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We study the maximal entropy per unit generator of push-point mapping classeson the punctured disk. Our work is motivated by fluid mixing by rods in aplanar domain. If a single rod moves among N-fixed obstacles, the resultingfluid diffeomorphism is in the push-point mapping class associated with theloop in \pi_1(D^2 - {N points}) traversed by the single stirrer. The collectionof motions in each of which the stirrer goes around a single obstacle generatethe group of push-point mapping classes, and the entropy efficiency withrespect to these generators gives a topological measure of the mixing per unitenergy expenditure of the mapping class. We give lower and upper bounds forEff(N), the maximal efficiency in the presence of N obstacles, and prove thatEff(N) -> log(3) as N -> \infty. For the lower bound we compute the entropyefficiency of a specific push-point protocol, HSP_N, which we conjectureachieves the maximum. The entropy computation uses the action on chains in a\Z-covering space of the punctured disk which is designed for push-pointprotocols. For the upper bound we estimate the exponential growth rate of theaction of the push-point mapping classes on the fundamental group of thepunctured disk using a collection of incidence matrices and then computing thegeneralized spectral radius of the collection.

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