In mathematics, the Conley–Zehnder theorem, named after
Charles C. Conley
Charles Cameron Conley (26 September 1933 – 20 November 1984) was an American mathematician who worked on dynamical systems.
The Conley index theory and the Conley–Zehnder theorem are named after him.
Early life and education
Conley was born ...
and
Eduard Zehnder, provides a lower bound for the number of
fixed points of
Hamiltonian diffeomorphisms of standard symplectic
tori in terms of the topology of the underlying tori. The lower bound is one plus the
cup-length of the torus (thus 2n+1, where 2n is the dimension of the considered torus), and it can be strengthen to the rank of the homology of the torus (which is 2
2n) provided all the fixed points are non-degenerate, this latter condition being generic in the C
1-topology.
The theorem was conjectured by
Vladimir Arnold
Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–A ...
, and it was known as the
Arnold conjecture
The Arnold conjecture, named after mathematician Vladimir Arnold, is a mathematical conjecture in the field of symplectic geometry, a branch of differential geometry.
Statement
Let (M, \omega) be a compact symplectic manifold. For any smooth func ...
on fixed points of
symplectomorphism
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the sym ...
s. Its validity was later extended to more general closed symplectic manifolds by
Andreas Floer
Andreas Floer (; 23 August 1956 – 15 May 1991) was a German mathematician who made seminal contributions to symplectic topology, and mathematical physics, in particular the invention of Floer homology. Floer's first pivotal contribution was a so ...
and several others.
References
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Dynamical systems
Fixed points (mathematics)
Theorems in analysis
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