Slow north-south dynamics on $\mathcal{P}\mathcal{ML}$

Abstract

Note by the authors: This article is in the public domain.

We consider the action of a pseudo-Anosov mapping class on $\mathcal{P}\mathcal{ML}(S)$. 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.