252 Chapter 5
the four points z 1 , z 2 , z 3 , z 4 lie on a circle (as well as their images). A
conjugate transformation carries circles into circles; for U = T J and since
both J and T map circles into circles, so does U.
As to the existence of fixed points, the situation is somewhat different
from that discussed for the bilinear transformation. In Section 5.6 we found
that the ordinary bilinear transformation has one or two fixed points, and if
the transformation has more than two fixed points, all points are fixed and
the transformation reduces to the identity. On the other hand, a conjugate
bilinear transformation may have no fixed point at all, one or two fixed
points, or infinitely many, in which case all lie on a generalized circle.
Examples
- w = -l/z has no fixed point.
2. w = (-4z -3)/(3z + 2) has one (double) fixed point, z = -1.
3. w = (3z + 2)/( 4z + 3) has two distinct fixed points,
z1 = 1/v12 and z2 = -1/v12.
- w = l/z has infinitely many fixed points, those of the circle lzl = 1..
In fact, the condition for a point z to be fixed under (5.12-1) isaz+ b
z=--
c2+d
(5.12-2)from which we get
_ az+ b
z=---
ez+ d
(5.12-3)Substituting (5.12-3) into (5.12-2) and reducing, we have
(ae + ed)z^2 + (ldl^2 - lal^2 +be - be)z - (ab+ bd) = 0 (5.12-4)
Hence if ae + ed =f. 0 there are two fixed points, which may coincide if the
discriminant
H = (lal^2 + ldl^2 +be+ be)^2 - 4
vanishes.
If ae + ed = 0 but ldl^2 - lal^2 + be - be =I 0, then the first equation
gives e = 0 or e =f. 0 and jaj = ldl. For e = 0 the second relation yields
la! =!' ldl, and since ad = 1 we have !al =f. 0, 1; Id! =f. 0, 1. Thus the
transformation becomes
a b
w = dz + d = a
2
z + abwhich has the fixed point z = (alal^2 b + ab)/(l - lal^4 ). In addition, the