# American Institute of Mathematical Sciences

October  2006, 14(4): 617-630. doi: 10.3934/dcds.2006.14.617

## Non-contractible periodic orbits of Hamiltonian flows on twisted cotangent bundles

 1 Department of Mathematics, University of California, Santa Cruz, Santa Cruz CA, 95064, United States

Received  October 2004 Revised  September 2005 Published  January 2006

For many classes of symplectic manifolds, the Hamiltonian flow of a function with sufficiently large variation must have a fast periodic orbit. This principle is the base of the notion of Hofer-Zehnder capacity and some other symplectic invariants and leads to numerous results concerning existence of periodic orbits of Hamiltonian flows. Along these lines, we show that given a negatively curved manifold $M$, a neigbourhood $U_{R}$ of $M$ in T*M, a sufficiently $C^{1}$-small magnetic field $\sigma$ and a non-trivial free homotopy class of loops $\alpha$, then the magnetic flow of certain Hamiltonians supported in $U_{R}$ with big enough minimum, has a one-periodic orbit in $\alpha$. As a consequence, we obtain estimates for the relative Hofer-Zehnder capacity and the Biran-Polterovich-Salamon capacity of a neighbourhood of $M$.
Citation: César J. Niche. Non-contractible periodic orbits of Hamiltonian flows on twisted cotangent bundles. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 617-630. doi: 10.3934/dcds.2006.14.617
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