Non-contractible periodic orbits of Hamiltonian flows on twisted cotangent bundles
César J. Niche
Discrete & Continuous Dynamical Systems - A 2006, 14(4): 617-630 doi: 10.3934/dcds.2006.14.617
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 $M$.
keywords: Floer homology. Periodic orbits Hamiltonian flows
Topological entropy of a magnetic flow and the growth of the number of trajectories
César J. Niche
Discrete & Continuous Dynamical Systems - A 2004, 11(2&3): 577-580 doi: 10.3934/dcds.2004.11.577
We prove formulae relating the topological entropy of a magnetic flow to the growth rate of the average number of trajectories connecting two points.
keywords: Magnetic flows topological entropy.

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