JournalscmhVol. 81 , No. 1DOI 10.4171/cmh/45

Applications of Hofer's geometry to Hamiltonian dynamics

  • Felix Schlenk

    Universität Leipzig, Germany
Applications of Hofer's geometry to Hamiltonian dynamics cover

Abstract

We prove that for every subset AA of a tame symplectic manifold (W,ω)(W,\omega) meeting a semi-positivity condition, the π1\pi_1-sensitive Hofer--Zehnder capacity of AA is not greater than four times the stable displacement energy of AA,

cHZ(A,W)4e(A×S1,W×TS1).c_{HZ}^\circ(A,W)\le 4e (A\times S^1, W\times T^*S^1).

This estimate yields almost existence of periodic orbits near stably displaceable energy levels of time-independent Hamiltonian systems. Our main applications are: \bullet The Weinstein conjecture holds true for every stably displaceable hypersurface of contact type in (W,ω)(W,\omega). \bullet The flow describing the motion of a charge on a closed Riemannian manifold subject to a non-vanishing magnetic field and a conservative force field has contractible periodic orbits at almost all sufficiently small energies. The proof of the above energy-capacity inequality combines a curve shortening procedure in Hofer geometry with the following detection mechanism for periodic orbits: If the ray {φFt}\{\varphi_F^t \}, t0t \ge 0, of Hamiltonian diffeomorphisms generated by a compactly supported time-independent Hamiltonian stops to be a minimal geodesic in its homotopy class, then a non-constant contractible periodic orbit must appear.