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doi: 10.3934/cpaa.2022009
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## Higher P-symmetric Ekeland-Hofer capacities

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

*Corresponding author

Received  February 2021 Revised  October 2021 Early access December 2021

Fund Project: Partially supported by the NNSF 11271044 of China

This paper is devoted to the construction of analogues of higher Ekeland-Hofer symplectic capacities for $P$-symmetric subsets in the standard symplectic space $(\mathbb{R}^{2n},\omega_0)$, which is motivated by Long and Dong's study about $P$-symmetric closed characteristics on $P$-symmetric convex bodies. We study the relationship between these capacities and other capacities, and give some computation examples. Moreover, we also define higher real symmetric Ekeland-Hofer capacities as a complement of Jin and the second named author's recent study of the real symmetric analogue about the first Ekeland-Hofer capacity.

Citation: Kun Shi, Guangcun Lu. Higher P-symmetric Ekeland-Hofer capacities. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2022009
##### References:
 [1] A. Akopyan and R. Karasev, Estimating symplectic capacities from lengths of closed curves on the unit spheres, preprint, arXiv: 1801.00242. Google Scholar [2] S. Artstein-Avidan and Y. Ostrover, Bounds for Minkowski billiard trajectories in convex bodies, Int. Math. Res. Not., 2014 (2014), 165-193.  doi: 10.1093/imrn/rns216.  Google Scholar [3] L. Baracco, M. Fassina and S. Pinton, On the Ekeland-Hofer symplectic capacities of the real bidisc, Pacific J. Math., 305 (2020), 423-446.  doi: 10.2140/pjm.2020.305.423.  Google Scholar [4] V. Benci, On the critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc., 274 (1982), 533-572.  doi: 10.2307/1999120.  Google Scholar [5] Y. Dong and Y. Long, Closed characteristics on partially symmetric compact convex hypersurfaces in $\mathbb{R}^{2n}$, J. Differ. Equ., 196 (2004), 226-248.  doi: 10.1016/S0022-0396(03)00168-2.  Google Scholar [6] I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics, Math. Z., 200 (1989), 355-378.  doi: 10.1007/BF01215653.  Google Scholar [7] I. Ekeland and H. Hofer, Symplectic topology and Haniltonian dynamics II, Math. Z., 203 (1990), 553-567.  doi: 10.1007/BF02570756.  Google Scholar [8] E. R. Fadell and P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45 (1978), 139-174.  doi: 10.1007/BF01390270.  Google Scholar [9] H. Hofer and E. Zehnder, Periodic solutions on hypersurfaces and a result by C. Viterbo, Invent. Math., 90 (1987), 1-9.  doi: 10.1007/BF01389030.  Google Scholar [10] H, Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, 1$^{st}$ edition, Birkhüser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8540-9.  Google Scholar [11] Rongrong Jin and Guangcun Lu, Generalizations of Ekeland-Hofer and Hofer-Zehnder symplectic capacities and applications, preprint, arXiv: 1903.01116v2. Google Scholar [12] R. R. Jin and G. C. Lu, Representation formula for symmetrical symplectic capacity and applications, Discrete Contin. Dyn. Syst., 40 (2020), 4705-4765.  doi: 10.3934/dcds.2020199.  Google Scholar [13] R. R. Jin and G. C. Lu, Coisotropic Ekeland-Hofer capacities, preprint, arXiv: 1910.14474. Google Scholar [14] V. G. B. Ramos, Symplectic embeddings and the Lagrangian bidisk, Duke Math. J., 166 (2017), 1703-1738.  doi: 10.1215/00127094-0000011X.  Google Scholar [15] J. C. Sikorav, Systémes Hamiltoniens et Topologie Symplectique, Ph.D thesis, Dipartimento di Matematica dell'Universitá di Pisa, 1990. Google Scholar [16] A. Szulkin, An index theory and existence of multiple brake orbits for star-shaped Hamiltonian systems, Math. Ann., 283 (1989), 241-255.  doi: 10.1007/BF01446433.  Google Scholar

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##### References:
 [1] A. Akopyan and R. Karasev, Estimating symplectic capacities from lengths of closed curves on the unit spheres, preprint, arXiv: 1801.00242. Google Scholar [2] S. Artstein-Avidan and Y. Ostrover, Bounds for Minkowski billiard trajectories in convex bodies, Int. Math. Res. Not., 2014 (2014), 165-193.  doi: 10.1093/imrn/rns216.  Google Scholar [3] L. Baracco, M. Fassina and S. Pinton, On the Ekeland-Hofer symplectic capacities of the real bidisc, Pacific J. Math., 305 (2020), 423-446.  doi: 10.2140/pjm.2020.305.423.  Google Scholar [4] V. Benci, On the critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc., 274 (1982), 533-572.  doi: 10.2307/1999120.  Google Scholar [5] Y. Dong and Y. Long, Closed characteristics on partially symmetric compact convex hypersurfaces in $\mathbb{R}^{2n}$, J. Differ. Equ., 196 (2004), 226-248.  doi: 10.1016/S0022-0396(03)00168-2.  Google Scholar [6] I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics, Math. Z., 200 (1989), 355-378.  doi: 10.1007/BF01215653.  Google Scholar [7] I. Ekeland and H. Hofer, Symplectic topology and Haniltonian dynamics II, Math. Z., 203 (1990), 553-567.  doi: 10.1007/BF02570756.  Google Scholar [8] E. R. Fadell and P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45 (1978), 139-174.  doi: 10.1007/BF01390270.  Google Scholar [9] H. Hofer and E. Zehnder, Periodic solutions on hypersurfaces and a result by C. Viterbo, Invent. Math., 90 (1987), 1-9.  doi: 10.1007/BF01389030.  Google Scholar [10] H, Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, 1$^{st}$ edition, Birkhüser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8540-9.  Google Scholar [11] Rongrong Jin and Guangcun Lu, Generalizations of Ekeland-Hofer and Hofer-Zehnder symplectic capacities and applications, preprint, arXiv: 1903.01116v2. Google Scholar [12] R. R. Jin and G. C. Lu, Representation formula for symmetrical symplectic capacity and applications, Discrete Contin. Dyn. Syst., 40 (2020), 4705-4765.  doi: 10.3934/dcds.2020199.  Google Scholar [13] R. R. Jin and G. C. Lu, Coisotropic Ekeland-Hofer capacities, preprint, arXiv: 1910.14474. Google Scholar [14] V. G. B. Ramos, Symplectic embeddings and the Lagrangian bidisk, Duke Math. J., 166 (2017), 1703-1738.  doi: 10.1215/00127094-0000011X.  Google Scholar [15] J. C. Sikorav, Systémes Hamiltoniens et Topologie Symplectique, Ph.D thesis, Dipartimento di Matematica dell'Universitá di Pisa, 1990. Google Scholar [16] A. Szulkin, An index theory and existence of multiple brake orbits for star-shaped Hamiltonian systems, Math. Ann., 283 (1989), 241-255.  doi: 10.1007/BF01446433.  Google Scholar
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