Entropy and the Burau Representation

The topological entropy of a braid is the infimum of the entropies of all homeomor-phisms of the disc 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 pseudo-Anosov 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. 1 INTRODUCTION Artin's braid group and its Burau representation have been extensively studied by many researchers from many points of view. In dynamical applications a braid is often used to describe the motion of a collection of points in the two-dimensional disk. Since the braid depends only on the motion of the points, it is describing an isotopy class of homeomorphisms on the complement of the points. Thus, the interpretation of the braid group on n-strings, B n , as a mapping class group of the n-punctured disk is frequently used, and so Thurston's classification theorem for surface isotopy classes is an important tool. The (reduced) Burau matrix, B(t), of a braid β ∈ B n , is an (n − 1) × (n − 1) matrix with entries in Z[t, t −1 ], i.e. the entries of the matrix are Laurent polynomials over the integers. In the early 1980's two different but closely related dynamical interpretations of the Burau matrix emerged. Using the construction in Franks' paper [Fra81], the Burau matrix can be interpreted as the signed, linking matrix of a certain Axiom A flow associated with the braid. The signed, linking matrix is an enhanced Markov transition matrix which records the linking of the Markov boxes with the strings of the braid as well as the orientations of their images. The second dynamical interpretation comes from the machinery in Fried's paper [Fri86]. In this case the Burau matrix of a braid …