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
fixed obstacles,
the resulting fluid diffeomorphism is in the point-push mapping class associated with the
loop in
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
, the maximal efficiency
in the presence of
obstacles, and prove that
as . For
the lower bound we compute the entropy efficiency of a specific point-push protocol,
, which we
conjecture achieves the maximum. The entropy computation uses the action on chains in a
–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.
