Non-autonomous curves on surfaces

Abstract

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

Incorporated referee remarks