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

Received  November 2021 Revised  March 2022 Published  August 2022 Early access  April 2022

Poincaré's last geometric theorem (Poincaré-Birkhoff Theorem [2]) states that any area-preserving twist map of annulus has at least two fixed points. We replace the area-preserving condition with a weaker intersection property, which states that any essential simple closed curve intersects its image under $ f $ at least at one point. The conclusion is that any such map has at least one fixed point. Besides providing a new proof to Poincaré's geometric theorem, our result also has some applications to reversible systems.

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.

Figure 1.  Carter's example
[1]

Julián López-Gómez, Eduardo Muñoz-Hernández, Fabio Zanolin. On the applicability of the poincaré–Birkhoff twist theorem to a class of planar periodic predator-prey models. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2393-2419. doi: 10.3934/dcds.2020119

[2]

William Clark, Anthony Bloch, Leonardo Colombo. A Poincaré-Bendixson theorem for hybrid systems. Mathematical Control and Related Fields, 2020, 10 (1) : 27-45. doi: 10.3934/mcrf.2019028

[3]

Denis Blackmore, Jyoti Champanerkar, Chengwen Wang. A generalized Poincaré-Birkhoff theorem with applications to coaxial vortex ring motion. Discrete and Continuous Dynamical Systems - B, 2005, 5 (1) : 15-33. doi: 10.3934/dcdsb.2005.5.15

[4]

B. Fernandez, E. Ugalde, J. Urías. Spectrum of dimensions for Poincaré recurrences of Markov maps. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 835-849. doi: 10.3934/dcds.2002.8.835

[5]

Eva Miranda, Romero Solha. A Poincaré lemma in geometric quantisation. Journal of Geometric Mechanics, 2013, 5 (4) : 473-491. doi: 10.3934/jgm.2013.5.473

[6]

Armengol Gasull, Víctor Mañosa. Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 651-670. doi: 10.3934/dcdsb.2019259

[7]

D. P. Demuner, M. Federson, C. Gutierrez. The Poincaré-Bendixson Theorem on the Klein bottle for continuous vector fields. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 495-509. doi: 10.3934/dcds.2009.25.495

[8]

V. Afraimovich, Jean-René Chazottes, Benoît Saussol. Pointwise dimensions for Poincaré recurrences associated with maps and special flows. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 263-280. doi: 10.3934/dcds.2003.9.263

[9]

Yanmin Niu, Xiong Li. An application of Moser's twist theorem to superlinear impulsive differential equations. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 431-445. doi: 10.3934/dcds.2019017

[10]

Betseygail Rand, Lorenzo Sadun. An approximation theorem for maps between tiling spaces. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 323-326. doi: 10.3934/dcds.2011.29.323

[11]

Dirk Pauly. On Maxwell's and Poincaré's constants. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 607-618. doi: 10.3934/dcdss.2015.8.607

[12]

Ferdinand Verhulst. Henri Poincaré's neglected ideas. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1411-1427. doi: 10.3934/dcdss.2020079

[13]

Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295

[14]

John Erik Fornæss. Periodic points of holomorphic twist maps. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 1047-1056. doi: 10.3934/dcds.2005.13.1047

[15]

Henk Broer, Konstantinos Efstathiou, Olga Lukina. A geometric fractional monodromy theorem. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 517-532. doi: 10.3934/dcdss.2010.3.517

[16]

Youming Wang, Fei Yang, Song Zhang, Liangwen Liao. Escape quartered theorem and the connectivity of the Julia sets of a family of rational maps. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5185-5206. doi: 10.3934/dcds.2019211

[17]

Claudio A. Buzzi, Jeroen S.W. Lamb. Reversible Hamiltonian Liapunov center theorem. Discrete and Continuous Dynamical Systems - B, 2005, 5 (1) : 51-66. doi: 10.3934/dcdsb.2005.5.51

[18]

Héctor E. Lomelí. Heteroclinic orbits and rotation sets for twist maps. Discrete and Continuous Dynamical Systems, 2006, 14 (2) : 343-354. doi: 10.3934/dcds.2006.14.343

[19]

John Hubbard, Yulij Ilyashenko. A proof of Kolmogorov's theorem. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 367-385. doi: 10.3934/dcds.2004.10.367

[20]

Rabah Amir, Igor V. Evstigneev. On Zermelo's theorem. Journal of Dynamics and Games, 2017, 4 (3) : 191-194. doi: 10.3934/jdg.2017011

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (155)
  • HTML views (63)
  • Cited by (0)

Other articles
by authors

[Back to Top]