Slow north-south dynamics on $\mathcal{PML}$

TL;DR

This paper investigates the convergence rate of pseudo-Anosov mapping classes on projective measured laminations, revealing that in some cases the convergence can be arbitrarily slow, and establishing this as the worst-case scenario.

Contribution

It provides a detailed analysis of the convergence rates, including examples where the rate approaches one, and proves this slow convergence is the maximal possible rate.

Findings

Convergence rate can decay exponentially with word length.

Examples show the slowest possible convergence rate is approaching one.

The worst-case convergence behavior is characterized and proven.

Abstract

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.