  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. The family includes pseudo-Anosov braids with minimum
dilatation for 3,4, and 5 strands; the latter according to a recent anouncement
of J.-Y. Ham and W.-T. Song [math.GT/0506295].
