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

Philip Boyland and Jason Harrington

Algebraic & Geometric Topology 11 (2011) 2265–2296

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 –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.

pseudo-Anosov, fluid mixing
Mathematical Subject Classification 2010
Primary: 37E30
Received: 8 April 2011
Revised: 23 July 2011
Accepted: 25 July 2011
Published: 25 August 2011
Philip Boyland
Department of Mathematics
University of Florida
Gainesville FL 32611-8105
Jason Harrington
Department of Mathematics
University of Florida
Gainesville FL 32611-8105