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), 27-45. doi: 10.1007/BF02621587.  Google Scholar

[2]

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

[3]

M. Entov and L. Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv., 81 (2006), 75-99. 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), Special Issue: In honor of Grigory Margulis, Part 1, 1037-1055. doi: 10.4310/PAMQ.2007.v3.n4.a9.  Google Scholar

[5]

V. L. Ginzburg, The Conley conjecture, Ann. of Math. (2), 172 (2010), 1127-1180. 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, 52, American Mathematical Society, Providence, RI, 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, Ensaios Mat., 7, Soc. Brasil. Mat., Rio de Janeiro, 2004, 51-91.  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, Progr. Math., 232, Birkhäuser Boston, Boston, MA, 2005, 525-570. 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, NATO Sci. Ser. II Math. Phys. Chem., 217, Springer, Dordrecht, 2006, 321-416. 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), Publ. Newton Inst., 8, Cambridge Univ. Press, Cambridge, 1996, 171-200.  Google Scholar

[11]

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

[12]

L. Polterovich, Symplectic geometry of quantum noise, Comm. Math. Phys., 327 (2014), 481-519. 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), 419-461. doi: 10.2140/pjm.2000.193.419.  Google Scholar

[14]

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

[15]

C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann., 292 (1992), 685-710. 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), 27-45. doi: 10.1007/BF02621587.  Google Scholar

[2]

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

[3]

M. Entov and L. Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv., 81 (2006), 75-99. 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), Special Issue: In honor of Grigory Margulis, Part 1, 1037-1055. doi: 10.4310/PAMQ.2007.v3.n4.a9.  Google Scholar

[5]

V. L. Ginzburg, The Conley conjecture, Ann. of Math. (2), 172 (2010), 1127-1180. 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, 52, American Mathematical Society, Providence, RI, 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, Ensaios Mat., 7, Soc. Brasil. Mat., Rio de Janeiro, 2004, 51-91.  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, Progr. Math., 232, Birkhäuser Boston, Boston, MA, 2005, 525-570. 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, NATO Sci. Ser. II Math. Phys. Chem., 217, Springer, Dordrecht, 2006, 321-416. 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), Publ. Newton Inst., 8, Cambridge Univ. Press, Cambridge, 1996, 171-200.  Google Scholar

[11]

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

[12]

L. Polterovich, Symplectic geometry of quantum noise, Comm. Math. Phys., 327 (2014), 481-519. 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), 419-461. doi: 10.2140/pjm.2000.193.419.  Google Scholar

[14]

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

[15]

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

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