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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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[7]

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

[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.  Google Scholar

[9]

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

[10]

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

[11]

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

[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.  Google Scholar

[13]

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

[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.   Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[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]. Google Scholar

[20]

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

[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.   Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[32]

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

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[7]

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

[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.  Google Scholar

[9]

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

[10]

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

[11]

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

[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.  Google Scholar

[13]

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

[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.   Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[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]. Google Scholar

[20]

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

[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.   Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[32]

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

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

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