ON BRILLOUIN ZONES

$\parbox{\awidth}{\makebox[\awidth]{J. J. P. Veerman}\\ {\footnotes... ...cs Department}\\ \makebox[\awidth]{UFPE}\\ \makebox[\awidth]{Recife, Brazil}} }$ $\parbox{\awidth}{\makebox[\awidth]{M. M. Peixoto}\\ {\footnotesize... ...PA}\\ \makebox[\awidth]{Rio de Janeiro, Brazil}\\ \makebox[\awidth]{\hbox{}}} }$ $\parbox{\awidth}{\makebox[\awidth]{A. C. Rocha}\\ {\footnotesize ... ...ematics Dept.}\\ \makebox[\awidth]{UFPE}\\ \makebox[\awidth]{Recife, Brazil}} }$ $\parbox{\awidth}{\makebox[\awidth]{S. Sutherland}\\ {\footnotesize... ...s Dept.}\\ \makebox[\awidth]{SUNY}\\ \makebox[\awidth]{Stony Brook, NY, USA}} }$

Abstract:

Given a metric space (X,d), a discrete set $S\subset X$, and a preferred point $x_0\in S$, the n-th Brillouin zone is defined (loosely speaking) as those points in X which are closest to exactly n elements of S, including x₀. This notion is a direct generalization of the concept introduced by Brillouin [Br] in the thirties, which describes some quantum mechanical properties of crystals.

We generalize a theorem of Bieberbach [Bi] asserting that the Brillouin zones tile the underlying space, and that each zone has the same area. We then use these ideas to discuss focusing of geodesics in orbifolds of constant curvature. In the particular case of the Riemann surfaces $\mbox{$\Bbb H$ }^2/\Gamma (N) \,\,(N=2,3, \;\mbox{or}\; 5)$, we count the number of geodesics of length t that connect the origin to itself.