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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
π1(D2 −{N points}) 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(N), the maximal efficiency in the
presence of N obstacles, and prove that Eff(N) → log(3) as N →∞. For
the lower bound we compute the entropy efficiency of a specific point-push
protocol, HSPN, 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 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.
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