The Burau Estimate for the Entropy of a Braid

The topological entropy of a braid is the infimum of the entropies of all homeomorphisms of the disk which have a finite invariant set represented by the braid. When the isotopy class represented by the braid is pseudo-Anosov or is reducible with a pseudoAnosov component, this entropy is positive. Fried and Kolev proved that the entropy is bounded below by the logarithm of the spectral radius of the braid’s Burau matrix, B(t), after substituting a complex number of modulus 1 in place of t. In this paper we show that for a pseudo-Anosov braid the estimate is sharp for the substitution of a root of unity if and only if it is sharp for t = −1. Further, this happens if and only if the invariant foliations of the pseudo-Anosov map have odd order singularities at the strings of the braid and all interior singularities have even order. An analogous theorem for reducible braids is also proved.