  This paper describes a family of pseudo-Anosov braids with small dilatation.
The smallest dilatations occurring for braids with 3, 4 and 5 strands appear in
this family. A pseudo-Anosov braid with 2g+1 strands determines a hyperelliptic
mapping class with the same dilatation on a genus-g surface. Penner showed that
logarithms of least dilatations of pseudo-Anosov maps on a genus-g surface grow
asymptotically with the genus like 1/g, and gave explicit examples of mapping
classes with dilatations bounded above by log 11/g. Bauer later improved this
bound to log 6/g. The braids in this paper give rise to mapping classes with
dilatations bounded above by log(2+sqrt(3))/g. They show that least dilatations
for hyperelliptic mapping classes have the same asymptotic behavior as for
general mapping classes on genus-g surfaces.
