Topological Entropy of Braids on the Torus

Abstract

A fast method is presented 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 sphere. Previous methods for computing topological entropies include the Bestvina--Handel train-track algorithm and matrix representations of the braid group. However, the Bestvina--Handel algorithm quickly becomes 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 appears to give 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. The efficiency of the method allows us to explore how much extra information about flow entropy is encoded in the braid as the number of trajectories becomes large.

19 pages, 44 figures. SIAM journal style

References in corpus (3)

• Topology, Braids, and Mixing in Fluids
• Topological Chaos in Spatially Periodic Mixers
• Alternative determinism principle for topological analysis of chaos

Cited by in corpus (6)

• Topology, Braids, and Mixing in Fluids
• Topology of Chaotic Mixing Patterns
• Is Chaotic Advection Inherent to Heterogeneous Darcy Flow?
• Braided mixing in confined chiral active matter
• On the topological entropy of families of braids
• Applications of the Dynnikov coordinate system on the boundary of Teichmüller space