# American Institute of Mathematical Sciences

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.

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##### 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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