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A NOTE ON HOFER'S NORM

  • Published : 2002.05.01

Abstract

We Show that When ($M,\;\omega$) is a closed, simply connected, symplectic manifold for all $\gamma\;\in\;\pi_1(Ham(M),\;id)$ the following inequality holds: $\parallel\gamma\parallel\;{\geq}\;sup_{\={x}}\;|A(\={x})|,\;where\;\parallel\gamma\parallel$ is the coarse Hofer's norm, $\={x}$ run over all extensions to $D^2$ of an orbit $x(t)\;=\;{\varphi}_t(z)$ of a fixed point $z\;\in\;M,\;A(\={x})$ the symplectic action of $\={x}$, and the Hamiltonian diffeomorphisms {${\varphi}_t$} of M represent $\gamma$.

Keywords

symplectic mnifold;Hamiltonian diffeomorphism;coarse Hofer′s norm;symplectic action;coupling form

References

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  2. D. McDuff and D. Salamon, Introduction to Symplectic Topology, Clarendon Press, Oxford, 1995.
  3. L. Polterovich, Hamiltonian loops and Arnold's principle, Amer. Math. Soc.Transl. Ser 2, 180 (1997), 181-187.