The entropy efficiency of point-push mapping classes on the punctured disk

Abstract

We study the maximal entropy per unit generator of point-push mapping classes on the punctured disk. Our work is motivated by fluid mixing by rods in a planar domain. If a single rod moves among $N$ fixed obstacles, the resulting fluid diffeomorphism is in the point-push mapping class associated with the loop in ${\pi }_1\left(D^2-\left\{N\text{ points}\right\}\right)$ traversed by the single stirrer. The collection of motions where each stirrer goes around a single obstacle generate the group of point-push mapping classes, and the entropy efficiency with respect to these generators gives a topological measure of the mixing per unit energy expenditure of the mapping class. We give lower and upper bounds for $Eff\left(N\right)$, the maximal efficiency in the presence of $N$ obstacles, and prove that $Eff\left(N\right)\to log\left(3\right)$ as $N\to \infty$. For the lower bound we compute the entropy efficiency of a specific point-push protocol, ${HSP}_N$, which we conjecture achieves the maximum. The entropy computation uses the action on chains in a $\mathbb{Z}$–covering space of the punctured disk which is designed for point-push protocols. For the upper bound we estimate the exponential growth rate of the action of the point-push mapping classes on the fundamental group of the punctured disk using a collection of incidence matrices and then computing the generalized spectral radius of the collection.