The entropy efficiency of point-push mapping classes on the punctured disk (Q640312)

The topological entropy \(h(f)\) of a map \(f\) is a standard measure of its dynamical complexity. The authors study ``the maximal entropy per unit generator of point-push mapping classes on the punctured disk. The work is motivated by fluid mixing by rods in a planar domain.'' ``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 of the mapping class.'' As the main result of the paper the authors ``give lower and upper bounds for \(\mathrm{Eff}(N)\), the maximal efficiency in the presence of \(N\) obstacles, and prove that \(\mathrm{Eff}(N)\rightarrow \log(3)\) as \(N\rightarrow \infty\). For the lower bound they compute the entropy efficiency of a specific point-push protocol, \(HSP_N\), which is conjectured to achieve the maximum.''