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

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 $π_{1} (D^{2} - {N points})$ 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 $Eff (N) \to log (3)$ as $N \to \infty$ . For the lower bound we compute the entropy efficiency of a specific point-push protocol, ${HSP}_{N}$ , 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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Keywords

pseudo-Anosov, fluid mixing

Mathematical Subject Classification 2010

Primary: 37E30

References

Publication

Received: 8 April 2011

Revised: 23 July 2011

Accepted: 25 July 2011

Published: 25 August 2011

Authors

Philip Boyland
Department of Mathematics University of Florida Gainesville FL 32611-8105 USA
http://www.math.ufl.edu/~boyland/
Jason Harrington
Department of Mathematics University of Florida Gainesville FL 32611-8105 USA
http://www.math.ufl.edu/~mathguy/

Volume 11, issue 4 (2011)

Philip Boyland and Jason Harrington

Abstract