# American Institute of Mathematical Sciences

July  2011, 5(3): 583-591. doi: 10.3934/jmd.2011.5.583

## A nondifferentiable essential irrational invariant curve for a $C^1$ symplectic twist map

 1 Université d’Avignon et des Pays de Vaucluse, Laboratoire d’Analyse non linéaire et Géométrie (EA 2151), F-84 018 Avignon, France

Received  April 2011 Revised  October 2011 Published  November 2011

We construct a $C^1$ symplectic twist map $f$ of the annulus that has an essential invariant curve $\Gamma$ such that:
•$\Gamma$ is not differentiable;
•$f$ ↾ $\Gamma$ is conjugate to a Denjoy counterexample.
Citation: Marie-Claude Arnaud. A nondifferentiable essential irrational invariant curve for a $C^1$ symplectic twist map. Journal of Modern Dynamics, 2011, 5 (3) : 583-591. doi: 10.3934/jmd.2011.5.583
##### References:
 [1] M.-C. Arnaud, Three results on the regularity of the curves that are invariant by an exact symplectic twist map, Publ. Math. Inst. Hautes Études Sci., 109 (2009), 1-17. [2] V. Arnol'd, Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, (Russian), Uspehi Mat. Nauk, 18 (1963), 13-40. [3] G. D. Birkhoff, Surface transformations and their dynamical applications, Acta Math., 43 (1922), 1-119. doi: 10.1007/BF02401754. [4] G. Bouligand, "Introduction à la Géométrie Infinitésimale Directe," Librairie Vuibert, Paris, 1932. [5] A. Chenciner, La dynamique au voisinage d'un point fixe elliptique conservatif: De Poincaré et Birkhoff à Aubry et Mather, (French) [The dynamics at the neighborhood of a conservative elliptic fixed point: From Poincaré and Birkhoff to Aubry and Mather], Seminar Bourbaki, Vol. 1983/84, Astérisque No., 121-122 (1985), 147-170. [6] A. Chenciner, Systèmes dynamiques différentiables, in "Encyclopedia Universalis," 1985. [7] L. H. Eliasson, S. Kuksin, S. Marmi and J.-C. Yoccoz, "Dynamical Systems and Small Divisors," Lectures from the C.I.M.E. Summer School held in Cetraro, June 13-20, 1998, Edited by Marmi and Yoccoz, Lecture Notes in Mathematics, 1784, Fondazione C.I.M.E., Springer-Verlag, Berlin; Centro Internazionale Matematico Estivo (C.I.M.E.), Florence, 2002. [8] M. R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math. No., 49 (1979), 5-233. [9] M.-R. Herman, "Sur les Courbes Invariantes par les Difféomorphismes de l'Anneau," Vol. 1, With an appendix by Albert Fathi, Asterisque, 103-104, Société Mathématique de France, Paris, 1983. [10] A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function, (Russian), Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527-530. [11] J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962 (1962), 1-20. [12] H. Rüssman, On the existence of invariant curves of twist mappings of an annulus, in "Geometric Dynamics" (Rio de Janeiro, 1981), 677-718, Lecture Notes in Math., 1007, Springer, Berlin, 1983.

show all references

##### References:
 [1] M.-C. Arnaud, Three results on the regularity of the curves that are invariant by an exact symplectic twist map, Publ. Math. Inst. Hautes Études Sci., 109 (2009), 1-17. [2] V. Arnol'd, Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, (Russian), Uspehi Mat. Nauk, 18 (1963), 13-40. [3] G. D. Birkhoff, Surface transformations and their dynamical applications, Acta Math., 43 (1922), 1-119. doi: 10.1007/BF02401754. [4] G. Bouligand, "Introduction à la Géométrie Infinitésimale Directe," Librairie Vuibert, Paris, 1932. [5] A. Chenciner, La dynamique au voisinage d'un point fixe elliptique conservatif: De Poincaré et Birkhoff à Aubry et Mather, (French) [The dynamics at the neighborhood of a conservative elliptic fixed point: From Poincaré and Birkhoff to Aubry and Mather], Seminar Bourbaki, Vol. 1983/84, Astérisque No., 121-122 (1985), 147-170. [6] A. Chenciner, Systèmes dynamiques différentiables, in "Encyclopedia Universalis," 1985. [7] L. H. Eliasson, S. Kuksin, S. Marmi and J.-C. Yoccoz, "Dynamical Systems and Small Divisors," Lectures from the C.I.M.E. Summer School held in Cetraro, June 13-20, 1998, Edited by Marmi and Yoccoz, Lecture Notes in Mathematics, 1784, Fondazione C.I.M.E., Springer-Verlag, Berlin; Centro Internazionale Matematico Estivo (C.I.M.E.), Florence, 2002. [8] M. R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math. No., 49 (1979), 5-233. [9] M.-R. Herman, "Sur les Courbes Invariantes par les Difféomorphismes de l'Anneau," Vol. 1, With an appendix by Albert Fathi, Asterisque, 103-104, Société Mathématique de France, Paris, 1983. [10] A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function, (Russian), Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527-530. [11] J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962 (1962), 1-20. [12] H. Rüssman, On the existence of invariant curves of twist mappings of an annulus, in "Geometric Dynamics" (Rio de Janeiro, 1981), 677-718, Lecture Notes in Math., 1007, Springer, Berlin, 1983.
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