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.