Spectral killers and Poisson bracket invariants

Previous Article
Topological full groups of minimal subshifts with subgroups of intermediate growth
JMD Home
This Volume
Next Article
On the rigidity of Weyl chamber flows and Schur multipliers as topological groups

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

Abstract
Related Papers

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.
[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.

[1]	Virginie Bonnaillie-Noël, Corentin Léna. Spectral minimal partitions of a sector. Discrete and 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]	Mónica Clapp, Angela Pistoia. Yamabe systems and optimal partitions on manifolds with symmetries. Electronic Research Archive, 2021, 29 (6) : 4327-4338. doi: 10.3934/era.2021088
[5]	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
[6]	Álvaro Pelayo, San Vű Ngọc. First steps in symplectic and spectral theory of integrable systems. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3325-3377. doi: 10.3934/dcds.2012.32.3325
[7]	Michael Entov, Leonid Polterovich, Daniel Rosen. Poisson brackets, quasi-states and symplectic integrators. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1455-1468. doi: 10.3934/dcds.2010.28.1455
[8]	Andrew James Bruce, Janusz Grabowski. Symplectic $ {\mathbb Z}_2^n $-manifolds. Journal of Geometric Mechanics, 2021, 13 (3) : 285-311. doi: 10.3934/jgm.2021020
[9]	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
[10]	Rafael de la Llave, Jason D. Mireles James. Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4321-4360. doi: 10.3934/dcds.2012.32.4321
[11]	Gusein Sh. Guseinov. Spectral method for deriving multivariate Poisson summation formulae. Communications on Pure and Applied Analysis, 2013, 12 (1) : 359-373. doi: 10.3934/cpaa.2013.12.359
[12]	Takeshi Saito, Kazuyuki Yagasaki. Chebyshev spectral methods for computing center manifolds. Journal of Computational Dynamics, 2021, 8 (2) : 165-181. doi: 10.3934/jcd.2021008
[13]	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
[14]	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
[15]	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
[16]	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.
[17]	Pierre-Damien Thizy. Schrödinger-Poisson systems in $4$-dimensional closed manifolds. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2257-2284. doi: 10.3934/dcds.2016.36.2257
[18]	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 and Heterogeneous Media, 2019, 14 (4) : 677-707. doi: 10.3934/nhm.2019027
[19]	Xing-Fu Zhong. Variational principles of invariance pressures on partitions. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 491-508. doi: 10.3934/dcds.2020019
[20]	Michal Kupsa, Štěpán Starosta. On the partitions with Sturmian-like refinements. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3483-3501. doi: 10.3934/dcds.2015.35.3483

2020 Impact Factor: 0.848

Tools

Metrics

Other articles
by authors

on AIMS
Sobhan Seyfaddini
on Google Scholar
Sobhan Seyfaddini

[Back to Top]