A classical property of pseudo-Anosov mapping classes is that they act on the space of projective measured laminations with north-south dynamics. This means that under iteration of such a mapping class, laminations converge exponentially quickly towards its stable lamination. We will discuss a new construction (joint with Saul Schleimer) of pseudo-Anosovs where this exponential convergence has base arbitrarily close to one and so is arbitrarily slow.