On hyperbolic points and periodic orbits of symplectomorphisms

Abstract

We prove the existence of infinitely many periodic orbits of symplectomorphisms isotopic to the identity if they admit at least one hyperbolic periodic orbit and satisfy some condition on the flux. Our result is proved for a certain class of closed symplectic manifolds and the main tool we use is a variation of Floer theory for symplectomorphisms, the Floer-Novikov theory. The proof relies on an important result on hyperbolic orbits, namely, that a \emph{Floer-Novikov} trajectory which converges to an iteration $k$ of the hyperbolic orbit and crosses its fixed neighborhood has energy bounded bellow by a strictly positive constant independent of $k$. The main theorem follows from this feature of hyperbolic orbits and certain properties of quantum homology on the class of symplectic manifolds we work with.

21 pages, last version. To appear in Journal of the London Mathematical Society

References in corpus (4)

• On Non-contractible Periodic Orbits of Hamiltonian Diffeomorphisms
• Hyperbolic Fixed Points and Periodic Orbits of Hamiltonian Diffeomorphisms
• A symplectic proof of a theorem of Franks
• Periodic Orbits of Hamiltonian Systems Linear and Hyperbolic at Infinity

Cited by in corpus (7)

• Hamiltonian Pseudo-rotations of Projective Spaces
• Non-contractible Periodic Orbits in Hamiltonian Dynamics on Closed Symplectic Manifolds
• On Periodic Points of Symplectomorphisms on Surfaces
• On non-contractible hyperbolic periodic orbits and periodic points of symplectomorphisms
• On the Barcode Entropy of Reeb Flows
• Periodic points of rational area-preserving homeomorphisms
• Invariant Sets and Hyperbolic Closed Reeb Orbits