Let \(f:D^2\to D^2\) be an orientation-preserving disk homeomorphism that is the identity map on the boundary \(\partial D^2\), and let \(\{p_i\}\) be a periodic orbit of \(f\). The points \(p_i\) move under an isotopy from the identity map on \(D^2\) to \(f\). Their trajectory forms a geometric braid \(\beta\), a collection of strands in \(D^2\times [0,1]\) connecting \(p_i\times 1\) to \(f(p_i)\times 0\). The isotopy class of \(\beta\) determines the homotopy class of \(f\) relative to \(\{p_i\}\cup\partial D^2\). An \(n\)-braid refers to the isotopy class of a geometric braid with \(n\) strands. When \(\beta\) is represented by a pseudo-Anosov homeomorphism \(f\) with dilatation factor \(\lambda_f=\lambda (f)\), the dilatation of the braid \(\lambda (\beta )=\lambda_f\).
In the paper, the minimum dilatation of pseudo-Anosov 5-braids is shown to be the largest zero \(\lambda_5\approx 1.72208\) of \(x^4-x^3-x^2-x+1\), which is attained for the braid \(\sigma_1\sigma_2\sigma_3\sigma_4\sigma_1\sigma_2\).