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

TL;DR

This paper investigates the maximum entropy efficiency of point-push mapping classes on punctured disks, providing bounds and showing it approaches log(3) as obstacles increase, with implications for fluid mixing efficiency.

Contribution

It establishes bounds for entropy efficiency in push-point mapping classes and introduces a specific protocol likely achieving maximal efficiency.

Findings

Eff(N) approaches log(3) as N increases.

A specific push-point protocol HSP_N is analyzed for efficiency.

Upper and lower bounds for Eff(N) are derived using spectral radius estimates.

Abstract

We study the maximal entropy per unit generator of push-point 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 push-point mapping class associated with the loop in \pi_1(D^2 - {N points}) traversed by the single stirrer. The collection of motions in each of which the stirrer goes around a single obstacle generate the group of push-point 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 -> \infty. For the lower bound we compute the entropy efficiency of a specific push-point 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 push-point protocols. For the upper bound we estimate the exponential growth rate of the action of the push-point 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.