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August  2020, 40(8): 4705-4765. doi: 10.3934/dcds.2020199

Representation formula for symmetrical symplectic capacity and applications

Rongrong Jin and  Guangcun Lu ,

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

* Corresponding author: Guangcun Lu

Received  June 2019 Revised  February 2020 Published  May 2020

Fund Project: The second author is partially supported by the NNSF 11271044 of China

This is the second installment in a series of papers aimed at generalizing symplectic capacities and homologies. We study symmetric versions of symplectic capacities for real symplectic manifolds, and obtain corresponding results for them to those of the first [19] of this series (such as representation formula, a theorem by Evgeni Neduv, Brunn-Minkowski type inequality and Minkowski billiard trajectories proposed by Artstein-Avidan-Ostrover).

Keywords: Representation formula, symmetrical Ekeland-Hofer symplectic capacity, symmetrical Hofer-Zehnder symplectic capacity, Brunn-Minkowski type inequality, Minkowski billiard trajectories.
Mathematics Subject Classification: Primary: 53D35, 53C23; Secondary: 70H05, 37J05, 57R17.
Citation: Rongrong Jin, Guangcun Lu. Representation formula for symmetrical symplectic capacity and applications. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4705-4765. doi: 10.3934/dcds.2020199
References:
[1]

P. Albers and U. Frauenfelder, The space of linear anti-symplectic involutions is a homogenous space, Arch. Math. (Basel), 99 (2012), 531-536.  doi: 10.1007/s00013-012-0461-4.

[2]

S. Artstein-Avidan, R. Karasev and Y. Ostrover, From symplectic measurements to the Mahler conjecture, Duke Math. J., 163 (2014), 2003-2022.  doi: 10.1215/00127094-2794999.

[3]

S. Artstein-Avidan and Y. Ostrover, A Brunn-Minkowski inequality for symplectic capacities of convex domains, Int. Math. Res. Not. IMRN 2008, (2008), Art. ID rnn044, 31 pp. doi: 10.1093/imrn/rnn044.

[4]

S. Artstein-Avidan and Y. Ostrover, Bounds for Minkowski billiard trajectories in convex bodies, Int. Math. Res. Not. IMRN 2014, (2014), 165–193. doi: 10.1093/imrn/rns216.

[5]

S. M. Bates, Some simple continuity properties of symplectic capacities, The Floer Memorial Volume, Progr. Math., Birkhäuser, Basel, 133 (1995), 185-193. 

[6]

S. M. Bates, A capacity representation theorem for some non-convex domains, Math. Z., 227 (1998), 571-581.  doi: 10.1007/PL00004394.

[7]

J. Blot, On the almost everywhere continuity, http://arXiv.org/abs/1411.3582v1[math.OC].

[8]

J. Bourgain, J. Lindenstrauss and V. D. Milman, Minkowski sums and symmetrizations, Geometric Aspects of Functional Analysis (1986/87), Lecture Notes in Math., Springer, Berlin, 1317 (1988), 44-66.  doi: 10.1007/BFb0081735.

[9]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equation, Universitext. Springer, New York, 2011.

[10]

F. H. Clarke, A classical variational principle for periodic Hamiltonian trajectories, Proc. Amer. Math. Soc., 76 (1979), 186-188.  doi: 10.2307/2042942.

[11]

F. H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.

[12]

I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Ergebnisse der Mathematik und Ihrer Grenzgebiete (3), 19. Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-74331-3.

[13]

I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics, Math. Z., 200 (1989), 355-378.  doi: 10.1007/BF01215653.

[14]

A. Figalli, J. Palmer and Á. Pelayo, Symplectic $G$-capacities and integrable systems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 18 (2018), 65-103. 

[15]

M. Ghomi, Shortest periodic billiard trajectories in convex bodies, Geometric and Functional Analysis, 14 (2004), 295-302.  doi: 10.1007/s00039-004-0458-7.

[16]

H. Hofer and E. Zehnder, A new capacity for symplectic manifolds, Analysis et Cetera, Academic Press, Boston, MA, (1990), 405–427.

[17]

H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser Advanced Texts: Basler Lehrbúcher., Birkhäuser Verlag, Basel, 1994.

