Journal of Modern Dynamics (JMD)

Hofer's length spectrum of symplectic surfaces

Pages: 219 - 235, Volume 9, 2015      doi:10.3934/jmd.2015.9.219

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Michael Khanevsky - Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, United States (email)

Abstract: Following a question of F. Le Roux, we consider a system of invariants $l_A : H_1 (M) \to \mathbb{R}$ of a symplectic surface $M$. These invariants compute the minimal Hofer energy needed to translate a disk of area $A$ along a given homology class and can be seen as a symplectic analogue of the Riemannian length spectrum. When M has genus zero we also construct Hofer- and $C^0$-continuous quasimorphisms $Ham(M) \to H_1(M)$ that compute trajectories of periodic non-displaceable disks.

Keywords:  Hofer’s metric, Hamiltonian dynamics, dynamics on surfaces, length spectrum.
Mathematics Subject Classification:  Primary: 53D05; Secondary: 37J05.

Received: February 2015;      Available Online: September 2015.