  We consider the action of a pseudo-Anosov mapping class on
$\mathcal{PML}(S)$. This action has north-south dynamics and so, under
iteration, laminations converge exponentially to the stable lamination.
  We study the rate of this convergence and give examples of families of
pseudo-Anosov mapping classes where the rate goes to one, decaying
exponentially with the word length. Furthermore we prove that this behaviour is
the worst possible.
