Curve obstruction for autonomous diffeomorphisms on surfaces

TL;DR

This paper investigates how the behavior of curves under Hamiltonian diffeomorphisms on surfaces can indicate whether the diffeomorphism is autonomous, providing new criteria based on curve obstructions.

Contribution

It introduces novel scenarios where the relationship between a curve and its image under a Hamiltonian diffeomorphism reveals non-autonomy.

Findings

Curve-image relationships serve as evidence of non-autonomy.

Specific geometric configurations obstruct autonomous behavior.

The approach offers a new method to analyze Hamiltonian diffeomorphisms.

Abstract

Consider a Hamiltonian diffeomorphism $g$ on a surface. We describe several scenarios where a curve $L$ and its image $g(L)$ provide a simple evidence that $g$ is not autonomous.