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

Marie-Claude Arnaud

doi:10.3934/jmd.2011.5.515

Abstract

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.

Highlights

The exact symplectic twist maps of the two-dimensional annulus1 were studied for a long time because they represent the dynamic of the generic symplectic diffeomorphisms of surfaces near their elliptic periodic points
In [9] and [1], some other examples are given of symplectic twist maps that have an non-differentiable essential invariant curve that contains some periodic points
Mather) Does there exist an example of a symplectic Cr twist map with an essential invariant curve that is not C1 and that contains no periodic point?

Summary

Introduction

The exact symplectic twist maps of the two-dimensional annulus were studied for a long time because they represent (via a symplectic change of coordinates) the dynamic of the generic symplectic diffeomorphisms of surfaces near their elliptic periodic points (see [5]). In [9] and [1], some other examples are given of symplectic twist maps that have an non-differentiable essential invariant curve that contains some periodic points. Mather) Does there exist an example of a symplectic Cr twist map with an essential invariant curve that is not C1 and that contains no periodic point (separate question for each r ∈ [1, ∞] ∪ {ω})?. Γ contains no periodic points; the restriction f|Γ is C0-conjugated to a Denjoy counter-example; if γ : T → R is the map whose Γ is the graph, γ is C1 at every point except along the projection of one orbit, along which γ has distinct right and left derivatives This lets open Mather’s question for r ≥ 2 and the following question: Question. Let us point out that things are not as simple as they seem to be, and the choice of the sequences (βkl ) and (βkr) is a delicate process as we will explain

Generalities about twist maps and other topics

Denjoy counter-example

Modification of the Denjoy counter-example g