Minimum dilatations of pseudo-Anosov braids

TL;DR

This paper determines the minimal dilatations of pseudo-Anosov braids with a large number of strands, confirming conjectures and solving the minimum dilatation problem for most cases on the n-punctured sphere.

Contribution

It explicitly computes the minimum dilatations for large n and confirms related conjectures, advancing the understanding of pseudo-Anosov braid dynamics.

Findings

Minimum dilatations $ o ext{attained by Hironaka-Kin and Venzke examples}$

Limit of $oxed{ ext{dilatation}^n}$ as $n o ext{large}$ approaches $(2+ ext{sqrt}(3))^2$

Confirms conjectures and solves the minimum dilatation problem for all but 6 values of n

Abstract

We determine the minimum dilatation $\delta_n$ among pseudo-Anosov braids with $n$ strands, for large enough values of $n$. These are the dilatations attained by the examples of Hironaka-Kin and Venzke, and they satisfy $\lim_{n \to \infty} \delta_n^n = (2+\sqrt{3})^2 \approx 13.928$. Together with previous work, this result confirms conjectures by Kin-Takasawa and Venzke, and solves the minimum dilatation problem on the $n$-punctured sphere, for all but $6$ values of $n$.