The asymptotic behavior of least pseudo-Anosov dilatations

Abstract

For a surface $S$ with $n$ marked points and fixed genus $g\ge 2$, we prove that the logarithm of the minimal dilatation of a pseudo-Anosov homeomorphism of $S$ is on the order of $log\left(n\right)∕n$. This is in contrast with the cases of genus zero or one where the order is $1∕n$.