A family of pseudo-Anosov braids with small dilatation

Eriko Hironaka, Eiko Kin

This paper concerns a family of pseudo-Anosov braids with dilatations arbitrarily close to one. The associated graph maps and train tracks have stable ``star-like" shapes, and the characteristic polynomials of their transition matrices form Salem-Boyd sequences. These examples show that the logarithms of least dilatations of pseudo-Anosov braids on $2g+1$ strands are bounded above by $\log(2 + \sqrt{3})/g$. It follows that the asymptotic behavior of least dilatations of pseudo-Anosov, hyperelliptic surface homeomorphisms is identical to that found by Penner for general surface homeomorphisms.