# American Institute of Mathematical Sciences

January  2018, 38(1): 131-154. doi: 10.3934/dcds.2018006

## Invariant curves of smooth quasi-periodic mappings

 1 School of Mathematics Sciences, Beijing Normal University, Beijing, 100875, China 2 School of Mathematical Sciences, Peking University, Beijing, 100871, China

* Corresponding author

Received  September 2016 Revised  August 2017 Published  September 2017

Fund Project: The second author is supported by the NSFC (11571041) and the Fundamental Research Funds for the Central Universities. The third author is supported by the NSFC (11231001)

In this paper we are concerned with the existence of invariant curves of planar mappings which are quasi-periodic in the spatial variable, satisfy the intersection property, $\mathcal{C}^{p}$ smooth with $p>2n+1$, $n$ is the number of frequencies.

Citation: Peng Huang, Xiong Li, Bin Liu. Invariant curves of smooth quasi-periodic mappings. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 131-154. doi: 10.3934/dcds.2018006
##### References:
 [1] L. Chierchia and D. Qian, Moser's theorem for lower dimensional tori, J. Differential Equations, 206 (2004), 55-93. doi: 10.1016/j.jde.2004.06.014. Google Scholar [2] M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau Ⅰ, Astérisque, (1983), 103-104. Google Scholar [3] M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau Ⅱ, Astérisque 144 (1986), 248pp. Google Scholar [4] P. Huang, X. Li and B. Liu, Quasi-periodic solutions for an asymmetric oscillation, Nonlinearity, 29 (2016), 3006-3030. doi: 10.1088/0951-7715/29/10/3006. Google Scholar [5] M. Levi and J. Moser, A Lagrangian proof of the invariant curve theorem for twist mappings, in Smooth Ergodic Theory and its Applications, (Seattle, WA, 1999) (Proc. Symp. Pure Math. 69), (Providence, RI: American Mathematical Society), 69 (2001), 733-746. doi: 10.1090/pspum/069/1858552. Google Scholar [6] B. Liu, Invariant curves of quasi-periodic reversible mapping, Nonlinearity, 18 (2005), 685-701. doi: 10.1088/0951-7715/18/2/012. Google Scholar [7] J. Moser, On invariant curves of area-perserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math. -Phys., 1962 (1962), 1-20. Google Scholar [8] J. Moser, A rapidly convergent iteration method and nonlinear differential equations Ⅱ, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 499-535. Google Scholar [9] J. Moser, A stability theorem for minimal foliations on a torus, Ergod Theory Dynam. Syst., 8 (1988), 251-281. doi: 10.1017/S0143385700009457. Google Scholar [10] H. Rüssmann, Kleine Nenner Ⅰ: Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. Ⅱ, 1970 (1970), 67-105. Google Scholar [11] H. Rüssmann, On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus, Dynamical systems theory and applications, 38 (1975), 598-624. Google Scholar [12] H. Rüssmann,On the existence of invariant curves of twist mappings of an annulus, Lecture Notes in Math., Springer, Berlin, 1007 (1983), 677-718. Google Scholar [13] C. Siegel and J. Moser, Lectures on celestial mechanics, Springer, Berlin, 1995. Google Scholar [14] E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems Ⅰ, Comm. Pure Appl. Math., 28 (1975), 91-140. doi: 10.1002/cpa.3160280104. Google Scholar [15] V. Zharnitsky, Invariant curve theorem for quasiperiodic twist mappings and stability of motion in the Fermi-Ulam problem, Nonlinearity, 13 (2000), 1123-1136. doi: 10.1088/0951-7715/13/4/308. Google Scholar

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##### References:
 [1] L. Chierchia and D. Qian, Moser's theorem for lower dimensional tori, J. Differential Equations, 206 (2004), 55-93. doi: 10.1016/j.jde.2004.06.014. Google Scholar [2] M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau Ⅰ, Astérisque, (1983), 103-104. Google Scholar [3] M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau Ⅱ, Astérisque 144 (1986), 248pp. Google Scholar [4] P. Huang, X. Li and B. Liu, Quasi-periodic solutions for an asymmetric oscillation, Nonlinearity, 29 (2016), 3006-3030. doi: 10.1088/0951-7715/29/10/3006. Google Scholar [5] M. Levi and J. Moser, A Lagrangian proof of the invariant curve theorem for twist mappings, in Smooth Ergodic Theory and its Applications, (Seattle, WA, 1999) (Proc. Symp. Pure Math. 69), (Providence, RI: American Mathematical Society), 69 (2001), 733-746. doi: 10.1090/pspum/069/1858552. Google Scholar [6] B. Liu, Invariant curves of quasi-periodic reversible mapping, Nonlinearity, 18 (2005), 685-701. doi: 10.1088/0951-7715/18/2/012. Google Scholar [7] J. Moser, On invariant curves of area-perserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math. -Phys., 1962 (1962), 1-20. Google Scholar [8] J. Moser, A rapidly convergent iteration method and nonlinear differential equations Ⅱ, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 499-535. Google Scholar [9] J. Moser, A stability theorem for minimal foliations on a torus, Ergod Theory Dynam. Syst., 8 (1988), 251-281. doi: 10.1017/S0143385700009457. Google Scholar [10] H. Rüssmann, Kleine Nenner Ⅰ: Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. Ⅱ, 1970 (1970), 67-105. Google Scholar [11] H. Rüssmann, On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus, Dynamical systems theory and applications, 38 (1975), 598-624. Google Scholar [12] H. Rüssmann,On the existence of invariant curves of twist mappings of an annulus, Lecture Notes in Math., Springer, Berlin, 1007 (1983), 677-718. Google Scholar [13] C. Siegel and J. Moser, Lectures on celestial mechanics, Springer, Berlin, 1995. Google Scholar [14] E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems Ⅰ, Comm. Pure Appl. Math., 28 (1975), 91-140. doi: 10.1002/cpa.3160280104. Google Scholar [15] V. Zharnitsky, Invariant curve theorem for quasiperiodic twist mappings and stability of motion in the Fermi-Ulam problem, Nonlinearity, 13 (2000), 1123-1136. doi: 10.1088/0951-7715/13/4/308. Google Scholar
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