# NORMALIZATION OF THE HAMILTONIAN AND THE ACTION SPECTRUM

• OH YONG-GEUN (Department of Mathematics University of Wisconsin, and Korea Institute for Advanced Study)
• Published : 2005.01.01

#### Abstract

In this paper, we prove that the two well-known natural normalizations of Hamiltonian functions on the symplectic manifold ($M,\;{\omega}$) canonically relate the action spectra of different normalized Hamiltonians on arbitrary symplectic manifolds ($M,\;{\omega}$). The natural classes of normalized Hamiltonians consist of those whose mean value is zero for the closed manifold, and those which are compactly supported in IntM for the open manifold. We also study the effect of the action spectrum under the ${\pi}_1$ of Hamiltonian diffeomorphism group. This forms a foundational basis for our study of spectral invariants of the Hamiltonian diffeomorphism in [8].

#### References

1. A. Banyaga, Sur la structure du groupe des diffeomorphismes qui preservent une forme symplectique, Comm. Math. Helv. 53 (1978), 174-227 https://doi.org/10.1007/BF02566074
2. I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics I, Math. Z. 200, (1989), 355-378 https://doi.org/10.1007/BF01215653
3. I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics II, Math. Z. 203, (1989), 553-569 https://doi.org/10.1007/BF02570756
4. M. Entov, K-area, Hofer metric and geometry of conjugacy classes in Lie groups, Invent. Math. 146 (2001), 93-141 https://doi.org/10.1007/s002220100161
5. A. Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989), 575-611 https://doi.org/10.1007/BF01260388
6. A. Floer and H. Hofer, Symplectic homology I, Math. Z. 215 (1994), 37-88 https://doi.org/10.1007/BF02571699
7. H. Hofer, On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh 115 (1990), 25-38
8. F. Lalonde, D. McDuff and L. Polterovich, Topological rigidity of Hamiltonian loops and quantum homology, Invent. Math. 135 (1999), 369-385 https://doi.org/10.1007/s002220050289
9. J. Marsden and J. Ratiu, Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds, in 'The Breadth of Stmplectic and Poisson Geometry', Progr. Math. 232 (2004), 525-570 ed., Birkhouser
10. Y.-G. Oh, Symplectic topology as the geometry of action functional, I, J. Differential Geom. 46 (1997), 499-577 https://doi.org/10.4310/jdg/1214459976
11. Y.-G. Oh, Symplectic topology as the geometry of action functional, II, Comm. Anal. Geom. 7 (1999), 1-55 https://doi.org/10.4310/CAG.1999.v7.n1.a1
12. Y.-G. Oh, Chain level Floer theory and Hofer's geometry of the Hamiltonian dif- feomorphism group, Asian J. Math. 6 (2002), 799-830, math.SG/0104243
13. L. Polterovich, The Geometry of the Group of Symplectic Diffeomorphisms, Birkhauser, 2001
14. P. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), 157-184 https://doi.org/10.1002/cpa.3160310203
15. M. Schwarz, On the action spectrum for closed symplectically aspherical mani- folds, Pacific J. Math. 193 (2000), 419-461 https://doi.org/10.2140/pjm.2000.193.419
16. P. Seidel, $\pi_1$ of symplectic diffeomorphism groups and invertibles in quantum homology rings, GAFA (1997), 1046-1095
17. C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992), 685-710 https://doi.org/10.1007/BF01444643

#### Cited by

1. Calabi quasi-morphisms for some non-monotone symplectic manifolds vol.6, pp.1, 2006, https://doi.org/10.2140/agt.2006.6.405
2. Hamiltonian Floer homology for compact convex symplectic manifolds vol.57, pp.2, 2016, https://doi.org/10.1007/s13366-015-0254-6
3. CONTINUOUS HAMILTONIAN DYNAMICS AND AREA-PRESERVING HOMEOMORPHISM GROUP OF D2 vol.53, pp.4, 2016, https://doi.org/10.4134/JKMS.j150288