Bilinear map z = 0,1,∞ to -3,-1,1 with fixed points

Let the transformation be w = (az + b)/(cz + d), with ad − bc ≠ 0.

Using the three conditions

• z = ∞ maps to w = 1: a/c = 1, so a = c.
• z = 0 maps to w = −3: b/d = −3, so b = −3d.
• z = 1 maps to w = −1: (a + b)/(c + d) = −1, so a + b = −(c + d).

Substitute a = c and b = −3d: c − 3d = −c − d, so 2c = 2d and c = d.

So a = c = d. Take a = c = d = 1 and b = −3.

Answer: w = (z − 3)/(z + 1)

Check: z = 0 gives −3 ✓. z = 1 gives −2/2 = −1 ✓. z → ∞ gives 1 ✓. Also ad − bc = 1 + 3 = 4 ≠ 0 ✓.

Fixed points

Fixed points satisfy w = z:

z = (z − 3)/(z + 1) → z² + z = z − 3 → z² = −3

Fixed points: z = ± i√3.