American Institute of Mathematical Sciences

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

Spectral killers and Poisson bracket invariants

Sobhan Seyfaddini 1,

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.
Keywords: spectral killers., Poisson bracket, partitions of unity, Symplectic manifolds.
Mathematics Subject Classification: Primary: 53D40; Secondary: 37J0.
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.  doi: 10.1007/BF02621587.  Google Scholar

[2]

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

[3]

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

[4]

M. Entov, L. Polterovich, and F. Zapolsky, Quasi-morphisms and the Poisson bracket,, Pure Appl. Math. Q., 3 (2007), 1037.  doi: 10.4310/PAMQ.2007.v3.n4.a9.  Google Scholar

[5]

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

[6]

D. McDuff and D. Salamon, $J$-Holomorphic Curves and Symplectic Topology,, American Mathematical Society Colloquium Publications, (2004).   Google Scholar

[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, (2004), 51.   Google Scholar

[8]

Y.-G. Oh, Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds,, in The Breadth of Symplectic and Poisson Geometry, (2005), 525.  doi: 10.1007/0-8176-4419-9_18.  Google Scholar

[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, (2006), 321.  doi: 10.1007/1-4020-4266-3_08.  Google Scholar

[10]

S. Piunikhin, D. Salamon and M. Schwarz, Symplectic Floer-Donaldson theory and quantum cohomology,, in Contact and Symplectic Geometry (Cambridge, (1994), 171.   Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

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.  doi: 10.1007/BF02621587.  Google Scholar

[2]

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

[3]

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

[4]

M. Entov, L. Polterovich, and F. Zapolsky, Quasi-morphisms and the Poisson bracket,, Pure Appl. Math. Q., 3 (2007), 1037.  doi: 10.4310/PAMQ.2007.v3.n4.a9.  Google Scholar

[5]

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

[6]

D. McDuff and D. Salamon, $J$-Holomorphic Curves and Symplectic Topology,, American Mathematical Society Colloquium Publications, (2004).   Google Scholar

[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, (2004), 51.   Google Scholar

[8]

Y.-G. Oh, Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds,, in The Breadth of Symplectic and Poisson Geometry, (2005), 525.  doi: 10.1007/0-8176-4419-9_18.  Google Scholar

[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, (2006), 321.  doi: 10.1007/1-4020-4266-3_08.  Google Scholar

[10]

S. Piunikhin, D. Salamon and M. Schwarz, Symplectic Floer-Donaldson theory and quantum cohomology,, in Contact and Symplectic Geometry (Cambridge, (1994), 171.   Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

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