2010, 17: 155-160. doi: 10.3934/era.2010.17.155

On almost Poisson commutativity in dimension two

1. 

Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany

Received  June 2010 Revised  October 2010 Published  December 2010

Consider the following question: given two functions on a symplectic manifold whose Poisson bracket is small, is it possible to approximate them in the $C^0$ norm by commuting functions? We give a positive answer in dimension two, as a particular case of a more general statement which applies to functions on a manifold with a volume form. This result is based on a lemma in the spirit of geometric measure theory. We give some immediate applications to function theory and the theory of quasi-states on surfaces with area forms.
Citation: Frol Zapolsky. On almost Poisson commutativity in dimension two. Electronic Research Announcements, 2010, 17: 155-160. doi: 10.3934/era.2010.17.155
References:
[1]

J. F. Aarnes, Quasi-states and quasi-measures,, Adv. Math., 86 (1991), 41. doi: 10.1016/0001-8708(91)90035-6.

[2]

L. Buhovski, The 2/3 - convergence rate for the Poisson bracket,, Geom. Funct. Anal., 19 (2010), 1620. doi: 10.1007/s00039-010-0045-z.

[3]

F. Cardin and C. Viterbo, Commuting Hamiltonians and Hamilton-Jacobi multi-time equations,, Duke Math. J., 144 (2008), 235. doi: 10.1215/00127094-2008-036.

[4]

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

[5]

M. Entov and L. Polterovich, ($C^0$)-rigidity of Poisson brackets,, Symplectic topology and measure preserving dynamical systems, (2010), 25.

[6]

M. Entov, L. Polterovich and D. Rosen, Poisson brackets, quasi-states and symplectic integrators,, preprint, ().

[7]

M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket,, Pure Appl. Math. Q., 3 (2007), 1037.

[8]

H. Federer, "Geometric Measure Theory,", Die Grundl. der math. Wiss., 153 (1969).

[9]

C. Pearcy and A. Shields, Almost commuting matrices,, J. Funct. Anal., 33 (1979), 332. doi: 10.1016/0022-1236(79)90071-5.

[10]

F. Zapolsky, Quasi-states and the Poisson bracket on surfaces,, J. Mod. Dyn., 1 (2007), 465.

show all references

References:
[1]

J. F. Aarnes, Quasi-states and quasi-measures,, Adv. Math., 86 (1991), 41. doi: 10.1016/0001-8708(91)90035-6.

[2]

L. Buhovski, The 2/3 - convergence rate for the Poisson bracket,, Geom. Funct. Anal., 19 (2010), 1620. doi: 10.1007/s00039-010-0045-z.

[3]

F. Cardin and C. Viterbo, Commuting Hamiltonians and Hamilton-Jacobi multi-time equations,, Duke Math. J., 144 (2008), 235. doi: 10.1215/00127094-2008-036.

[4]

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

[5]

M. Entov and L. Polterovich, ($C^0$)-rigidity of Poisson brackets,, Symplectic topology and measure preserving dynamical systems, (2010), 25.

[6]

M. Entov, L. Polterovich and D. Rosen, Poisson brackets, quasi-states and symplectic integrators,, preprint, ().

[7]

M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket,, Pure Appl. Math. Q., 3 (2007), 1037.

[8]

H. Federer, "Geometric Measure Theory,", Die Grundl. der math. Wiss., 153 (1969).

[9]

C. Pearcy and A. Shields, Almost commuting matrices,, J. Funct. Anal., 33 (1979), 332. doi: 10.1016/0022-1236(79)90071-5.

[10]

F. Zapolsky, Quasi-states and the Poisson bracket on surfaces,, J. Mod. Dyn., 1 (2007), 465.

[1]

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

[2]

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

[3]

Sobhan Seyfaddini. Spectral killers and Poisson bracket invariants. Journal of Modern Dynamics, 2015, 9: 51-66. doi: 10.3934/jmd.2015.9.51

[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]

Ermal Feleqi, Franco Rampazzo. Integral representations for bracket-generating multi-flows. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4345-4366. doi: 10.3934/dcds.2015.35.4345

[6]

Nancy López Reyes, Luis E. Benítez Babilonia. A discrete hierarchy of double bracket equations and a class of negative power series. Mathematical Control & Related Fields, 2017, 7 (1) : 41-52. doi: 10.3934/mcrf.2017003

[7]

Kariane Calta, John Smillie. Algebraically periodic translation surfaces. Journal of Modern Dynamics, 2008, 2 (2) : 209-248. doi: 10.3934/jmd.2008.2.209

[8]

Anton Petrunin. Metric minimizing surfaces. Electronic Research Announcements, 1999, 5: 47-54.

[9]

Yong Lin, Gábor Lippner, Dan Mangoubi, Shing-Tung Yau. Nodal geometry of graphs on surfaces. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1291-1298. doi: 10.3934/dcds.2010.28.1291

[10]

Alfonso Artigue. Expansive flows of surfaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 505-525. doi: 10.3934/dcds.2013.33.505

[11]

Boris Kolev. Poisson brackets in Hydrodynamics. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 555-574. doi: 10.3934/dcds.2007.19.555

[12]

Alexandre Girouard, Iosif Polterovich. Upper bounds for Steklov eigenvalues on surfaces. Electronic Research Announcements, 2012, 19: 77-85. doi: 10.3934/era.2012.19.77

[13]

Seung Won Kim, P. Christopher Staecker. Dynamics of random selfmaps of surfaces with boundary. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 599-611. doi: 10.3934/dcds.2014.34.599

[14]

Robert S. Strichartz. Average error for spectral asymptotics on surfaces. Communications on Pure & Applied Analysis, 2016, 15 (1) : 9-39. doi: 10.3934/cpaa.2016.15.9

[15]

Gareth Ainsworth. The attenuated magnetic ray transform on surfaces. Inverse Problems & Imaging, 2013, 7 (1) : 27-46. doi: 10.3934/ipi.2013.7.27

[16]

François Béguin. Smale diffeomorphisms of surfaces: a classification algorithm. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2/3) : 261-310. doi: 10.3934/dcds.2004.11.261

[17]

Gareth Ainsworth. The magnetic ray transform on Anosov surfaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1801-1816. doi: 10.3934/dcds.2015.35.1801

[18]

Leo T. Butler. A note on integrable mechanical systems on surfaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1873-1878. doi: 10.3934/dcds.2014.34.1873

[19]

Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space of surfaces. Journal of Geometric Mechanics, 2011, 3 (4) : 389-438. doi: 10.3934/jgm.2011.3.389

[20]

Roberto Paroni, Podio-Guidugli Paolo, Brian Seguin. On the nonlocal curvatures of surfaces with or without boundary. Communications on Pure & Applied Analysis, 2018, 17 (2) : 709-727. doi: 10.3934/cpaa.2018037

2016 Impact Factor: 0.483

Metrics

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

Other articles
by authors

[Back to Top]