Hostname: page-component-cb9f654ff-fg9bn Total loading time: 0 Render date: 2025-08-23T11:11:51.463Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-differentiable irrational curves for $C^1$ twist map

Part of: Low-dimensional dynamical systems

Published online by Cambridge University Press:  02 December 2021

ARTUR AVILA  and
BASSAM FAYAD Open the ORCID record for BASSAM FAYAD [Opens in a new window]
ARTUR AVILA
Affiliation:
Institut fϋr Mathematik Universität Zϋrich, Winterthurerstrasse 190, CH-8057 Zϋrich, Switzerland
BASSAM FAYAD*
Affiliation:
Institut de Mathématiques de Jussieu-Paris Rive Gauche, 58-56, avenue de France, Boite Courrier 7012, 75205 Paris Cedex 13, France
*
e-mail: bassam.fayad@imj-prg.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct a $C^1$ symplectic twist map g of the annulus that has an essential invariant curve $\Gamma $ such that $\Gamma $ is not differentiable and g restricted to $\Gamma $ is minimal.

Keywords

twist mapsinvariant curvesBirkhoff theory

MSC classification

Primary: 37E40: Twist maps 37E10: Maps of the circle

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 2: Anatole Katok Memorial Issue Part 1: Special Issue of Ergodic Theory and Dynamical Systems , February 2022 , pp. 491 - 499
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Arnaud, M.-C.. 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.CrossRefGoogle Scholar
Arnaud, M.-C.. A non-differentiable essential irrational invariant curve for a ${C}^1$ symplectic twist map. J. Mod. Dyn. 5(3) (2011), 583–591.10.3934/jmd.2011.5.583CrossRefGoogle Scholar
Arnaud, M.-C.. Boundaries of instability zones for symplectic twist maps. J. Inst. Math. Jussieu 13(1) (2013), 19–41.CrossRefGoogle Scholar
Birkhoff, G. D.. Surface transformations and their dynamical application. Acta Math. 43 (1920), 1–119.CrossRefGoogle Scholar
Chenciner, A.. La dynamique au voisinage d’un point fixe elliptique conservatif: de Poincaré et Birkhoff à Aubry et Mather [The dynamics at the neighborhood of a conservative elliptic fixed point: from Poincaré and Birkhoff to Aubry and Mather]. Sem. Bourbaki 1983/84(121–122) (1985), 147–170 (in French).Google Scholar
Eliasson, L. H., Kuksin, S., Marmi, S. and Yoccoz, J.-C.. Dynamical Systems and Small Divisors: Lectures from the C.I.M.E. Summer School held in Cetraro (13–20 June 1998) (Lecture Notes in Mathematics, 1784). Eds. Marmi, S. and Yoccoz, J.-C.. Springer, Berlin, 1998.Google Scholar
Herman, M.. Sur les courbes invariantes par les difféomorphismes de l’anneau (Asterisque, 103–104). Vol. 1. Société Mathématique de France, Paris, 1983.Google Scholar