Estimating topological entropy from the motion of stirring rods

TL;DR

This paper investigates how well the topological features of periodic rod motions in a viscous fluid can estimate the flow's topological entropy, analyzing the gap between lower bounds and actual entropy through simulations.

Contribution

It introduces a method to evaluate the accuracy of topological entropy estimates based on rod motion topology, highlighting the role of homology sign in the estimation gap.

Findings

Numerical simulations show the gap varies with rod motion topology.

The sign of the action on homology influences the entropy estimate accuracy.

The study provides insights into the limitations of topological bounds in fluid mixing.

Abstract

Stirring a two-dimensional viscous fluid with rods is often an effective way to mix. The topological features of periodic rod motions give a lower bound on the topological entropy of the induced flow map, since material lines must `catch' on the rods. But how good is this lower bound? We present examples from numerical simulations and speculate on what affects the 'gap' between the lower bound and the measured topological entropy. The key is the sign of the rod motion's action on first homology of the orientation double cover of the punctured disk.