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August
2022, 42(8): 4051-4059.
doi: 10.3934/dcds.2022045
A fixed point theorem for twist maps
| 1. | Department of Mathematics, Northwestern University, Evanston, IL 60208 USA |
| 2. | Department of Mathematics, Southern University of Science and Technology, Shenzhen, China |
Poincaré's last geometric theorem (Poincaré-Birkhoff Theorem [
Keywords:
Twist maps,
Poincaré's last geometric theorem,
Poincaré-Birkhoff theorem,
reversible maps.
Mathematics Subject Classification:
Primary: 37E40, 37E30, 37J12.
Citation:
Zhihong Xia, Peizheng Yu. A fixed point theorem for twist maps. Discrete and Continuous Dynamical Systems,
2022, 42
(8)
: 4051-4059.
doi: 10.3934/dcds.2022045
![]()
References:
| [1] |
R. B. Barrar,
Proof of the fixed point theorems of Poincaré and Birkhoff, Canadian J. Math., 19 (1967), 333-343.
doi: 10.4153/CJM-1967-024-5. |
| [2] |
G. D. Birkhoff,
Proof of Poincaré's geometric theorem, Trans. Amer. Math. Soc., 14 (1913), 14-22.
doi: 10.2307/1988766. |
| [3] |
G. D. Birkhoff,
An extension of Poincaré's last geometric theorem, Acta Math., 47 (1926), 297-311.
doi: 10.1007/BF02559515. |
| [4] |
M. Brown,
A new proof of Brouwer's lemma on translation arcs, Houston J. Math., 10 (1984), 35-41.
|
| [5] |
P. H. Carter,
An improvement of the Poincaré-Birkhoff fixed point theorem, Trans. Amer. Math. Soc., 269 (1982), 285-299.
doi: 10.2307/1998604. |
| [6] |
F. Chen and D. Qian,
An extension of the Poincaré-Birkhoff theorem for Hamiltonian systems coupling resonant linear components with twisting components, J. Differential Equations, 321 (2022), 415-448.
doi: 10.1016/j.jde.2022.03.016. |
| [7] |
S.-N. Chow and M. L. Pei,
Aubry-Mather theorem and quasiperiodic orbits for time dependent reversible systems, Nonlinear Anal., 25 (1995), 905-931.
doi: 10.1016/0362-546X(95)00087-C. |
| [8] |
C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38. American Mathematical Society, Providence, R.I., 1978. |
| [9] |
R. L. Devaney,
Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc., 218 (1976), 89-113.
doi: 10.1090/S0002-9947-1976-0402815-3. |
| [10] |
A. Fonda and A. J. Ureña,
A higher dimensional Poincaré-Birkhoff theorem for Hamiltonian flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 679-698.
doi: 10.1016/j.anihpc.2016.04.002. |
| [11] |
J. Franks,
Generalizations of the Poincaré-Birkhoff theorem, Ann. of Math., 128 (1988), 139-151.
doi: 10.2307/1971464. |
| [12] |
J. Franks,
Geodesics on $S^2$ and periodic points of annulus homeomorphisms, Invent. Math., 108 (1992), 403-418.
doi: 10.1007/BF02100612. |
| [13] |
J. Franks, Notes on Chain Recurrence and Lyapunonv Functions, (2017), 1–8, http://arXiv.org/abs/1704.07264. |
| [14] |
J. Kang,
On reversible maps and symmetric periodic points, Ergodic Theory Dynam. Systems, 38 (2018), 1479-1498.
doi: 10.1017/etds.2016.71. |
| [15] |
P. Le Calvez and J. Wang,
Some remarks on the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc., 138 (2010), 703-715.
doi: 10.1090/S0002-9939-09-10105-3. |
| [16] |
M. B. Sevryuk, Reversible Systems, Lecture Notes in Mathematics, 1211. Springer-Verlag, Berlin, 1986.
doi: 10.1007/BFb0075877. |
| [17] |
C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Springer-Verlag, New York-Heidelberg, 1971. |
| [18] |
E. E. Slaminka,
Removing index $0$ fixed points for area preserving maps of two-manifolds, Trans. Amer. Math. Soc., 340 (1993), 429-445.
doi: 10.1090/S0002-9947-1993-1145963-5. |
show all references
References:
| [1] |
R. B. Barrar,
Proof of the fixed point theorems of Poincaré and Birkhoff, Canadian J. Math., 19 (1967), 333-343.
doi: 10.4153/CJM-1967-024-5. |
| [2] |
G. D. Birkhoff,
Proof of Poincaré's geometric theorem, Trans. Amer. Math. Soc., 14 (1913), 14-22.
doi: 10.2307/1988766. |
| [3] |
G. D. Birkhoff,
An extension of Poincaré's last geometric theorem, Acta Math., 47 (1926), 297-311.
doi: 10.1007/BF02559515. |
| [4] |
M. Brown,
A new proof of Brouwer's lemma on translation arcs, Houston J. Math., 10 (1984), 35-41.
|
| [5] |
P. H. Carter,
An improvement of the Poincaré-Birkhoff fixed point theorem, Trans. Amer. Math. Soc., 269 (1982), 285-299.
doi: 10.2307/1998604. |
| [6] |
F. Chen and D. Qian,
An extension of the Poincaré-Birkhoff theorem for Hamiltonian systems coupling resonant linear components with twisting components, J. Differential Equations, 321 (2022), 415-448.
doi: 10.1016/j.jde.2022.03.016. |
| [7] |
S.-N. Chow and M. L. Pei,
Aubry-Mather theorem and quasiperiodic orbits for time dependent reversible systems, Nonlinear Anal., 25 (1995), 905-931.
doi: 10.1016/0362-546X(95)00087-C. |
| [8] |
C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38. American Mathematical Society, Providence, R.I., 1978. |
| [9] |
R. L. Devaney,
Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc., 218 (1976), 89-113.
doi: 10.1090/S0002-9947-1976-0402815-3. |
| [10] |
A. Fonda and A. J. Ureña,
A higher dimensional Poincaré-Birkhoff theorem for Hamiltonian flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 679-698.
doi: 10.1016/j.anihpc.2016.04.002. |
| [11] |
J. Franks,
Generalizations of the Poincaré-Birkhoff theorem, Ann. of Math., 128 (1988), 139-151.
doi: 10.2307/1971464. |
| [12] |
J. Franks,
Geodesics on $S^2$ and periodic points of annulus homeomorphisms, Invent. Math., 108 (1992), 403-418.
doi: 10.1007/BF02100612. |
| [13] |
J. Franks, Notes on Chain Recurrence and Lyapunonv Functions, (2017), 1–8, http://arXiv.org/abs/1704.07264. |
| [14] |
J. Kang,
On reversible maps and symmetric periodic points, Ergodic Theory Dynam. Systems, 38 (2018), 1479-1498.
doi: 10.1017/etds.2016.71. |
| [15] |
P. Le Calvez and J. Wang,
Some remarks on the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc., 138 (2010), 703-715.
doi: 10.1090/S0002-9939-09-10105-3. |
| [16] |
M. B. Sevryuk, Reversible Systems, Lecture Notes in Mathematics, 1211. Springer-Verlag, Berlin, 1986.
doi: 10.1007/BFb0075877. |
| [17] |
C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Springer-Verlag, New York-Heidelberg, 1971. |
| [18] |
E. E. Slaminka,
Removing index $0$ fixed points for area preserving maps of two-manifolds, Trans. Amer. Math. Soc., 340 (1993), 429-445.
doi: 10.1090/S0002-9947-1993-1145963-5. |
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