A non-differentiable essential irrational invariant curve for a $C^1$ symplectic twist map

TL;DR

This paper constructs a $C^1$ symplectic twist map with a non-differentiable invariant curve exhibiting complex dynamics similar to a Denjoy counter-example, highlighting new phenomena in low-regularity symplectic maps.

Contribution

It provides the first example of a $C^1$ symplectic twist map with a non-differentiable invariant curve displaying Denjoy-like dynamics.

Findings

Existence of a non-differentiable invariant curve in $C^1$ symplectic twist maps.

The invariant curve's dynamics are conjugate to a Denjoy counter-example.

Demonstrates complex invariant structures in low-regularity symplectic maps.

Abstract

We construct a $C^1$ symplectic twist map f of the annulus that has an essential invariant curve C such that: --C is not differentiable; -- the dynamic of f restricted to C is conjugated to the one of a Denjoy counter-example.