2015, 9: 51-66. doi: 10.3934/jmd.2015.9.51

Spectral killers and Poisson bracket invariants

1. 

Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, United States

Received  May 2014 Revised  October 2014 Published  May 2015

We find optimal upper bounds for spectral invariants of a Hamiltonian whose support is contained in a union of mutually disjoint displaceable balls. This gives a partial answer to a question posed by Leonid Polterovich in connection with his recent work on Poisson bracket invariants of coverings.
Citation: Sobhan Seyfaddini. Spectral killers and Poisson bracket invariants. Journal of Modern Dynamics, 2015, 9: 51-66. doi: 10.3934/jmd.2015.9.51
References:
[1]

K. Cieliebak, A. Floer, H. Hofer and K. Wysocki, Applications of symplectic homology. II. Stability of the action spectrum, Math. Z., 223 (1996), 27-45. doi: 10.1007/BF02621587.

[2]

M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not., (2003), 1635-1676. doi: 10.1155/S1073792803210011.

[3]

M. Entov and L. Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv., 81 (2006), 75-99. doi: 10.4171/CMH/43.

[4]

M. Entov, L. Polterovich, and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., 3 (2007), Special Issue: In honor of Grigory Margulis, Part 1, 1037-1055. doi: 10.4310/PAMQ.2007.v3.n4.a9.

[5]

V. L. Ginzburg, The Conley conjecture, Ann. of Math. (2), 172 (2010), 1127-1180. doi: 10.4007/annals.2010.172.1129.

[6]

D. McDuff and D. Salamon, $J$-Holomorphic Curves and Symplectic Topology, American Mathematical Society Colloquium Publications, 52, American Mathematical Society, Providence, RI, 2004.

[7]

A. Oancea, A survey of Floer homology for manifolds with contact type boundary or symplectic homology, in Symplectic Geometry and Floer Homology. A Survey of the Floer Homology for Manifolds with Contact Type Boundary or Symplectic Homology, Ensaios Mat., 7, Soc. Brasil. Mat., Rio de Janeiro, 2004, 51-91.

[8]

Y.-G. Oh, Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds, in The Breadth of Symplectic and Poisson Geometry, Progr. Math., 232, Birkhäuser Boston, Boston, MA, 2005, 525-570. doi: 10.1007/0-8176-4419-9_18.

[9]

Y.-G. Oh, Lectures on Floer theory and spectral invariants of Hamiltonian flows, in Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, NATO Sci. Ser. II Math. Phys. Chem., 217, Springer, Dordrecht, 2006, 321-416. doi: 10.1007/1-4020-4266-3_08.

[10]

S. Piunikhin, D. Salamon and M. Schwarz, Symplectic Floer-Donaldson theory and quantum cohomology, in Contact and Symplectic Geometry (Cambridge, 1994), Publ. Newton Inst., 8, Cambridge Univ. Press, Cambridge, 1996, 171-200.

[11]

L. Polterovich, Quantum unsharpness and symplectic rigidity, Lett. Math. Phys., 102 (2012), 245-264. doi: 10.1007/s11005-012-0564-7.

[12]

L. Polterovich, Symplectic geometry of quantum noise, Comm. Math. Phys., 327 (2014), 481-519. doi: 10.1007/s00220-014-1937-9.

[13]

M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math., 193 (2000), 419-461. doi: 10.2140/pjm.2000.193.419.

[14]

M. Usher, The sharp energy-capacity inequality, Commun. Contemp. Math., 12 (2010), 457-473. doi: 10.1142/S0219199710003889.

[15]

C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann., 292 (1992), 685-710. doi: 10.1007/BF01444643.

show all references

References:
[1]

K. Cieliebak, A. Floer, H. Hofer and K. Wysocki, Applications of symplectic homology. II. Stability of the action spectrum, Math. Z., 223 (1996), 27-45. doi: 10.1007/BF02621587.

[2]

M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not., (2003), 1635-1676. doi: 10.1155/S1073792803210011.

[3]

M. Entov and L. Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv., 81 (2006), 75-99. doi: 10.4171/CMH/43.

[4]

M. Entov, L. Polterovich, and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., 3 (2007), Special Issue: In honor of Grigory Margulis, Part 1, 1037-1055. doi: 10.4310/PAMQ.2007.v3.n4.a9.

[5]

V. L. Ginzburg, The Conley conjecture, Ann. of Math. (2), 172 (2010), 1127-1180. doi: 10.4007/annals.2010.172.1129.

[6]

D. McDuff and D. Salamon, $J$-Holomorphic Curves and Symplectic Topology, American Mathematical Society Colloquium Publications, 52, American Mathematical Society, Providence, RI, 2004.

[7]

A. Oancea, A survey of Floer homology for manifolds with contact type boundary or symplectic homology, in Symplectic Geometry and Floer Homology. A Survey of the Floer Homology for Manifolds with Contact Type Boundary or Symplectic Homology, Ensaios Mat., 7, Soc. Brasil. Mat., Rio de Janeiro, 2004, 51-91.

[8]

Y.-G. Oh, Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds, in The Breadth of Symplectic and Poisson Geometry, Progr. Math., 232, Birkhäuser Boston, Boston, MA, 2005, 525-570. doi: 10.1007/0-8176-4419-9_18.

[9]

Y.-G. Oh, Lectures on Floer theory and spectral invariants of Hamiltonian flows, in Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, NATO Sci. Ser. II Math. Phys. Chem., 217, Springer, Dordrecht, 2006, 321-416. doi: 10.1007/1-4020-4266-3_08.

[10]

S. Piunikhin, D. Salamon and M. Schwarz, Symplectic Floer-Donaldson theory and quantum cohomology, in Contact and Symplectic Geometry (Cambridge, 1994), Publ. Newton Inst., 8, Cambridge Univ. Press, Cambridge, 1996, 171-200.

[11]

L. Polterovich, Quantum unsharpness and symplectic rigidity, Lett. Math. Phys., 102 (2012), 245-264. doi: 10.1007/s11005-012-0564-7.

[12]

L. Polterovich, Symplectic geometry of quantum noise, Comm. Math. Phys., 327 (2014), 481-519. doi: 10.1007/s00220-014-1937-9.

[13]

M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math., 193 (2000), 419-461. doi: 10.2140/pjm.2000.193.419.

[14]

M. Usher, The sharp energy-capacity inequality, Commun. Contemp. Math., 12 (2010), 457-473. doi: 10.1142/S0219199710003889.

[15]

C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann., 292 (1992), 685-710. doi: 10.1007/BF01444643.

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