Applications of Hofer's geometry to Hamiltonian dynamics

By Urs Frauenfelder, F. Schlenk. 2003

88 citations recorded by the source corpus.

We prove that for every subset $A$ of a tame symplectic manifold $(W,\omega)$ meeting a semi-positivity condition, the $\pi_1$-sensitive Hofer--Zehnder capacity of $A$ is not greater than four times the stable displacement energy of $A$, $$ c_{HZ}^\circ(A,W)\le 4e (A\times S^1, W\times T^*S^1). $$ 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 $(W,\omega)$. $\bullet$ 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 $\{\varphi_F^t \}$, $t \ge 0$, 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

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/755f4593851789408dbb5117ebd55b5868e77811

How researchers see this paper

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In Geometry and Topology

1. Basak Z. Gurel
2. V. Ginzburg
3. Michael Usher
4. G. Contreras
5. Alberto Abbondandolo
6. L. Polterovich
7. E. Shelukhin
8. Sheila Sandon

In Mathematical Physics

1. V. Ginzburg
2. Basak Z. Gurel
3. G. Contreras
4. Alberto Abbondandolo
5. Michael Usher
6. L. Polterovich
7. Sheila Sandon
8. E. Shelukhin

Citations in the corpus

• THE SHARP ENERGY-CAPACITY INEQUALITY (2008, methodology): “We also have the following improvement of the stable energy-capacity inequality of [25]:”
• Boundary depth in Floer theory and its applications to Hamiltonian dynamics and coisotropic submanifolds (2009, methodology): “Using the stable energy-capacity inequality ([36] Theorem 1.”
• The Palais-Smale condition on contact type energy levels for convex Lagrangian systems (2003, background): “Very low energy levels which do not project onto M are displaceable, and then, by results of Frauenfelder and Schlenk [9], [27], they have finite Hofer-Zehnder capacity.”
• Lectures on the free period Lagrangian action functional (2013, background): “n diffeomorphism of T∗Mwhich displaces H−1(κ) from itself (see [Con06, Proposition 8.2]). Sets which are displaceable by a Hamiltonian diffeomorphism have finite π1-sensitive Hofer-Zehnder capacity (see [Sch06] and [FS07]) and this fact implies the almost everywhere existence result for periodic orbits”
• Quantitative symplectic geometry (2005, unknown): “M; !/, which has many applications [87].”
• Periodic Orbits of Twisted Geodesic Flows and The Weinstein-Moser Theorem (2007, background): “, [CGK], [Co], [CIPP], [FS], [GG2], [Gü], [Ke3], [Ma], [Lu1], [Lu2], [Sc], following the original work of Hofer and Zehnder and of Struwe, [FHW], [HZ1], [HZ2], [HZ3], [St].”