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.

[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), 18 (1963), 13.

[3]

G. D. Birkhoff, Surface transformations and their dynamical applications,, Acta Math., 43 (1922), 1. doi: 10.1007/BF02401754.

[4]

G. Bouligand, "Introduction à la Géométrie Infinitésimale Directe,", Librairie Vuibert, (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], 121-122 (1985), 121.

[6]

A. Chenciner, Systèmes dynamiques différentiables,, in, (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, 1784 (1998), 13.

[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.

[9]

M.-R. Herman, "Sur les Courbes Invariantes par les Difféomorphismes de l'Anneau,", Vol. 1, 103-104 (1983), 103.

[10]

A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function,, (Russian), 98 (1954), 527.

[11]

J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962 (1962), 1.

[12]

H. Rüssman, On the existence of invariant curves of twist mappings of an annulus,, in, 1007 (1981), 677.

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.

[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), 18 (1963), 13.

[3]

G. D. Birkhoff, Surface transformations and their dynamical applications,, Acta Math., 43 (1922), 1. doi: 10.1007/BF02401754.

[4]

G. Bouligand, "Introduction à la Géométrie Infinitésimale Directe,", Librairie Vuibert, (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], 121-122 (1985), 121.

[6]

A. Chenciner, Systèmes dynamiques différentiables,, in, (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, 1784 (1998), 13.

[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.

[9]

M.-R. Herman, "Sur les Courbes Invariantes par les Difféomorphismes de l'Anneau,", Vol. 1, 103-104 (1983), 103.

[10]

A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function,, (Russian), 98 (1954), 527.

[11]

J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962 (1962), 1.

[12]

H. Rüssman, On the existence of invariant curves of twist mappings of an annulus,, in, 1007 (1981), 677.

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