360 AUDREY TERRAS, ARITHMETICAL QUANTUM CHAOS
- . -. •
H
- • • • • •
T
Figure 3. Geodesics in the 3 Symmetric Spaces H, T , and H3.H
3
3
5 5 4 •
2 1
T
0 -1 -2
1 1 0
Figure 4. Horocycles in the 3 Symmetric Spaces H, T, and H3. For the tree,
points in the nth horocycle are labeled n.value 1 at the origin. Since the power function y^8 = Im(z)^8 is an eigenfunction
of the Laplacian on H corresponding to the eigenvalue s(s - 1), we can build up
spherical functions from the power function by integration over K. Here dk denotes
the Haar measure given. by dB, If. k = ( cos. e e sin e e ). See Terras [83] for more
-sm cos
information on spherical functions for continuous symmetric spaces.