[Betti numbers|ST71ZV3FES]] $\sum_k \dim H_k(M; \F)$ for a field $\F$.
By the Morse inequalities this sum is at most the number of critical points of any Morse function, so the homological version is weaker.
A degenerate version, without the nondegeneracy assumption, asks for at least as many fixed points as the minimal number of critical points of an arbitrary smooth function.

For $\phi = \id$ generated by $H = 0$ the hypothesis fails, and indeed every point is fixed.
If $H$ does not depend on $t$ and is a $C^2$-small Morse function, the fixed points of $\phi$ are exactly the critical points of $H$.
Taking $H$ with the minimal number of critical points shows that the bound is sharp.

Conley and Zehnder proved the conjecture for the standard torus $\mathbb{T}^{2n}$ in 1983.
Floer proved the homological version for monotone symplectic manifolds by building [[Floer homology|NCKFPAGHXF]], whose chain complex is generated by the fixed points of $\phi$.
The homological version with rational coefficients was proved for all closed symplectic manifolds in the late 1990s, by Fukaya and Ono and by Liu and Tian, among others.
The version with the minimal number of critical points of a Morse function is open in general.

## Explanations

### L1 human

A symplectic manifold is the phase space of a mechanical system.
A Hamiltonian diffeomorphism is what happens to the whole phase space after running the system for one unit of time.
The conjecture says that such a map must have many fixed points: at least as many as a smooth function on the space must have critical points.
For a two-dimensional torus, a generic such map has at least four fixed points, much more than topology alone would force.
The conjecture led to Floer homology, one of the main tools of modern symplectic geometry.