Article Contents
Article Contents

# Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants

• We use the Hofer norm to show that all Hamiltonian diffeomorphisms with compact support in $\mathbb{R}^{2n}$ that displace an open connected set with a nonzero Hofer-Zehnder capacity move a point farther than a capacity-dependent constant. In $\mathbb{R}^2$, this result is extended to all compactly supported area-preserving homeomorphisms. Next, using the spectral norm, we show the result holds for Hamiltonian diffeomorphisms on closed surfaces. We then show that all area-preserving homeomorphisms of $S^2$ and $\mathbb{RP}^2$ that displace the closure of an open connected set of fixed area move a point farther than an area-dependent constant.
Mathematics Subject Classification: Primary: 57R17.

 Citation:

•  [1] S. Seyfaddini, The displaced disks problem via symplectic topology, arXiv:1307.5704. [2] H. Hofer, Estimates for the energy of a symplectic map, Comment. Math Helv., 68 (1993), 48-72.doi: 10.1007/BF02565809. [3] S. Smale, Diffeomorphisms of the 2-sphere, Proc. Amer. Math. Soc., 10 (1959), 621-626. [4] M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347.doi: 10.1007/BF01388806. [5] H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 1994.doi: 10.1007/978-3-0348-8540-9. [6] Y. G. Oh, $C^0$-coerciveness of Moser's problem and smoothing area preserving homeomorphism, arXiv:math/0601183v5. [7] S. Seyfaddini, $C^0$-limits of Hamiltonian flows and Oh-Schwarz spectral invariants, arXiv:1109.4123v2. [8] Y.-G. Oh, Spectral invariants, analysis of the Floer moduli space, and geometry of the Hamiltonian diffeomorphism group, Duke Math. J., 130 (2005), 199-295. [9] M. Usher, The sharp energy-capacity inequality, Comm. Contemp. Math., 12 (2010), 457-473.doi: 10.1142/S0219199710003889. [10] S. Seyfaddini, Descent and $C^0$-rigidity of spectral invariants on monotone symplectic manifolds, J. Topol. Anal., 4 (2012), 481-498.doi: 10.1142/S1793525312500215.