American Institute of Mathematical Sciences

Advanced
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

[1]

Virginie Bonnaillie-Noël, Corentin Léna. Spectral minimal partitions of a sector. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 27-53. doi: 10.3934/dcdsb.2014.19.27
[2]

Frol Zapolsky. Quasi-states and the Poisson bracket on surfaces. Journal of Modern Dynamics, 2007, 1 (3) : 465-475. doi: 10.3934/jmd.2007.1.465
[3]

Karina Samvelyan, Frol Zapolsky. Rigidity of the ${{L}^{p}}$-norm of the Poisson bracket on surfaces. Electronic Research Announcements, 2017, 24: 28-37. doi: 10.3934/era.2017.24.004
[4]

Francisco Crespo, Francisco Javier Molero, Sebastián Ferrer. Poisson and integrable systems through the Nambu bracket and its Jacobi multiplier. Journal of Geometric Mechanics, 2016, 8 (2) : 169-178. doi: 10.3934/jgm.2016002
[5]

Michael Entov, Leonid Polterovich, Daniel Rosen. Poisson brackets, quasi-states and symplectic integrators. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1455-1468. doi: 10.3934/dcds.2010.28.1455
[6]

Álvaro Pelayo, San Vű Ngọc. First steps in symplectic and spectral theory of integrable systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3325-3377. doi: 10.3934/dcds.2012.32.3325
[7]

Gusein Sh. Guseinov. Spectral method for deriving multivariate Poisson summation formulae. Communications on Pure & Applied Analysis, 2013, 12 (1) : 359-373. doi: 10.3934/cpaa.2013.12.359
[8]

Martin Pinsonnault. Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. Journal of Modern Dynamics, 2008, 2 (3) : 431-455. doi: 10.3934/jmd.2008.2.431
[9]

Rafael de la Llave, Jason D. Mireles James. Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4321-4360. doi: 10.3934/dcds.2012.32.4321
[10]

Chi-Kwong Fok. Picard group of isotropic realizations of twisted Poisson manifolds. Journal of Geometric Mechanics, 2016, 8 (2) : 179-197. doi: 10.3934/jgm.2016003
[11]

Manuel de León, David Martín de Diego, Miguel Vaquero. A Hamilton-Jacobi theory on Poisson manifolds. Journal of Geometric Mechanics, 2014, 6 (1) : 121-140. doi: 10.3934/jgm.2014.6.121
[12]

Daniel N. Dore, Andrew D. Hanlon. Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants. Electronic Research Announcements, 2013, 20: 97-102. doi: 10.3934/era.2013.20.97
[13]

Pierre-Damien Thizy. Schrödinger-Poisson systems in $4$-dimensional closed manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2257-2284. doi: 10.3934/dcds.2016.36.2257
[14]

Dmitry Jakobson and Iosif Polterovich. Lower bounds for the spectral function and for the remainder in local Weyl's law on manifolds. Electronic Research Announcements, 2005, 11: 71-77.

[15]

Yuhua Zhu. A local sensitivity and regularity analysis for the Vlasov-Poisson-Fokker-Planck system with multi-dimensional uncertainty and the spectral convergence of the stochastic Galerkin method. Networks & Heterogeneous Media, 2019, 14 (4) : 677-707. doi: 10.3934/nhm.2019027
[16]

Michal Kupsa, Štěpán Starosta. On the partitions with Sturmian-like refinements. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3483-3501. doi: 10.3934/dcds.2015.35.3483
[17]

Bernard Helffer, Thomas Hoffmann-Ostenhof, Susanna Terracini. Nodal minimal partitions in dimension $3$. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 617-635. doi: 10.3934/dcds.2010.28.617
[18]

Xing-Fu Zhong. Variational principles of invariance pressures on partitions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 491-508. doi: 10.3934/dcds.2020019
[19]

Samuel T. Blake, Thomas E. Hall, Andrew Z. Tirkel. Arrays over roots of unity with perfect autocorrelation and good ZCZ cross-correlation. Advances in Mathematics of Communications, 2013, 7 (3) : 231-242. doi: 10.3934/amc.2013.7.231
[20]

Santiago Cañez. Double groupoids and the symplectic category. Journal of Geometric Mechanics, 2018, 10 (2) : 217-250. doi: 10.3934/jgm.2018009

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences