Topological entropy of braids on the torus

We present a fast method for computing the topological entropy of braids on the torus. This work is motivated by the need to analyze large braids when studying two-dimensional flows via the braiding of a large number of particle trajectories. Our approach is a generalization of Moussafir's technique for braids on the disk. Previous methods for computing topological entropy include the Bestvina-Handel train-track algorithm and matrix representations of the braid group. However, the Bestvina-Handel algorithm is computationally intractable for large braid words, and matrix methods give only lower bounds, which are often poor for large braids. Our method is computationally fast and gives exponential convergence towards the exact entropy. As an illustration we apply our approach to the braiding of both periodic and aperiodic trajectories in the sine flow.