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

Abstract

We study the maximal entropy per unit generator of push-point 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 push-point mapping class associated with the loop in π_1(D^2 - {N points}) traversed by the single stirrer. The collection of motions in each of which the stirrer goes around a single obstacle generate the group of push-point 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(N), the maximal efficiency in the presence of N obstacles, and prove that Eff(N) -> log(3) as N -> \infty. For the lower bound we compute the entropy efficiency of a specific push-point protocol, HSP_N, which we conjecture achieves the maximum. The entropy computation uses the action on chains in a \Z-covering space of the punctured disk which is designed for push-point protocols. For the upper bound we estimate the exponential growth rate of the action of the push-point 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.