Abstract: In the 20's Birkhoff proved that any invariant curve of a symplectic twist map of the annulus which is not homotopic to a point (essential curve) is the graph of a Lipschitz function. An open question of Mather asks for an example of a $C^r$ symplectic twist map with an invariant essential curve that is not $C^1$ and that contains no periodic point. In this talk we will discuss the construction of such example when $r=1$. The example is due to Marie-Claude Arnaud. The cases $r>1$ and $r=\omega$ remain open.