5.8 Fixed Points
A fixed point of \(\, w = f(z)\,\) is a point \(z_0\) such that \(\, z_0 = f(z_0)\).
If \(\, w = az + b,\,\) the fixed points are determined by \(\, z_0 = az_0 + b \,\) or \(\, z_0 = \frac {b}{ 1 - a}\,\) for \(\, a\neq 1\).
So for \(a\neq 1\), \(\, w = az + b\,\) has a finite fixed point \(\, z_0 = \frac {b}{1 - a}\).
If \(a = 1,\, w = z + b\,\) has no fixed points in \(\mathbb {C}\).
We can then extended \(\mathbb {C}\) to \(\, \overline {\mathbb {C}} = \mathbb {C}\cup \{\pm \infty \}.\,\) Thus in the extended plane, the point at infinity \(\, w = \infty \,\) is a fixed point of the
translation \(\, w = z + b\).
Solution. A fixed point of \(w=f(z)\) is a solution of \(f(z)=z\).
For the identity \(w=z\) the equation holds for every \(z\), so every point is fixed — a mapping may have infinitely many fixed points.
For the inversion \(w=\dfrac {1}{z}\), \[\frac {1}{z}=z\implies z^2=1\implies z=\pm 1 ,\] so there are exactly two, \(z=1\) and \(z=-1\). The point \(z=0\) is not in the domain; on the Riemann sphere the inversion exchanges \(0\) and \(\infty \) rather than fixing them.
The contrast matters for linear fractional transformations in general: a non-identity one has at most two fixed points, so any such transformation agreeing with the identity at three points is the identity.
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