A fixed point of a transformation w=f(z) is a point Zo such that f(Zo)=Zo, Show that every linear transformation, with the exception of the identity transformation w=z, has at most two fixed points in the extended plane.
A fixed point of a transformation w=f(z) is a point Zo such that f(Zo)=Zo, Show that every linear transformation, with the exception of the identity transformation w=z, has at most two fixed points in the extended plane.
If $\displaystyle f$ is a linear transformation then $\displaystyle z\mapsto az+b$. Say that $\displaystyle f(z) = z$. If $\displaystyle z\not = \infty$ then $\displaystyle az+b = z \implies z(a-1) = b$.
- If $\displaystyle a=1$ and $\displaystyle b\not = 0$ then the equation is not solvable and so $\displaystyle f$ has no fixed points in $\displaystyle \mathbb{C}$. It does however have $\displaystyle \infty$ as a fixed point on the Riemann sphere.
- If $\displaystyle a=1$ and $\displaystyle b=0$ then $\displaystyle f$ is the identity.
- If $\displaystyle a\not = 1$ then $\displaystyle \tfrac{b}{a-1}$ is a fixed point in $\displaystyle \mathbb{C}$ the otherfixed point is $\displaystyle \infty$ because $\displaystyle f(\infty) = \infty$ if $\displaystyle a\not = 0$ because otherwise $\displaystyle f(\infty) = 0$ is not a fixed point.
The question asks about linear transformations, but the thread is headed "linear fractional transformations", in other words those of the form $\displaystyle z\to\frac{az+b}{cz+d}$. In that case, a fixed point must satisfy $\displaystyle z=\frac{az+b}{cz+d}$. This is clearly a quadratic equation for z, except when b=c=0 and a=d, when it becomes identically true.