Minimum dilatations of pseudo-Anosov braids
We determine the minimum dilatation $δ_n$ 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 $\lim_{n \to \infty} δ_n^n = (2+\sqrt{3})^2 \approx 13.928$. 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 $n$.
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