# Actions of symplectic homeomorphisms/diffeomorphisms on foliations by curves in dimension 2

@article{Arnaud2020ActionsOS, title={Actions of symplectic homeomorphisms/diffeomorphisms on foliations by curves in dimension 2}, author={Marie-Claude Arnaud and Maxime Zavidovique}, journal={Ergodic Theory and Dynamical Systems}, year={2020} }

The two main results in this paper concern the regularity of the invariant foliation of a
$C^0$
-integrable symplectic twist diffeomorphism of the two-dimensional annulus, namely that (i) the generating function of such a foliation is
$C^1$
, and (ii) the foliation is Hölder with exponent
$\tfrac 12$
. We also characterize foliations by graphs that are straightenable via a symplectic homeomorphism and prove that every symplectic homeomorphism that leaves invariant…

## One Citation

### Weak K.A.M. solutions and minimizing orbits of twist maps

- Mathematics
- 2022

. For exact symplectic twist maps of the annulus, we etablish a choice of weak K.A.M. solutions u c = u ( · ,c ) that depend in a Lipschitz-continuous way on the cohomology class c . This allows us…

## References

SHOWING 1-10 OF 20 REFERENCES

### Invariant Manifolds

- Mathematics
- 1961

0. Introduction. Let M be a finite dimensional Riemann manifold without boundary. Kupka [5], Sacker [9], and others have studied perturbations of a flow or diffeomorphism of M leaving invariant a…

### Torsion of instability zones for conservative twist maps on the annulus

- Mathematics
- 2020

For a twist map f of the annulus preserving the Lebesgue measure, we give sufficient conditions to assure the existence of a set of positive measure of points with non-zero asymptotic torsion. In…

### A C^1 Arnol'd-Liouville theorem

- MathematicsAstérisque
- 2020

In this paper, we prove a version of Arnol'd-Liouville theorem for C 1 commuting Hamiltonians. We show that the Lipschitz regularity of the foliation by invariant Lagrangian tori is crucial to…

### A C1 Arnol'd-Liouville theorem

- Mathematics
- 2020

In this paper, we prove a version of Arnol'd-Liouville theorem for C 1 commuting Hamiltonians. We show that the Lipschitz regularity of the foliation by invariant Lagrangian tori is crucial to…

### Hyperbolicity for conservative twist maps of the 2-dimensional annulus

- Mathematics
- 2015

These are notes for a minicourse given at Regional Norte UdelaR in Salto, Uruguay for the conference CIMPA Research School-Hamiltonian and Lagrangian Dynamics. We will present Birkhoff and…

### Ergodic Theory and Differentiable Dynamics

- Mathematics
- 1986

0. Measure Theory.- 1. Measures.- 2. Measurable Maps.- 3. Integrable Functions.- 4. Differentiation and Integration.- 5. Partitions and Derivatives.- I. Measure-Preserving Maps.- 1. Introduction.- 2.…

### Surface transformations and their dynamical applications

- Mathematics
- 1922

A state of motion in a dynamical system with two degrees of freedom depends on two space and two velocity coiirdinates, and thus may be represented by means of a point in space of four dimensions.…

### A necessary and sufficient condition for a twist map being integrable

- Mathematics
- 1996

It is shown that for an area preserving twist map on cylinder the phase space is foliated by invariant circles not null-homotopic if and only if Jacobi fields do not have conjugate points.

### Principles of mathematical analysis

- Mathematics
- 1964

Chapter 1: The Real and Complex Number Systems Introduction Ordered Sets Fields The Real Field The Extended Real Number System The Complex Field Euclidean Spaces Appendix Exercises Chapter 2: Basic…

### Sur la Conjugaison Différentiable des Difféomorphismes du Cercle a des Rotations

- Mathematics
- 1979

© Publications mathématiques de l’I.H.É.S., 1979, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http://…