The asymptotic behavior of least pseudo-Anosov dilatations

TL;DR

This paper establishes that for surfaces of fixed genus at least 2, the minimal dilatation of pseudo-Anosov homeomorphisms decreases roughly as ( n)/n, contrasting with genus 0 or 1 cases.

Contribution

It provides the asymptotic behavior of minimal pseudo-Anosov dilatations for surfaces with fixed genus and varying marked points.

Findings

Logarithm of minimal dilatation scales as ( n)/n for genus   g  2.

Contrasts with genus 0 or 1 where the order is 1/n.

Shows the asymptotic behavior of dilatations as the number of marked points grows.

Abstract

For a surface $S$ with $n$ marked points and fixed genus $g\geq2$, we prove that the logarithm of the minimal dilatation of a pseudo-Anosov homeomorphism of $S$ is on the order of $(\log n)/n$. This is in contrast with the cases of genus zero or one where the order is $1/n$.