# Non-autonomous curves on surfaces

## TL;DR

This paper investigates how the dynamical properties of Hamiltonian diffeomorphisms on symplectic surfaces can be inferred from pairs of isotopic curves, revealing conditions under which the map exhibits chaos or positive entropy.

## Contribution

It introduces methods to detect chaotic behavior and positive entropy of Hamiltonian diffeomorphisms from geometric data of isotopic curves on surfaces.

## Key findings

- Identification of conditions implying chaos in Hamiltonian diffeomorphisms
- Detection of positive entropy through curve pairs
- Analysis of non-autonomous behavior on symplectic surfaces

## Abstract

Consider a symplectic surface $\Sigma$ with two properly embedded Hamiltonian isotopic curves $L$ and $L'$. Suppose $g \in Ham (\Sigma)$ is a Hamiltonian diffeomorphism which sends $L$ to $L'$. Which dynamical properties of $g$ can be detected by the pair $(L, L')$? We discuss two cases where one can deduce that $g$ is `chaotic': non-autonomous or even of positive entropy.