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

TL;DR

This paper constructs a $C^1$ symplectic twist map with a non-differentiable invariant curve on which the dynamics are minimal, challenging assumptions about smoothness and invariant structures in dynamical systems.

Contribution

It provides the first example of a $C^1$ twist map with a non-differentiable invariant curve exhibiting minimal dynamics.

Findings

Existence of a $C^1$ symplectic twist map with a non-differentiable invariant curve.

The invariant curve supports minimal dynamics despite its non-differentiability.

The construction challenges previous beliefs about smoothness constraints on invariant curves.

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.