Boundaries of instability zones for symplectic twist maps

TL;DR

This paper constructs a $C^2$ symplectic twist map with a non-differentiable invariant curve exhibiting Denjoy-type dynamics, located at the boundary of an instability zone, highlighting complex boundary behaviors in such systems.

Contribution

It introduces a novel symplectic twist map with a non-differentiable invariant curve conjugated to a Denjoy example, expanding understanding of boundary phenomena in instability zones.

Findings

Existence of a non-differentiable invariant curve with Denjoy dynamics

The invariant curve lies at the boundary of an instability zone

Demonstrates complex boundary behavior in symplectic twist maps

Abstract

We construct a $C^2$ symplectic twist map f of the annulus that has an essential invariant curve C such that: - C is not differentiable; - the dynamics of f restricted to C is conjugated to the one of a Denjoy counter-example; - C is at the boundary of an instability zone for f.