Ham, Ji-Young; Song, Won Taek The minimum dilatation of pseudo-Anosov 5-braids. (English) Zbl 1151.37037 Exp. Math. 16, No. 2, 167-179 (2007). 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 \(\sigma_1\sigma_2\sigma_3\sigma_4\sigma_1\sigma_2\). Reviewer: Georgy Osipenko (St. Peterburg) Cited in 17 Documents MSC: 37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces 37B40 Topological entropy 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 57M60 Group actions on manifolds and cell complexes in low dimensions Keywords:dilatation; train track; braid; topological entropy; pseudo-Anosov disk homeomorphisms × Cite Format Result Cite Review PDF Full Text: DOI arXiv