[18]

K. Irie, Periodic billiard trajectories and Morse theory on loop spaces, Comment. Math. Helv., 90 (2015), 225-254.  doi: 10.4171/CMH/352.

[19]

R. R. Jin and G. C. Lu, Generalizations of Ekeland-Hofer and Hofer-Zehnder symplectic capacities and applications, (2019), arXiv: 1903.01116v2[math.SG].

[20]

S. G. Krantz, Convex Analysis, Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2015.

[21]

A. F. Künzle, Singular Hamiltonian systems and symplectic capacities, Singularities and Differential Equations, Banach Center Publications, Polish Acad. Sci. Inst. Math., Warsaw, 33 (1996), 171-187. 

[22]

S. Lisi and A. Rieser, Coisotropic Hofer-Zehnder capacities and non-squeezing for relative embeddings, arXiv: 1312.7334[math.SG].

[23]

C. G. Liu and Q. Wang, Symmetrical symplectic capacity with applications, Discrete Contin. Dyn. Syst., 32 (2012), 2253-2270.  doi: 10.3934/dcds.2012.32.2253.

[24]

J. Moser and E. J. Zehnder, Notes on Dynamical Systems, Courant Lecture Notes in Mathematics, 12. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/cln/012.

[25]

E. Neduv, Prescribed minimal period problems for convex Hamiltonian systems via Hofer-Zehnder symplectic capacity, Math. Z., 236 (2001), 99-112.  doi: 10.1007/PL00004828.

[26]

R. S. Palais, The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322.

[27]

A. Rieser, Lagrangian blow-ups, blow-downs, and applications to real packing, Journal of Symplectic Geometry, 12 (2014), 725-789.  doi: 10.4310/JSG.2014.v12.n4.a4.

[28]

R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28 Princeton University Press, Princeton, N.J., 1970.

[29]

R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511526282.

[30]

R. Schneider, Stability for some extremal properties of the simplex, Journal of Geometry, 96 (2009), 135-148.  doi: 10.1007/s00022-010-0028-0.

[31]

J.-C. Sikorav, Systémes Hamiltoniens et Topologie Symplectique, Dipartimento di Matematica dell'Universitá di Pisa, 1990.

[32]

C. Viterbo, Symplectic real algebraic geometry, preprint, (1999).

[33] Y. C. Xu, Linear Algebra and Matrix Theory, Higher Education Press, Beijing, 1992. 
[34]

F. C. Yang and Z. Wei, Generalized Euler identity for subdifferentials of homogeneous functions and applications, J. Math. Anal. Appl., 337 (2008), 516-523.  doi: 10.1016/j.jmaa.2007.04.008.

show all references

References:
[1]

P. Albers and U. Frauenfelder, The space of linear anti-symplectic involutions is a homogenous space, Arch. Math. (Basel), 99 (2012), 531-536.  doi: 10.1007/s00013-012-0461-4.

[2]

S. Artstein-Avidan, R. Karasev and Y. Ostrover, From symplectic measurements to the Mahler conjecture, Duke Math. J., 163 (2014), 2003-2022.  doi: 10.1215/00127094-2794999.

[3]

S. Artstein-Avidan and Y. Ostrover, A Brunn-Minkowski inequality for symplectic capacities of convex domains, Int. Math. Res. Not. IMRN 2008, (2008), Art. ID rnn044, 31 pp. doi: 10.1093/imrn/rnn044.

[4]

S. Artstein-Avidan and Y. Ostrover, Bounds for Minkowski billiard trajectories in convex bodies, Int. Math. Res. Not. IMRN 2014, (2014), 165–193. doi: 10.1093/imrn/rns216.

[5]

S. M. Bates, Some simple continuity properties of symplectic capacities, The Floer Memorial Volume, Progr. Math., Birkhäuser, Basel, 133 (1995), 185-193. 

[6]

S. M. Bates, A capacity representation theorem for some non-convex domains, Math. Z., 227 (1998), 571-581.  doi: 10.1007/PL00004394.

[7]

J. Blot, On the almost everywhere continuity, http://arXiv.org/abs/1411.3582v1[math.OC].

[8]

J. Bourgain, J. Lindenstrauss and V. D. Milman, Minkowski sums and symmetrizations, Geometric Aspects of Functional Analysis (1986/87), Lecture Notes in Math., Springer, Berlin, 1317 (1988), 44-66.  doi: 10.1007/BFb0081735.

[9]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equation, Universitext. Springer, New York, 2011.

[10]

F. H. Clarke, A classical variational principle for periodic Hamiltonian trajectories, Proc. Amer. Math. Soc., 76 (1979), 186-188.  doi: 10.2307/2042942.

[11]

F. H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.

[12]

I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Ergebnisse der Mathematik und Ihrer Grenzgebiete (3), 19. Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-74331-3.

[13]

I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics, Math. Z., 200 (1989), 355-378.  doi: 10.1007/BF01215653.

[14]

A. Figalli, J. Palmer and Á. Pelayo, Symplectic $G$-capacities and integrable systems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 18 (2018), 65-103. 

[15]

M. Ghomi, Shortest periodic billiard trajectories in convex bodies, Geometric and Functional Analysis, 14 (2004), 295-302.  doi: 10.1007/s00039-004-0458-7.

[16]

H. Hofer and E. Zehnder, A new capacity for symplectic manifolds, Analysis et Cetera, Academic Press, Boston, MA, (1990), 405–427.

[17]

H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser Advanced Texts: Basler Lehrbúcher., Birkhäuser Verlag, Basel, 1994.

[18]

K. Irie, Periodic billiard trajectories and Morse theory on loop spaces, Comment. Math. Helv., 90 (2015), 225-254.  doi: 10.4171/CMH/352.

[19]

R. R. Jin and G. C. Lu, Generalizations of Ekeland-Hofer and Hofer-Zehnder symplectic capacities and applications, (2019), arXiv: 1903.01116v2[math.SG].

[20]

S. G. Krantz, Convex Analysis, Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2015.

[21]

A. F. Künzle, Singular Hamiltonian systems and symplectic capacities, Singularities and Differential Equations, Banach Center Publications, Polish Acad. Sci. Inst. Math., Warsaw, 33 (1996), 171-187. 

[22]

S. Lisi and A. Rieser, Coisotropic Hofer-Zehnder capacities and non-squeezing for relative embeddings, arXiv: 1312.7334[math.SG].

[23]

C. G. Liu and Q. Wang, Symmetrical symplectic capacity with applications, Discrete Contin. Dyn. Syst., 32 (2012), 2253-2270.  doi: 10.3934/dcds.2012.32.2253.

[24]

J. Moser and E. J. Zehnder, Notes on Dynamical Systems, Courant Lecture Notes in Mathematics, 12. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/cln/012.

[25]

E. Neduv, Prescribed minimal period problems for convex Hamiltonian systems via Hofer-Zehnder symplectic capacity, Math. Z., 236 (2001), 99-112.  doi: 10.1007/PL00004828.

[26]

R. S. Palais, The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322.

[27]

A. Rieser, Lagrangian blow-ups, blow-downs, and applications to real packing, Journal of Symplectic Geometry, 12 (2014), 725-789.  doi: 10.4310/JSG.2014.v12.n4.a4.

[28]

R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28 Princeton University Press, Princeton, N.J., 1970.

[29]

R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511526282.

[30]

R. Schneider, Stability for some extremal properties of the simplex, Journal of Geometry, 96 (2009), 135-148.  doi: 10.1007/s00022-010-0028-0.

[31]

J.-C. Sikorav, Systémes Hamiltoniens et Topologie Symplectique, Dipartimento di Matematica dell'Universitá di Pisa, 1990.

[32]

C. Viterbo, Symplectic real algebraic geometry, preprint, (1999).

[33] Y. C. Xu, Linear Algebra and Matrix Theory, Higher Education Press, Beijing, 1992. 
[34]

F. C. Yang and Z. Wei, Generalized Euler identity for subdifferentials of homogeneous functions and applications, J. Math. Anal. Appl., 337 (2008), 516-523.  doi: 10.1016/j.jmaa.2007.04.008.

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